Source identification ofincoherent sound effects

F. Kamalizadeh

DCT 2008.080

Master’s thesis

Coaches: ir. R. Scholtedr. ir. I. Lopez Arteaga

Supervisor: prof. dr. H. Nijmeijer

Committee: prof. dr. H. Nijmeijerprof. dr. ir. N.B. Roozendr. ir. I. Lopez Arteagadr. ir. H.J. Martin

Eindhoven University of TechnologyDepartment Mechanical EngineeringDynamics and Control Group

Eindhoven, June, 2008

Abstract

We are constantly exposed to sounds in our daily life. These could be desired such as music,but also undesired sounds such as traffic noise. In both cases it is desired to have control overthe present sound. To gain this control, research into the behavior of sound sources is essential.Planar Nearfield Acoustic Holography (PNAH) is considered an important part in this research.This method makes it possible to visualize and even propagate soundfields to any desired planevia pressure measurements in the nearfield of sound sources . In practice soundfields commonlyconsists of multiple sound sources. This makes interpretation of acoustic holograms and indi-vidual analysis of sound sources difficult. An important step is therefore to identify and separatethe different sound sources within a combined soundfield. Consequently, the goal of this grad-uation project is to investigate the existing sound source identification methods and additionallyintroduce and implement improvements.

A literature study has provided insight into the existing source identification method andshows the current status quo. From this investigation it transpires that almost all identificationmethods are based on so-called coherence analysis methods. In this thesis two identificationmethods are highlighted and analyzed in more detail.

The first methods is the renowned and widely used Principal Component Analysis (PCA)method. Here a x amount of references are placed in a combined soundfield. With a singularvalue decomposition, the independent directions/sound sources in the reference set are deter-mined. The results of such a PCA analysis are very dependent on the correct placement ofthe references and is therefore not unique. The other highlighted method is the identificationmethod of Nam. This relatively new method does not use references, but instead localizes inde-pendent sound sources based on maxima in the acoustical holograms. Here it is assumed thateach maximum originates from a single sound source and subsequently the independent soundsources are iteratively identified via a coherence analysis. This assumption is not trivial and willlead to significant errors in the identification results if it does not hold. The result of the de-tailed investigation of these two methods is the introduction of a new method called PCA withAdvanced Soundfield Observation (PCAASO). By combining the powerful qualities of the abovetwo identification methods, an attempt is made to introduce a more robust and improved soundsource identification method.

The characteristics of PCA, Nam and PCAASO are investigated and highlighted by perform-ing Matlab simulations and experiments. The identification performance of PCAASO with re-spect to PCA has also been analyzed. The simulation and experiment show that applicationof PCAASO results in an improved source identification and demonstrates the potential of themethod.

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Samenvatting

In ons dagelijks leven worden wij continu blootgesteld aan geluiden. Deze geluiden kunnengewenst zijn zoals muziek, maar ook ongewenst zoals verkeerslawaai. Voor beide gevallen is hetgewenst om controle te hebben over het aanwezige geluid. Om deze control te verkrijgen is onder-zoek naar het gedrag van geluidsbronnen essentieel. Een belangrijk hulpmiddel hierbij is PlanarNearfield Acoustic Holography (PNAH). Deze methode maakt het mogelijk om geluidsvelden tevisualiseren en ook te propageren naar elk ander gewenst vlak via drukmetingen in het nabijhei-dsveld van geluidsbronnen. In de praktijk bestaan geluidsvelden meestal uit verschillende gelu-idsbronnen. Dit maakt interpretatie van akoestische hologrammen en individuele analyse naargeluidsbronnen moeilijk. Een belangrijke stap is daarom om de verschillende geluidsbronnen ineen gecombineerd geluidsveld te kunnen scheiden. Het doel van deze afstudeeropdracht is danook om onderzoek te doen naar de aanwezige geluidsbron identificatie technieken en daarnaastverbeteringen aan te dragen en te implementeren.

Met een literatuur studie is er inzicht verkregen in de bestaande bron scheidingstechniekenen is de huidige stand van zaken in kaart gebracht. Uit dit onderzoek blijkt dat nagenoeg allescheidingstechnieken gebaseerd zijn op zogenaamde coherentie analyse methodes. In dit verslagzijn er twee scheidingstechnieken uitgelicht en verder onderzocht.

De eerste methode is de gerenommeerde en breed toegepaste Principal Component Analy-sis (PCA) methode. Hierbij worden er een x aantal referenties in een gecombineerd geluidsveldgeplaatst. Via een singuliere waarde decompositie worden vervolgens de onafhankelijke richtin-gen c.q. geluidsbronnen in deze referentie set bepaald. De resultaten van een dergelijke PCAanalyse zijn erg afhankelijk van de correcte plaatsing van de referenties en is daardoor dus nietuniek. De andere methode, die uitgebreid is geanalyseerd, is de scheidingstechniek van Nam.Deze relatief nieuwe methode maakt geen gebruik van referenties, maar localiseert geluidsbron-nen gebaseerd op maxima in akoestiche hologrammen. Hierin wordt aangenomen dat elk maxi-mum afkomstig is van een enkele geluidsbron en voorts worden via een coherentie analyse iter-atief de geluidsbronnen geïdentificeerd. Deze aanname is niet triviaal en wanneer deze niet geldtleidt dit tot incorrecte scheidingsresultaten. Resultaat van de gedetailleerde analyse van deze tweemethodes is de introductie van een nieuwe methode genaamd PCA with Advanced SoundfieldObservation (PCAASO). Door de krachtige eigenschappen van bovenstaande scheidingsmeth-odes te combineren, is getracht een robuuster en verbeterde scheidingstechniek te introduceren.

Met Matlab simulaties en experimenten zijn de verschillende kenmerkende eigenschappenvan PCA, Nam en PCAASO onderzocht. De scheidingsprestatie van PCAASO ten opzichte vanPCA is ook onderzocht. De simulatie en het experiment laten zien dat toepassing van PCAASOleidt tot verbeterde bron identificatie en toont het potentieel van de methode.

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Contents

Abstract i

Samenvatting iii

1 Introduction 31.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 History of source identification methods . . . . . . . . . . . . . . . . . . . . . . . 41.4 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Planar Nearfield Acoustic Holography 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Basics of Acoustics Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Angular Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 PNAH method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Theoretical analysis of sound source identification methods 173.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.1 Basics of PCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.2 Application of PCA for sound identification . . . . . . . . . . . . . . . . . 203.2.3 Virtual Coherence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.4 Influence of reference placement on sound identification . . . . . . . . . 25

3.3 Partial field decomposition with NAH . . . . . . . . . . . . . . . . . . . . . . . . 273.4 Spatial Transformation of Sound Fields . . . . . . . . . . . . . . . . . . . . . . . 323.5 Principal Component Analysis with Advanced Soundfield Observation . . . . . . 333.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Simulations of identification methods 374.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Simulation scenarios and results . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

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5 Experiments with baffle setup 535.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.3 Experiment scenarios and results . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6 Conclusions and Recommendations for future research 696.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.2 Recommendations for future research . . . . . . . . . . . . . . . . . . . . . . . . 71

Bibliography 74

A Spectra smoothing 75

B STSF analysis 77

C Eigen and Singular Value Decomposition relation 83

D Matlab scripts 85D.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85D.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100D.3 PacImNAH changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

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Chapter 1

Introduction

1.1 Motivation

In the past, relatively little attention was paid to the influence of acoustics during the designprocess of a wide variety of products. This field was considered less important. Nowadays theimportance and impact of acoustics is receiving more and more acknowledgement. The ever in-creasing performance demands of customer and industrial applications go hand in hand with agrowing acoustical contribution for meeting these requirements. Here, the impact of acousticsis divided into two main areas: noise pollution and impact on product operation. Noise pollu-tion is considered the largest area, which encompasses examples as high noise levels in factories,tyre-road noise and sound contamination of MRI scanners and computer parts (e.g. harddisk).Generally, noise also leads to decreased operational performance of products. Acoustical pres-sure fronts are able to induce mechanical vibrations, which result into performance loss of, forexample, high precision machines. These examples demonstrate that acoustics has influence ona large field of products. The combination of a relatively young research area and the increas-ing importance of acoustic knowhow makes acoustics a very interesting research field where stillmany topics are unexplored and much knowledge is to gain.Research in acoustics is mainly concerned with understanding the behavior of acoustic sourcesand finding methods which make it possible to alter this behavior. An important step for achiev-ing the aforementioned goals is determination and visualization of acoustical soundfields. Withacoustic imaging the foundation was laid for Nearfield Acoustical Holography (NAH), which isa well respected acoustic soundfield visualization technique introduced in 1980 [21]. NAH de-termines the emitted soundfield by measuring over a grid layout, at a predefined distance fromthe sound sources. The obtained soundfield is consequently backpropagated to the sourceplaneor any other desired plane. The introduction of NAH enabled fast, high resolution soundfieldimaging. In most cases a soundfield is not made up from a single sound source, but consistsof the contributions of multiple sound sources. A visualization of such a composite soundfieldprovides a total view of the encountered soundfield, but is unable to distinguish between the indi-vidual contributions of the present sound sources to the total soundfield. Obtaining the individualcontributions of sound sources is important for several reasons. First, by individually analyzingthe sound sources much insight is gained on the behavior of the isolated sound sources. Theidentification of sound sources also assists in determining the location of the sound sources andhereto tracing back the physical origin of emitted soundfields. Finding the independent contribu-tors of a composite soundfield makes specialized action per sound source possible and the overall

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sound control more effective.An improved NAH method [15] is utilized for creating all soundfield holograms shown in thisthesis. The determination of soundfields with NAH can be performed by a full-array or movingarray/single point microphone setup. With a full array setup all grid positions are determined si-multaneously, whereas a moving array has to perform several steps to measure the complete gridlayout. The research in this thesis is focussed on single point hologram measurements. Thismeasurement method provides a much more flexible setup and enables high spatial resolutionholography.

1.2 Problem Statement

This Master thesis focusses on identifying and obtaining the contributions of individual sourceswithin a composite soundfield using a moving array setup.From the literature study performed for this thesis two distinguished sound source identifica-tion methods are highlighted. The first method, Principal Component Analysis (PCA), is a wellrespected identification technique, which utilizes reference microphones to perform the decom-position of a soundfield. The technique which has recently been introduced by Nam [11] is con-sidered the second method. The Nam method does not use reference microphones, but insteadperforms source identification based on the NAH measurements itself. In this thesis the im-pact of the different approaches on sound source identification and their suitability for movingarray measurements is investigated. To be applicable for moving array measurements, Nam re-quires the use of Spatial Transformation of Sound Fields (STSF) [5], which constructs incoherentsoundfield holograms and is therefore a source identification method itself. STSF is analyzedintensively to investigate its implications on source identification in combination with Nam.The goal of this thesis is to thoroughly investigate the two aforementioned identification meth-ods. From the analysis of their strong qualities and weaknesses an attempt is made to optimizethe identification results for a single point measurement setup. By performing simulations andexperiments in a controlled environment the analytical outcomes and statements are investigated.

1.3 History of source identification methods

The first methods for identifying and separating sound sources date back to the late 1960s, early1970s. These techniques were, in general, based on coherence functions and nowadays theseapproaches still form the foundation of the current identification techniques; Coherence analysisis a widely accepted and practised procedure in this field. Over the years, sound source identi-fication investigations included the use of ordinary, partial and virtual coherence functions andprincipal component analysis. An overview of the development of these sound source identifica-tion techniques is presented.

Coherence analysis techniques aim at uncovering linear relationships between reference sig-nals, followed by an estimation of the relation between references and so-called target locations.The reference measurements are done at locations representative for the effects that build up thesound source field of the objects. Often, microphones are used as reference signal transducersand these microphones are ideally positioned at the exact locations of the present sound sources.Target locations on the other hand are defined as the locations where the analysis of the proper-ties of the emitted sound sources is desired. In our case the target locations would represent the

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location of the hologram plane grid positions. The coherence analysis then relates the referenceand target spectra. The general objective of coherence analysis consists of analyzing and explain-ing the relation between phenomena observed in the target and reference set of measurements.Depending upon the intercorrelation between the reference signals, ordinary coherence, partialcoherence or virtual coherence analysis is applied.

The use of the ordinary coherence functions shows the ability to qualitatively decomposesound sources in an incoherent sound field (see Brown and Halvorsen [2]). The ordinary coher-ence function makes it possible to identify sound sources with the assumption that the presentreference signals are mutually incoherent. The power is then allocated to the different referencespectra, where each reference transducer is supposed to represent a single source. However,when reference signals pick up more than one incoherent source and thus become partially co-herent to eachother, the decomposition leads to considerable errors. Therefore, the applicationof ordinary coherence functions is not satisfactory in a sound field with inter-source coherences.Since such scenarios are very common in practice, this is generally acknowledged as a shortcom-ing of the use of ordinary coherence functions [12].

In case of partially coherent reference signals, source identification is achieved by using par-tial coherence functions. Partially coherent reference signals are often found in structurally cou-pled systems, e.g. a combustion engine where the vibrations of most individual parts are causedby several sources. Reference transducers positioned on or close to such an object are seldomable to pick up the influence of a single source. In general, partial coherence analysis derivesa set of uncorrelated reference spectra by means of a gradual elimination method [12]. Thesereference spectra are then correlated to the target spectra, e.g. an acoustic hologram. A resultingdrawback is that the uncorrelated reference spectra do not represent the present physical soundsources per definition, which makes the interpretation of the results less straightforward. Also,the elimination method requires a certain ranking of the signals to be successful [19]. This re-quires a priori knowledge about the measured signals. Another restriction of partial coherenceanalysis is that numerical problems arise when the different signals are highly correlated (> 0.5correlation), leading to an ill-conditioned problem. The results of the early applications of partialcoherence analysis could be considered disappointing [1, 3, 13]. This was mainly due to the lack ofdecent data acquisition and data handling. Therefore, the researchers were restricted to use, forexample, low frequency resolutions for FFTs and were unable to use many reference transducers.Nowadays however, partial coherence analysis, if applied correctly, is considered a strong method,which is widely accepted and applied in the sound source identification field.Since partial coherence analysis is not able to identify sources in case of omni-directional energyflow, another adaptation was made to the coherence technique to account for these specific casesby Price and Bernhard [14]. This method, the virtual coherence analysis, transforms a datasetformed by n independent sound processes and measured by m > n reference signals, into a setof n incoherent virtual sources. Simulations with random signals show an accurate identifica-tion of the incoherent processes [14]. The virtual sources represent the incoherent processes ina soundfield, which are often unequal to the physical sound sources. This makes analysis of theresults somewhat more difficult, because of the lack of a clear link of the virtual sources with thephysical sources.

A new way of looking at the sound identification problem was provided by Leuridan et al. [9].Instead of coherence analysis, Leuridan used a statistical procedure, called Principal ComponentAnalysis (PCA), to examine the linear correlations between reference signals. PCA is a dataseparation technique which dates back to the early 20th century (Pearson (1901) and Hotelling(1933)). The basic idea of the founders of PCA is that within a data set consisting of p variables,

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there may be a smaller fundamental set of independent variables which encompass the wholedata set. These variables, called principal components, are defined as the maximized successivecontributions to the total of the variances of the original variables. Although the basics of PCA areavailable since the beginning of the 20th century, the application of the technique is only widelyimplemented from the 1980s onward. The boost of interest was triggered by the theoretical,yet predominantly by the technical, progress leading to the widespread availability of computers.Leuridan et al. used PCA to determine the principal components of the cross-spectral matrix ofthe reference signals. The number of significant principal components represented the numberof incoherent sound source processes. Kompella [7] introduced a somewhat similar method toderive the number of incoherent processes in a composite sound field. Otte [12] worked out thevirtual coherence analysis of Price and Bernhard more extensively and combined it with the PCA-based technique by Leuridan. This led to the introduction of a versatile method to identify soundsources within a composite soundfield.

1.4 Outline of Thesis

This thesis consists of 6 chapters. After the introduction, chapter 2 treats the theory of PlanarNearfield Acoustic Hologram (PNAH). This method is used throughout the work to visualizesoundfields with so-called sound holograms. Chapter 3 is considered the core of this thesis. Herethe principles of the two existing sound identification methods of PCA and Nam are extensivelyanalyzed. Also, STSF comes into the picture here. The findings of this analysis have served asa foundation for the development of a new identification method named Principal ComponentAnalysis with Advanced Soundfield Observation (PCAASO). In chapter 4 and 5, respectively, theresults of simulations and experiments with the existing identification methods and the newPCAASO method are discussed. Finally the conclusions of this thesis and recommendations forfuture work are presented in chapter 6.

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Chapter 2

Planar Nearfield Acoustic Holography

2.1 Introduction

First, an introduction into acoustics is given. In the following section an overview of the basic andmost important equations of acoustics are presented. This acoustic theory forms the foundationof near-field acoustic holography (NAH). With NAH, a captured sound field is back propagatedin time and space, which results in an accurate representation of the actual sound field and its be-havior. Different sources require different implementations of NAH, being planar, cylindrical orspherical. In this thesis planar near-field acoustic holography (PNAH) is utilized. The foundingacoustical theory is followed by a section dedicated to the PNAH strategy, which forms the coreof the chapter. This chapter finishes with a discussion of the presented theory.

2.2 Basics of Acoustics Theory

The homogenous wave equation forms the foundation of the majority of acoustics studies. Thisequation is also an essential part in PNAH:

∇2p− 1c2

∂2p

∂t2= 0. (2.1)

Here p [Pa] defines the pressure, ∇2 is the three dimensional Laplacian and c [m/s] representsthe speed of sound in the medium (for this project we assume the medium is air). Solving thisequation gives the sound pressure, as a function of time and space, in the absence of soundsources, which is useful in the reminder of this section. To analyze the pressure in the frequencydomain a Fourier transformation is performed. The Fourier transformed pressure distribution isdefined as

p(ω) =∫ ∞

−∞p(t)e−jωtdt, (2.2)

where ω = 2πf stands for the angular frequency and f [Hz] is the sound frequency. The depen-dence of p on space (x, y, z) is omitted for clarity. Hereto a Fourier transformation is performedon (2.1), resulting in the homogenous wave equation in the frequency domain (also known as theHelmholtz equation)

∇2p + k2p = 0. (2.3)

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Here k = ωc [ rad

m ] represents the spatial wave frequency or the acoustic wavenumber. This newvariable is essential and very useful for analyzing acoustical problems and is used intensively inthis report. The acoustic wavenumber represents the number of waves per unit distance. This isproperly visualized by looking at a ’picture’ of a sound wave at a certain time t and determiningthe number of waves. The wavenumber k can be seen as the spatial counterpart of the angularfrequency ω. Consequently the following analogy is suitable:

ω → t

as (2.4)

k → λ,

where λ = cf [m] represents the acoustic wavelength, corresponding to the length of a single

wave. Using k = ωc , the acoustic wavelength is rewritten as λ = 2π

k . Note the striking analogywith ω and t.A possible solution of the Helmholtz equation (2.3) for the pressure distribution is

p(ω) = A(ω)ej(kxx+kyy+kzz), (2.5)

where A(ω) is the frequency and position dependent pressure amplitude and kx, ky, kz are thewavenumbers in the respective directions. The direction of any plane wave is given by ~k =kxi + kyj + kzl [1]. In the analysis it is also assumed that the z-direction is divided into two parts;

zxy

sourceplane

hologram

plane

Figure 2.1: definition of coordinate base

the first part for which z ≤ 0 contains possible sound sources and the second part (z > 0) isconsidered source-free. It must be noted that (2.5) is only a valid solution of (2.3) if

k2 = k2x + k2

y + k2z . (2.6)

Since k is per definition constant for a single frequency, (2.6) shows that the directional wavenum-bers are mutually dependent. In the remainder of this section the assumption that ky = 0 isutilized. Note that this assumption does not affect the analysis in any way, but merely makes theanalysis easier and more clear. The assumption that ky = 0 implies that there are no pressure

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variations in the y-direction. Consequently the transition is made from a three dimensional to atwo dimensional problem;

k2 = k2x + k2

z (2.7)

Now we have one equation and two variables and we assume that from these wavenumbers kx isindependent and that kz is dependent of kx.Looking at the pressure distribution of an acoustic wave in the independent x-direction, we canclassify a wave as either being subsonic or supersonic. An acoustic wave is classified as a super-sonic wave if the wave velocity in a direction is larger than c, the speed of sound. Note that thesedirectional velocities or trace velocities are projections of the wave velocity on the coordinate axesand do not imply the speed of travel of the wave in that specific direction. A subsonic wave, on theother hand, represents a wave with its trace velocity being smaller than c. The characteristic of asupersonic wave, cx > c, directly results into a wavenumber in the x-direction, kx = ω

cx, smaller

than k. In case of a subsonic wave kx > k, substituting the subsonic case in (2.6) results into

kz =√

k2 − k2x = j

√k2

x − k2 = jk′z, (2.8)

which means that kz is complex in case of subsonic waves. These findings are very important,since they have a large influence on the solution of the Helmholtz equation:

p(ω) = A(ω)ej(kxx+kzz) supersonic wave, propagating, (2.9)

p(ω) = A(ω)e−k′zzej(kxx) subsonic wave, evanescent.

From (2.9) it follows that the pressure amplitude of supersonic waves remain constant, resultingin a plane wave. However subsonic wave amplitudes decay exponentially as the acoustic wavetravels in the positive z-direction. This effect is caused by a complex kz as seen in (2.8), whichresults in a real decaying exponential function after substitution in (2.5). These evanescent wavesare considered essential in PNAH analysis, which becomes clear further on.

2.2.1 Angular Spectrum

In [1] Williams states that any pressure distribution in the source-free region can be expressed as

p(x, y, z) =1

4π2

∫ ∞

−∞dkx

∫ ∞

−∞dkyP (kx, ky)ei(kxx+kyy+kzz), (2.10)

where P (kx, ky), a complex quantity, is the pressure distribution in the wavenumber-domain.Using (2.10) and Fourier theory, Williams concludes that any pressure distribution can be deter-mined when a pressure distribution at a certain distance z is known. This statement is written as

P (kx, ky, z) = P (kx, ky, z′)eikz(z−z′). (2.11)

This implies that if the pressure distribution in one plane z′ is known, one can simply calculatethe pressure in another arbitrary plane z. The exponential function in (2.11) is also known as apropagator (in this case the pressure propagator). Similar propagators exist, which relate the par-ticle velocity distribution in one plane to another plane or the pressure distribution in one planeto the particle velocity distribution in another plane. In [2] all the propagators are determined and

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tabulated. Propagators provide a method to extrapolate pressure fields with relative ease. Herethe analysis of the data in the wavenumber-domain incredibly simplifies the calculation. Useof spatial data to find the pressure distribution in another plane requires a convolution integral.These propagators form the basis of PNAH.Now look at the propagator Gp = eikz(z−z′); if we are dealing with plane waves the multiplicationwith the propagator only leads to a phase change, but in case of evanescent waves somethingelse happens. For evanescent waves, kz is imaginary and this results in the following pressurepropagator

Gp = ek′z(z′−z), (2.12)

which demonstrates that for z′ > z evanescent waves result in a propagator with an increasingexponential. Consequently, when z and z′ differ the amplitude of the propagator changes. Thisis explained by looking at the behavior of these acoustic waves. Evanescent waves decay expo-nentially as they move away from the source (see (2.9)). If the pressure is known in one plane(z′) and it is desired to find the pressure distribution of a plane closer to the source (z < z′), thepropagator must contain an exponentially rising term to compensate for the decay. In theory theexponentially rising term in the propagator is perfectly in balance with the exponentially decay-ing term (2.9) and gives the correct solution. Unfortunately this inverse problem is not directlyapplicable in practice; presence of noise or disturbances, even small, can lead to a blow-up ofthe inverse problem leading to an erroneous solution. Therefore modifications must be madeto this ill-posed inverse problem to be able to correctly extrapolate the pressure fields. With so-called L-curve regularization discussed in the next section, PNAH is able to account for thesedisturbances.

2.3 PNAH method

The basic idea behind PNAH is to determine the inverse solution of the wave equation by mak-ing use of planar acoustic information at a certain distance from the source. The previous sectionshows that the inverse propagators are very useful for this cause. Unfortunately, these propaga-tors cannot be applied directly in practice which is due to the general presence of measurementnoise. The back propagation of contaminated pressure measurements lead to degraded resultsdue to the amplification of the measurement noise. Other measurement errors are mainly as-cribed to discretization effects. Acoustic pressure is measured on discrete positions and on afinite plane, and moreover FFT algorithms lead to possible aliasing and signal leakage. Thereforeseveral signal processing techniques and regularization procedures must be applied to minimizethese disturbing effects before the back propagation of a hologram plane is performed. In [2]Scholte has revised the basic PNAH method and has shown significant improvement over thebasic strategy. Therefore, this revised PNAH method is adopted for the remainder of this report.

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Step Domain

Frequency

Wavenumber

Frequency

border paddingwindowing

k-space filteringzero padding

actionsData processing

Time1

3

4

pzh(x, y, t)

pzh(kx, ky, ω)

pzs(kx, ky, ω)

pzs(x, y, ω)

2 pzh(x, y, ω) FFTt

FFTx,y

2D

Gp(kx, ky, ω)x

Transformation

IFFTx,y

2D

Table 2.1: schematic view of PNAH strategy

Table 2.1 shows the road map to performing PNAH. The basic steps, together with their dataprocessing features for accurate PNAH measurements, are provided in this table. These stepswill be discussed gradually in this section. In figure 2.2 the setup for PNAH measurements isshown schematically. Here zs, the source plane, defines the boundary between the source halfspace (z ≤ 0) and the source-free half space (z > 0). The pressure distribution from a plane inthe source-free half space can therefore be maximally extrapolated to the source plane zs. At acertain distance zh measurements are made parallel to the source plane. In thet hologram planeat zh, an array of points is defined in the x and y direction as shown in the figure. For each ofthese points a pressure measurement is performed for a certain period of time.

︸︷︷︸ ︸︷︷︸

∆x

∆y

pzh(x, y, t)zs zh

z

y

x

y

a) b)

Figure 2.2: Basic setup for PNAH measurements. a) side-view, b) front view

The analogue acoustic signals are sampled to process the data digitally. To prevent under-sampling of the signal, also known as aliasing, the sampling frequency must be chosen at leasttwo times the maximum frequency present in the signal (fs ≥ 2fmax). This statement is basedon the Nyquist Sampling Criterium [4]. Each individual point is then Fourier transformed re-sulting in the pressure distribution as a function of frequency. From this pressure distributionpzh

(x, y, ω) several interesting frequencies are chosen, which are further analyzed. But beforewe proceed with the PNAH method one has to account for the effects of the transition from thetime-domain to the frequency-domain. A common problem of the FFT (fast Fourier transform)is the introduction of signal leakage. Leakage describes the loss of power of a given frequency toother frequencies in the FFT spectrum. When performing FFT, measurement data is repeated aninfinite amount of time. FFT is based on the fact that the measured data is fully periodic, whichis often not the case. If large differences between the end and the beginning of a measurementexist, high frequencies (non-physical) are introduced into the frequency-domain data. Signal leak-age is reduced by filtering the beginning and end of a measurement, which results in periodic

11

measurement data. This smoothening of the endpoints is accomplished by applying a windowon the measured data. A drawback of this method is that the data at the borders is affected, sinceit does not represent the actual acoustic data after windowing.

−2

0

2Original data

0

0.5

1Hanning window

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−2

0

2

Time Domain [s]

Windowed data

(a) Influence of windowing on original data

0 5 10 15 20 25 30 35 40 45 500

20

40

60

80

100

120

140

160

180

200

Frequency Domain [Hz]

Original data

Windowed data

leakage effects

(b) Frequency plot of original and windowed data re-spectively

Figure 2.3: Consequences of windowing technique on measurement data

However, this disadvantage is decreased by introducing so-called border-padding (see [2]).With this technique, which is based on zero-padding, the border values of a measurement arepadded outward resulting into a largermeasurement area. Now, by placing the part of the window,which does not affect the data over the actual measured data and the side tapering of the windowover the border-padded parts, one has the result of reducing leakage without affecting the actualmeasured data (see figure 2.4). The increased measurement grid also affects the size of thewavenumber increments ∆kx and ∆ky. Again, the reader is reminded of the analogy shownin (2.4). The sampling wavenumber and the wavenumber increment are defined as kx,s = 2π

∆x ,∆kx = 2π

X and ky,s = 2π∆y , ∆ky = 2π

Y for x and y respectively. Here X and Y represent the axeslenghts of the measurement grid. Consequently, increasing the size of the measurement gridreduces the wavenumber increment. Hereto, a more detailed wavenumber-domain is obtained.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

−1

0

1

2Original data

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4−2

−1

0

1

2Borderpadded data

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4−2

−1

0

1

2

Time Domain [s]

Windowed borderpadded dataWindow

Figure 2.4: Borderpadding

12

After the border-padding and windowing techniques are applied, the data is again Fouriertransformed. Unlike the common transformation over time, now the spatial data is Fourier trans-formed over space. This results in the transition from the spatial domain to another domain,called the wavenumber domain. The measurements are made in a two dimensional plane (x-and y-direction) and therefore a two dimensional Fourier transformation over space is necessary.As a result of this transformation one ends up in the wavenumber domain. This transformationis made to be able to make use of the propagators mentioned earlier, which are defined in thewavenumber domain. In the spatial domain, the propagators cannot be used and convolutionintegrals are required to determine extrapolated pressure fields. However, in the wavenumberdomain the extrapolated sound fields are determined by a relatively easy multiplication.In the wavenumber-domain, the noise level in themeasurements has to be accounted for. A smallsignal-to-noise ratio (SNR) easily leads to a blow-up of the noise, because of the multiplicationof this pressure distribution in the wavenumber-domain with a propagator (rising exponential).Small SNRs mainly occur at high wavenumbers (possibly evanescent waves) since these waveshave relatively small acoustic level amplitudes compared to low wavenumbers, which leads tosmaller SNR assuming a constant noise level (see figure 2.5).

propagating wave

evanescent wavedistance from source

acou

sticlevelamplitude

noise level

nearfield farfield

Figure 2.5: Plot of acoustic level amplitudes as a function of the distance from the source forpropagating and evanescent waves [15]

Because relatively high wavenumbers are more sensitive to noise, this data has to undergo an-other filter operation to prevent erroneous results. By applying a k-space lowpass-filter, the higherwavenumbers are suppressed and consequently only data with a relatively large SNR is analyzed.Unforunately, determining the cutoff-frequency of the k-space filter is not straightforward. Onthe one hand a higher cut-off wavenumber results into a higher resolution, but on the other handa smooth tapering must be used to prevent unwanted FFT effects, e.g. sinc-effects. A method tofind the ideal cut-off wavenumber is called L-curve regularization [16]. A typical L-curve shape ispresented in figure 2.7.By plotting the norm of the k-space filtered pressure distribution at plane z and the norm of thedifference between the actual distribution and the filtered distribution at the hologram plane zh,the effects of too much/little filtering is clearly observed. On the one hand, choosing the cut-off wavenumber kco relatively small makes sure that the backpropagated pressure field does notblow up, but also leads to a distorted representation of the pressure field at the hologram planezh (large difference between the actual distribution and the filtered distribution at zh). On theother hand, a large kco (low filtering) results in a good representation of the measured pressure

13

k[rad/m]kco−kco 0

1

0.5

Figure 2.6: Example of a possible k-space filter

over filtering

ideal kco

underfiltering

||p(kx, ky, zh) − pf (kx, ky, zh)||

||pf(k

x,k

y,z

)||

Figure 2.7: L-curve used for finding ideal cut-off wavenumber

field, but leads to blow-up of the backpropagated pressure field at z. Therefore, a compromisemust be made and the optimal cut-off frequency is found at the bend of the L-curve. Hitherto theoptimal cut-off frequency cannot be found beforehand. L-curve regularization provides an area inwhich the optimal cut-off wavenumber can be found. Analysis of the results for different valuesof kco in this region is needed to fine-tune the search for the ideal kco. After applying all thesetechniques the pressure distribution at the hologram plane zh in k-space is finally multiplied bythe propagator Gp to calculate the pressure distribution in another plane z. An additional way toimprove the acoustical image is to apply k-space zero-padding; by adding higher wavenumberswith zero energy, the k-space is made larger (increase of kx,max), resulting in an up-samplingof the spatial data (decrease of ∆x). Note that this action effects the spatial domain, similar asborder-padding effects the wavenumber domain. The pressure distribution in the spatial domainis ultimately obtained by performing a two dimensional inverse fourier transformation on thepressure distribution in the wavenumber-domain.

2.4 Discussion

Evanescent sound waves contain a lot of acoustical information. A property of these waves istheir fast exponential decay in space. Consequently measurements must be performed in closevicinity, the nearfield, of sound sources to capture evanescent waves. PNAH utilizes this informa-tion by measuring the sound in the nearfield of sound sources, which results in high resolution

14

holograms of the soundfield. Also, a large quality of the PNAH method is the relatively easysoundfield propagation to any desired plane. This is achieved by performing the calculations inthe wavenumber domain. Additionally, the PNAH analysis has shown that proper post process-ing of acoustical data can influence the end result significantly. This is exactly where a lot ofimprovement is achieved for the PNAH strategy (see for example Scholte [2]). Data processingsteps as border and zero-padding, k-space filtering and very important L-curve regularization areable to significantly improve the soundfield visualization quantitatively as well a qualitatively.

15

16

Chapter 3

Theoretical analysis of sound sourceidentification methods

3.1 Introduction

NAH is a powerful tool to investigate the characteristics of acoustic sources. Performing acous-tic holography on an object results in an estimation of the present sound field. In case multiplesound sources share the same frequency bands, a sound field measured or estimated by near-fieldacoustic holography is a summation of the contributions of the individual sources. Therefore,acoustic holography, provides a total view of a soundfield and is unable to estimate the individualsource contributions which make up a composite soundfield.The ability of locating individual sources and distinguishing their contributions is of great valueto a better understanding of sound fields. Being able to extract individual contributions from acomposed sound field can for instance result into linking separate sound contributions to specificphysical parts of an object (connection between acoustics and dynamics).This chapter covers three major topics. In section 3.2 the principles of PCA and its specific ap-plication for sound source identification is discussed. Hereafter the partial field decompositionmethod of Nam is analyzed in section 3.3. For Nam to be applicable for single point hologrammeasurements the method of Spatial Transformation of Soundfields (STSF) is needed and thismethod is discussed in section 3.4. The analysis of these two identification methods has re-sulted in the development of a newmethod named Principal Component Analysis with AdvancedSoundfield Observation (PCAASO). This method combines the qualities of both PCA and Namand is introduced in section 3.5. Finally, the chapter is concluded with a discussion of all threesound identification methods.

3.2 Principal Component Analysis

As mentioned in section 1.3 the basics of PCA are introduced by the works of Hotelling. In[6] the concept is described and discussed extensively. This concept is discussed here briefly todemonstrate the origins of PCA.

The main goal of Hotelling is to find a selection of data, which would represent a minimalrepresentation of the entire data set. Consequently a significantly smaller data set is used todescribe the original data set. Now suppose that x is a vector of p random variables. If the numberof variables is large, it is time-consuming to look at all p variances and 1

2p(p − 1) covariances to

17

analyze the properties of the data set. Hotelling therefore introduced a method with which a largedata set could be qualitatively described by using a lot less variables. The first step in this methodis to look for a linear function αT

1 x of the elements of x which has maximum variance.

αT1 x = α11x1 + α12x2 + ...α1pxp =

p∑j=1

α1jxj . (3.1)

Here α1 is a vector of p constants α11, α12, ..., α1p The second step is to find a function αT2 x,

uncorrelated to αT1 x, which has maximum variance. And so the iteration goes on until the pth

time. αTp x, defined as the pth principal component, has maximum variance, but is uncorrelated

with αTi x for i = 1, 2, . . . , p, i 6= p. Up to p principal components are found, but in general

most of the information in x will be accounted for by less principal components. In principle, thefirst few principal components will account for most of the variation in the original variables andthe last few principal components add very little information about the data set. Now considerthe case where the vector x has a known covariance matrix Σ. This matrix has the variancesof the variables on the diagonal and the covariances between the variables outside the diagonal.Mathematically it is proven that for the principal components αT

k x (k = 1, 2, ..., p), αk is theeigenvector of the covariance matrix Σ, which corresponds with the kth largest eigenvalue λk.This idea forms the foundation for PCA. Hotelling considered PCA predominantly as a methodto reduce the complexity of a data set. However, the next passages show that PCA is also used forother purposes.

3.2.1 Basics of PCA

To illustrate the basic ideas of PCA, an example of a spring-mass system shown in figure 3.1(a)is used throughout this section. By moving the mass down, outside its equilibrium position andletting go, the mass should translate continuously up and down assuming no friction forces. Themovement of the mass is our main interest and we try to analyse this by capturing its movementusing a camera. Of course we know that the line of action is along the z-axis, but we pretendwe do not have any information about the system a priori. Therefore we position the cameraarbitrarily in a plane parallel to the mass. The slightly tilted camera captures the position of themass with a certain sampling frequency and a possible data set of a measurement with this con-figuration is shown in figure 3.1(b). Here the circles represent the position of the mass duringthe measurement.Basically the goal of PCA is to clarify the data by re-expressing the data by means of a new coordi-nate basis. Now consider an [e× t] original data matrix E, where e is the number of dimensionsmeasured and t represents the number of time samples taken. The matrix E can be converted toa new representation of the data, C, by an [e× e] transformation matrix T ;

C = TE. (3.2)

The goal is now to find a transformation matrix T such that C becomes the most clear coordinatebasis. Evidently from figure 3.1(b), the directions with most variances contain the most valuableinformation about the movement of the spring-mass system. We consequently assume that thedirections with largest variances in our data set contain the dynamics of interest. Note that thisassumption is in line with the Hotelling theory treated in the introduction of this section. Ttherefore must act as a rotation of the initial axes to a new axis orientation, which contains themaximum variances along the axes.

18

cameraorientation

m

xy

z

(a) spring-mass system with camera ori-entation

cx

cy

(b) position data about with camera

Figure 3.1: Case example for PCA explanation

Furthermore, the data in figure 3.1(b) can also be used to investigate possible linear relation-ships between the axes of the initial coordinate base. If there would exist a linear relationshipbetween the axes, then data can be reduced to only encompass the independent axis directions.Also in case of figure 3.1(b) it is clear that the data is predominantly present along one specificdirection. Since we have used a two dimensional frame (cx, cy) for the measurement, one dimen-sion can be discarded without losing much information. This possibility of data reduction is inliterature called redundancy and is very helpful to analyze and clarify large data sets. Redundancyis considered a very strong quality of PCA. A statistical function tomeasure the linear dependencebetween signals is the covariance. Therefore with the transformation matrix T ultimately a maxi-mization of the variances (finding the new axes) and a minimization of the covariances (utilizingthe fewest number of dimensions for a measurement) is desired. The [e × e] covariance matrixCc has the variances and covariances of C on the respective diagonal and off-diagonal positions.Thus, with the desire to maximize the variances and minimize the covariances, the matrix Cc

would ideally become a diagonal matrix. The relationship between E and C is C = TE, whereT is chosen such that covariance matrix Cc is diagonalized. The rows of T , pi for i = 1, 2, . . . , e,then become a set of new axes for expressing the original dataset E. These rows are also calledthe principal axes of E.

The covariance matrix of the original data set Ce = EET is a symmetric matrix1. When werewrite the covariance matrix Cc in terms of Ce using (3.2) we find that

Cc = CCT = (TE)(TE)T = T (EET )T T = TCeTT . (3.3)

Since the matrix T merely acts as a rotation matrix, the covariance matrix Cc is also proven to bea symmetric matrix. From linear algebra theory [8] it follows that any square symmetric matrix isorthogonally diagonalizable. This means that a symmetric matrix, for example Ce, can be writtenin the following form

Ce = UDUT , (3.4)

where the columns of U are the eigenvectors of Ce and D is a diagonal matrix containing the

1A matrix A is symmetric if A = AT

19

eigenvalues. Substituting (3.4) into (3.3) gives

Cc = TCeTT = T (UDUT )T T = (TU)D(TU)T . (3.5)

Since the columns of U are eigenvectors and thus orthogonal, U is a orthogonal matrix. A prop-erty of orthogonal matrices is that the transpose of a matrix is equal to the inverse, i.e. UT = U−1.This property can be very well exploited by choosing T = UT and substituting this in (3.6);

Cc = (TU)D(TU)T = (UT U)D(UT U)T = (U−1U)D(U−1U)T = D. (3.6)

It is evident that the choice for T diagonalizes Cc, since D is a diagonal matrix. The decomposi-tion of (3.4) can be achieved by several ways, such as eigenvalue or Cholesky decomposition, buthere a singular value decomposition (SVD) is used. SVD is preferred over other methods sinceSVD also ranks the eigenvalues/singular values in descending order in the diagonal matrix D.This implies that the first x principal components contain a relatively larger amount of the infor-mation compared to the lower positioned principal components. The lower positioned principalcomponents consequently represent less of the total data set and are less important. ThereforePCA is seen as a data reduction possibility, since only a few yet significant , principal componentscould suffice to describe the total data set. The effect of performing PCA for the spring-massexample is shown in figure 3.2. Here is shown that the original bases (cx, cy) is rotated and thedata set is represented by a new coordinate base (p1, p2), which are the rows of the matrix T . Theeigenvalues di for i = 1, 2, . . . , e in D determine the significance of the new axes p1, p2. Note thatp1 has a significantly larger corresponding eigenvalue d1 compared to p2 and thus is relativelymore important for representing the data. Hereto the axes of p2 could be considered unimpor-tant for the movement of the mass and therefore be disregarded in the further analysis. HerePCA is used for data reduction, but in this thesis PCA is predominantly used to determine theindependent axes of a data set.

cx

cy

fitted linecx

cy

p2

p1

e1

e2

PCAtransformation

Figure 3.2: Result of PCA operation on initial data set. e1 and e2 are the axes scaled to importancebased on the respective eigenvalues d1 and 2. Left: initial data, right: PCA representation of initial

data set. e1 = d1p1 and e2 = d2p2

3.2.2 Application of PCA for sound identification

Now the use and interpretation of PCA for the sound source identification problem is discussed.Note that all variables in the following sections are observed in the frequency domain, whereas

20

the variables in the previous subsection are in the time domain. Utilizing PCA for sound sourceidentification requires reference and target signals. Reference signals are commonly obtained bymicrophones or particle velocity transducers and are ideally placed in the near environment ofthe sound sources that make up the soundfield. The locations of these references are essential fora correct outcome of the PCA procedure. Therefore, in section 3.2.4 the focus is on the influenceof the reference locations on the identification results. Target signals are considered to be themeasurements made at the location of interest for the sound identification, which for PNAH arethe grid measurements at the hologram plane. In brief, PCA for sound source identification isperformed as follows: The reference signals are considered as the original data analogously toE in the previous subsection. By performing a SVD of the original data the linear dependenciesbetween the reference signals are identified and eliminated. This operation consequently intro-duces a new set of independent variables, called the principal components. Now, by correlatingthese principal components to the target signal measurements, the independent signals at thetarget locations are extracted and consequently the independent partial sound effects at the placeof interest are identified. The theoretical foundation of PCA for sound identification is discussedbelow.Consider R as a [r × f ]-matrix, where row i represents the f -line frequency data of referencesignal i, i = 1, 2, ..., r. Our aim is now to find a new set of uncorrelated signal spectra R′ withinthe measured reference spectra R, using a set of unit linear combinations T :

R′ = TR. (3.7)

Note the analogy with (3.2). In literature, the rows of R′ are called the principal componentsand are defined as the mutually independent signals contained in the reference matrix R. Asstated before PCA aims at finding these independent signals and achieves this by performinga singular value decomposition on the spectral matrix SRR. The main reason for choosing thisdecomposition method is that SVD or the highly similar eigenvalue decomposition (ED)2 is theideal linear separationmethod when only a single frequency is considered. SVD is also commonlyused in control to decouple multi-input multi-output (MIMO) - systems [20]. The SVD of thespectral matrix SRR gives

SRR = RRH = UΛV H . (3.8)

Note that SRR and all other matrices in this section correspond to a single frequency-line. SinceSRR is a positive definite Hermitian matrix, V is equal to U as shown in Appendix A. The spectralmatrix SRR has the autopower spectra of the reference signals on its main diagonal and thecross spectra between the reference signals on the off diagonal positions. Λ is a diagonal matrix,containing the eigenvalues/singular values λi, i = 1, 2, . . . , r, of SRR. The columns of U , Ui,i = 1, 2, . . . , r, are the eigenvectors of SRR corresponding to the respective eigenvalues λi. Byusing (3.7), the spectral matrix of R′ is written as

SR′R′ = R′R′H = (TR)(TR)H = T RRH︸ ︷︷ ︸SRR

TH . (3.9)

The eigenvectors Ui span the data-space of SRR and are defined as the independent directionswithin the data-space. These eigenvectors are useful when determining the independent spectrawithin a data set. By substituting the singular value decomposed SRR in (3.9) we find

SR′R′ = R′R′H = TSRRTH = TUΛUHTH = (TU)Λ(TU)H . (3.10)2See appendix C for more details about SVD and ED relation

21

The principal components R′ are mutually independent, which implies that the off-diagonalterms of SR′R′ should be zero. By choosing the transformation matrix T as T = U−1 = UH

(orthogonality property) and substituting this in (3.10) we finally find that

SR′R′ = R′R′H = (TU)Λ(TU)H = (U−1U)︸ ︷︷ ︸I

Λ (U−1U)H︸ ︷︷ ︸I

= Λ, (3.11)

where I is defined as the identity matrix. As a result of the above choice for T the spectral matrixSR′R′ becomes the eigenvalue matrix Λ. Note that since SR′R′ now is a diagonal matrix, the crosspower spectra between the principal components R′

i are all zero. This deduction confirms theearlier mentioned independence of the principal components. Also, these eigenvalues λi of Λare considered as the autopower spectra of the principal components R′

i.Now that the principal components are found, the identification of the sources at the hologramplane is initiated. Consider P as a [N × f ]-matrix, where row i represents the f -line frequencydata of target signal i, i = 1, 2, ..., N . The spectral matrix SPP , with the autospectra of the targetsignals on the diagonal and the cross spectra on the off-diagonal positions, is calculated as

SPP = PPH = SPRS−1RRSRP , (3.12)

where SPR represents the cross spectral matrix between the target signals and the referencesignals. Proof for the definition of SPP in (3.12) is written as

SPP = SPRS−1RRSRP = (PRH)(R−HR−1)(RPH) = P (RHR−H)︸ ︷︷ ︸

I

(R−1R)︸ ︷︷ ︸I

PH , (3.13)

where again I represents the identity matrix. Now by substituting the SVD of SRR from (3.8) into(3.12) we find

SPP = PPH = SPRUΛ−1UHSRP = (SPRUΛ−1/2)(SPRUΛ−1/2)H . (3.14)

From (3.14) it is clear to see that the contributions of the principal components to the total sound-field are defined as P ′ = SPRUΛ−1/2. Consequently the individual contributions of the incoher-ent sources to the plane of interest are determined as

P ′i = SPRU(i)Λ(i, i)−1/2 for i = 1, 2, . . . , r, (3.15)

where U(i) is the ith column of U . Finally note that the total sound field is obtained by adding theindividual sources together on a energy basis. This concludes the theory of finding the incoherentsound effects using PCA. The following paragraphs are aimed at investigating the implicationsand results of the identification method.

Before applying PCA, it must be clear what information the analysis exactly provides. Theindividual principal components are the independent sound effects observed in a data set. Trans-forming this idea to the sound identification problem, the mistake that the principal componentsrepresent the individual sound sources in the sound field is easily made. It must be noted thatthis is not the case and that principal components generally do not have a direct relationship withphysical sources. However, each of the principal components correspond with independent, un-correlated sound effects observed in the measured reference set. The difference between incoher-ent sound sources and incoherent sound effects must be emphasized, since this understanding isessential for understanding the results of PCA. To illustrate this important property an example

22

is discussed. Take case A where there are two physical sound sources b1 = s1 and b2 = s2, whichare uncorrelated. Application of PCA leads to the decomposition of the uncorrelated effects inthe sound field. Since, by chance, the uncorrelated effects are equal to the two sound sources, aseparation of sound sources is performed. Now define three sound sources b1 = s1, b2 = s2 andb3 = s1 for case B. Contrary to the previous case, here there are two mutually coherent sources b1

and b3. In this case PCA does not lead to a separation of the three individual sound sources, butagain to the decomposition of the uncorrelated effects s1 and s2. This is shown schematically infigure 3.3. Therefore this property is very important in the application of PCA and the subsequentanalysis of the results.Section 3.2.1 demonstrated that the principal components from a PCA analysis are ordered based

Case A

Case B

totalsoundfield source 1 source 2

incoherent incoherent

b1 = s1

b3 = s1

b2 = s2

b1 = s1 b2 = s2

Figure 3.3: Identification of sound sources using PCA. For case A the incoherent effects corre-spond to the incoherent sources, which (by chance) results in the identification of the individual

sources. In case B this is not the case.

on significance. Since the trace of SR′R′ = Λ, being the sum of the eigenvalues λi, is considereda measure for the total power in the measured reference set ([12]) PCA therefore distributes thetotal power to progressively smaller incoherent sound effects. Consequently, the first number ofprincipal component spectra with a significant amplitude level determine the number of domi-nant independent effects present in the reference signal data set. Hereto the study of a complexnoise and vibration problem is reduced to the study of independent, uncorrelated sound effects.The main problem that exists here lies in the meaning of the word significant. There are no clearguidelines to label a principal component significant or not. Therefore, the labeling of significantprincipal components must be evaluated case by case and by using engineering insight. In sec-tion 3.2.3 virtual coherence methods are discussed, which facilitates the problem of determining’significant’ principal components.The theoretical discussion of PCA assumes that the number of principal components is equal tothe number of uncorrelated sound effects. This assumption does not generally hold in practice,since artificial uncorrelated effects can be introduced by measurement noise and/or data acquisi-tion effects. The application of Fourier analysis, for example, possibly leads to bias errors in thedata, which introduces additional low principal components. Some low principal componentsmay be due to low power sources that are less clearly represented in the reference signals, but

23

they can also be generated by uncorrelated signal noise. Virtual coherences, discussed in the nextsection, are helpful in distinguishing between real and artificial sound effects.

3.2.3 Virtual Coherence Analysis

With PCA, it is possible to obtain an uncorrelated set of principal components from a set ofcorrelated references. However, allocation of the energy to the principal components is onlyuseful for the source identification problem if the principal components have a direct physicalmeaning. In this section coherence functions are used to correlate the principal componentswith the physically measured reference signals. This is done to analyze the relationship betweenthe principal components and the physical sound sources. The method which linearly relatesthe reference signals to the principal components is called the virtual coherence analysis. Herethe term ’virtual’ is used, since the reference signals are linked to the mathematically obtainedprincipal components. These principal components generally do not represent physical soundsources, yet they represent the independent sound effects, the virtual sources, instead.The basic statistical relationship between two signals in the frequency domain is given by thecross power spectrum. The cross power spectrum is the frequency domain equivalent of thestatistical covariance between signals in time. By introducing the virtual cross power spectrum asthe cross power spectrum between the reference signals and the principal components as

SRR′ = RR′H , (3.16)

the mutual relationship between signals is investigated. Often the measure of dependence be-tween two signals is determined by the statistical correlation. A correlation of 1 between twosignals means that the signals are totally correlated, where a correlation of 0 implies two com-pletely independent signals. The frequency domain equivalent of the correlation function is thecoherence function. The coherence function measures the degree of linear dependence betweenthe spectra of two signals in a linear least squares sense [12]. Analogue to the normal coherencefunction, the virtual coherence function γ2

RR′ is defined as the ordinary coherence between thereference signals and the principal components;

γ2RR′ =

|SRR′ |2

SRR · SR′R′. (3.17)

The principal components are, by definition, uncorrelated. This signifies that the coherence func-tions between the principal components are also uncorrelated and consequently can be simplyadded. The coherence functions of all the principal components with a single reference signal,should eventually add up to one;

r∑j=1

γ2RiR′

j= 1, (3.18)

where r represents the total number of principal components. Equation (3.17) proves to be ahelpful tool in estimating the dimension of the observed sound field and judging the importanceof the calculated principal components.With PCA the principal components are ordered according to themagnitude of the correspondingprincipal values. As mentioned earlier the low principal components can represent low poweredsound sources but can also be induced by independent noise effects in the soundfield. A low

24

principal component is considered a representation of a weak sound source if it has a high virtualcoherence with one reference. But if the virtual coherences are significant over all the referencesthe low principal component most likely represents a noise effect. Also, in general a high virtualcoherence between one principal component and one physical reference locations gives qualita-tive information on the location of sources and render some physical significance to the principalcomponent. Virtual coherences are also used for determining the number of significant principalcomponents. If for example, the summation of the first x virtual coherence functions of a refer-ence signal approximately equals one, then the sound field at the location of the reference signalis predominately defined by x independent sound effects. Ideally for PCA the references each re-ceive a single incoherent sound effect. By using virtual coherences, a validity check is performedon the references. References that capture several incoherent sound effects are less useful forthe sound identification and consequently these references can be relocated or neglected for theanalysis based on the virtual coherences. The virtual coherence functions therefore are a valuableaddition to PCA by providing better interpretable information about the relationship betweenthe reference signals and the derived principal components and qualitative information on thelocation of the physical sources.

3.2.4 Influence of reference placement on sound identification

For PCA to be successful as a source identification technique, the placement of the references isessential. Source identification can only be performed properly if the locations of the referencesare well chosen and the contribution of one of the sound sources is dominant. If these conditionsare satisfied, PCA can be used as a source identification technique and the principal componentsare given a physical significance. Unfortunately it is difficult to make these conditions generallyapplicable by stating a clear criterion. Therefore its validity must be evaluated case by case. Inthis section the influence of reference positioning on the identification results is discussed.Consider the relation between a set of physical sources S and reference signals R as

R = HS. (3.19)

HereH is the [r×s]-transfer matrix between the r references and the s sources. Also consider thatthe sources S are mutually incoherent. This implies that the [s×s] spectral matrix SSS = SSH isof full rank. With (3.19) the relationship between the spectral matrices SRR and SSS is obtainedas

SRR = RRH = (HS)(SHHH) = HSSSHH , (3.20)

where SSS is a [r × r] diagonal matrix. To be able to associate the principal components di-rectly with the physical sources S, the transfer function matrix H should be proportional to theeigenvector matrix U from equation (3.8). The following equations clarify this statement;

SRR = HSSSHH

H = αU

}SRR = α2USSSUH ,

SRR = UΛUH equation (3.8)(3.21)

where α is a scalar. Note that the construction of U is dependent on the reference locations. Un-fortunately, H = αU is a very difficult condition to fulfill and is hardly ever met in practice. Thismakes the direct linking of principal components and physical sources difficult if not impossible.

25

Despite this difficulty, the analysis can be simplified by trying to find the relationship betweenprincipal components and incoherent sound effects instead of the physical sources.It is clear that to represent the total sound field, the rank of the transfer function H must be at

X

Y

X

Z

Y

(a) XY axis

Z

X

X

Z

Y

(b) XZ axis

X

Z

Y

(c) XY Z axis

Figure 3.4: Analogy between space and references

least equal to the number of present incoherent sources s. Therefore the number of referencesshould always be equal or more than the present incoherent sources (r ≥ s) for correct determi-nation of the present soundfield. This criterion can be clearly visualized by considering a threedimensional space. The number of principal components found from a SVD is equal to the num-ber of used references. Here a scenario of three references and three incoherent sound sourcesis considered and, consequently, three principal components are obtained from the decomposi-tion. These principal components represent the incoherent/independent sources within the totalsoundfield. Transposing these principal components to a spatial domain, the principal compo-nents form an orthogonal basis that spans a three dimensional space. Each principal componentthus characterizes an orthogonal axis of the three dimensional space. Assume now that instead ofthree references, two references are used. For the sound identification problem, two incoherentsound effects would be found instead of the three present effects. Consequently the sound fieldformed by three incoherent sources cannot be captured using two references. The spatial analogyfor this case is that a two dimensional basis is used to represent a three dimensional space. Thisis shown in figure 5.4. The two dimensional basis (X, Y ) finds two coordinates of the box object

26

correctly, but is unable to find the third coordinate; the object could be positioned anywhere alongthe bar in figure 5.4. Similarly a different two dimensional basis (X, Z) is used to represent thethree dimensional space (figure 5.4). Again only two coordinates are obtained. As is the casefor the sound identification problem, it is clear that a 2D orthogonal basis is unable to representa three dimensional space completely. The three dimensional orthogonal basis (X, Y, Z) is ob-tained by combining the two two dimensional bases (X, Y ) and (X, Z) . The exact location of theobject is now obtained as is shown in figure 5.4. The addition of an extra orthogonal axis resultsin a complete representation of the three dimensional spatial field by the three orthogonal axes.Therefore, to represent an incoherent soundfield completely, the number of references must beat least equal to the number of incoherent sources.Note the assumption of correctly placed references is made in the analogy between the spatialaxes and the references. Using three references when three incoherent sources are present doesnot guarantee a correct representation of the sound field. The references should also be placedcorrectly such that for each reference the contribution of one of the respective sound sources isdominant. Let us go back to the spatial analogy to accentuate this property. Consider an objectplaced at a two dimensional plane. First, take the axes X, Y , which correspond with the orthog-onal basis shown in figure 5.4. With this basis the location of the object is correctly found, sinceeach axes (predominantly) represents a independent direction within the two dimensional plane.Now take as axis X and X + 0.2Y . The construction of a new orthogonal axes with these axes(SVD operation) results in an undervaluing of the independent direction of Y . Consequently thepositioning of the object in this direction is not correctly represented.Whether the number of dominant principal components, and hence the rank of spectral refer-ence matrix SRR, denotes the number of sources, depends on the rank of the transfer matrix. Ifthe transfer matrix is full rank, the rank of SRR equals the rank of SSS . This rank then equalsthe number of uncorrelated sources. On the other hand, if H is rank-deficient, there are linearrelationships between the columns of H ; The sources in combination with the references causecorrelated responses. For example, suppose two referencemicrophones and two acoustic sources,which are at equal distance of the microphones. The transfer functions of each source to the mi-crophones are identical and the rank of H as well as of SRR are one. Hence, only one incoherenteffect is recognized. The same conclusion hold for reference transducers, which are placed at thesame location. For the spatial analogy, this is considered similar to taking one dimension twicefor a basis of a two dimensional plane. Ideally, the reference transducers should be as close to thesound sources as physically possible in order to perform a proper decomposition and relate theprincipal components with the physical incoherent effects.

3.3 Partial field decomposition with NAH

The literature study in section 1.3 shows the many attempts made to identify sources in a com-posite sound field. The majority of the proposed identification methods are based on partial orvirtual coherence analysis. An essential part of these coherence based approaches is the correctplacement of the reference transducers; Ideally, the reference transducers are placed at the exactlocations of the respective present sound sources. This condition implies that one requires infor-mation about the locations of the sound sources before carrying out ameasurement. This require-ment of a priori knowledge stands perpendicular to the purpose of the identification technique,since finding the locations of sound sources is exactly one of the main reasons of performingsound source identification. Moreover, in practice it could be difficult to place the references at

27

the earlier obtained source locations. Nam et al. [10, 11] introduced a sound source identificationmethod that does not need references to be placed near the physical sound sources. Calculated,rather than measured, source signals are used for the identification. The method is also strictlyfocussed on source identification using NAH and therefore could be an interesting alternativefor sound source identification besides the coherence based methods. The Nam identificationmethod is discussed here in detail.

The main principle of Nam is based on the theory of multi-input multi-output (MIMO) sys-tems, discussed in [7]. First, consider the system shown in figure 3.3. Here x and y are the

input output

noise

x(f)y(f)

n(f)

y(f)+

transferfunction

measuredoutput

Hxy(f)

Figure 3.5: a single-input/single-output system with output noise

respective input and output of the system. Due to a noise signal n the actual output y is con-taminated. This results in the measured output y = y + n. The noise signal n is assumed to beincoherent to the input signal x. Also, let w be a signal coherent to the input x. That is,

w = cx, (3.22)

where c is a constant scalar. Then, according to [7], the true output spectrum syy is expressed as

syy = γ2wy syy =

|swy|2

sww, (3.23)

where γ2wy is the coherence function between w and y, syy is the measured output autospectrum,

sww is the autospectrum of w and swy is the cross-spectrum between w and y. This means thata true output spectrum can be calculated from a distorted output if a signal coherent to an input,w, is known. Now, consider the two-input/single-output system in figure 3.3. The x1 and x2 aretwo inputs, y1 and y2 are the respective outputs and y is considered the total output. Furthermorethe two inputs are assumed to be mutually incoherent. The latter assumption brings about thatthe total output spectrum is formed by the summation of the partial outputs y1 and y2;

syy = sy1y1 + sy2y2 . (3.24)

Since the inputs are incoherent to each other, the output due to one input is regarded as uncorre-lated noise to the other input. Consequently, if a signal wl coherent to the lth input is given, thelth contribution sylyl

to the total output spectrum is calculated by

sylyl= γ2

wlysyy =

|swly|2

swlwl

. (3.25)

The above equation holds since the covariance of wl and the outputs yi, i = 1, 2, . . . , n, i 6= l, areall zero.

28

inputs outputs

x2(f)y2(f)

y(f)+

transferfunctions

totaloutput

Hx2y2(f)

x1(f) Hx1y1(f)y1(f)

Figure 3.6: a two-input/single-output system

The analytical discussion in the previous paragraph can be translated to the sound sourceidentification problem. Consider the incoherent system inputs xi as incoherent sound sourceson the source plane. These sources propagate their sound waves into the surrounding space. Byperforming NAH3 at an arbitrary distance from the source plane one can construct a hologramof the propagated sound field. The respective contributions of the sources xi to the hologramcan than be considered as the system outputs from the previous paragraph, yi. Consequently, thesystems in figure 3.3 represent the respective propagations of the sound sources xi to their con-tributions on the hologram plane yi. The theory now claims that if signals coherent to the soundsources xi are available, their respective contributions on the hologram plane yi are obtained asin equation (3.25). Therefore the main challenge with this technique is to find these signals, wi,coherent to the sound sources. Most conventional methods attempt to obtain the coherent signalswi by placing reference transducers close to the actual sound sources. Nam on the other hand,searches for wi within the data sets of the measured hologram planes. The identification withNam is discussed in the next paragraphs.A first glance at a hologram of a soundfield can already provide useful information about thepresent sources. For example, locations of high pressure on a hologram can intuitively be as-sociated with the whereabouts of existing sound sources. Nam basically uses this potency of ahologram to identify the present sound sources. Let N be the number of measurement points ona hologram plane and pHn the pressure at the nth point. Let M be the number of measurementpoints on a source plane and pSm the pressure at the mth point. Note that all below equations arein the frequency-domain. Define two row vectors as

PH = [pH1, · · · , pHN ], PS = [pS1, · · · , pSM ]. (3.26)

These two vectors satisfy the relationship

Ps = HPH , (3.27)

where H is the transfer matrix of propagating pressure from the hologram to the source plane.Note that the matrix H mentioned here has the frequency equivalent of the wavenumber propa-gator Gp (chapter 2) per point on its diagonal. The spectral matrices on the hologram and sourceplane are defined as

SHH = PHPHH SSS = PSPH

S , (3.28)

where the E and H represent the expectation and conjugate transpose or hermitian, respectively.Rewriting SSS using (3.26), (3.27) and (3.28) leads to

SSS = HPHPHH HH = HSHHHH . (3.29)

3Conventional NAH using one reference microphone is not able to reconstruct a composite incoherent sound field.Hereto a resort to other methods have to be made as in section 3.4

29

Let us assume a soundfield consisting of L sources, which are mutually incoherent. Then thespectral matrix SSS on the source plane is expressed as

SSS =L∑

l=1

SSlSl. (3.30)

Note that (3.30) represents the same relationship as (3.24). Here SSlSlexpresses the contribution

of the lth source to the total spectral matrix SSS . Then, if wl is a signal coherent to the lth source,SSlSl

is rewritten as

SSlSl=

SwlSSHwlS

swlwl

, (3.31)

where

SwlS = wlPHS . (3.32)

The above derivation shows that signals coherent to individual sources are essential for the iden-tification of incoherent sources. Note that, until now, the technique of Nam et al. draws the sameconclusions as PCA, namely that signals coherent to the physical sources are required for thesound source decomposition.From here on the techniques diverge from each other. An assumption that the Nam methodmakes is that the sound fields from individual sources hardly overlap on the source plane. Thatis, the method regards maximum pressure on a source plane as a signal coherent to one source.This is essentially equivalent to the idea of placing sensors near the sources; the sensor ideallyonly captures the sound field from one source. The sound source identification technique of Namis composed of five steps, as is illustrated in figure 3.3.In the first step the pressure on the hologram grid positions are measured. Herewith the holo-gram spectral matrix SHH is calculated. Secondly, the spectral matrix SSS on the source planeis determined, using equation (3.29). These two steps essentially construct the data for acousticholography. In the third step the coherent source signals wl are determined. The maximumvalue of the autospectra of the source plane, the diagonal entries of SSS , is isolated. Here theassumption is made that this signal is constructed from only one source and thus is consideredas coherent to the specific source. Ultimately, the signal is regarded as the autospectrum swlwl

of the signal wl. Then, in step four, the autospectrum swlwlis utilized to estimate the contribu-

tion of the first source to the spectral matrix SSS . With equation (3.31) the data coherent to theautospectrum swlwl

is extracted from the total spectral matrix SSS . This results in obtaining thespectral matrix SSlSl

, which is regarded as the contribution of source l on the total soundfield.The identified source contribution SSlSl

can now be plotted. The calculated SSlSlis removed

from the spectral matrix SSS in step five. The remaining spectral matrix SSS·l, which consists ofthe initial matrix SSS minus SSlSl

, now encompasses all the source contributions except for thesource l. This SSS·l is then again used in step three to find the next source contribution. Stepsthree to five are repeated to estimate the contributions of all the present sources. Note that thenumber of iterations is equal to the rank K of the SHH , since the remaining spectral matrix isempty after the Kth repetition. Note that the most valuable property of the Nam method is thatno references are used and thus its identification results are independent of reference locations.The quality of the proposed method depends on how strongly the soundfields, generated by in-dividual sources, overlap one another. This is because the signals wl are assumed to be coherent

30

Figure 3.7: The sound source identification technique consists of five consecutive steps. Source:Nam et al. [11]

to solely one source and incoherent to all other sources. As shown in the previous paragraph, foreach iteration Nam considers the maximum of a hologram as the coherent source signal wl. Anerroneous sound source identification occurs if the pressure at this position is constructed frommore than one source. The spatial overlap then generates a decomposition error. Unfortunatelyit is difficult to know the amount of spatial overlap or decomposition error beforehand, becausethe spatial overlap is directly related to the locations of individual sources, which are of courseunknown a priori. Clearly the overlap on the source plane will be less than that on the hologramplane as the sound pressure diverges from the sources into the environment. Consequently theestimated contributions become better as the decomposition plane goes to the source plane, be-cause of the reduction of the overlap. [11] even discusses the use of a ’virtual’ decompositionplane, which is positioned behind the source plane. Although physically debatable, experimentsshow better decomposition with this plane as opposed to decomposition on the source plane.In the first step of this method the hologram spectral matrix SHH is determined. This matrixcan only be obtained when measuring with a full array setup. Calculation of this matrix witha single point measurement system is not possible, since phase mismatch will occur betweenthe grid points of a hologram. Consequently, no incoherent sound effects can be identified. Incase of single point hologram measurements sound source identification using Nam can only beperformed when it is preceded by a Spatial Transformation on Sound Fields (STSF) procedure toconstruct an incoherent sound hologram. Since the research of this thesis is focussed on sound

31

source identification with single point hologram measurement systems, Nam therefore has to beperformed in combination with STSF. In the next section the conclusions of the analysis of STSFare discussed. A thorough STSF analysis is provided in appendix B.

3.4 Spatial Transformation of Sound Fields

The NAHmethod as implemented in [15] and discussed in chapter 2 , is very effective in capturingsoundfields consisting of coherent partial sound sources. However, when a soundfield composedof mutually incoherent sound sources is to be measured NAH is unable to determine a represen-tative composite acoustical hologram when performing scan-based measurements. STSF makesit possible to determine composite soundfields.The analysis in appendix B demonstrates that STSF actually consists of two predominant parts.The first part of the technique is dedicated to finding the mutually incoherent sound sources inthe soundfield. Although obtained via a different mathematical path, ultimately a decompositionexactly analogue to the PCA method discussed in section 3.2.2 is used. This becomes clear bycomparing the equations (B.27)-(B.30) of STSF and the equations (3.8),(3.12)-(3.15) of PCA. Theseequations are shown side by side in figure 3.8. The columns of Ah from (B.30) are mathemati-

SRR = UΛUH ,

S−1RR = (UΛUH)−1 = U−HΛ−1U−1.

SPP = SHRP U−HΛ−1U−1SRP

SPP = SHRP U−HΛ−1/2︸ ︷︷ ︸

Ah

Λ−1/2U−1SRP︸ ︷︷ ︸AH

h

.

SRR = RRH = UΛV H

SPP = PPH = SPRS−1RRSRP

SPP = PPH = SPRUΛ−1UHSRP

Pi = SPRU(:, i)Λ(i, i)−1/2

STSF PCA

(E.27)

(E.28)

(E.27)

(E.30)

(3.8)

(3.12)

(3.14)

(3.15)

Figure 3.8: Comparison of equations between STSF and PCA

cally equal to the partial holograms Pi from (3.15). Thus it is concluded that the first part of STSFis analogue to the previously discussed PCA technique. In the second part of STSF, the identifiedincoherent sources are back-propagated to the source plane. After a summation of the back prop-agated partial holograms, the goal of STSF, an incoherent source plane hologram, is obtained.The procedure of hologram plane back propagation is done by using PNAH. This technique isalso thoroughly discussed in chapter 2. Consequently, STSF is considered as a combination ofthe, in the field of acoustics, very well respected techniques of PCA and PNAH.In figure 3.9 a representation is given of the mutual relations between the discussed techniquesand methods in this thesis. Initially, a PCA operation is performed on the reference measure-ments. By correlating the obtained principal components of the reference field with the holo-gram grid measurements, the mutually incoherent holograms are constructed. These partialholograms can consequently be back-propagated to the source plane by means of PNAH. Hereto,the incoherent source planes are acquired. An incoherent source plane is then found by addingthe partial sources together. With this incoherent source plane Nam is then able to identify themutually incoherent sources. It is clear from figure 3.9 that for single point hologram measure-ments the identification results with Nam are in the best case (no source overlap) equal to theresults of PCA.

32

hologramplane

PCAgrid pointmeasurements

referenceplane

PNAH

sourceplane

+

NAM

identified source planes

identified hologram planes

STSF

incoherentsourceplane

referencemeasurements

Figure 3.9: schematic figure showing connections between PCA, Nam STSF and NAH

3.5 Principal Component Analysis with Advanced Soundfield Observa-tion

The analysis of the two identification methods of PCA and Nam in this chapter has provided in-sight into the qualities and weaknesses of the methods. The identification algorithm of PCA isvery powerful, but the results are not unique and dependent on reference positioning. Nam, onthe other hand, does not use reference sensors, but then the identification method has its flaws.In this section a new developed identification implementation, Principal Component Analysiswith Advanced Soundfield Observation (PCAASO), is presented, which combines the strong fea-tures of both methods to obtain a more robust and generic identification method.In this thesis a single point holography measurement system is preferred. This has several rea-sons. Besides practical reasons as the additional costs of a full array and large data handling, a gridmeasurement system is also much more flexible and can be used for many practical scenarios.For example, with a single point measurement system one is able to adjust the spatial resolutionof the grid based on the specific measurement scenario. Therefore this new implementation isdesigned especially for source identification using point to point grid measurements. Mind thatthe implementation presented here must be performed per frequency.The first step of PCAASO is to perform a standard PCA identification procedure. Commonly, thereference microphones are positioned based on the available knowledge of the specific situation.This results in the identification of incoherent sound sources expressed in partial holograms.However, it is not known if this identification is correct and corresponds with the actual sound-field. An insufficient amount of references or incorrectly placed references will lead to the ab-sence or wrongly represented sound sources in the incoherent holograms. In step two of the newimplementation the representation of all sound sources in the PCA analysis is evaluated. Duringa point to point measurement the sound pressure is measured with a measurement microphone

33

Step 1source 1 source 2incoherent incoherent

Perform PCA measurement

Step 2Construct autospectral hologram

Step 3Perform

source 1 source 3incoherent incoherent

source 2incoherent

updatedPCA measurement

Figure 3.10: Steps of PCAASO method, where the crosses represent the reference locations.

at a certain distance from the the source plane. After a point measurement is finished, the mea-surement microphone moves to another location of the grid and then performs another pointmeasurement. This way, the whole grid layout is covered. The consequence of the movementof the measurement microphone along the grid positions, is that each measured location hasdifferent phase information. Hereto the mutually distinguishable property of incoherent soundsources, phase, is lost and therefore no identification can be performed based on these grid mea-surements. However these grid measurements do hold a different quality, which can be exploited.During the point to point measurements, the measurement microphone measures differences inpressure magnitudes. High pressure measurements at a certain location indicate the local pres-ence of sound sources, whereas low pressure areas likely point towards the absence of soundsources. Consequently the grid measurements can serve as a localization of sources in the mea-sured soundfield. For the new implementation the autospectra per grid position are calculated

full array setup

singe point setup

reference positioning

partial source overlap issues

PCA Nam PCAASO

- - n/a ++

yes

no

no no

yes yes

yesyesyes

Figure 3.11: Comparison of several features and applicabilities of PCA, Nam and PCAASO

and together constitute a so-called autospectra hologram. The calculation of the autospectra holo-gram is step two of the new implementation method. With the autospectra hologram two aspectsare evaluated. First, the autospectra hologram is compared with the identified incoherent holo-

34

grams. If the presence of a high pressure area in the autospectra hologram is not encounteredin the incoherent holograms, this indicates that a present source is not captured by the initialPCA analysis. Therefore, the PCA analysis is not considered to be a complete representation ofthe actual present sound scenario. A response to such a case would be to perform an additionalPCA analysis for which extra references are placed at the high pressure locations observed on theautospectra hologram. Also, the autospectra hologram is of use even if all high pressure areasare captured by the initial PCA analysis. The high pressure areas indicate the locations of soundsources and consequently the references are preferred to be located at these locations. Therefore,the autospectra hologram is also exploited as a way to evaluate the positioning of the references.Repositioning of the references based on the autospectra hologram for a consecutive PCA analy-sis will consequently lead to an optimized measurement setup. The second PCA measurementwith the extra and/or optimized reference locations is considered as the third and final step of thenew implementation method. Note that performing an initial PCA measurement as in the firststep of PCAASO is not essential. However it does provide a reference to determine the amountof improvements PCAASO delivers with respect to the conventional PCA method.

3.6 Discussion

In this chapter two existing sound source identification methods are highlighted and discussedextensively. PCA provides a powerful method to identify incoherent sound sources. With PCAit is possible to transform a set of correlated references into a new set of uncorrelated princi-pal components. In our case, these individual principal components represent the independentsound effects observed in a data set. Representing independent sound effects does not auto-matically mean that the principal components represent the physical sound sources. To providethe principal components with any physical significance, virtual coherence functions are used.Unfortunately, in practice the application of PCA is not straightforward, since the identificationresults are not unique due to their dependence on reference locations.The second method of Nam does not need reference microphones for identification. Instead itsolely uses NAH holograms to discriminate between sound effects. Hereto, a precondition ofthe Nam method is the availability of an incoherent sound hologram. In our case of single pointholographymeasurements the method of STSF is needed to obtain such an incoherent hologram.The analysis of STSF in section 3.4 showed that STSF is a combination of PCA and PNAH. Theoutcome of this is that for single point hologram measurements the Nam method is in best casesimilar to the PCAmethod and could be even worse in case of source overlap. Therefore the Nammethod is not considered as an alternative sound identification method for single point hologrammeasurements.The analysis of PCA and Nam in this chapter have functioned as a foundation for the newlydeveloped PCAASO method. The new method is based on PCA and additionally utilizes so-called autospectra holograms. These autospectra holograms provide accurate information forthe ideal locations of the reference microphones. To validate PCAASOs improved identificationperformance, PCAASO is further analyzed and compared with PCA and Nam in the followingsimulations and experiments chapters.

35

36

Chapter 4

Simulations of identification methods

4.1 Introduction

The analysis of the identification methods from the last chapter is continued in this chapter bya discussion of their simulations. The simulations, carried out in Matlab, aim at investigatingthe analytical findings and provide qualitative and quantitative results. The main purpose ofthese simulations is to demonstrate the properties of the identificationmethods and consequentlyuncover their specific strengths and weaknesses. Also, these simulations will serve as a steppingstone to the experiments discussed in the next chapter.In the following section the scripts, which define the simulation environment, are discussed.Subsequently, several simulation scenarios and their respective results are presented. The chapteris concluded with a discussion regarding the outcome of the simulations.

4.2 Simulation setup

The simulations of the PCA and Nam identification methods are performed usingMatlab. Firstly,the generation of the simulation set-up is discussed. Here the definition of the source field andpropagation of sources to other planes of interest are introduced. Hereafter, the scripts for thetwo identification methods are discussed. The complete Matlab scripts are presented in appendixD for reference.The simulations are based on the practical baffle setup1 in the lab, which is utilized for the exper-iments in chapter 5. This makes mutual comparison between the simulations and experimentspossible and more straightforward. The sound sources are modeled as acoustical point sourcesin accordance with the experimental setup. In the simulations, the number of incoherent soundsources is limited to a maximum of two. This has several reasons. The practical setup providesa maximum of two incoherent sources as mentioned in Chapter 5. Since a comparison betweenthe simulations and experiments will be carried out, the simulation setups are based on the prac-tical situation. An additional advantage of using few incoherent sources is that less calculationresources are needed and that the analysis is not made unnecessary complicated. In principle ofcourse, the number of sources can be increased to any desired value.Now, a summary of the Matlab script, with which the simulations are performed, is presented.Initially, a source plane at z = 0 m is defined as a NxM spatial grid of lxs by lys m. The source

1See chapter 5 for details of the baffle setup

37

source planesource plane reference plane hologram plane

lxs

lys

Figure 4.1: schematic representation source field for simulations

signals are positioned by allocating them to any of these grid squares. Physically this can be seenas the definition of a square piston of lxs

N by lys

M m. The sound sources are defined as normally dis-tributed random time signals. This signal type is commonly used to generate incoherent sources.Incoherence is achieved by segment averaging of the random time signals to average the signalinaccuracies. For a detailed discussion of generating incoherent sources see Appendix A.In practice, with PNAH a hologram plane at a certain distance of the source is measured and backpropagated to obtain a soundfield representation closer to the sound sources. However for thesimulations this is not the case, since by definition the soundfield is known. After defining thesound field on the source plane, the sound fields on the reference plane and hologram plane arecalculated respectively. This implies that the sound field on the source plane should be forwardpropagated. In general, Rayleigh’s second integral is used to accurately propagate a sound field.Nevertheless PNAH propagation as in chapter 2 is preferred for this operation. An importantadvantage of PNAH is the significantly faster calculation algorithm. With the NAH forward prop-agation the reference and hologram plane can be determined at any desired location. From thecalculated reference plane at z = zref several grid positions are chosen. These selected positionsare considered as the locations of the used reference microphones for the PCA procedure. Thesignals at the selected positions are defined as the respective reference signals.As demonstrated in Appendix A, the source signals must be averaged to generate incoherentsources. Therefore practically the whole simulation process must be iterated t times to pro-duce these signals. Note that, as an exception, the source field definition only has to be formedonce. The source time signals are divided into t equally large segments. Each signal segment isthen used in a loop procedure, which develops as follows; The soundfield Si(x, y, t) for the seg-ments i of the source time signals are Fourier transformed to the frequency domain, resulting inSi(x, y, f). Since NAH propagation is performed by a multiplication in the wavenumber domain,Si(x, y, f) subsequently undergoes a two-dimensional spatial Fourier transform. This yields thesource plane Si(kx, ky, f) in the wavenumber domain. Si(kx, ky, f) is then multiplied by twowavenumber propagators, leading to the reference Ri(kx, ky, f) and hologram plane Pi(kx, ky, f)in the wavenumber domain respectively. Since we are interested in a frequency analysis, thesetwo planes are inverse Fourier transformed to the frequency domain. By selecting the desiredreference locations from Ri(x, y, f) eventually all the desired data is obtained . Now the spectralmatrix Si

rr between the reference signals, the spectral matrix Sipp between the hologram points

and the spectral matrix Sipr between the hologram points and the reference signals is obtained.

38

This is the end of the loop and the procedure is repeated for the segments i = i + 1, ..., t. Afterthis procedure all the segment spectral matrices Si

rr and Sipr are summed and averaged to obtain

the smoothed averaged spectral matrices;

Srr = 1t

∑ti=1 Si

rr

Spp = 1t

∑ti=1 Si

pp

Spr = 1t

∑ti=1 Si

pr.

(4.1)

These spectral matrices are used as inputs to the scripts for the two identification methods.The first act of the PCA script is a SVD of Srr;

Srr = UDUH , (4.2)

With the decomposition, the principal components and eigenvectors are obtained. Together withthe cross spectrum Spr the incoherent sound effects and their partial holograms are calculated;

Sipp = SprU(:, i)D(i, i)−1U(:, i)HSrp = PiP

Hi (4.3)

Pi = SprU(:, i)√

D(i, i)−1, (4.4)

(4.5)

where Pi is the contribution of incoherent sound effects i on the hologram plane. Alongsidefinding the incoherent effects the virtual coherences are determined. The absolute value of thetotal source field is obtained by summing the partial holograms of the principal components.For the Nam method the availability of a full array measurement system is assumed. Hereto,Spp is used for the method. Spp is back-propagated to acquire the spectral matrix on the sourceplane, Sss. Now the maximum absolute value within Sss is collected. The first incoherent partialfield is consequently obtained by all the data within Sss that is correlated to this maximum value.The next incoherent partial field is found by subtracting the latter data from the initial Sss andgoing through above procedure again with the new Sss. This iteration continues until Sss isempty. With the Nam method the total source field is obtained by taking the square root ofthe diagonal terms of Spp. For comparison of Nam with PCA, the incoherent partial fields areforward-propagated to the hologram plane.

4.3 Simulation scenarios and results

In this section the executed simulation scenarios and their respective results are presented. Sixdifferent simulations are performed. In each simulation a single specific feature of the identifi-cation methods is isolated. identification criterium, reference positioning, incoherent sources/-effects, source overlap, PCAASO. All simulations are performed for an ideal situation where nonoise contribution is present in the soundfield. Note that in some scenarios the Nam methodis not mentioned. This implies that it is expected that results of this method are not affectedby these scenario’s and correct sound source identification is achieved. Also note that if in thediscussion of the results the Nam method is not mentioned, the results of the PCA and Nammethod are identical.Before discussing the simulation results, an additional aspect used during the simulations is

39

treated. As mentioned in the previous chapters, the number of reference microphones and theirlocations are essential to the outcome of a PCA-based experiment. For the simulations the ac-tual sound source signals are known. Consequently a method can be derived, which qualifies towhat extent the reference signals are placed correctly. With this method the minimal number ofreferences needed to capture the entire present soundfield and their optimal locations are deter-mined. Assume the availability of the source and reference signals S and R, respectively, thenthe spectral matrices is determined as

Srr = RRH

Ssr = SRH ,(4.6)

where Srr is the spectral matrix between the reference signals and Ssr is the cross spectral matrixbetween the source and reference signals. With these matrices the spectral matrix Sss of thesource plane is reconstructed;

Sss = SsrS−1rr Srs = S XHX−H︸ ︷︷ ︸

I

X−1X︸ ︷︷ ︸I

SH . (4.7)

If the reference signals R have captured the entire source sound field, it must hold that

SSH = SsrS−1rr Srs. (4.8)

However, if this is not the case, it implies that the reference signals do not represent the totalpresent sound field. Consequently, the number of used references and/or their locations must bereconsidered. The correctness of the reference locations is determined by examining the normal-ized principal components. The normalized principal components are defined as pcn

i = pci

pc1for

i = 1, ..., p. By comparing pcni on the source plane with pcn

i on the reference plane conclusionscan be drawn on how characteristic the received sound effects by the references are for the actualsoundfield. The reference microphones are ideally positioned if the normalized principal compo-nents on the two planes are equal. Additionally, this method consequently provides a quantitativecomparison of the performance for different reference locations.

simulation scenario 1: Best-case source identification

In the first scenario a soundfield consisting of two incoherent sources, with equal amplitudelevels is defined. These sound sources will be placed relatively far apart. This is done to preventpossible issues as sound short cutting and source overlap. The reference signals are positionedat the exact locations of the sources. This is to obtain ideal reference signals. This simulation isconsidered as a best-case scenario. The purpose of this simulation is to examine the functioningof the identification methods in case of ideal circumstances.In this scenario also two additional reference microphones are used besides the two at the ideallocations. Since the soundfield consists of two incoherent sound effects the expectation is that aPCA analysis will provide two dominant and two negligible principal components.The PCA identification results for the first simulation scenario are shown in figure 4.3. Theupper figure represents the total hologram plane, whereas the other four figures define the con-tributions to the hologram plane of the respective principal components. The identification ofthe two incoherent sound sources is clearly observed in partial hologram 1 and 2. The emptypartial holograms 3 and 4 also show that no additional incoherent sound effects are observed.Also, the significance of the identified partial holograms is indicated with the amplitudes of their

40

s1 s1

r1 r2

r3

r4

Figure 4.2: schematic representation of source and reference setup for simulation scenario 1.The squares represent the sound sources, whereas the circles are the reference microphones.

respective principal components; pc1 and pc2, corresponding to the first two partial holograms,are dominant compared to the principal components pc3 and pc4 for the empty partial holograms.The above conclusion is supported by the virtual coherence analysis of this scenario provided intable 4.1. Note the high correlation of reference microphone 1 and 2 with principal component 1and 2 respectively. These principal components are both only highly correlated to a single refer-ence microphone and thus location. Hence the incoherent sound effects represented by pc1 andpc2 are positioned at the locations of these respective reference microphone locations. This is incorrespondence with the simulation setup provided in figure 4.3. Also, the zero-valued virtual co-herences γ2

1,3, γ21,4 and γ2

2,3, γ22,4 in table 4.1 indicate that the last two principal components do not

represent uncorrelated noise and are considered as calculation induced errors. Note that a soundidentification analysis should not be based solely on a virtual coherence investigation. If this isdone for this case, pc3 could be considered as representation of a third incoherent sound effect(γ2

3,3), which is not the case. By also considering the amplitudes of the principal componentsthese faults are avoided.

Virtual Coherences γ2i,j

pc1 pc2 pc3 pc4

ref1 0.9943 0.0057 0 0ref2 0.0065 0.9935 0 0ref3 0.0036 0.0158 0.8028 0.1777ref4 0.0042 0.0092 0.5045 0.4821

Table 4.1: Virtual coherences of simulation 1

41

Simulation PCA

Total Hologram

pc1 = 1.0413 pc

2 = 1.0064 pc

3 = 2.85e−017 pc

4 = 1.93e−017

1

2

3

x 10−3 |p| Pa

Partial Hologram 4Partial Hologram 3Partial Hologram 2Partial Hologram 1

Figure 4.3: Identification results of PCA for simulation 1

simulation scenario 2: Reference positioning influence

This scenario is predominantly focussed on the PCA identification method. For the outcomeof PCA it is essential that each reference microphone dominantly observes a single incoherentsound effect from the sound field. With this scenario the influence of reference locations on theresults of PCA is investigated. The source field for this simulation scenario consists of two in-coherent sources, which is captured with two reference microphones positioned in front of thesound sources at a distance zref = 0.015m from the source plane. The initial horizontal distancebetween the sources is 0.16 m. Hereafter, in several sub simulations reference microphone 1,initially in front of source 1, is moved subsequently 0.02 m in the direction of reference micro-phone 2, which stays fixed. This results in a situation where reference microphone 1 graduallyreceives a stronger signal from source 2 than from source 1. Hence it is expected that the signalof source 1 will become less and less represented in the reference set.In figure 4.5 the source identification results for each successive sub simulation are shown. Thesecond row shows the partial holograms of the first principal component for each sub scenarioand the third row the partial holograms of the second principal component. Also note that eachcolumn represents the obtained partial holograms per sub scenario. The reference microphonepositioning for all the scenarios are provided in the first row for reference. These figures donot demonstrate any dependence of source identification on the reference locations; The par-tial holograms are identical for all cases. This unexpected result is explained by the absence ofnoise in the simulations. The soundfield solely consists of two incoherent sources without anyadditional noise terms. This implies a infinite signal-to-noise ratio (SNR). Hereto, any sourceeffect, independent of its amplitude, is correctly observed by the references. Consequently, thesources are correctly identified for all scenarios. However, the influence of reference position-ing is demonstrated quantitatively. In figure 4.6 the development of the principal components

42

s1r2

r1s2

s1r2

s2r1

Figure 4.4: schematic representation of source and reference setup for simulation scenario 2.The squares represent the sound sources, whereas the circles are the reference microphones.left: from initial reference locations (black circles) r1 is moved to source s2. right: setup of last

subscenario.

for the sub simulations is provided. The development of the virtual coherences is also shownin fig 4.7. In sub scenario 1 the respective reference microphones are positioned each in frontof the two incoherent sources. This results in obtaining two significant principal components.Additionally, the virtual coherences show total correlation of principal component 1 and 2 withreference microphones 2 and 1 respectively, i.e. γ2

2,1 = 1 and γ21,2 = 1. The values of the other

virtual coherences are zero, which indicates that each reference microphone captures a singleincoherent effect. Hereafter, figure 4.6 also shows that for each successive simulation scenarioprincipal component 2 rapidly decreases to zero, while principal component 1 stays large andfairly constant. This is explained by the fact that the left source s1 is decreasingly observed as ref-erence 1 is moving away from s1. This leads to a less significant part of s1 in the total sound fieldand consequently a smaller principal component. Here is demonstrated that due to the distanc-ing of reference microphone 1 the source field consisting of two incoherent sources is eventuallywrongly observed as consisting of only a single significant incoherent sound effects. The virtualcoherences development in figure 4.7 also provides an interesting view. The virtual coherencesrelated to reference microphone 2 (γ2

2,1 = 1, γ22,2 = 0) are equal for all sub scenarios. This in-

dicates that reference microphone 2 is positioned at the same location for all sub scenarios. Asexpected, the virtual coherences of reference microphone 1 are influenced by the changing po-sitioning of reference microphone 1. γ2

1,1, the correlation between reference microphone 1 andprincipal component 1, is zero for the first scenario and gradually increases. For scenario 5 γ2

1,1

has become 0.5 and eventually it is 1 for scenario 10. γ21,2 goes through the same development in

reverse order. This implies that during the sub scenarios reference microphone 1 is increasinglycorrelated to principal component 1 ,representing the right incoherent source, and decreases itscorrelation with principal component 2, left incoherent source. Note that the virtual coherences,γ2

1,1 and γ21,2 respectively, become larger and smaller than 0.5 after scenario 5. For this scenario

reference microphone 1 is positioned equally far from source 1 and source 2. Consequently, with-out a priori knowledge of the reference locations the virtual coherence analysis can also be usedto determine the locations of the reference microphones.

simulation scenario 3: Incoherent sound effects

In this scenario three sound sources consisting of two incoherent effects are used. Sources 1and 2 are considered to be coherent whereas source 3 is totally incoherent with respect to source

43

scenario 1 scenario 2scenario 1 scenario 3 scenario 4 scenario 5 scenario 6 scenario 7 scenario 8 scenario 9 scenario 10

r2

r1 r1 r1 r1 r1 r1 r1 r1 r1 r1

r2 r2 r2 r2 r2 r2 r2 r2 r2

0

0.02

0.04

0.06

0.08

0.1

0.12

scenario 1 scenario 2 scenario 3 scenario 4 scenario 5 scenario 6 scenario 7 scenario 8 scenario 9 scenario 10

|p| Pa

Figure 4.5: Identification of sources using PCA. Each column shows the decomposed partialholograms for the specific sub scenarios.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Prin

cipa

l Com

pone

nts

[]

pc1

pc2

scenario 1 scenario 2 scenario 3 scenario 4 scenario 5 scenario 6 scenario 7 scenario 8 scenario 9 scenario 10

Figure 4.6: Principal component development through sub scenarios of simulation 2

0

0.2

0.4

0.6

0.8

1

Virt

ual c

oher

ence

γij []

γ11

γ21

γ12

γ22

scenario 1 scenario 2 scenario 3 scenario 4 scenario 5 scenario 6 scenario 7 scenario 8 scenario 9 scenario 10

Figure 4.7: Virtual coherence development through sub scenarios of simulation 2

1 and 2. The purpose of this simulation scenario is to show and emphasize the fact that theidentification methods identify incoherent sound effects, which is not by definition equal to iden-

44

tifying individual sound sources. This important property has not become clear in the previousscenario’s. It is expected that two dominant principal components will be found.The outcome of the PCA analysis for this scenario is provided in figure 4.9. Two significant prin-

s1

r2

s2

r3

s1

r1

Figure 4.8: schematic representation of source and reference setup for simulation scenario 3.The squares represent the sound sources, whereas the circles are the reference microphones.

cipal components are obtained, whereas the third principal component is negligible. Additionally,an interesting PCA property is observed in the partial holograms. In the previous simulations,each principal component corresponded to a single sound source. This is not the case for thissimulation, since the partial hologram of principal component 1 consists of two sound sources.The correlations between the sound sources are also observed in the virtual coherences in table4.2. Reference microphone 1 and 2, which are positioned in front of the top left and bottommiddle sound source respectively, are strongly correlated to principal component 1, whereas ref-erence microphone 3 in front of the middle right source is correlated to principal component 2.Also note that all reference microphones have no correlation to the third principal component.This implies that the third principal component does not represent a general noise contributionto the soundfield. Therefore this principal component does not encompass any information ofthe soundfield and is considered negligible.

Total Hologram Partial Hologram 1 Partial Hologram 2 Partial Hologram 3

pc1 = 1.2639 pc

2 = 0.9005 pc

3 = 1.7202e−008

0.020.040.060.080.10.12

|p| Pa

Figure 4.9: Identification results of PCA for simulation 3

45

Virtual Coherences γ2i,j

pc1 pc2 pc3

ref1 0.9999 0.0001 0.0ref2 0.9999 0.0001 0.0ref3 0.0003 0.9997 0.0

Table 4.2: Virtual coherences of simulation 3

simulation scenario 4: Sound source overlap

With this scenario the main weakness of the Nam method is investigated. Chapter 3 showedthat application of Nam results in erroneous results when sources (partially) overlap. Startingwith no overlap of two incoherent sound sources, the sources are subsequently moved closer toeach other as shown in figure 4.10. This gradually results in partial overlap of the sources. TheNam method is expected to incorrectly identify sound sources in case of partial overlap, whereasPCA should perform correct sound source identification.Initially, the point sound sources do not overlap on the source field. In figure 4.11 the identifi-

s1

s2s2

s1s1

s2

Figure 4.10: schematic representation of source and reference setup for simulation scenario 4.The squares represent the sound sources, whereas the circles are the reference microphones.From initial source positions (left) the sources are moved closer to eachother. The overlapping

area is indicated by the black color.

cation results for the first scenario are presented. The left most hologram in figure 4.11 showsthe total soundfield, whereas the two additional rows represent the results of PCA and Nam re-spectively. As is shown in figure 4.11 both methods are able to correctly identify the incoherentsources. Hereafter the sound sources are positioned according to the second setup in figure 4.10.Now the identification results of PCA and Nam are different. PCA again performs a correct iden-tification of the point sources. Both partial holograms show clear circular point sources withthe correct amplitudes. Nam however fails to obtain correct partial holograms. The first partialhologram contains, besides the actual right point source, an additional side lobe at the left of thispoint source. The position and size of this side lobe coincides with the amount of overlap ofthe sound sources. Also, the added side lobe in the first partial hologram is subtracted from thesecond partial hologram. This effect is best observed as a reduced amplitude of the point sourcein the second partial hologram. In the next step the sound sources are positioned as shown in theright setup in figure 4.10, where the source overlapping area is increased. The results are againpresented in figure 4.13. PCA once again provides a correct identification of the point sources.

46

Total Hologram

Partial Hologram 1 Partial Hologram 2

0.02

0.04

0.06

0.08

0.1

0.12

|p| Pa

Nam

PCA

Figure 4.11: Identification results of PCA and Nam for first setup of simulation 4

Total Hologram

Partial Hologram 1 Partial Hologram 2

0

0.02

0.04

0.06

0.08

0.1

0.12

|p| Pa

PCA

Nam

Figure 4.12: Identification results of PCA and Nam for second setup of simulation 4

Also, the effect of source overlap on the identification results of Nam are observed here moreprofoundly. Next to the right sound source Nam also attributes the overlapping source area to thefirst partial hologram. As a consequence, the representation of the left sound source is heavilyaffected, which results to the erroneous second partial hologram.

47

Total Hologram

Partial Hologram 1 Partial Hologram 2

0

0.02

0.04

0.06

0.08

0.1

0.12

|p| Pa

PCA

Nam

Figure 4.13: Identification results of PCA and Nam for third setup of simulation 4

simulation scenario 5: Coherent sound sources

This simulation scenario simply acts as a check to the functioning of the identification methods.The analysis in chapter 3 has demonstrated that PCA and Nam are able to identify incoherentsound effects. The sound field for this scenario consists of two fully coherent sound sourceswith two ideally located reference microphones. It is expected that both methods only find oneincoherent effect and consequently no source identification is performed. Figure 4.15 confirms

s1 s1

r1 r2

Figure 4.14: schematic representation of source and reference setup for simulation scenario 5.The squares represent the sound sources, whereas the circles are the reference microphones.

this expectation. Only one significant principal component is obtained from the PCA analysis.The partial holograms also show that the total sound field is represented in the first partial holo-gram and consequently partial hologram 2 is empty. The virtual coherences in table 4.3 reconfirmabove conclusions by the complete correlation of both reference microphones to the first princi-pal component.

48

Total Hologram Partial Hologram 1 Partial Hologram 2

pc1 = 1.196 pc

2 = 0

0

0.05

0.1

|p| Pa

Figure 4.15: Identification results of PCA for simulation 5

Virtual Coherences γ2i,j

pc1 pc2

ref1 1 0ref2 1 0

Table 4.3: Virtual coherences of simulation 5

simulation scenario 6: Evaluation of PCAASO

In this scenario the method of PCAASO introduced in section 3.5 is compared with the PCAmethod. For the PCA method the source locations are not known a priori, whereas PCAASOprovides a good indication of the source locations. Therefore, for the PCA case the referencesare distributed evenly over the source plane and for PCAASO the references are positioned atthe source locations. Hereto, the potential of the new implementation method is investigated.Although the importance of reference locations is already demonstrated in simulation scenario

s1

r2

r1

s2

s1 s2

r2r1

Figure 4.16: schematic representation of source and reference setup for simulation scenario 4.The squares represent the sound sources, whereas the circles are the reference microphones.left:

configuration for initial PCA, right: configuration of new implementation.

49

2, here a more thorough and side-by-side analysis is provided.The identification results for both methods are shown in figure 4.17. The first and second rowshow the results of the PCA and PCAASO method respectively. No identification differences areobserved. This is attributed to the same reasons mentioned in the results of simulation 3, be-ing the noise-free simulation environment. The principal components, however, demonstrate a

Total Hologram Partial Hologram 1 Partial Hologram 2

0.02

0.04

0.06

0.08

0.1

0.12

Initial PCA

Updated PCA

|p| Pa

Figure 4.17: Identification results of initial and updated PCA for simulation 6

significant difference between the two methods. In figure 4.18 the amplitudes of the principalcomponents are shown side-by-side. Here a large difference is observed between the values ofthe original PCA method and the newly developed PCAASO. Since the principal components are

0

0.5

1

1.5

2

2.5

3

3.5

4

Con

trib

utio

ns o

f PC

s

pc1

pc2

I U

Figure 4.18: Principal component amplitudes for initial and updated PCA analysis

seen as a measure for observed sound source energy, the new implementation method capturesa larger amount of the emitted energy of the soundfield. This does not directly mean that the

50

new implementation method provides a better identification of the soundfield. To investigatethis aspect the normalized principal components introduced in section 4.3 are compared. The

source plane reference plane, initial reference plane, updated

pc1 13.440 0.1480 1.8437pc2 11.883 0.0798 1.6301pcn

1 1 1 1pcn

2 0.8841 0.5394 0.8842

Table 4.4: normalized principal components of PCAASO simulation

representativeness of the soundfield on the reference plane is quantified by the difference of thenormalized principal components on the source and reference plane. A large mutual correspon-dence represents a correct representation of the soundfield on the reference plane, whereas alarge mutual difference implies an incorrect display of the soundfield by the reference micro-phones. In table 4.4 the normalized principal components for both methods are shown. It isclear that PCAASO corresponds much better with the normalized principal components of thesourceplane and thus provides an improved identification method compared to PCA.In some of the simulation results nothing is mentioned about the results of the Nam identifica-tion method. This is because the situations in these respective simulation scenarios did not affecttheir results. The identification results in these cases are identical to the PCA method.

4.4 Discussion

The simulations in this chapter provide insight into the properties of the sound source identi-fication methods. Several important properties discussed in chapter 3 have been acknowledgedduring the simulations. An important finding from simulations 1 and 3 is the notion that theidentification methods provide discrimination of incoherent sound effects, which is not per defi-nition similar to identification of sound sources. Also, the results from simulation 2 demonstratethe influence of reference microphone locations on the PCA identification results. For soundeffects to be represented correctly in the identification results the reference microphones mustbe located in close vicinity of the sound sources. Hereto, the source identification with PCA isnot unique and can differ according to a different reference set-up.The sensitivity of PCA and Nam to source overlap is demonstrated in simulation 4. It shows thatPCA, assuming correct reference locations, is able to perform correct incoherent source identi-fication in case of sound source overlap. Nam is, however, unable to cope with this effect andconsequently provides erroneous identification results.In simulation 6 PCAASO provides an improved source identification compared to the PCAmethod. According to the normalized principal components, PCAASO is able to represent thesourceplane more accurately than PCA. But since the simulations are performed in ideal circum-stances its practical potential must still be investigated. Therefore, the next chapter discusses theresults of the practical implementation of PCAASO and the existing identification methods.

51

52

Chapter 5

Experiments with baffle setup

5.1 Introduction

In the previous chapter the identification techniques are simulated in a Matlab environment. Toanalyse the properties of sound source identification in practice, also experiments are performedand their results are discussed in this chapter. For the experiments a robot traverse system isutilized, which performs single point grid measurements. The entire experimental setup is pre-sented in the next section. As stated earlier, the Nam method is only fully exploited for fullarray hologram measurements. For single point measurements the identification results of Namare in the best case (disregarding source overlap issues) identical to the identification results ofPCA. Hereto Nam does not provide any additional insight into source identification in the caseof single point hologram measurements and therefore Nam analyses are not performed for theexperiments.For the experiments the same scenarios as in the simulations are used. Additionally, a so-calledblind experiment is carried out to investigate the identification results with PCAASO and to as-sess its performance with respect to conventional PCA.

5.2 Experimental setup

The complete experimental setup is shown in figure 5.1, which consists of three main parts,a measurement robot system, a baffle setup and a speaker system. For sound generation twospeaker setups are available. The speakers are isolated by placing them in a double wooden boxconstruction with a small opening in the front from which the generated sound can exit. With aSiglab data acquisition system the speakers are directly excited. To generate point sources for theexperiments a so-called baffle is constructed. This baffle consists of a large rectangular woodenpanel. In the middle of this wooden panel an aluminium grid plate is inserted, which is depictedin figure 5.2.

This grid plate contains 33 openings, which are positioned in a specific grid. With specialtubes and pins the openings are opened and closed respectively. Now, by connecting an isolatedspeaker box with a hose to an opening on the grid plate a point source is created. By sendingrandom noise signals to the openings incoherent sound sources are created.In front of the baffle a measurement robot system is located. This system consists of a frameworkwith three motors, which can accurately position a measurement microphone on a rod in threedimensions. Hereto, a NAH hologram measurement is possible at any desired plane position by

53

Figure 5.1: Pictures of experimental setup. left: front view showing measurement robot andbaffle, middle: back view showing speaker system, right: closeup of aluminium grid plate.

20mm

20mm

4.8 mm hole diameter33 baffle openings

200mm

Figure 5.2: Dimensions of square aluminium grid plate, which is inserted into the baffle.

moving the measurement microphone over a user-defined grid. The reference microphones usedfor the PCA method can be either placed in front or at the back of the baffle. When placed at theback of the baffle themicrophones are supported by specially constructed connection parts, whichfit into the hoses from the speaker boxes and isolate the microphones. Herewith, the referencemicrophones are able to measure the sound pressure very accurately when placed at the back ofthe baffle.This experimental setup is designed specially for research into stationary sound sources. Therobot traverse systemmeasures one grid position at a time. When the measurement microphonemoves to the next grid location and performs another measurement this measurement is out ofphase with respect to the previous measurement. This eventually leads to an erroneous soundhologram. To deal with this phase effect a fixed reference microphone is used alongside the gridpoint measurements. Also, the signals to the speakers are triggered to their initial state every timethe measurement microphone moves to another grid location. Hereto, all grid positions measurethe soundfield in the same phase and a correct sound hologram is constructed. The control of themeasurement robot, excitation of the speakers, triggering of the sound signals, measurement ofall reference microphones and grid positions are fully integrated and automated by the in-housedeveloped NAH software package. The identification post-processing of the data is performed byusing self developed Matlab scripts. These scripts are provided in Appendix D.

54

5.3 Experiment scenarios and results

In this section the several performed experiment scenario’s are introduced. The first four sce-narios are similar to simulations performed in the previous chapter. Consequently the analogiesbetween the simulations and experiments are investigated and analyzed. In the fifth experimentscenario the identification results of the newly introduced PCAASO method are discussed andcompared with a conventional PCA method.

experiment scenario 1: Best-case scenario identification

In the first scenario an experiment is performed consisting of two incoherent point sources. Thesound sources are placed on the same vertical height and have a horizontal distance of 8 cm. Bothsources are individually constructed by a separate Siglab output. This eventuates into obtainingtwo mutually incoherent sound sources. In both connection parts at the back of the baffle micro-phones are placed. These microphones serve as references for the PCA method. Furthermore, athird reference microphone is used. This microphone is positioned 1.5 cm in front of the baffleand in the middle of the two point sources (see figure 5.3).The identification results of PCA for this scenario are shown in figure 5.4. The total captured

s1 s2r3r1 r2

Figure 5.3: Frontal view of point sources and reference locations for scenario 1. ri (square) and si

(circle) represent the reference microphones and point sources respectively.

sound field is represented by the left figure, whereas the additional figures represent the identi-fied incoherent partial holograms. Figure 5.4 clearly demonstrates the ability of PCA to identifyand separate incoherent sources in a composite sound field. The quantitative data of the principalcomponents and virtual coherences is also very powerful. The amplitudes of the first two princi-pal components corresponding to the two identified incoherent sources are significant, whereasthe third principal component is relatively small. This implies that the obtained partial field fromthe third principal component contributes relatively less to the total sound field than the otherpartial fields. This corresponds with the notion that only two incoherent sources are present; thethird partial field does not present a third incoherent source component. That the third principalcomponent is not zero as would be the case in theory, can be attributed to multiple causes: Errorsin incoherent sound source generation, post processing or presence of a noise source introducesmeasurement errors.The virtual coherences, which quantify the correlations between the references and principal

components, are provided in table 5.1. Note the high correlation between reference microphoneone and two with principal component one and two respectively. Since the referencemicrophonesare placed in the connection parts at the back of the baffle the reference microphones solely cap-ture a single incoherent signal from the speaker boxes. The small values of γ2

1,2, γ22,1 are assigned

to the earlier mentioned experiment errors.

55

Total Hologram Partial Hologram 1 Partial Hologram 2 Partial Hologram 3

2468101214

x 10−6|p| Pa

pc1 = 0.7173e−3 pc

2 = 0.5010e−3 pc

3 = 0.0003e−3

Figure 5.4: f = 1206.25 Hz, z = 0.0 m, Identification of sources using PCA. The left figure repre-sents the total hologram, whereas the other figures are the decomposed partial holograms.

Virtual Coherences γ2i,j

pc1 pc2 pc3

ref1 0.9927 0.0073 0ref2 0.0299 0.9701 0ref3 0.3741 0.6167 0.0092

Table 5.1: Virtual coherences of two incoherent sources experiment

experiment scenario 2: Reference positioning influence

The second experiment corresponds to simulation scenario 2 and analyzes the importance ofthe reference positions with respect to the sound sources is analyzed. This experiment consistsof ten sub experiments. The locations of the two incoherent sound sources are fixed for eachof these sub experiments and again have a mutual horizontal distance of 8 cm. The differencebetween the sub experiments is the positioning of the reference microphones. Initially the tworeference microphones are positioned 1.5 cm in front of the respective sound sources. For eachsub experiment, reference microphone two, r2, remains fixed whereas the first reference micro-phone, r1, is gradually moved to the second sound source s2. These sub scenarios are also shownin figure 5.5. The expected outcome of this experiment is that for each consecutive step sound

r1

s2s1

r2

Figure 5.5: Frontal view of point sources and reference locations for scenario 2. ri (square) andsi (circle) represent the reference microphones and point sources respectively. In experiment

scenario 2 r1 gradually moves away from s1.

source s1 is captured less by the reference microphone set. This results in an increasingly less

56

r2r1

r2r1

r2r1

r2r1

r2r1 r2

r1r2

r1r2

r1r1 r1

r2 r2

scenario 1 scenario 2 scenario 3 scenario 4 scenario 5 scenario 6 scenario 7 scenario 8 scenario 9 scenario 10

0

0.5

1

1.5

2

x 10−5

0

1

2

3

scenario 1 scenario 2 scenario 3 scenario 4 scenario 5 scenario 6 scenario 7 scenario 8 scenario 9 scenario 10

x 10−5

x 10−5PC

1

PC2

|p| Pa

|p| Pa

Figure 5.6: f = 1793.75 Hz, z = 0.0 m, Identification of sources using PCA. Each column showsthe decomposed partial holograms for the specific sub scenarios.

represented sound source s1 in the total captured sound field and consequently an erroneousidentification of sound source s1. The PCA results of this experiment are shown in figure 5.6.The second row shows the partial holograms of the first principal component for each sub sce-nario and the third row the the partial holograms of the second principal component. Also notethat each column represents the obtained partial holograms per sub scenario. The reference mi-crophone positioning for all the scenarios are provided in the first row for reference.

Although the amplitude of the second partial holograms do decrease over the sub experiments,the expected dependency on reference locations is not clearly demonstrated. It is considered thatthis can be assigned to the quality of the experimental setup. The semi-anechoic chamber pro-vides very high signal-to-noise ratios. The reference microphone set can therefore still clearlycapture the signal from sound source s1 when positioned as in scenario 10. Fortunately, dis-tinct reference positioning dependency is observed when analyzing the accompanying principalcomponents and virtual coherences. Figure 5.7 demonstrates the development of the principalcomponent amplitudes during the sub scenarios. Note that comparisons between principal com-ponents of different experiments should be performed with care [12]. Since the measurementenvironment is equal for all sub scenarios, comparisons between principal components is al-lowed and meaningful in this case. As is shown in figure 5.7, principal component one, whichrepresents the right sound source s2 in the sound field, remains large and fairly constant duringall sub scenarios. This is induced by the location of the fixed reference microphone r2. For all subscenarios r2 is positioned in the close vicinity of s2 and therefore this source is captured equallywell during the entire experiment. The situation is clearly different for principal component two,which gradually decreases during the sub experiments. The step by step distancing of referencemicrophone one with respect to sound source one, leads to a decreasing contribution of s1 to thetotal intensity level of the capture sound field. Consequently the left sound source is considereddecreasingly significant in the total soundfield, which is physically incorrect. The virtual coher-ences development, figure 5.8, also provides corresponding results. γ21, the coherence between

57

0

1

2

3

4

5

6x 10

−5

Prin

cipa

l Com

pone

nts

[]

pc1

pc2

scenario 1 scenario 2 scenario 3 scenario 4 scenario 5 scenario 6 scenario 7 scenario 8 scenario 9 scenario 10

Figure 5.7: Principal components development during the experiment sub scenarios of experi-ment 2.

reference microphone two and principal component one is practically constant and almost oneduring all sub experiments. On the other hand, the coherence between r2 and principal com-ponent two, γ22, is zero during the entire experiment. This implies that r2 is fully correlated toprincipal component one, which represents the right sound source in the sound field. Conse-quently, it is stated that reference microphone two is positioned in the close vicinity of the rightsound source. The virtual coherences of reference microphone one experience a more dynamicdevelopment. At scenario 1, r1 is predominantly correlated to principal component two whereasat scenario 10 the situation is completely opposite. Reference microphone r1 is now totally cor-related to principal component one. This clearly shows the influence of the relative positions ofthe reference microphones. With the knowledge that all sub experiments comprise of the samesound source setup, the movement of reference microphone r1 during the entire experiment canbe determined. Also note that the development of the principal components and virtual coher-ences (figure 5.7 and 5.8) is in large accordance with the obtained simulation results in chapter 4(figure 4.6 and 4.7).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Vir

tual

co

her

ence

γij [

]

γ11

γ21

γ12

γ22

scenario 1 scenario 2 scenario 3 scenario 4 scenario 5 scenario 6 scenario 7 scenario 8 scenario 9 scenario 10

Figure 5.8: Virtual coherences development during the experiment sub scenarios of experiment2.

experiment scenario 3: Incoherent sound effects

The identification method of PCA is meant to identify incoherent sound effects within a com-posite sound field. Often the mistake is made that PCA is able to separate sound sources in gen-

58

eral. However, PCA is unable to identify and separate the individual sound sources if the soundsources are coherent. This feature is analyzed in this third experiment. During this experimentthree sound sources are constructed and positioned as shown in the schematic representation offigure 5.9. The point sources s1 and s3 are excited by a single Siglab output and thus are mutually

s1

s2

s3

r1

r3

r2

Figure 5.9: Frontal view of point sources and reference locations for scenario 3. ri (square) andsi (circle) represent the reference microphones and point sources respectively.

coherent. Point source s3 is constructed from another Siglab output and is incoherent to the othertwo sound sources. Each sound source is accompanied by a respective reference microphone ri

for i = 1, 2, 3. From theory, the identification should lead to a principal component representings1 and s3, a principal component representing s2 and a redundant principal component sincethere are only two incoherent sound effects present.Figure 5.10 shows the results of PCA identification for this scenario. As expected two signifi-cant principal components corresponding to the two incoherent effects are obtained. The thirdprincipal component is an order of 102 smaller than the second principal component and istherefore considered negligible. Still a sound component is visible in the third partial hologram.Besides the previously mentioned possibilities in scenario 1, a possible explanation could be thatthe sound sources of s1 and s3 create a vibration of the baffle at the exits of the source loca-tions resulting in a very small third component. In accordance with the theory and simulationsthe partial holograms show that the first principal component represents the mutually coherentsource s1 and s3 and the second principal component represents the other incoherent effects cor-responding to s2. Therefore, once again the ability of PCA to identify incoherent sound effectsis demonstrated. In table 5.2 the virtual coherences of this experiment are presented. These pa-

Total Hologram Partial Hologram 1 Partial Hologram 2 Partial Hologram 3

0

1

2

3

4x 10

−5|p| Pa

pc1 = 0.5973e−4 pc

2 = 0.3377e−4 pc

3 = 0.0033e−4

Figure 5.10: f = 1781.3 Hz, z = 0.0 m, Identification of sources using PCA.

rameters provides a similar view of the results. Reference microphone one and three are stronglycorrelated to principal component one, whereas reference microphone two is correlated to prin-cipal component two. This corresponds with the partial holograms in figure 5.10. Also note that

59

none of the reference microphones have a correlation with principal component three. Hereto,principal component three most likely does not represent a vibration of the baffle as suggestedearlier and consequently is considered as calculation error residue.

Virtual Coherences γ2i,j

pc1 pc2 pc3

ref1 0.9599 0.0401 0ref2 0.2544 0.7456 0ref3 0.9721 0.0279 0

Table 5.2: Virtual coherences of third experiment

experiment scenario 4: Coherent sound sources

The experimental setup for the fourth experiment consists of two coherent sound sources, whichare on the same vertical height and have a mutual horizontal distance of 8 cm on the source planeas shown in figure 5.11. Also, two reference microphones are placed in the respective connectionparts located behind the baffle. From the previous analysis it is expected that the identification

s1 s2r1 r2

Figure 5.11: Frontal view of point sources and reference locations for scenario 4. ri (square) andsi (circle) represent the reference microphones and point sources respectively. s1 and s2 are

mutually coherent point sources.

method is unable to separate the two coherent sound sources.The identification results of this experiment scenario are shown in figure 5.12. Again, the leftfigure represents the total reconstructed soundfield, whereas the two consecutive figures corre-spond to the identified incoherent effects. The identified sound effects are accompanied by theirrespective principal components. Figure 5.12 clearly shows the inability of PCA to separate coher-ent sound sources. The two present sound sources are captured completely by the first principalcomponent. This effect could be introduced by multiple causes, such as the presence of back-ground noise or erroneous data handling. The large size difference between principal componentone and principal component two indicates that contribution of the second partial hologram tothe complete sound field is very small or even negligible. Furthermore, the virtual coherences intable 5.3 show that both reference microphones are fully correlated to the first principal compo-nent pc1. This again implies that the captured sound field consists of a single coherent soundeffect.

60

Partial Hologram 1 Partial Hologram 2Total Hologram

0

0.5

1

1.5x 10

−5

pc1 = 0.339e−4 pc

2 = 0.002e−4

|p| Pa

Figure 5.12: f = 1750 Hz, z = 0.0 m, Identification of sound effects using PCA.

Virtual Coherences γ2i,j

pc1 pc2

ref1 1 0ref2 0.9999 0.0001

Table 5.3: Table of virtual coherences of experiment 4

experiment scenario 5: Evaluation of PCAASO

The potential of PCAASO is analyzed by performing a so-called blind experiment. For thisscenario no a-priori information is available about the locations, the incoherent properties andthe number of sound sources. Consequently, for PCA the reference microphone are distributedevenly over the measured hologram area (figure 5.13) to cover an as large as possible area. WithPCAASO an autospectra hologram is constructed, which among other things provides informa-tion about the preferred locations of the reference microphones. It is expected that a second PCAexperiment with the newly obtained reference locations from PCAASO results into an improvedsource identification.In figure 5.19 the PCA results for four frequencies are presented. Note that the frequencies arechosen pseudo-random; The main criteria for these frequencies is a high signal to noise ratio(SNR). Also, for practical reasons it is decided to not analyze the higher frequencies. The fasterdecaying evanescent waves of higher frequencies would require closer hologram measurementsand consequently a much higher spatial resolution. This would significantly increase the mea-surement time. A drawback of choosing lower frequencies is that due to the relatively large wave-lengths, local impedances could influence the sound propagation of the near by point sources into free space.The four analyzed frequencies in figure 5.19 shows that there are two incoherent sound effectscaptured during the measurements. The third partial holograms display randomly distributedhigh pressure areas, which are considered to be noise. This conclusion is also drawn from theamplitudes of the third principal components, which are relatively negligible. Now for step twoof the PCAASO method the autospectra hologram is observed. Note that the grid measurementsare performed at the hologram plane. These grid measurements are then backpropagated to the

61

r3

r1

Initialreference setup

r2

Figure 5.13: Frontal view of point sources and reference locations for scenario 5. ri (square) and si

(circle) represent the reference microphones and point sources respectively. Here the referencemicrophones are evenly distributed over the measurement plane, since no a priori information is

available.

source plane where the autospectra hologram is eventually calculated. The autospectra of eachgrid point are qualitatively identical and is shown in figure 5.14. Five distinct frequencies, all

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1000010

−14

10−13

10−12

10−11

10−10

10−9

10−8

Autospectra grid measurement microphone

Frequency [Hz]

Mag

itude

Noise floor

Figure 5.14: autospectra of measurement microphone at the middle of grid layout. The autospec-tra at different positions within the measurement grid are qualitatively similar.

with high SNR, are selected and their respective autospectra holograms are calculated and pre-sented in figure 5.15. These frequencies are also denoted by the circles in the autospectra plot offigure 5.14. Comparing the high pressure areas in the autospectra holograms in figure 5.15 withthe PCA results in figure 5.19 do not show any unrepresented sources in the PCA analysis. Thisindicates that all sound sources are captured by the reference microphones in the initial PCAmeasurements. However, the autospectra holograms are still useful to optimize the locations ofthe reference microphones. The autospectra holograms display four distinct high autospectraareas and consequently the reference microphones for the successive PCA measurement are re-located to those positions. Unfortunately, only three reference microphones are available in thecurrent hardware setup. Hereto, the three initial reference microphones are relocated to the po-sitions shown in figure 5.16.

62

Figure 5.15: Several selected frequencies of the autospectra holograms, which display distinctivesource layouts. The distinct observed source locations are encircled. The autospectra magnitudes

of the frequencies are provided in figure 5.14 (circles).

r3

Updatedreference setup

r2

r1

Figure 5.16: New location of reference microphones obtained by PCAASO measurement. Thesereference microphone locations are used for the updated PCA measurement.

Now a consecutive PCA measurement is performed with the adjusted reference microphonepositions. This is step three of the PCAASO method. Figure 5.20 displays the initial and secondPCA measurements side by side for each analyzed frequency. Also, figure 5.17 shows the actualsource setup for this experiment, which was not known beforehand. Having this figure in mind,the second PCA measurement is considered as an improvement compared to the initial PCAmeasurement. The identification for all four frequencies show improvement when PCAASOis used and this is most clearly observed for the frequencies of 1231.25 and 3118.75 Hz. Forinstance, the identification performed at 1231.25 Hz with the initial PCA in figure 5.20 is notcorrect. The first partial hologram claims that the three sources are coherent to eachother. How-ever from the actual source setup in figure 5.17 is observed that the bottom middle sound sourceis incoherent to the other sound sources. The updated PCA measurement does show the correctincoherences between the sound sources. Also, at 3118.75 Hz figure 5.20 presents an improve-ment of the updated PCAmeasurement over the initial PCA measurement. The relocation of thereferences for the updated PCA measurements results in less cross talk between the physicallyincoherent sources. Additionally , for all investigated frequencies the third partial hologramsdisplay less noise contributions when PCAASO is utilized. The improvement of the updatedPCA measurement is also demonstrated quantitatively in figure 5.18. This figure displays theamplitudes of the principal components for all the measurements. Here is demonstrated that theprincipal components of the updated PCA measurement are more than 300 percent larger than

63

s1(12) s1(

14)

s1(14)

s2(12)

s2(12)

Figure 5.17: The actual locations of the sound sources. The subscript denotes the identificationof the incoherent sources, whereas the number between brackets represents the relative power of

the sources.

the principal components of the initial PCA measurement. And since principal components areconsidered as a measure for energy, it is concluded that for the same measurement scenario theupdated PCA measurement captures a significantly larger portion of the emitted sound energyfrom the actual soundfield. Consequently, the updated PCAmeasurement represents the presentsoundfield more accurately.

0

1

2

3

4

5

6

7

8x 10

−5

Con

trib

utio

ns o

f PC

s

pc1

pc2

pc3

f1 f

2f3

f4

I U I U I U I U

Figure 5.18: The amplitudes of the principal components for the initial and updated PCA mea-surements. I= initial, U= upgraded. f1, ..., f4 correspond with the frequencies from figure 5.20

5.4 Discussion

With the measurement setup consisting of a robot measurement system and a baffle and speakersetup several experiments are performed. The main reason for these experiments is to verify theearlier made statements and findings of the identification methods and analyse their identifica-tion performance in practice.

64

Qualitatively similar to the results obtained from the simulation 2 in chapter 4, are the findingsfrom experiment 2. The influence of the reference microphones positions is clearly observed indevelopment of the principal components and virtual coherences. Since in practice commonlythe position of the sound sources are not known a priori, this reference positioning can highlyinfluence the results obtained with PCA. Therefore, results of PCA identification must always beanalyzed with reference dependency in mind.Experiments 1 and 3 provide two important findings: First, these experiments demonstrate theability of performing sound identification with the current measurement setup and software. Butmost importantly, the property of separating incoherent sound effects rather than sound sourcesis again demonstrated. Also, note the large qualitative correspondence of between all the experi-mental results and the simulation results from chapter 4.The identification performance of the PCAASO method is also put to the test with experiments.With a blind experiment in experiment 6, where no a priori source information is available, theidentification results of PCA and PCAASO are compared. This experiment shows that PCAASOhas potential as an alternative source identification method. The reference microphones can beplaced more ideally and consequently for this experiment PCAASO yields an improved sourceidentification compared to PCA. With PCAASO the captured sound effects are more represen-tative for the actual soundfield and this results into a more accurate and correct source iden-tification. The results in this section provide a first glance at the potential power of the newimplementation method of PCAASO.

65

Total Hologram Partial Hologram 1 Partial Hologram 2 Partial Hologram 3

0

1

2

3

4

5

x 10−5

f = 1200 Hz

|p| Pa

Total Hologram Partial Hologram 1 Partial Hologram 2 Partial Hologram 3

0

1

2

3

4

5x 10

−5

f = 1231.25 Hz

|p| Pa

Total Hologram Partial Hologram 1 Partial Hologram 2 Partial Hologram 3

0

0.5

1

1.5

2

2.5x 10

−5

f = 2725 Hz

|p| Pa

Total Hologram Partial Hologram 1 Partial Hologram 2 Partial Hologram 3

0

0.5

1

1.5

2

2.5

x 10−5

f = 3118.75 Hz

|p| Pa

Figure 5.19: Identification results of initial PCA measurement for four frequencies.

66

67

Figure 5.20: Comparison of identification results of initial and updated PCA measurement forfour frequencies.

68

Chapter 6

Conclusions and Recommendations forfuture research

6.1 Conclusions

An essential step for gaining more insight into the behavior of acoustical sources and their rootcauses is the ability to isolate individual sound sources. Therefore, a study into identification ofsound sources within a composite soundfield is performed. The research has resulted into devel-oping an improved source identification method, PCAASO, which demonstrates large potentialwith respect to the existing identification methods.The literature study performed for this thesis provides an insight into the existing source identi-fication methods. Coherence analysis is considered the foundation of the majority of the soundsource identification methods. From the literature study two distinct source identification meth-ods are selected, which are investigated in more detail.Principal Component Analysis (PCA) is a well respected and widely applied identificationmethod.With PCA it is possible to transform a set of correlated references into a new set of uncorrelatedprincipal components. These individual principal components represent the independent effectsobserved in a data set. An important aspect of PCA is the correct use of the reference signals. Theoutcome of source identification with PCA is not unique and is dependent on the positioning ofthe reference signals. This sensitivity on reference positioning is considered as the weakest char-acteristic of PCA.The other method which is analyzed in detail is developed by Nam [11]. The Nammethod does notuse reference microphones to perform source identification, but rather utilizes the informationcaptured within NAH holograms to separate sound effects. By selecting the maximum pressureon a constructed hologram and taking all data points within a hologram coherent to the signalat the maximum pressure location a sound effect is identified. This process is iterated until allhologram points are assigned to a source effect. The independence of reference microphone po-sitioning is considered a powerful quality of Nam. However, Nam has its own drawbacks, whereerroneous source identification in case of source overlap is one of the largest. To investigate theproperties of the PCA and Nam method several simulations are performed. The reference posi-tioning dependency of PCA and the source overlap issues of Nam are several qualities that havebeen clearly demonstrated with the executed simulations.For source identification with Nam an incoherent sound hologram is required. Such a compositesound hologram is obtained either by performing full array hologram measurements or using

69

Spatial Transformation of Soundfields (STSF) in case of single point hologram measurements.For this thesis we are interested in source identification methods applicable for single point holo-gram measurements and therefore further analysis of the STSF technique is carried out.A thorough analysis of the STSF technique showed that STSF is a combination of PCA andPNAH. By utilizing references, the incoherent sound sources within a soundfield are obtainedsimilarly to the singular value decomposition of PCA. The identified sound sources are then back-propagated to the source plane using PNAH. The independent sound sources on the source planeare finally added to obtain the incoherent sound hologram. The analysis of the STSF methodstates that for single point hologram measurements source identification with PCA and Nam isbased on the same mathematical background and therefore their respective identification resultsare identical. Note that the Nam method does not provide any added value with respect to PCAin case of single point measurements and that in case of soundfields with overlapping soundsources the identification results even deteriorate.The gained knowledge from the analysis of PCA and Nam eventuates into a newly developedidentification method named PCA with Advanced Sourcefield Observation (PCAASO). PCA usesa powerful data separation technique, but its performance is largely dependent on reference mi-crophone positioning. The Nam method is able to locate sound sources accurately, but the sepa-ration of incoherent sources is incorrect in case of overlapping source effects and its applicationis only useful for full array hologram measurements. PCAASO combines the powerful quali-ties of the two methods and attempts to eliminate the drawbacks. The new method is based onPCA and additionally utilizes so-called autospectra holograms to improve the identification re-sults. With the autospectra hologram two important identification qualities are evaluated. First,by comparing the identified sources obtained from PCA with the high pressure areas of the au-tospectra hologram one can check if all sound sources have been captured with the PCA analysis.Secondly, the autospectra hologram demonstrates where the high pressure areas within a sourceplane are and consequently provides the locations of the sound sources. Hereto, the autospectraholograms present the optimized locations for the reference microphones and PCA is performedwith (near) ideal reference microphone locations. With the initial PCA method the reference mi-crophones are positioned according to the available information on the source distribution. Inpractice this information is minimal and the reference microphones are positioned based on theinsight of the engineer. Now the autospectra hologram provides a way to place the microphonesmore generically, which improves the robustness of the implementation.In the simulations and experiments the conventional PCAmethod and the new PCAASOmethodare analyzed and compared side-by-side. The simulations demonstrate that PCAASO is able ac-curately place the reference microphones and consequently result into an improved sourcefieldrepresentation and better source identification in comparison with PCA. A blind-experiment isused to evaluate PCAASO in practice. Using PCAASO, the captured sound effects are more rep-resentative for the actual soundfield compared with the results from PCA. This eventuates into amore accurate and correct source identification. PCAASO shows great potential and is considereda promising alternative source identification method.

70

6.2 Recommendations for future research

The work in this Master thesis has resulted into presenting an alternative source identificationmethod. Despite the presented work, several aspects are still open for future research. Therefore,several recommendations for future research are presented in this section.The simulations presented in this thesis are executed in Matlab. With several Matlab scripts asource plane consisting of a spatial grid is created. Hereafter, a random time signal is assigned toa desired grid block, which consequently represent a source on the source plane. These sourcesare then forward-propagated to create a soundfield. The simulation results demonstrated that theincoherent sound effects are always identified correctly even when there is a minimal capturing ofthe sound effects. This occurs since the simulations environment is noise-free and therefore rep-resents an ideal clean situation. To approach practical situations more closely a noise term shouldbe introduced. The noise term results into creating a certain noise level during the simulations.Consequently experiments are simulated more accurately and the outcome of practical applica-tions of the identification methods are better approximated. Therefore, it is recommended toextend the applicability of the simulations by introducing additional noise terms into the sound-field. Also, the possible definitions of the source shapes and types are limited. Now only squaresources are generated with the simulations and this is not sufficient when simulations of morecomplex soundfields are desired. Also, currently only simulations with monopole sources arepossible. The possibility of simulations with different source effects, e.g. dipole or quadripolesources, is desired.The sound sources for the experiments are generated by exciting a speaker through a Siglab sys-tem. The soundfield from the speaker are transported to an opening on the baffle resulting in apoint source. The siglab output used for creating sound is user-defined and thus the soundfieldfrom the speaker is known. However, the effects of transportation of the sound from speaker tothe baffle are unknown. For example no information is available on the amount of power loss,(destructive) interaction of soundwaves and the influence of the length of the connection tubes. Arecommendation for future work would therefore be to model these effects to obtain an accurateestimation of the soundfield at the openings on the baffle, the sourceplane. Tijdeman [18, 17] hasdeveloped a powerful modeling method for these effects. With a transfer matrix method based ona continuous analytical expression for the perturbations of the pressure and the velocity severalinfluences are modeled. By acquiring an accurate representation of the soundfield at the source-plane, an improved qualification of the identification methods is possible.PCAASO has demonstrated its potential in simulations and the executed experiments. The sim-ulations provided an ideal noise-free environment, whereas the experiments are performed in acontrolled environment. To truly demonstrate the effectiveness of the identification abilities ofPCAASO, the method should be analyzed in more complex and practical situations. The baf-fle setup has shown that PCAASO is able to improve the identification results in comparisonwith PCA, but no information is available about the implementation effectiveness of PCAASOwhen actual noise-emitting objects are observed. It is recommended to perform experiments withPCAASO on real life objects to demonstrate its added value in comparison to PCA.The postprocessing analysis of identification with PCA/PCAASO is now performed manually byutilizing several Matlab scripts. These identification methods, however, are very suitable for auto-mated application. A recommendation is to add the identification functionality of PCA/PCAASOto the existing NAH software package to create a completely automated source identification soft-ware application.

71

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[2] D. Brown and W. Halvorsen, Application of the coherence function to acoustic noise measure-ments, Proceedings Inter-Noise 72 (1972), 417–422, Washington DC, U.S.A.

[3] C.J. Dodds and J.D. Robson, Partial coherence in multivariate random processes, Journal ofSound and Vibration 42 (1975), no. 2, 243–249.

[4] G.F. Franklin, J. David Powell, and A. Emami-Naeini, Feedback control of dynamic systems,third edition, Addison-Wesley Publishing Company, Inc., U.S.A., 1994.

[5] J. Hald, STSF - an unique technique for scan-based near-field acoustic holography without restric-tions on coherence, Technical Review, Brüel & Kjær (1989), no. 1.

[6] I.T. Jolliffe, Principal component analysis, Springer-Verlag New York Inc., New York, U.S.A.,1968.

[7] M.S. Kompella, P. Davies, R.J. Bernhard, and D.A. Ufford, A technique to determine the num-ber of incoherent sources contributing to the response of a system, Mechanical Systems and SignalProcessing (1994), Purdue University, U.S.A.

[8] Lay, Linear algebra and its applications, second edition, Addison Wesley Longman, Inc., U.S.A.,1997.

[9] J. Leuridan, D. Roesems, and D. Otte, The use of principal component analysis for correlationanalysis between vibration and acoustical signals, Proceedings 16th International Symposiumon Automotive Technology and Automation (1987), 487–504, Florence, Italy.

[10] K.U. Nam and Y.H. Kim, Visualization of multiple incoherent sources by the backward predictionof near-field acoustic holography, Journal Acoustical Society America 109 (2001), no. 5, 1808–1816.

[11] K.U. Nam and Y.H. Kim, A partial field decomposition algorithm and its examples for near-fieldacoustic holography, Journal Acoustical Society America 116 (2004), no. 1, 172–185.

[12] D. Otte, H.v.d. Auweraer, and J. Leuridan, The use of multivariate correlation techniques forvibro-acoustic experimental data analysis, Proceedings 17th International Seminar on ModalAnalysis (1992), Leuven, Belgium.

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74

Appendix A

Spectra smoothing

Calculating a (cross) spectral matrix results in spectra with a certain variance. To produce spectrawhich have smaller variance Bartletts smoothing procedure is used. Here it is suggested to,instead of computing a ’normal’ autospectrum Sxx over the whole time range N of a randomsignal x = [x1, x2, ..., xN ], divide the signal into k equal blocks of length M = N/k and computethe section spectra Si

xx, i = 1, 2, ..., k. The mean of these section spectra is then

Sxx =1k

k∑i=1

Sixx (A.1)

and is called a smoothed autospectrum. It is shown in [1] that the averaging procedure shownabove leads to a reduction of the variance. The variance of the spectral matrix Sxx is σ2 and sincethe signal sections are independent for a random signal the variance for the matrices Si

xx are alsoσ2. Consequently the variance of the smoothed spectra Sxx is σ2/k.

0 20 40 60 80 1000.55

0.6

0.65

0.7

0.75

0.8

frequency [Hz]

Sxx

[−]

k = 400

0 20 40 60 80 100

0.5

0.6

0.7

0.8

0.9

frequency [Hz]

Sxx

[−]

k = 100

0 20 40 60 80 1000

0.5

1

1.5

frequency [Hz]

Sxx

[−]

k = 10

0 20 40 60 80 1000

1

2

3

4

5

6

7

frequency [Hz]

Sxx

[−]

k = 1

Figure A.1: frequency - Sxx plot. Sxx shown for different values of k, where clearly the variancedecrease is visible

75

Hence by averaging the spectra, the variance is significantly reduced. However, there is a down-side to the averaging procedure; averaging leads to a reduction of the variance, but also to anincrease of the bias. The bias is defined as the difference between the calculated spectra and theexpected theoretical spectra and is desired to be as small as possible. Thus one is forced to makea compromise between the variance and the bias. Here a short discussion will be presented forthe optimization of the spectral matrices.To estimate the spectral matrices as accurately as possible two requirements must be met.

100

101

102

103

10−3

10−2

10−1

100

varia

nce

of S

xx [−

]

k [−]

computed datatheoretical function

Figure A.2: k - variance plot. The varianceof the smoothed spectra are defined as vari-ance/k, which gives a linear line on a loglog

axis.

0 50 100 150 200 250 300 350 4000.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

bias

of S

xx [−

]

k [−]

Figure A.3: k - bias plot. Here the relationbetween the number of averages k and the

bias of Sxx is shown.

The first requirement demands that the smoothed spectra is as close as possible to the theo-retically expected spectra. That is, the bias should be as small as possible. If this is the caseone speaks of high fidelity. The other condition is that the variance of the smoothed spectra issmall. This is called high stability. For a more detailed discussion about the interaction betweenhigh fidelity and high stability [1] is recommended, since that discussion lies beyond the scopethis report. [1] claim that smoothing of the spectral matrices is completely determined by theshape and bandwidth of the windows (e.g. Hanning, Boxcar) used during averaging, where band-width has the largest influence. For achieving a high fidelity it is suggested to use a windowwith a bandwidth in the same order as the width of the smallest important detail in the spec-trum. Where for a high stability, small variance, a large N/M ratio is required. It is shown thatbandwidth ∗ variance = constant, which coincides with the aformentioned interaction betweenvariance and bias. Ultimately an ideal spectral analysis is one, where M is sufficiently large forhigh fidelity and N/M is sufficiently large for high stability. The auto and cross spectral matricesin the remainder of this report are estimated by using the above mentioned frequency domainaveraging. From the detailed analysis in [1], the choice is made to use a Tukey-window.

[1] G.M. Jenkins and D.G. Watts, Spectral analysis and its applications. Holden-Day, San Fran-sisco, 1968

76

Appendix B

STSF analysis

In the discussion of the STSF holography method a scan-based (point to point) measurementenvironment is taken as a reference point, since this is where the qualities of STSF come intosight. Also, all variables in the next section are function of frequency.Consider a typical measurement setup as shown in figure B.1. The hologram plane consists of

zxy so

urceplane

p1s

pMs

hologram

plane

p1h

pNh

Figure B.1: schematic figure of measurement setup

a N-point grid array, where the local pressure for each gridpoint , phn for n = 1, 2, ..., N , ismeasured. These grid measurements are arranged in a column matrix ph resulting in

ph =[

ph1 ph2 ... phN

]T(Nx1). (B.1)

Similarly, the source plane is defined as a grid of M points, psm for m = 1, 2, ...,M , and arrangedinto the row matrix ps:

ps =[

ps1 ps2 ... psM

]T(Mx1). (B.2)

For theoretical convenience it is considered that the number of grid points N on the hologramplane equals the number of grid points M on the source plane. The transformation from thepressure on the hologram plane to the pressure on the source plane is denoted as

ps = Hph, (B.3)

77

where H is the transfer function between the two planes. The transfer function H is definedand obtained in several ways, in this case the propagation method described in section 2.2.1 isutilized. Here, the pressure on the hologram plane is transformed from the frequency domainto the wavenumber domain by performing a 2D FFT, z2. Hereafter, the pressure is multipliedby the inverse propagator Gp to obtain the pressure on the source plane. To obtain the pressureon the source plane in the spatial domain, the pressure is transformed using an inverse 2D FFT,z−1

2 . The transfer function for this case can therefore be defined as

H = z−12 Gpz2. (B.4)

The key feature of STSF is to use cross spectra over the planes to constitute a complete repre-sentation of the present sound field. The cross spectra of the hologram and source plane arerespectively defined as follows

SPP = phpHh , (B.5)

SSS = pspHs . (B.6)

Now consider a specific soundfield and assume that there is a set of L mutually incoherent partialsources present. The individual cross spectra on the hologram and source plane can then bedetermined by

SlPP = pl

hplHh ,

SlSS = pl

splHs ,

(B.7)

where plh for l = 1, 2, ..., L (N x 1) are the incoherent partial sources on the hologram plane and pl

s

for l = 1, 2, ..., L (M x 1) are the theoretical counterparts on the sourceplane. Since these partialsources are incoherent, there is no linear relationship between the partial sources and thereforethe cross spectra in equation (B.7) contribute independently to the total cross spectra SPP ,SSS .This means that the total cross spectra are a summation of the respective individual cross spectra

SPP =∑L

l=1 SlPP ,

SSS =∑L

l=1 SlSS .

(B.8)

For further analysis, a matrix representation of (B.8) is preferred. This is achieved by defining

PLh =

[p1

h p2h . . . pL

h

](NxL),

PLs =

[p1

s p2s . . . pL

s

](MxL).

(B.9)

Note that each row of the matrices in (B.9) shows all the individual contributions of the L partialsources at one grid position. By redefining (B.8) using (B.9) the matrix representation of (B.8) isobtained:

SPP = PLh PLH

h ,

SSS = PLs PLH

s .(B.10)

78

Since each column of PLh consists of a partial source pl

h and all partial sources are mutually in-coherent, the rank of the hologram cross spectra SPP is equal to L. Consequently, one can statethat the rank of the hologram cross spectra SPP is equal to the number of incoherent partialsources pl

h. Besides this conclusion, the above derivation is not very useful for the applicationof STSF. This is because, in general, the independent partial sources are difficult to identify inpractice. For this thesis, finding these partial sources is actually the main reason for implement-ing this method. Therefore, an alternative representation of the hologram cross spectra must beobtained.To be able to discriminate between all the incoherent sound sources present in a soundfield, therank of a new representative matrix must at least equal to the rank of the hologram spectra SPP ,i.e the rank of the new matrix is larger than or equal to rank(SPP )=L. This condition makes surethat all the partial sources are represented in the calculated sound field. Now, consider a matrixAh with J columns and N rows, where J ≥ L, then the hologram cross spectra are rewritten as

SPP = AhAHh . (B.11)

The columns of matrix Ah, ahi for i = 1, 2, ..., J are considered a linear combination of the partialsound sources pl

h.This self-defined matrix Ah is a substitute for a real pressure field. Thus equation (B.3) is usedto propagate Ah to the source plane, which results in As = HAh. To validate this statement,consider the following. From equation (B.3) follows that

PLs = HPL

h . (B.12)

Substituting equation (B.12) into the source hologram cross spectra SSS gives

SSS = PLs PLH

s = (HPLh )(HPL

h )H = H (PLh PLH

h )︸ ︷︷ ︸SPP

HH . (B.13)

Combining equation (B.11) and equation (B.13) finally results in

SSS = HSPP HH = HAhAHh HH = (HAh)(HAh)H = AsA

Hs . (B.14)

Equation (B.14) clearly shows the validity of the earlier statement, that As = HAh.Although the following derivation is focussed on the variables related to the hologram plane,

the derivation on the source plane would be completely analogue. Since the cross spectra SPP

potentially becomes a very large matrix, in practice the cross spectra are not calculated directlybut instead a representation by Ah is used. The main goal is therefore to fully reconstruct thecross spectra SPP by defining a matrix Ah, which is constructed using a minimum amount ofdata acquisition. To address this challenge, STSF introduces a principal component techniquequite similar to the technique discussed in section 3.2.2.Hald et al.[5] considers a set of K reference transducers, which will be used simultaneously to thehologram grid measurements. Consequently, a new signal rl

h is introduced, which is a K-elementcolumn matrix of signals from partial source l measured by the K reference transducers. Similarto pl

h, rlh is difficult to obtain in practice, but prove to be helpful in the further theoretical analysis.

By grouping the signals rlh a new matrix Rh is defined as

Rh =[

r1h r2

h . . . rlh

](KxL). (B.15)

79

Note that the construction of rlh is similar to the signals pl

h. This shared quality is used to definea new set of signals ql

h;

qih =

[rih

pih

]for i = 1, 2, . . . , l (KNx1). (B.16)

Equation (B.16) combines the pressure on the hologram grid with the pressure measured on thereference grid. These ql

h are considered as columns of a matrix Qh, which then becomes

Qh =[

qlh q2

h . . . qlh

]=

[Rh

Ph

](KNxL). (B.17)

Hereafter, the cross spectra between the hologram grid and references are defined as

S+PP = QhQH

h =[

SRR SRP

SHRP SPP

](KNxKN), (B.18)

where SPP is the hologram cross spectra as in (B.6) and SRR, SRP are the cross spectra betweenthe references and between the references and hologram grid measurements respectively:

SRR = RhRHh (KxK), (B.19)

SRP = RhPHh (KxN). (B.20)

Obtaining the matrix Ah dependens on one very important assumption; The set of K ref-erence transducers must represent all the incoherent partial sources L in the sound field. Thisimplies that all the sound field information is represented by the set of K references and conse-quently

rank(SRR) = rank(S+PP ). (B.21)

Now, by subdividing S+PP into two parts

S+PP =

[S1

... S2

]=

SRR... SRP

SHRP

... SPP

, (B.22)

it clarifies that the rank of S1 equals the rank of SRR, since SRR already contains all the possibleinformation of the soundfield. This, on itself, also means that S2 does not add any additionalinformation to S+

PP and thus the columns of S2 are seen as a linear combination of the columnsof S1. This brings about that there exists a matrix E (KxN), which connects S1 and S2;

S2 = S1E. (B.23)

By replacing S1,S2 in the above equality by their respective definitions denoted in equation (B.22)a solvable system of linear equations arises,

SRP = SRRE, (B.24)

SPP = SHRP E. (B.25)

80

From equation (B.24) follows that E = S−1RRSRP . When this is substituted into equation (B.25) a

new definition for the cross spectra matrix SPP is found:

SPP = SHRP S−1

RRSRP . (B.26)

Equation (B.26) forms the basis for obtaining the cross spectral matrix SPP in STSF.Here the principal component technique becomes prominent. Using a singular value decompo-sition, The reference cross spectra matrix is divided into three parts:

SRR = UΛUH , (B.27)

where Λ contains the eigenvalues of SRR and U contains the respective eigen vectors. Equation(B.26) encompasses the inverse of SRR, which are now denoted as

S−1RR = (UΛUH)−1 = U−HΛ−1U−1. (B.28)

Substitution of equation (B.28) into the definition of SPP in equation (B.26) gives

SPP = SHRP U−HΛ−1U−1SRP . (B.29)

Comparing the previously defined SPP in equation (B.11) with equation (B.29) we finally arriveat the notation for the matrix Ah:

SPP = SHRP U−HΛ−1/2︸ ︷︷ ︸

Ah

Λ−1/2U−1SRP︸ ︷︷ ︸AH

h

. (B.30)

Since U−1 = UH [8] for orthogonal matrices, Ah is finally defined as Ah = SHRP UΛ−1/2. Conse-

quently the definition from equation (B.11) is found. To obtain this notation of Ah the referencecross spectra SRR and the cross spectra between the references and hologram points SRP areutilized. Presuming that the number of references K is far less than the number of grid pointsN , (B.30) reduces the number of calculations significantly compared to a direct calculation of thehologram cross spectra. Also Ah now fully represents the incoherent soundfield. Each columnof Ah, Ai

h for i = 1, 2, . . . ,K, corresponds to the representation of incoherent sound effects onthe hologram plane. The ultimate goal of STSF, to obtain a hologram of an incoherent sound-field, is then found by adding the columns of Ah together. The obtained incoherent sound effectsAi

h are individually back propagated to the source plane and subsequently added to acquire thesource plane of the incoherent soundfield. The PNAH method described in chapter 2 is utilizedfor this purpose. Ultimately, STSF serves as a technique for scan-based measurement scenarios.From a principal component representation of the soundfield using references and hologrammeasurements, the incoherent sound sources are extracted. Hereafter, the identified individualsources are backpropagated (using NAH) to the source plane and finally, an incoherent sourceplane representation of the soundfield is obtained.

81

82

Appendix C

Eigen and Singular Value Decompositionrelation

Eigen value decompositions (EVD) are only applicable to a certain class of square matrices. AnEVD of a [M ×M ] square matrix E would result in

E = CΛCH , (C.1)

withC being an [M×M ] orthonormal matrix with eigenvector columns andΛ a [M×M ] diagonalmatrix with the eigenvalues of E.The singular value decomposition (SVD) uses the EVD of a positive semi-definite matrix in orderto derive a similar decomposition applicable to all rectangular matrices. The SVD of a [M × N ]matrix S leads to a factorization form

S = UΣV H , (C.2)

where U is an [M ×M ] orthogonal matrix, V is an [N ×N ] orthogonal matrix and Σ is a [M ×N ]diagonal matrix. The values on the diagonal of matrix Σ are called the singular values of P andU, V are called the left and right singular vectors respectively.By applying a singular value decomposition to a semi-positive definite matrix, the matrices ofan EVD and SVD are similar. A positive semi-definite matrix can be obtained by multiplying amatrix with its hermitian. With the matrix S from (C.2) two positive semi-definite matrices arepossible, namely SSH and SHS. By substituting the SVD from (C.2) into these matrices we get

SSH = UΣV H(UΣV H)H = UΣV HV ΣUH = UΣ2UH (C.3)

SHS = (UΣV H)HUΣV H = V ΣUHUΣV H = V Σ2V H (C.4)

Note that V HV = UHU = I is a condition of an orthogonal matrix. The right equations in (C.3)are the EVDs of SSH and SHS respectively. Consequently, the squares of the singular values ofS are equal to the eigenvalues of SSH or SHS, the columns of U are the eigenvectors of SSH

and the columns of V are the eigenvectors of SHS.

83

84

Appendix D

Matlab scripts

In this appendix the Matlab files used during the simulations, experiments and post-experimentanalyses are provided.

D.1 Simulations

1 function [v,ns] = BaffledPistons(x_p,y_p,r_p,v_p,M,N,dx,dy)2

3 %%%%%%%%%%%%%%%%%%%%//<<11−01−2004>>\\%%%%%%%%%by :%Rick %Scholte %%%%%%%%4 %5 % BaffledPistons.m6 % With this script a sourcefield is created7 %8 % [ v] = BaffledPistons ( x_p , y_p , r_p , v_p , M, N, dx , dy )9 %10 % input :11 % x_p : x−coordinate ( s) centre of piston ( s)12 % y_p : y−coordinate ( s) centre of piston ( s)13 % r_p : radius of piston ( s)14 % M: number of columns ( x−direction )15 % N: number of rows ( y−direction )16 % dx : point spacing [ m] between two neighbouring samples in17 % x−direction18 % dy : point spacing [ m] between two neighbouring samples in19 % y−direction20 %21 % output :22 % v: normal velocities of the baffle with pistons [ m/ s]23 %24 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%25

26 Mx = M;27 Ny = N;28

29 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%30 % Setup of the pistons in the x, y−plane.31 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%32 v = zeros(Ny,Mx);33 ns=0;

85

34 for (piston = 1:length(v_p)),35 for (m = (x_p(piston) −r_p(piston)):dx:(x_p(piston)+r_p(piston))),36 for (n = (y_p(piston) −r_p(piston)):dy:(y_p(piston)+r_p(piston))),37 if (sqrt((m −x_p(piston))^2 + (n −y_p(piston))^2) ≤ r_p(piston)),38 v(floor(n/dy),floor(m/dx)) = v_p(piston);39 ns=ns+1;40 else41 v(ceil(n/dy),ceil(m/dx)) = 0;42 end ;43

44 end ;45 end ;46 end ;47

48 % figure ;49 % imshow ( v,[])50 % colorbar51 % set ( gca ,' YDir ',' normal ')

1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%2 %3 % Total_Bafflesimulation_1sfield.m.m4 %5 % This script6 % − defines and generates the sound source plane7 % − calculates the spectral matrices Sss. This is the spectra between8 % the source plane measurements. This spectra is used to compare the9 % principal components on the sourceplane with the principal10 % components of the reference plane. See section normalised principal11 % components in thesis. Note the spectral matrix of the reference12 % plane is calculalted with Total_Bafflesimulation_2spec.m13 %14 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%15 clear all16 close all17 clc18

19 dx = 0.01; % point spacing [ m] between two neighbouring samples in x−direction20 dy = 0.01; % point spacing [ m] between two neighbouring samples in y−direction21 M=32; % number of columns in x−direction22 N=32; % number of columns in y−direction23

24 %%%%%%%%%%%%%%%%%%%25 %%%Defining Acoustic Sources %%%26 %%%%%%%%%%%%%%%%%%%27

28 n_avg=200; % Number of averaging steps ( see appendix Spectra smoothing )29 calsize=300 * 256* 2; % Number of samples time signal30 randn( ' state ' ,125689) % Define two random time signals ( from different states ) .31 x1=randn(calsize,1);32 randn( ' state ' ,6348)33 x2=randn(calsize,1);34 Stime=[x1 x2]; % Stime = [ M* N x tlength ], rows : time , columns : positions35 [p_z_s,n] = BaffledPistons([0.08 0.24],[0.16 0.16],[0.0091 * ones(1,2)], ...36 [Stime(1,1) Stime(1,2)],M,N,dx,dy); %x y37 [py,px,Xs]=find(p_z_s);

86

38 figure;imshow(p_z_s,[])39

40 %%%%%%%%%%%%%%%%%%%%%%%%41 %%%Defining spectra matrix ( averaging ) %%%42 %%%%%%%%%%%%%%%%%%%%%%%%%43

44 %%%45 % Parameters frequency domain46 %%%47 Fs=2^(10); % Sampling frequency48 nfftfreq=256;49 freqaxis=Fs/nfftfreq * (0:nfftfreq/2);50 % freq axis = ∆ f * (1: N/2), ∆ f = Fs/ N, Fs = 1/ ∆ t , ∆ t = T/ N51 Ssstot=zeros(n * n,1);52 fchoose = 15; % choose specific frequency for spectra calculation53 windowfr = tukeywin(nfftfreq);54

55 for mm=1:1:n_avg;56 Stimewin=Stime(1+(mm −1) * nfftfreq:mm * nfftfreq,:). * (windowfr * ones(1,size(Xs,1)));57 % Windowed section of source signals in time58 Sfreq=fft(Stimewin).'; %Note : frequency spectrum correct for frequency f > Fs/ nfftfreq59

60 %%%61 % Calculate powerspectra matrices Sss62 %%%63 Sss=[];64 for d=1:1:size(Sfreq,1)65 for e=1:size(Sfreq,1)66 sss=conj(Sfreq(d,fchoose)). * Sfreq(e,fchoose);67 Sss=[Sss;sss];68 end69 end70

71 Ssstot=Ssstot+Sss; % summation72 display([ ' end of iteration ' num2str(mm)])73

74 end75 Sss_avg=Ssstot/mm; % final calculation of averaged spectra76

77 Sss=[];78 for e = 1:n79 Sssrow = Sss_avg((e −1) * n+1:e * n,1).';80 Sss = [Sss;Sssrow];81 end82

83 % pc_source =svd ( Sss ) % examine principal components of source plane84 % [ Uref , Sref , Vref ]= svd ( Sxx );85

86 Total_Bafflesimulation_2spec

1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%2 %3 % Total_Bafflesimulation_2spec.m4 %5 % With this script one :6 % − calculates the spectral matrices Sxx and Syx.

87

7 % x = reference plane , y = hologram plane8 %9 % Note : This script should ALWAYSbe preceded by10 % Total_Bafflesimulation_1sfield.m11 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%12

13

14 [py,px,Xs]=find(p_z_s); % px , py are grid coordinates of sound sources on the source plane15 n=size(Xs,1); % number of sources16 % −−> px _____17 % | | |18 % v | ____ |19 % py20

21 dxs = 0.01; % point spacing [ m] between two neighbouring samples in x−direction...22 dys = 0.01; % point spacing [ m] between two neighbouring samples in y−direction23 % ( dxs , dys are used to change sourceplane size )24 c = 343; % speed of sound [ m/ s]25

26

27

28 %%%29 % Parameters wavenumber domain30 %%%31 %x = 0: dxs : xtot , ∆ k = 2* pi / xtot or ∆ k = ks / nfftk , ks =2* pi / dxs ; [ rad / m]32 nfftkx=M;33 nfftky=N;34 xlength=nfftkx * dxs;35 kxaxis= −2* pi/dxs/2 : 2 * pi/dxs/nfftkx : 2 * pi/dxs/2 −2* pi/dxs/nfftkx; % [ rad / m]36 kyaxis= −2* pi/dys/2 : 2 * pi/dys/nfftky : 2 * pi/dys/2 −2* pi/dys/nfftky; % [ rad / m]37

38 m=nfftkx * nfftky; % number of hologram measurement points39

40 % new coordinates41 pxn=[];42 pyn=[];43 factor = dx/dxs; % gridsize change44 dlength=M * dx; % length of sourceplane45

46 for g = 0:factor − 147 pxna = (xlength/dlength −1) * factor * 1/(xlength/dlength) * M+px* factor − (factor −1) + g;48 pxn=[pxn pxna];49 pyna = (xlength/dlength −1) * factor * 1/(xlength/dlength) * M+py* factor − (factor −1) + g;50 pyn=[pyn pyna]; % pxn , pyn =( new) grid coordinates of sources on source plane51 end52

53 % reference locations54 xref = [8;26] ;55 yref = [18;14]; % grid positions of reference microphones56 zref = 0.01; % z position of reference plane [ m]57 numref = size(xref,1); % number of reference microphones58 ref_notused=[]; % unused references for scenario59 zknown = 0; % z−location source plane [ m]60 znew=0.04; % z−location hologram plane [ m]61 zsource=zknown;62 zholo=znew;63

88

64

65 %%66 Sxxtot=zeros(numref,numref);67 Syxtot=zeros(m,numref);68 Syytot=zeros(m,m);69

70 windowfr = tukeywin(nfftfreq);71

72 for mm=1:1:n_avg; %calsize / nfftfreq73 Stimewin=Stime(1+(mm −1) * nfftfreq:mm * nfftfreq,:). * (windowfr * ones(1,size(Xs,1)));74 % Windowed section of source signals in time75

76 %%%77 % Transformation to frequency spectrum P( x, y, z, t ) −−> P( x, y, z, f )78 %%%79 Sfreq=fft(Stimewin).';80 % Note : frequency spectrum correct for frequency f > Fs/ nfftfreq81 Refmic=[];82 Hpoint=[];83 for b = fchoose84 Sfreqplane=zeros(nfftkx,nfftky);85 Sfreqselect=Sfreq(:,b);86 for d=1:size(px,1)87 for t=pyn(d,:)88 for u=pxn(d,:);89 Sfreqplane(t,u)=Sfreqselect(d); % Sourceplane in frequency domain90 end91 end92 end93

94 %%%95 % Transformation to wavenumber domain P( x, y, z, f ) −−> P( kx , ky , z, f )96 %%%97 Swave=fft2(Sfreqplane,nfftkx,nfftky);98 Swave=fftshift(Swave);99 %%%100 % NAH Angular spectrum , P( kx , ky , zs , f ) −−> P( kx , ky , zh , f )101 %%%102 k=2 * pi * freqaxis(b)/c;103 Kztot=[];104 for s = kyaxis105 Kz=[];106 for t = kxaxis107 kz=sqrt(k^2 −t^2 −s^2);108 Kz=[Kz kz];109 end110 Kztot=[Kztot;Kz];111 end112

113 Swaveprop_refs=Swave. * exp(Kztot * i * (zref − zknown));114 Swaveprop_refsi=ifftshift(Swaveprop_refs);115 Sfreqplaneprop_refs=ifft2(Swaveprop_refsi);116 % Reference plane in frequency domain117

118 Swaveprop=Swave. * exp(Kztot * i * (znew − zknown));119 Swavepropi=ifftshift(Swaveprop);120 Sfreqplaneprop=ifft2(Swavepropi);

89

121 % Hologram plane in frequency domain122

123 % % Display reference locations124 % Reflocations =zeros ( nfftkx , nfftky );125 % for d=1: size ( xref ,1)126 % for t =yref ( d,:)127 % for u=xref ( d,:);128 % Reflocations ( t , u)=−100;129 % end130 % end131 % end132

133

134

135 %%%136 % Defining reference microphoness137 %%%138 Refmica=[];139 for d = 1:numref140 refmica=Sfreqplaneprop_refs(yref(d),xref(d));141 Refmica=[Refmica;refmica];142 end143 Refmic=[Refmic Refmica];144 % each row represents a reference microphone signal145

146 %%%147 % Defining Hologram measurement points148 %%%149 % Hpoint = frequency development of the measurement points ( hologram150 % plane ) [ M* N x freqaxis ] , rows : positions columns : frequency development151

152 Hpointa=[];153 for j = 1:nfftkx154 Hpointrow = Sfreqplaneprop(j,:);155 Hpointa = [Hpointa;Hpointrow.'];156 end157 Hpoint = [Hpoint Hpointa];158

159 end160

161

162 %%%163 % Calculate powerspectra matrices Sxx , Syx and Syy164 %%%165 Sxx=[];166 Syx=[];167

168 Sxx = Refmic * Refmic';169 Syx = Hpoint * Refmic';170 Syy=Hpoint * Hpoint';171

172 Sxxtot=Sxxtot+Sxx; % summation173 Syxtot=Syxtot+Syx; % summation174 Syytot=Syytot+Syy; % summation175

176 display([ ' end of iteration ' num2str(mm)])177

90

178 end179

180 Sxx=Sxxtot/mm;181 Syx=Syxtot/mm;182 Syy=Syytot/mm; % final calculation of averaged spectra183

184

185 % %%% %%%186 % % Define actual spectral matrices %187 % %%% %%%188 % This part is used when less references are desired for the PCA measurement189 ref_notusedsort=sort(ref_notused,2, ' descend ' ) ;190 for l = ref_notusedsort191 Sxx(l,:)=[];192 Sxx(:,l)=[];193 Syx(:,l)=[];194 end195

196 numref=numref −size(ref_notused,2);197

198 save simdata199

200 PCAsims

1 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%2 % %%% PCA method %%%%3 % This script4 % − performs a PCA measurement on the loaded data set5 % − calculates the virtual coherences6 % − plots the obtained incoherent sound holograms7 % %%%

%%%%8 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%9

10 clear all11 clc12

13 % load mat−file14 [filename, pathname] = uigetfile( ' * .mat ' , ' Select data −file with spectra ' );15 load([pathname filename]);16

17 M=nfftkx;18 N=nfftky;19 %%% %%%20 % Singular Value Decomposition %21 % to determine principal components %22 %%% %%%23

24 [U,S,V]=svd(Sxx); % Singular value decomposition U=V eigenvectors , S=eigenvalues25 PC=sqrt(diag(S)); % principal components of loaded data set26

27

28 %%%29 % Partial Hologram determination30 %%%31

91

32 Syv=Syx * U; % Spectra between hologram measurement points and principal components33 Sinv=inv(S);34 Pinv=sqrt(Sinv);35

36 Qinv=Pinv * Syv'; % Partial holograms ( each row ) on hologram plane37 % Qinv i =[ pc_1 all positons fi ]38 % [ ... ]39 % [ pc_n all positons fi ]40

41 Qholototplanep=zeros(N,M);42 for j = 1:size(Qinv,1)43 Qholo=Qinv(j,:);44 Qholoplanep=[];45 for b = 1:N46 Qholo_row=Qholo(1+(b −1) * M:b* M);47 Qholoplanep=[Qholoplanep;Qholo_row];48 end49 eval([ ' Qholoplane_ ' num2str(j) ' =Qholoplanep ; ' ]);50 % Partial holograms in plane on hologram plane51

52 end53

54 S_QQ=Qinv' * Qinv; % Spectra of hologram plane55 Qholototplanep=zeros(N,M);56 Qholo=diag(S_QQ);57 Qholototplanep=[];58 for b = 1:N59 STSFholo_col = Qholo(1+(b −1) * M:b* M);60 Qholototplanep = [Qholototplanep;STSFholo_col.'];61 end62 Qholototplane=sqrt(Qholototplanep); % Total sound hologram in plane on hologram plane63

64

65 %%%66 % Virtual coherences67 %%%68

69 Sxxac=Sxx * U; % Spectra between references and principal components70 Sxacxac=S; % Spectra of principal components71

72

73 VIRCOH=[];74 for r=1:numref75 vircohrow=[];76 for s=1:numref77 vircoh = abs(Sxxac(r,s))^2/(Sxx(r,r) * Sxacxac(s,s));78 vircohrow=[vircohrow vircoh];79 end80 VIRCOH=[VIRCOH;vircohrow]; % Virtual coherences81 end82 % VIRCOH =83 % [ gamma_i, j gamma_i, j +1 ... gamma_i, n]84 % [ gamma_i+1, j ]85 % [ ... ]86 % [ gamma_n, j gamma_n, n ]87 % n = # of references88

92

89 %%%90 % Plotting of results91 %% %92

93 %Create figure94 figure1 = figure( ' Units ' , ' centimeters ' , ' Position ' ,[8 15 11 13]);95 %% Create axes96 axes1 = axes( ...97 ' Ytick ' ,[], ...98 ' Units ' , ' centimeters ' , ...99 ' Xtick ' ,[], ...100 ' CLim' ,[0 max(max(abs(Qholototplane)))], ...101 ' Layer ' , ' top ' , ...102 ' Position ' ,[4 8 3 3], ...103 ' TickDir ' , ' out ' , ...104 ' YDir ' , ' reverse ' , ...105 ' Parent ' ,figure1);106 axis(axes1,[0.5 32.5 0.5 32.5]);107 title(axes1, ' Simulation PCA' );108 box(axes1, ' on' );109 hold(axes1, ' all ' );110

111 %% Create image112 image1 = image( ...113 ' CData ' ,abs(Qholototplane), ...114 ' CDataMapping ' , ' scaled ' , ...115 ' Parent ' ,axes1);116

117 %% Create axes118 axes2 = axes( ...119 ' Ytick ' ,[], ...120 ' Units ' , ' centimeters ' , ...121 ' Xtick ' ,[], ...122 ' CLim' ,[0 max(max(abs(Qholototplane)))], ...123 ' Layer ' , ' top ' , ...124 ' Position ' ,[1.5 3 3 3], ...125 ' TickDir ' , ' out ' , ...126 ' YDir ' , ' reverse ' , ...127 ' Parent ' ,figure1);128 axis(axes2,[0.5 32.5 0.5 32.5]);129 xlabel(axes2,[ ' pc_1 = ' num2str(PC(1))]);130 box(axes2, ' on' );131 hold(axes2, ' all ' );132

133 %% Create image134 image2 = image( ...135 ' CData ' ,abs(Qholoplane_1), ...136 ' CDataMapping ' , ' scaled ' , ...137 ' Parent ' ,axes2);138

139 %% Create axes140 axes3 = axes( ...141 ' CLim' ,[0 max(max(abs(Qholototplane)))], ...142 ' Ytick ' ,[], ...143 ' Units ' , ' centimeters ' , ...144 ' Xtick ' ,[], ...145 ' Layer ' , ' top ' , ...

93

146 ' Position ' ,[6.5 3 3 3], ...147 ' TickDir ' , ' out ' , ...148 ' YDir ' , ' reverse ' , ...149 ' Parent ' ,figure1);150 axis(axes3,[0.5 32.5 0.5 32.5]);151 xlabel(axes3,[ ' pc_2 = ' num2str(PC(2))]);152 box(axes3, ' on' );153 hold(axes3, ' all ' );154

155 %% Create image156 image3 = image( ...157 ' CData ' ,abs(Qholoplane_2), ...158 ' CDataMapping ' , ' scaled ' , ...159 ' Parent ' ,axes3);160

161 save PCAdata PC VIRCOH Qholoplane * Qholototplane

1 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%2 % %%% NAM method %%%%3 % This script4 % − performs a NAM measurement on the loaded data set5 % − plots the obtained incoherent sound holograms6 % %%%

%%%%7 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%8

9 clear all10 clc11

12 % load mat−file13 [filename, pathname] = uigetfile( ' * .mat ' , ' Select data −file with spectra ' );14 load([pathname filename]);15

16 zsource=zknown; % source plane z position17 zholo=znew; % hologram plane z position18 M=nfftkx; % # of gridlocations in X19 N=nfftky; % # of gridlocations in Y20

21 NAMholo=diag(Syy);22 NAMholoplanre=[];23 for b = 1:M24 NAMholo_col = NAMholo(1+(b −1) * M:b* M);25 NAMholoplanre = [NAMholoplanre;NAMholo_col.'];26 end27 NAMholototplane=sqrt(NAMholoplanre); % Hologram plane28

29

30 [S,D,SS] = svd(Syy); % Note : heavy calculation31 Y = S(:,1:n) * sqrt(D(1:n,1:n)); % n = # of incoherent sound effects32 %% note : if D( i , i ) << 1 make D( i , i )=0 . This is to bypass back propagating33 %% errors34

35

36 %%%%%%%%%%%%%%37 %%%Back propagate hologram measurements to source plane38 %%%%%%%%%%%

94

39

40 Ybprop=[];41 for b = 1:size(Y,2)42 Yplane=[];43 Yselect=Y(:,b);44 for j = 1:N45 Yplanerow = Yselect((j −1) * M+1:j * M,1).';46 Yplane = [Yplane;Yplanerow]; %partial hologram measurements in plane −form47 end48

49

50 %%%51 % Transformation to wavenumber domain P( x, y, z, f ) −−> P( kx , ky , z, f )52 %%%53 Ywave=fft2(Yplane,nfftkx,nfftky);54 Ywave=fftshift(Ywave);55

56 %%%57 % NAH Angular spectrum , P( kx , ky , zs , f ) −−> P( kx , ky , zh , f )58 %%%59

60 Ywavebprop=Ywave. * exp(Kztot * i * (zsource −zholo));61 Yfreqplanebprop=ifft2(ifftshift(Ywavebprop));62 eval([ ' Ybpropplane_ ' num2str(b) ' =Yfreqplanebprop ; ' ]);63 % incoherent hologram plane b propagated to source plane64

65 Ybpropa=[];66 for j = 1:N67 Ybproprow = Yfreqplanebprop(j,:).';68 Ybpropa = [Ybpropa;Ybproprow];69 end70 Ybprop=[Ybprop Ybpropa];71

72

73 end74

75 %%%76 % Calculate total sound hologram on source plane77 %%%78 Sssnam=Ybprop * Ybprop'; %Spectra at source plane79

80 NAMsource=diag(Sssnam);81 NAMsourceplanre=[];82 for b = 1:M83 NAMsource_col = NAMsource(1+(b −1) * N:b * N);84 NAMsourceplanre = [NAMsourceplanre;NAMsource_col.'];85 end86 NAMsourcetotplane=sqrt(NAMsourceplanre);87 %Total sound hologram on source plane88

89

90 %%%91 %% NAM identification method on source plane92 %%%93 Ybpropo=Ybprop;94 sy=size(Ybpropo,2);95 for s=1:sy

95

96 autos=Ybprop. * Ybprop;97 S_aapdiag=sum(abs(autos),2); % Autospectra of Sssnam98

99

100 [I,J]=find(S_aapdiag == max(S_aapdiag));101 % Find maximum of spectra of source plane102

103 if size(I,1) > 1104 Ir=I(1);105 else106 Ir=I;107 end % In case more maxima, take first maximum for calculation108

109 G_smaxr=Ybprop(Ir,:); % [1 x K]110 S_wS=Ybprop * G_smaxr'; % [ N x K] [ K x 1] = [ N x 1]111 s_ww=G_smaxr* G_smaxr'; % [1 x K] [ K x 1] = [1 x 1] == max( S_aapdiag )112

113 G_s1 = S_wS/sqrt(s_ww); % [ N x 1]114 Ybprop = Ybprop * (eye(sy,sy) − (G_smaxr' * G_smaxr)/s_ww);115 % [ N x K] Total source hologram minus sth contribution116

117 Saa_pr=[];118 for b = 1:M119 Saa_prow = G_s1((b −1) * M+1:b * M).'; %G_s1120 Saa_pr = [Saa_pr;Saa_prow];121 end122 eval([ ' NAMsourceplane_ ' num2str(s) ' =Saa_pr ; ' ]);123 % Partial holograms on source plane124

125 if max(max(abs(Ybprop))) == 0126 sy=s127 break128 end129 % If total remaining source hologram is empty , stop loop130 end131

132 %%%133 % Plotting of results134 %% %135

136 %Create figure137 figure1 = figure( ' Units ' , ' centimeters ' , ' Position ' ,[8 15 11 13]);138 %% Create axes139 axes1 = axes( ...140 ' Ytick ' ,[], ...141 ' Units ' , ' centimeters ' , ...142 ' Xtick ' ,[], ...143 ' CLim' ,[0 max(max(abs(NAMsourcetotplane)))], ...144 ' Layer ' , ' top ' , ...145 ' Position ' ,[4 8 3 3], ...146 ' TickDir ' , ' out ' , ...147 ' YDir ' , ' reverse ' , ...148 ' Parent ' ,figure1);149 axis(axes1,[0.5 32.5 0.5 32.5]);150 title(axes1, ' Simulation NAM sourceplane ' );151 box(axes1, ' on' );152 hold(axes1, ' all ' );

96

153

154 %% Create image155 image1 = image( ...156 ' CData ' ,abs(NAMholototplane), ...157 ' CDataMapping ' , ' scaled ' , ...158 ' Parent ' ,axes1);159

160 %% Create axes161 axes2 = axes( ...162 ' Ytick ' ,[], ...163 ' Units ' , ' centimeters ' , ...164 ' Xtick ' ,[], ...165 ' CLim' ,[0 max(max(abs(NAMsourcetotplane)))], ...166 ' Layer ' , ' top ' , ...167 ' Position ' ,[1.5 3 3 3], ...168 ' TickDir ' , ' out ' , ...169 ' YDir ' , ' reverse ' , ...170 ' Parent ' ,figure1);171 axis(axes2,[0.5 32.5 0.5 32.5]);172 %xlabel ( axes2 ,[' pc_1 = ' num2str ( PC(1))]);173 box(axes2, ' on' );174 hold(axes2, ' all ' );175

176 %% Create image177 image2 = image( ...178 ' CData ' ,abs(NAMsourceplane_1), ...179 ' CDataMapping ' , ' scaled ' , ...180 ' Parent ' ,axes2);181

182 if sy ≥ 2183 %% Create axes184 axes3 = axes( ...185 ' CLim' ,[0 max(max(abs(NAMsourcetotplane)))], ...186 ' Ytick ' ,[], ...187 ' Units ' , ' centimeters ' , ...188 ' Xtick ' ,[], ...189 ' Layer ' , ' top ' , ...190 ' Position ' ,[6.5 3 3 3], ...191 ' TickDir ' , ' out ' , ...192 ' YDir ' , ' reverse ' , ...193 ' Parent ' ,figure1);194 axis(axes3,[0.5 32.5 0.5 32.5]);195 %xlabel ( axes3 ,[' pc_2 = ' num2str ( PC(2))]);196 box(axes3, ' on' );197 hold(axes3, ' all ' );198

199 %% Create image200 image3 = image( ...201 ' CData ' ,abs(NAMsourceplane_2), ...202 ' CDataMapping ' , ' scaled ' , ...203 ' Parent ' ,axes3);204 end205

206 %%%%%%%%%%%207 %%%Forward propagate source hologram measurements to hologram plane208 %%%%%%%%%%%209

97

210 Yhprop=[];211 for b = 1:sy212 Yplane=[];213 Ys_select=eval([ ' NAMsourceplane_ ' num2str(b);]);214

215

216 %%%217 % Transformation to wavenumber domain P( x, y, z, f ) −−> P( kx , ky , z, f )218 %%%219 Ys_wave=fft2(Ys_select,nfftkx,nfftky);220 Ys_wave=fftshift(Ys_wave);221 %%%222 % NAH Angular spectrum , P( kx , ky , zs , f ) −−> P( kx , ky , zh , f )223 %%%224 % k=2* pi * freqaxis ( b)/ c;225 % Kztot =[];226 % for s = kyaxis227 % Kz=[];228 % for t = kxaxis229 % kz =sqrt ( k^2−t ^2−s^2);230 % Kz=[ Kz kz ];231 % end232 % Kztot =[ Kztot ; Kz];233 % end234

235 Yhwaveprop=Ys_wave. * exp(Kztot * i * (zholo −zsource));236 Yhfreqplaneprop=ifft2(ifftshift(Yhwaveprop));237 %figure ; imshow ( abs ( Yhfreqplaneprop ), [])238 eval([ ' Yhpropplane_ ' num2str(b) ' =Yhfreqplaneprop ; ' ]);239 % incoherent hologram plane b propagated to source plane240

241 Yhpropa=[];242 for j = 1:N243 Yhproprow = Yhfreqplaneprop(j,:).';244 Yhpropa = [Yhpropa;Yhproprow];245 end246 Yhprop=[Yhprop Yhpropa];247

248 end249

250 % total hologram251 Shhnam=Yhprop * Yhprop'; % Spectra at hologram plane252

253 NAMholoprop=diag(Shhnam);254 NAMholoplanre=[];255 for b = 1:M256 NAMholo_col = NAMholoprop(1+(b −1) * N:b * N);257 NAMholoplanre = [NAMholoplanre;NAMholo_col.'];258 end259 NAMholototplaneprop=sqrt(NAMholoplanre); %Total sound hologram on source plane260

261 % Plotting262 %% Create figure263 figure1 = figure( ' Units ' , ' centimeters ' , ' Position ' ,[8 15 11 13]);264 %% Create axes265 axes1 = axes( ...266 ' Ytick ' ,[], ...

98

267 ' Units ' , ' centimeters ' , ...268 ' Xtick ' ,[], ...269 ' CLim' ,[0 max(max(abs(NAMholototplaneprop)))], ...270 ' Layer ' , ' top ' , ...271 ' Position ' ,[4 8 3 3], ...272 ' TickDir ' , ' out ' , ...273 ' YDir ' , ' reverse ' , ...274 ' Parent ' ,figure1);275 axis(axes1,[0.5 32.5 0.5 32.5]);276 title(axes1, ' Simulation NAM holoplane ' );277 box(axes1, ' on' );278 hold(axes1, ' all ' );279

280 %% Create image281 image1 = image( ...282 ' CData ' ,abs(NAMholototplaneprop), ...283 ' CDataMapping ' , ' scaled ' , ...284 ' Parent ' ,axes1);285

286 %% Create axes287 axes2 = axes( ...288 ' Ytick ' ,[], ...289 ' Units ' , ' centimeters ' , ...290 ' Xtick ' ,[], ...291 ' CLim' ,[0 max(max(abs(NAMholototplaneprop)))], ...292 ' Layer ' , ' top ' , ...293 ' Position ' ,[1.5 3 3 3], ...294 ' TickDir ' , ' out ' , ...295 ' YDir ' , ' reverse ' , ...296 ' Parent ' ,figure1);297 axis(axes2,[0.5 32.5 0.5 32.5]);298 %xlabel ( axes2 ,[' pc_1 = ' num2str ( PC(1))]);299 box(axes2, ' on' );300 hold(axes2, ' all ' );301

302 %% Create image303 image2 = image( ...304 ' CData ' ,abs(Yhpropplane_1), ...305 ' CDataMapping ' , ' scaled ' , ...306 ' Parent ' ,axes2);307

308 if sy ≥ 2309 %% Create axes310 axes3 = axes( ...311 ' CLim' ,[0 max(max(abs(NAMholototplaneprop)))], ...312 ' Ytick ' ,[], ...313 ' Units ' , ' centimeters ' , ...314 ' Xtick ' ,[], ...315 ' Layer ' , ' top ' , ...316 ' Position ' ,[6.5 3 3 3], ...317 ' TickDir ' , ' out ' , ...318 ' YDir ' , ' reverse ' , ...319 ' Parent ' ,figure1);320 axis(axes3,[0.5 32.5 0.5 32.5]);321 %xlabel ( axes3 ,[' pc_2 = ' num2str ( PC(2))]);322 box(axes3, ' on' );323 hold(axes3, ' all ' );

99

324

325 %% Create image326 image3 = image( ...327 ' CData ' ,abs(Yhpropplane_2), ...328 ' CDataMapping ' , ' scaled ' , ...329 ' Parent ' ,axes3);330 end331

332 save NAMdata Y NAMsourcetotplane ...333 NAMsourceplane_ * NAMholototplaneprop Yhpropplane_ *

D.2 Experiments

1 %%%%%%%%%%%%%%%%%%%%%%%2 %%%3 %%%param_load.m4 %%%Load inital parameters for experiments5 %%%6 %%%%%%%%%%%%%%%%%%%%%%%7 clear8 close all9 clc10

11 cd D:\Experiments\Experiments_vfgtrig_130612

13 global freqdes dim numf14 % load mat−file15 [filename, pathname] = uigetfile( ' * .mat ' , ' Select data −file ( measplanepres ) ' );16 load([pathname filename]);17 cd(pathname);18

19 % Choose desired frequencies20 freqdes = 388; % Desired frequencies for analysis (= freqdes * freq resolution Hz)21 numf=size(freqdes,2);22 dim = [rows cols];23

24 PCAexperimental;

1 %%% %%%2 % PCAexperimental.m %3 % Script to identify incoherent %4 % sound effects using PCA−method %5 %%% %%%6

7 % Input Variables8 %9 % Srr : Spectral matrix of reference microphones. Autospectra on diagonal ,10 % cross spectra off diagonal. numref x numref matrix , where numref is the11 % number of references used.12 %13 % Spr : Spectral matrix between hologram measurements and reference14 % microphones. nummeas x numref matrix , where nummeas is the number of15 % gridpoint measurements.16 %

100

17 % freqvec : Frequency axis of measurements18 %19 % freqdes : Desired frequencies to perform PCA−technique on20 %21 % numref : Number of used references during measurement22 %23 % dim : Size of measurement grid , vector notation where first number is24 % number of rows and second number is number of columns , [ rows cols ]25 %26 %27 % Outputs28 %29 % PCAholototplane : Total sound holograms of selected frequencies on hologram plane.30 %31 % PCAholoplane : Partial sound holograms of selected frequencies on hologram plane.32 %33 % PCAsourcetotplane : Total sound holograms of selected frequencies on source plane.34 %35 % PCAsourceplane : Partial sound holograms of selected frequencies on source plane.36 %37 % PC: Principal components for each selected frequency.38 %39 % freqdes : Output of selected frequencies for which PCA is done.40 %41 % VIRCOHtot : Virtual coherences of selected frequencies.42 % numref x numref * size ( freqsel ,2) matrix. For example ,43 % VIRCOH=VIRCOHtot (:,1: numref ) contains the virtual coherences for the first44 % selected frequency. VIRCOH( i , j ) is virtual coherence between reference i45 % and principal component j.46

47 %%%48 % Ferhad Kamalizadeh49 % TU Eindhoven , Mechanical Engineering50 %%%51

52

53 % %%%%%%%%%%%%%%%%%%%%%%%%%54 % %%% Start PCA−method %%%55 % %%%%%%%%%%%%%%%%%%%%%%%%%56 % clear all57 % close all58 % clc59

60 global Srr Spr freqvec freqdes numref dim numf pathname expnm rs261 %dim =[ rows cols ];62 load( ' MyColormaps ' , ' colorgraymap ' ) % Colormap Rick Scholte63

64 syms y n65 dstor = input( ' Do you want to store data ? [ y/ n] ' );66 if dstor == y67 dstor = 1;68 display( ' >>> Data will be stored <<<' )69 elseif dstor == n70 dstor = 0;71 display( ' >>>Data will not be stored <<<' )72 end73

101

74 qstsf = 1;75

76 PC=[];77 VIRCOHtot=[];78

79 stopper = 1;80 for stopper = 1;81 kl=1;82 for fa = freqdes83

84 Sxx = Srr(:,:,fa);85 Syx = Spr(:,:,fa);86

87 [S,D,SS] = svd(Sxx);88 % Singular Value Decomposition of spectral matrix of references89 for r = 1:numref90 if D(r,r) ≤ 1e−1391 display([ ' >>>WARNING: rank Sxx < numref @ ' num2str((fa −1) * freqvec(2)) ' Hz<<<' ])92 else93 end94 end95 % Loop to bypass singularity issues when Srr is not full rank96

97 PC=[PC sqrt(diag(D))]; % Principal components98

99

100 %%%101 % Partial Hologram determination102 %%%103

104 %%% Virtual Cross spectrum Syv105

106 Syv=Syx * S;107 % Spectra between hologram measurement points and principal components108 Sinv=inv(D);109 Pinv=sqrt(Sinv);110

111 %%%112 % Virtual Pressure fields qi ( f )113 %%%114

115 Syy= Syx * inv(Sxx) * Syx'; % Spectra on hologram plane116 Syydiag=diag(Syy);117 Syydiagp=[];118 for b = 1:dim(1)119 Syydiag_row=Syydiag(1+(b −1) * dim(2):b * dim(2),1).';120 Syydiagp=[Syydiagp;Syydiag_row];121 end122 eval([ ' PCAholototplane ' num2str(fa) ' =sqrt ( Syydiagp ); ' ]);123 % Total sound hologram on hologram plane124

125

126 Qinv=Pinv * Syv';127 % actually Qinv = Pinv * inv ( U) * Sxy , but since inv ( U)= U' also holds Qinv =Pinv * ( Syx* U)'128 % Qinv i =[ pc_1 all positons fi ]129 % [ ... ]130 % [ pc_n all positons fi ]

102

131

132 Qholototplanep=zeros(dim(1),dim(2));133 for a = 1:size(Qinv,1)134 Qholo=Qinv(a,:);135 Qholoplanep=[];136 for b = 1:dim(1)137 Qholo_row=Qholo(1+(b −1) * dim(2):b * dim(2));138 Qholoplanep=[Qholoplanep;Qholo_row];139 end140 eval([ ' PCAholoplane ' num2str(fa) ' _' num2str(a) ' =Qholoplanep ; ' ]);141 % Partial holograms on hologram plane142 end143

144

145 %%%Virtual Coherences %%%146 Sxxac=Sxx * S; % Spectra between references and principal components147 Sxacxac=D; %Spectra of principal components148

149 VIRCOH=[];150 for r=1:numref151 vircohrow=[];152 for s=1:numref153 vircoh = abs(Sxxac(r,s))^2/(Sxx(r,r) * Sxacxac(s,s));154 vircohrow=[vircohrow vircoh];155 end156 VIRCOH=[VIRCOH;vircohrow];157 end158 VIRCOHtot=[VIRCOHtot VIRCOH]; % Virtual coherences159 % VIRCOH = [ gamma_i, j gamma_i, j +1 ... gamma_i, n]160 % [ gamma_i+1, j ]161 % [ ...

]162 % [ gamma_n, j gamma_n, n ]163 % n = # of references164

165 %%%166 % Plotting of results167 %% %168 if qstsf169 max_axistot = max(max(abs(eval([ ' PCAholototplane ' num2str(fa)]))));170 figure;set(gcf, ' Position ' ,[260 248 4 * 189 4 * 128]);171 subplot(2,3,2);172 imshow(abs(eval([ ' PCAholototplane ' num2str(fa)])),[0 max_axistot]);173 set(gcf, ' Colormap ' ,colorgraymap)174 title([ ' f_s =' num2str((fa −1) * freqvec(2)) ...175 ' Hz, ' ' z=0.015m , | p| [ Pa] Complete Hologram PCA, holoplane ' ])176

177 subplot(2,3,4);178 imshow(abs(eval([ ' PCAholoplane ' num2str(fa) ' _' num2str(1)])),[0 max_axistot])179 colormap(jet);set(gcf, ' Colormap ' ,colorgraymap)180 title( ' Partial Hologram 1' )181 if numref ≥ 2182 subplot(2,3,5);183 imshow(abs(eval([ ' PCAholoplane ' num2str(fa) ' _' num2str(2)])),[0 max_axistot])184 colormap(jet);set(gcf, ' Colormap ' ,colorgraymap)185 title( ' Partial Hologram 2' )186 end

103

187

188 if numref ≥ 3189 subplot(2,3,6);190 imshow(abs(eval([ ' PCAholoplane ' num2str(fa) ' _' num2str(3)])),[0 max_axistot])191 colormap(jet);set(gcf, ' Colormap ' ,colorgraymap)192 title( ' Partial Hologram 3' )193 end194 end195

196

197 for lp = 1:numref198 Measdatast(:,:,(kl −1) * (numref+1)+lp)= ...199 eval([ ' PCAholoplane ' num2str(fa) ' _' num2str(lp)]);200 end201 Measdatast(:,:,(kl −1) * (numref+1)+lp+1)= ...202 eval([ ' PCAholototplane ' num2str(fa)]);203 kl=kl+1;204 end205

206 clear Measdata207 Measdata=Measdatast;208 freqvec = 1:size(freqdes,2) * (numref+1);209 save measplanepres_holo Measdata freqvec210 % Measdata = partial holograms and total hologram , used for PAcImNAH via nah_init_id.m211

212 % ask if pac PCA is available213

214 syms y n215 rs = input( ' Has the hologram plane been propagated to sourceplane ? [ y/ n] ' );216 if rs == y217 rsst = input( ' Do you want to plot backpropagated holograms ? [ y/ n] ' );218 if rsst == y219 % load pac PCA mat−file ( hologram plane has been propagated )220 cd(pathname)221 [filename2, pathname2] = ...222 uigetfile( ' * .mat ' , ' Select PAcImNAH output file ( pac_out ) ' );223 load([pathname2 filename2]);224 cd(pathname2)225

226 dimac = [size(nah.freq(1).p_planes_p2p,1) size(nah.freq(1).p_planes_p2p,2)];227 %%%%%%%%228 %%%%%hologram plane transformed to source plane using PAcImNAH229 %%%%%%%%230

231 kl=1;232 for fa = freqdes233

234 fr = 0;235 A_ac=[];236 for f =1+(kl −1) * size(nah.freq_selection,2)/numf : ...237 (kl −1) * size(nah.freq_selection,2)/numf+numref+1238 fr = fr + 1;239 Sholo = nah.freq(f).p_planes_p2p(:,:,1);240

241 if fr ≤numref242 eval([ ' PCAsourceplane ' num2str(fa) ' _' num2str(fr) ' =Sholo ; ' ])243 % Partial holograms on source plane

104

244 else245 eval([ ' PCAsourcetotplane ' num2str(fa) ' =Sholo ; ' ]);246 % Total hologram on source plane247 end248 end249 %%%250 % Plotting of results251 %% %252 %% Plotting PCA on source plane253 if qstsf254 max_axistot = max(max(abs( eval([ ' PCAsourcetotplane ' num2str(fa)]))));255 figure;set(gcf, ' Position ' ,[260 248 4 * 189 4 * 128]);256 subplot(2,3,2);257 imshow(abs( eval([ ' PCAsourcetotplane ' num2str(fa)])),[0 max_axistot]);258 set(gcf, ' Colormap ' ,colorgraymap)259 title([ ' f_s =' num2str((fa −1) * freqvec(2)) ...260 ' Hz, ' ' z=0.015m , | p| [ Pa] Complete Hologram PCA, source plane ' ])261

262 subplot(2,3,4);263 imshow(abs( eval([ ' PCAsourceplane ' num2str(fa) ' _' num2str(1)])), ...264 [0 max_axistot])265 colormap(jet);set(gcf, ' Colormap ' ,colorgraymap)266 title( ' Partial Hologram 1' )267

268 if numref ≥ 2269 subplot(2,3,5);270 imshow(abs( eval([ ' PCAsourceplane ' num2str(fa) ' _' num2str(2)])), ...271 [0 max_axistot])272 colormap(jet);set(gcf, ' Colormap ' ,colorgraymap)273 title( ' Partial Hologram 2' )274 end275

276 if numref ≥ 3277 subplot(2,3,6);278 imshow(abs( eval([ ' PCAsourceplane ' num2str(fa) ' _' num2str(3)])), ...279 [0 max_axistot])280 colormap(jet);set(gcf, ' Colormap ' ,colorgraymap)281 title( ' Partial Hologram 3' )282 end283 end284 kl=kl+1;285 end286 if dstor287 save( [ ' PCAdatasource ' num2str(expnm)], ' PCAholototplane * ' , ...288 ' PCAholoplane * ' , ' PCAsourcetotplane * ' , ' PCAsourceplane * ' , ...289 ' freqvec ' , ' PC' , ' VIRCOHtot ' )290 display( ' Data stored in PCAdatasource.mat ' )291 end292 elseif rsst == n293 if dstor294 save( [ ' PCAdataholo ' num2str(expnm)], ' PCAholototplane * ' , ...295 ' PCAholoplane * ' , ' freqvec ' , ' PC' , ' VIRCOHtot ' )296 display( ' Data stored in PCAdataholo ' )297 end298

299 end300 elseif rs == n

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301 rs2 = input( ' Do you want to backpropagate holograms to source plane ? [ y/ n] ' );302 if rs2 == n303 if dstor304 save( [ ' PCAdataholo ' num2str(expnm)], ' PCAholototplane * ' , ...305 ' PCAholoplane * ' , ' freqvec ' , ' PC' , ' VIRCOHtot ' )306 display( ' Data stored in PCAdataholo ' )307 end308 return309 elseif rs2 == y310 display( ' PAcImNAH loaded ' )311 display( ' Backpropagation of partial holograms to source plane ' )312 PAcImNAH313 keyboard314 load([pathname filename]);315 % With PAcImNAH partial holograms and total hologram316 %will be propagated to sourceplane317

318 % load pac PCA mat−file ( hologram plane has been propagated )319 cd(pathname)320 [filename2,pathname2] =uigetfile( ' * .mat ' , ' Select PAcImNAH output file ( pac_out ) ' );321 load([pathname2 filename2]);322 cd(pathname2)323

324 dimac = [size(nah.freq(1).p_planes_p2p,1) size(nah.freq(1).p_planes_p2p,2)];325 %%%%%%%%326 %%%%%hologram plane transformed to source plane using PAcImNAH327 %%%%%%%%328

329 kl=1;330 for fa = freqdes331

332 fr = 0;333 A_ac=[];334 for f = 1+(kl −1) * size(nah.freq_selection,2)/numf : ...335 (kl −1) * size(nah.freq_selection,2)/numf+numref+1336 fr = fr + 1;337 Sholo = nah.freq(f).p_planes_p2p(:,:,1);338

339 if fr ≤numref340 eval([ ' PCAsourceplane ' num2str(fa) ' _' num2str(fr) ' =Sholo ; ' ])341 % Partial holograms on source plane342 else343 eval([ ' PCAsourcetotplane ' num2str(fa) ' =Sholo ; ' ])344 % Total hologram on source plane345 end346 end347

348

349 %%%350 % Plotting of results351 %% %352

353 %% Plotting PCA on source plane354 if qstsf355 max_axistot = max(max(abs( eval([ ' PCAsourcetotplane ' num2str(fa)]))));356 figure;set(gcf, ' Position ' ,[260 248 4 * 189 4 * 128]);357 subplot(2,3,2);

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358 imshow(abs( eval([ ' PCAsourcetotplane ' num2str(fa)])), ...359 [0 max_axistot]);360 set(gcf, ' Colormap ' ,colorgraymap)361 title([ ' f_s =' num2str((fa −1) * freqvec(2)) ...362 ' Hz, ' ' z=0.015m , | p| [ Pa] Complete Hologram PCA, source plane ' ])363

364 subplot(2,3,4);365 imshow(abs( eval([ ' PCAsourceplane ' num2str(fa) ' _' num2str(1)])), ...366 [0 max_axistot])367 colormap(jet);set(gcf, ' Colormap ' ,colorgraymap)368 title( ' Partial Hologram 1' )369

370 if numref ≥ 2371 subplot(2,3,5);372 imshow(abs( eval([ ' PCAsourceplane ' num2str(fa) ' _' num2str(2)])), ...373 [0 max_axistot])374 colormap(jet);set(gcf, ' Colormap ' ,colorgraymap)375 title( ' Partial Hologram 2' )376 end377

378 if numref ≥ 3379 subplot(2,3,6);380 imshow(abs( eval([ ' PCAsourceplane ' num2str(fa) ' _' num2str(3)])), ...381 [0 max_axistot])382 colormap(jet);set(gcf, ' Colormap ' ,colorgraymap)383 title( ' Partial Hologram 3' )384 end385 end386 kl=kl+1;387 end388 if dstor389 save( [ ' PCAdatasource ' num2str(expnm)], ' PCAholototplane * ' , ...390 ' PCAholoplane * ' , ' PCAsourcetotplane * ' , ' PCAsourceplane * ' , ...391 ' freqvec ' , ' PC' , ' VIRCOHtot ' )392 display( ' Data stored in PCAdatasource.mat ' )393 end394

395 end396 end397

398 end

1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%2 %%% %%%3 % Autospectra_holo.m %4 % Script to determine autospectra holograms ( PCAASO) %5 %%% %%%6

7 clear all8 close all9 clc10 global pb11

12 row = 16; % # of rows of hologram grid13 col = row; % # of colums of hologram grid14

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15 [filename, pathname] = uigetfile( ' * .vna ' , ' Select first data −file ( file_ma001 −001) ' );16 cd(pathname);17

18

19 f = 1;20 for freq = [194] % desired frequencies for autospectra holograms21

22 tmr=0;23 for i = 1:row24 for j = 1:col25 if i <1026 string1 = [ ' file_ma00 ' num2str(i)];27 else28 string1 = [ ' file_ma0 ' num2str(i)];29 end30

31 if j <1032 string2 = [ ' −00' num2str(j) ' .vna ' ];33 else34 string2 = [ ' −0' num2str(j) ' .vna ' ];35 end36

37 string=[string1 string2];38 load(eval([ ' string ' ]), ' −mat ' )39 % input : matfiles generated during hologram measurement40

41 Measdata(i,j,f)=SLm.scmeas(2).aspec(freq); % Autospectra holograms42 tmr = tmr +1;43 end44 end45 display(num2str(f));46 f = f +1;47 end48 freqvec = [1:f −1];49 save aspec_holo Measdata freqvec50

51 syms y n52 pb=input( ' Do you want to backpropagate the autospectra holograms to the source plane ?[ y/ n] ' );53 if pb == y54 display( ' PAcImNAH loaded ' )55 display( ' Backpropagation of autospectra holograms ' )56 PAcImNAH57 keyboard58 load([pathname filename]);59 elseif pb == n60 return61 end

D.3 PacImNAH changes

Several changes are made to the original matlabfiles of the PacImNAH software package. Seeaccompanying CD-rom for these files.

nah_init.m : Adjustments to support SPR and SRR calculation in nah_unpackvna.m.Renamed to nah_init_id.m.

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nah_unpackvna.m : Added functionality to calculate SPR and SRR matrices.

vna_multiref.m : Added support for correct signal triggering during vna independent (us-ing vfg), added functionality to store several additional Siglab calculations (time history, auto-spectrum, cross-spectrum, transfer function, coherence, auto-correlation, cross-correlation). Re-named to vna_multireffer.m.

Note: vna_multiref.m is part of Siglab software.

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