study of fast fourier sampling algorithm for ultra...

Study of fast-Fourier sampling algorithm for ultra wideband digital receiver applications

Chen Wu Sreeraman Rajan

Defence R&D Canada – Ottawa Technical Report

DRDC Ottawa TR 2013-139 November 2013

Study of fast Fourier sampling algorithm for ultra wideband digital receiver applications

Chen Wu Sreeraman Rajan DRDC Ottawa

Defence R&D Canada – Ottawa Technical Report DRDC Ottawa TR 2013-139 November 2013

Principal Author

Original signed by Chen Wu

Chen Wu

Senior Defence Scientist

Approved by

Original signed by Julie Tremblay-Lutter

Julie Tremblay-Lutter

Head Radar Electronic Warfare Section

Approved for release by

Original signed by Chris McMillan

Chris McMillan

Chief Scientist

© Her Majesty the Queen in Right of Canada, as represented by the Minister of National Defence, 2013

© Sa Majesté la Reine (en droit du Canada), telle que représentée par le ministre de la Défense nationale, 2013

DRDC Ottawa TR 2013-139 i

Abstract ……..

The compressive sensing (CS) concept is a new paradigm in sampling of signals. The key concept of CS is to sample and obtain as much information about the signal with limited observations. The Fast Fourier Sampling (FFS) algorithm is a novel technique that exploits the idea of sparsity like CS and effectively provides a spectral representation using minimal sampled observations. This report provides a Matlab-based algorithm implementation and explores the applicability of the FFS to signal interception and detection in ultra-wide instantaneous bandwidth radio frequency/microwave digital receivers. The aim of the study is to quickly detect sparsely distributed carrier frequencies in an ultra-wide frequency band.

Résumé ….....

L’acquisition comprimée constitue un nouveau modèle d’échantillonnage de signaux. Cette technique consiste essentiellement à échantillonner un signal et à obtenir le plus d’information possible à partir d’un nombre limité de valeurs observées. L’algorithme de transformation de Fourier rapide (TFR) est une nouvelle technique fondée sur le concept de dispersion, comme l’acquisition comprimée, qui fournit une représentation spectrale à l’aide d’un minimum de valeurs échantillonnées. Le présent rapport porte sur l’utilisation d’un algorithme MATLAB et étudie l’applicabilité de la TFR dans l’interception et la détection de signaux à l’aide de récepteurs numériques à bande ultralarge de radiofréquences ou de faisceaux hertziens instantanés. L’étude vise à permettre la détection rapide de fréquences porteuses dispersées sur une bande ultralarge.

ii DRDC Ottawa TR 2013-139

This page intentionally left blank.

DRDC Ottawa TR 2013-139 iii

Executive summary

Study of fast Fourier sampling algorithm for ultra wideband digital receiver applications:

Chen Wu; Sreeraman Rajan; DRDC Ottawa TR 2013-139; Defence R&D Canada – Ottawa; November 2013.

Compressive sensing (CS) [1]-[3] is a new sampling paradigm that is currently widely researched in the area of signal processing. The key concept of the CS is to successfully sense in a frugal manner, yet retain as much relevant information as possible. The ‘Fugitive Signal Detection’ project extends the CS concept to electronic warfare (EW), specifically for the future radar electronic warfare (REW) systems and system-concept developments. Using the idea of sparsity, the fast Fourier sampling (FFS) method [4]-[6] was proposed by researchers from computer science and mathematics for obtaining spectral information from a sparse sample set of observations. The fast Fourier transform (FFT) is one of the most widely used signal processing algorithms in modern digital radio frequency (RF)/microwave receiver design as it can be effectively implemented in digital signal processing (DSP) devices such as field programmable gate arrays (FPGA). Currently only a small number of sampled data points, for example, 1024 (1K) data points, can be processed by DSP devices. This imposes limitations on the frequency resolution for applications that require estimation of frequencies, especially when the digital receiver is designed to have an ultra-wide instantaneous frequency band. In order to cover a wide instantaneous frequency band with fixed frequency resolution, the number of FFT points may be increased beyond what can be currently implemented in DSP devices. On the other hand, if the intercepted signal has only a few dominant frequencies, for example, less than four carrier frequencies, using 1K time-domain sampled data to find these spectra may be viewed as oversampled from the CS point of view, since the majority of the frequency-domain bins are equal to zero. This leads to wasteful use of power, memory and computing resources. It is necessary to use an algorithm that can provide good resolution using fewer sampled data. One such algorithm is FFS. The FFS algorithm enables digital receivers to detect signals with fewer sampled data, i.e. if an arbitrary signal of length 𝑁 can be represented by 𝑀 frequency components; the algorithm can probabilistically estimate these frequencies using random samples from the signal and produce detection results [6].

From the EW perspective, in this report, we have studied the use of the FFS algorithm for detection and estimation of signals in an ultra-wideband receiver. This report summarizes and presents the findings of this study, with a special focus on the extraction of spectral information using fewer sampled data. This report treats the FFS algorithm as an effective algorithm that exploits the concepts of compressive sensing and demonstrates through simulations its applicability for fast detection and estimation of frequencies in an ultra-wideband digital EW receiver using simple test cases under certain assumptions. The Matlab version of the algorithm implementation is also presented in the report.

iv DRDC Ottawa TR 2013-139

Sommaire .....

Study of fast Fourier sampling algorithm for ultra wideband digital receiver applications:

Chen Wu; Sreeraman Rajan; DRDC Ottawa TR 2013-139; R & D pour la défense Canada – Ottawa; novembre 2013.

L’acquisition comprimée [1]-[3] est un nouveau modèle d’échantillonnage qui fait l’objet de nombreuses études dans le domaine du traitement des signaux. Il s’agit essentiellement de détecter un signal avec efficience tout en conservant un maximum d’information pertinente. Le projet de détection de signaux fugitifs applique le concept d’acquisition comprimée à la guerre électronique (GE), notamment aux éventuels systèmes de guerre radioélectronique et à la conception des systèmes. Fondée sur le concept de dispersion, la méthode de transformation de Fourier rapide (TFR) [4]-[6] a été proposée par des chercheurs en informatique et en mathématiques pour obtenir de l’information spectrale à partir d’un ensemble dispersé de valeurs échantillonnées.

La TFR est l’un des algorithmes de traitement des signaux les plus utilisés dans la conception de récepteurs numériques de radiofréquences ou de faisceaux hertziens. En effet, elle peut être intégrée à des dispositifs de traitement numérique des signaux, comme les réseaux logiques programmables (FPGA). Pour le moment, les dispositifs de traitement numérique des signaux ne peuvent traiter que quelques points de données échantillonnés, par exemple les points à 1 024 octets (1 Kio). Par conséquent, la résolution en fréquence est limitée pour les applications nécessitant l’évaluation des fréquences, en particulier lorsque le récepteur numérique est conçu pour fonctionner sur une bande ultralarge de fréquences instantanées. Pour couvrir une large bande de fréquences instantanées à résolution fixe, il est possible d’augmenter le nombre de points TFR au-delà de la capacité actuelle des dispositifs de traitement numérique des signaux. Toutefois, si le signal n’est intercepté que sur quelques fréquences dominantes (p. ex. moins de quatre fréquences porteuses), l’utilisation des données à 1 Kio du domaine temporel pour détecter ce spectre peut être considérée comme du suréchantillonnage du point de vue de l’acquisition comprimée, car la majorité des cellules du domaine fréquentiel égalent zéro. Cette situation entraîne un gaspillage d’énergie, de mémoire et de ressources informatiques. Il faut un algorithme offrant une bonne résolution avec une quantité réduite de données échantillonnées. Cet algorithme est la TFR; il permet aux récepteurs numériques de détecter les signaux en utilisant moins de données échantillonnées. Par exemple, si un signal arbitraire d’une certaine longueur peut être représenté par des composants de fréquence, l’algorithme peut, selon une méthode probabiliste, calculer ces fréquences à partir d’échantillons aléatoires du signal et produire des données de détection [6].

Dans le contexte de la GE, le présent rapport étudie l’utilisation de l’algorithme de TFR pour détecter et analyser les signaux d’un récepteur à bande ultralarge. Il résume les conclusions de l’étude, en mettant l’accent sur l’extraction d’information spectrale à partir d’une quantité réduite de données échantillonnées. Le rapport traite la TFR comme un algorithme efficace fondé sur le concept de l’acquisition comprimée et démontre, à l’aide de simulations, de scénarios d’essai simples et de certaines hypothèses, son applicabilité dans la détection et l’analyse rapides des

DRDC Ottawa TR 2013-139 v

fréquences d’un récepteur numérique de GE à bande ultralarge. La version MATLAB de l’algorithme est également présentée.

vi DRDC Ottawa TR 2013-139


DRDC Ottawa TR 2013-139 vii

Table of contents

Abstract …….. ................................................................................................................................. i 1. Résumé …..... ............................................................................................................................ i Executive summary ........................................................................................................................ iii Sommaire ..... .................................................................................................................................. iv

Table of contents ........................................................................................................................... vii List of figures ................................................................................................................................. ix

List of tables .................................................................................................................................... x

Acknowledgements ........................................................................................................................ xi 1 Introduction ............................................................................................................................... 1

2 Introduction to fast Fourier sampling method .......................................................................... 3

2.1 Signal model and the goal of signal detection and estimation ....................................... 3

2.2 Theorem of the Fourier sampling algorithm .................................................................. 4

2.3 The Algorithm and its implementation .......................................................................... 4

2.4 Concepts used in the algorithm ..................................................................................... 5

2.4.1 Frequency-domain permutation through time dilation ........................................ 5

2.4.2 Selection of data using arithmetic-progression .................................................... 5

2.4.3 Spectral translation through permutation and shift .............................................. 6

2.4.4 Spectral-band separation using filter bank .......................................................... 6

2.4.5 Fast ‘significant’ frequency detection using group-testing or bit-testing ............ 6

2.4.5.1 The special case when 𝑵 is equal to power of two .................................... 7

2.4.5.2 General case ............................................................................................... 7

2.5 Identification of “significant” frequency bins ............................................................. 10

2.5.1 Isolation of “significant” frequencies: Method 1 .............................................. 10

2.5.2 Isolation of “significant” frequencies: Method II ............................................. 11

2.5.3 Using FFT in K-shattering ................................................................................. 11

2.6 The Fourier sampling algorithm .................................................................................. 12

3 Application of FFS in wideband digital receiver .................................................................... 15

3.1 The UWB receiver model and assumptions for the study ........................................... 15

3.2 Signal detection using FFS .......................................................................................... 15

3.2.1 Probability of detection of signal having one frequency ................................... 15

3.2.2 POD of single frequency using sparse sampling ............................................... 18

3.3 Double tone instantaneous dynamic range (IDR) study .............................................. 25

4 Conclusions ............................................................................................................................. 31

References ….. .............................................................................................................................. 32

Annex A The concept of ‘Bulk’ sampling .................................................................................. 33

A.1 Reformulation of the original model ........................................................................... 33

A.2 Fast computation ......................................................................................................... 34

viii DRDC Ottawa TR 2013-139

A.3 ‘Bulk’ sampling for coefficient (energy) estimation in section 2.6 ............................. 34

A.4 Summary...................................................................................................................... 36

Annex B Even and odd filters for bit-testing when the signal length equals power of two ........ 37

Parameters ............................................................................................................................... 37

The time-domain signal .......................................................................................................... 37

Set figure size ......................................................................................................................... 37

Bit testing to find frequency ................................................................................................... 37

Annex C Bit-testing frequency if signal length does not equal power of two ............................ 43

Parameters ............................................................................................................................... 43

Loop for Monte Carlo method and using for cores for parallel computing ............................ 43

Plot result ................................................................................................................................ 44

Annex D Matlab code to identify frequency bin in original signal ............................................. 45

Parameters ............................................................................................................................... 45

To generate signal ................................................................................................................... 45

To permute the original signal ................................................................................................ 46

Low pass filter ........................................................................................................................ 47

The permuted signal filtered by the filter-bank ...................................................................... 47

Annex E 𝑲 -shattering method to isolate frequncy bins in a signal ........................................... 57

Parameters ............................................................................................................................... 57

To Generate signal .................................................................................................................. 57

To get permutation parameters ............................................................................................... 58

Low pass filter ........................................................................................................................ 58

To filter original signal using permuted filters in the filter-bank ........................................... 58

Annex F The Matlab code of the Algorithm-1 in [6] ................................................................. 65

Parameters ............................................................................................................................... 65

Generate signal ....................................................................................................................... 65

Parameters for greedy-pursuit method .................................................................................... 67

Retain m pairs from Lambda .................................................................................................. 69

Compare the estimated results to input bins and their amplitudes .......................................... 69

Check time-domain results ..................................................................................................... 70

List of symbols/abbreviations/acronyms/initialisms ..................................................................... 72

Distribution list ............................................................................... Error! Bookmark not defined.

DRDC Ottawa TR 2013-139 ix

List of figures

Figure 1: POD of signal frequency vs. SNR for FFS using different sets of parameters. ............. 17

Figure 2: POD with 𝐾 = 2 and fixed 𝜎 ........................................................................................ 20



Figure 5: POD with 𝐾 = 16 and fixed 𝜎 ...................................................................................... 23

Figure 6: POD with 𝐾 = 2 and 𝑡𝑖𝑛𝑖𝑡𝑖𝑎𝑙 = 1 with fixed 𝜎 ............................................................ 24

Figure 7: IDR study: frequencies of two tones are randomly selected. ........................................ 26

Figure 8: IDR study: Randomly positioned strong first tone frequency and the second tone frequency is positioned one bin less than the first one. ............................................... 27

Figure 9: IDR study :Randomly positioned strong first tone frequency and the second tone frequency is positioned five bins less than the first one. ............................................. 28

Figure 10: IDR study: Randomly positioned strong first tone frequency and the second tone frequency is ten bins less than the first one. ................................................................ 29

x DRDC Ottawa TR 2013-139

List of tables

Table 1: The frequency can be found by the bit-testing filter pair once N and 𝛾 are given, and N does not equal power of two. ..................................................................................... 9

Table 2: Parameters used in FFS for single frequency detections ................................................. 16

Table 3: Examples of a portion of time samples used in the estimations with 100% POD .......... 18

DRDC Ottawa TR 2013-139 xi

Acknowledgements

The authors would like thank Mr. Bruce Liao, for reviewing the report of its technical content and Dr. Brad Jackson, for providing feedback that led to improvement in the presentation of the document. Authors thank both the reviewers for providing suggestions for improvement.

xii DRDC Ottawa TR 2013-139


DRDC Ottawa TR 2013-139 1

1 Introduction

In many ultra-wideband (UWB) digital receiver applications in electronic warfare (EW), fast detection of signals is an essential requirement. The fast Fourier transform (FFT) is widely used in modern digital receivers to detect and extract the spectral information of digitized signals. Limited by the memory size and computational power of digital signal processing (DSP) devices, trade-offs between the frequency resolution and instantaneous frequency bandwidth in FFT-based receivers are always a design compromise. If a system requires a fine frequency resolution and a wide instantaneous bandwidth, the length of FFT has to increase. This not only slows signal detection but also requires higher end DSP devices. The power requirements of these DSP devices are generally higher as they are required to quickly execute complicated mathematical operations.

In order to develop a method that is computationally faster than the traditional FFT method, researchers from mathematics and computer science have embraced principles of compressive sensing (CS) [1-5] and have developed the fast Fourier sampling (FFS) method [6]. In order to demonstrate the efficacy of the proposed method, in [6] the detection of 60 frequencies was undertaken and detection results were compared using FFS and the traditional FFT method. It was reported that the FFS was faster than the FFT method when the length of data points in the FFT was more than one million points, and the calculation time of the FFS was almost independent of the length of data used in the computation (see Figure 2 of [6]). In contrast to the FFS algorithm, the computational time required for the FFT exponentially increased with the length of data.

As discussed in [7], in many applications, UWB digital receivers are required to detect signals that contain sparse carrier frequencies; for example, a received electromagnetic signal may have only a few frequencies. The traditional spectral estimation methods, such as the FFT and Multiple Signal Classification (MUSIC), require much more time-domain sample data than the number of frequencies in the signal. From the CS perspective, these spectral estimation methods use oversampled time-domain data, since the majority of the bins in the frequency-domain are empty. This leads to unnecessary computations. If a traditional algorithm is implemented in an FPGA, it will also require a larger footprint. Realizing the advantage of FFS over traditional methods of spectral estimation, this report is aimed at understanding this new Fourier sampling algorithm and adapting it to EW scenarios.

This report summarizes various aspects of the FFS algorithm for UWB digital receiver applications, and focuses particularly on the fast detection of carrier frequencies. The focus is on computing the content of the signal in a resource efficient manner. The main principle of this sampling procedure is that the signal is uniformly sampled into a discrete block and then an algorithm is applied on this block. The sampling algorithm uses only a small number of correlated yet random samples from the block to produce an approximate representation in the Fourier domain to identify the significant components. This is done taking memory storage and computation time into consideration. This falls under the notion of CS as the number of samples finally used for extracting the frequency information is significantly less than the Nyquist requirement.

2 DRDC Ottawa TR 2013-139

From a CS point of view, this report presents the application of the FFS algorithm for ultra-wideband digital receiver application using sparsity principles. The report is organized as follows: Section 2 introduces the main concepts of the FFS algorithm. A sketch of the algorithm and the corresponding Matlab code is included in the Appendices for completeness. Section 3 presents a few examples of the detection of signals. Detection of signals with different number of carrier frequencies and under different signal strength and SNR scenarios are studied through several examples. Section 4 concludes this report.


2 Introduction to fast Fourier sampling method

In this section, we introduce the FFS algorithm, and present some details of the derivations that are necessary to understand and implement the FFS algorithm in Matlab. More comprehensive derivation details of the FFS method can be found in references [4] and [5].

2.1 Signal model and the goal of signal detection and estimation

Consider a length-𝑁 discrete time-domain signal 𝑥 that is linearly composed of 𝑀 frequency components. This signal can be decomposed in the following manner:

𝑥[𝑡] =1

√𝑁∑ 𝑐𝑙𝑒

𝑗2𝜋𝜔𝑙𝑡/𝑁 ,

𝑀−1

𝑙=0

(1)

where the sample index 𝑡 = 0, 1, 2 … 𝑁 − 1, 𝑀 is the number of frequency components, and 𝜔𝑙 , 𝑐𝑙 is 𝑙th frequency component and its Fourier coefficient. The intent of this report is to detectthe components of the signal 𝑥[𝑡] using a minimum number of discrete time-domain samples.

Theoretically, an 𝑁-point discrete Fourier transform (DFT), as given in (2), can be used to obtain 𝜔𝑙 , 𝑐𝑙𝑙∈𝑀. However, the majority of the frequency bins, ω, in (2) will be empty,when 𝑀 ≪ 𝑁,

𝑋[𝜔] =1

√𝑁∑ 𝑥[𝑡]𝑒−𝑗2𝜋ω𝑡/𝑁𝑁−1

𝑡=0

, (2)

where 𝜔 = 0, 1, 2, …𝑁 − 1, and 𝜔𝑙, 𝑐𝑙 ∈ 𝑋. This now leads to a fundamental question: Howmany time domain samples should be used to obtain 𝑋[𝜔]? Fortunately, the discrete uncertainty principle provides the answer to the question: if the spectrum of a discrete signal with 𝑁 samples has only 𝑀 non-zero frequency components, then the time domain signal has at least 𝑁/𝑀 non zero samples. This suggests that we need only 𝑁/𝑀 samples in the time domain to characterize the signal completely in the frequency domain. Interestingly in the EW area, most of the signal detection is done in a blind fashion. One does not have the luxury of knowing the number of frequency components in the signal and hence the question of how to compressively sample the signal blindly continues to be a challenging research problem. However, if some assumptions on the signals are made, some reprieve is in order.


2.2 Theorem of the Fourier sampling algorithm

To find or estimate the frequency components 𝜔𝑙 , 𝑐𝑙𝑙∈𝑀′ in (1), the FFS algorithm randomlytakes 𝑀 ∙ 𝑝𝑜𝑙𝑦 (1 , log (1

𝛿) , log(𝑁)) samples from 𝑥[𝑡] that has 𝑁 equal-spaced sampled data, and

returns an approximation of the signal 𝑦[𝑡] that can be written as:

𝑦[𝑡] =1


𝑗2𝜋𝜔𝑙𝑡/𝑁

𝑀′−1

𝑙=0

(3)

with probability at least 1 − 𝛿 and satisfies the error bound

‖𝑥 − 𝑦‖2 ≤ (1 + 휀)‖𝑥 − 𝑥𝑜𝑝𝑡‖2 + 휀, (4)

where 𝑥𝑜𝑝𝑡 is the best approximation to 𝑥 of the form (1). The algorithm calculates the signalestimate in (3) for which the time and storage requirements are a polynomial function of logarithm of N. So as N increases several folds, the increase in computational time and storage is not drastic.

Note that, the number of estimated frequencies 𝑀′ may not be equal to 𝑀. ε and 𝛿 are positiveconstants.

2.3 The Algorithm and its implementation

At a high level, the algorithm proceeds iteratively in greedy pursuit fashion. Given a signal 𝑥, the algorithm determines a representation signal 𝑦 and then uses the residual signal 𝑥 − 𝑦 to better the representation. Initially, the algorithm sets the representation signal 𝑦 to zeros, and considers the residual signal 𝑥 − 𝑦. The aim of the algorithm is to reduce the ‖𝑥 − 𝑦‖2 iteratively byrepeating the following: 1) randomly sampling the residual signal 𝑥 − 𝑦 using approximately 𝑂𝑀 ∙ 𝑝𝑜𝑙𝑦 (1/휀, 𝑙𝑜𝑔 (1/𝛿), 𝑙𝑜𝑔 (𝑁) ) points; 2) identifying a set of ‘significant’ frequencies in the spectrum of the residual signal; 3) estimating the Fourier coefficients of those ‘significant’ frequencies; 4) adding the contribution of those ‘significant’ frequencies to signal 𝑦, and 5) doing the next iteration on the residual signal 𝑥 − 𝑦 [5].

The Algorithm-1 given on page 61 of [6] gives a practical implementation of the greedy pursuit process. The derivation of the algorithm is based on a number of technical lemmas whose descriptions are out of scope for this report. Interested readers can find the details of the lemmas, theorems and proofs in [4] and [5]. The rest of this section and the annexes at the end of the report, present enough details to understand the implementation of Algorithm-1 in [6].


2.4 Concepts used in the algorithm

This section discusses a number of important concepts that are used in FFS for Algorithm-1 in [6].

2.4.1 Frequency-domain permutation through time dilation

One of the challenges for detection and estimation of man-made signals is that significant tones in a signal can be clustered, for example, several X-band navigation radars may be operating around 9.4 GHz in the area of surveillance. One of the innovations of the FFS is the random permutation of the spectrum that helps to isolate significant tones from each other, and then the isolated tones can be extracted using an appropriately designed filter-bank (a set of band-pass filters).

Given a Fourier transform pair 𝑥(𝑡) ⇔ 𝑋(𝜔), the time and frequency scaling property of the

transform [8] can be expressed as:

𝑥(𝑎𝑡) ⇔ 𝑎−1𝑋(𝑎−1𝜔) (5)

where 𝑎 is a positive scaling constant. Equation (5) may be interpreted in the following way: the spectrum of signal 𝑥(𝑡) can be permuted, if one uses time-domain data differently, for example: using dilation 𝑑𝜎: 𝑡

𝑚𝑎𝑝 𝑡𝑜→ 𝜎𝑡 𝑚𝑜𝑑 𝑁, and 𝜎 and 𝑁 are relatively prime. The number 𝜎∗ is the

multiplicative inverse that satisfies 𝜎𝜎∗ ≡ 1 𝑚𝑜𝑑 𝑁.

2.4.2 Selection of data using arithmetic-progression

The first step of the greedy pursuit approach is to randomly choose 𝐾 data points from a signal of length 𝑁. Arithmetic progression (AP) is used to randomly choose the time-domain sampled data 𝑥(𝑡𝑘), where

𝑡𝑘 = (𝑡0 ± 𝜎𝑘) 𝑚𝑜𝑑 𝑁, (6)

where 𝑘 = 0, 1, 2, 3, … , 𝐾 − 1 and

𝑡0 is selected randomly within [0, 𝑁 − 1); 𝜎 has to be a integer relatively prime to 𝑁, if 𝑁 is a power of two, 𝜎 can be an odd number.

It should be noted that once the parameters of (6) are chosen, the sequence is not random. However from one realization to another, these parameters are random.

The following are the reasons to use (6) to select the data:

1. Randomly selecting data points is a key factor that allows the method to use subsets of original sampled time-domain data for the spectrum estimation. It is guaranteed that, by randomly picking 𝑡0 and 𝜎, the resulting subset of data is AP-independent [5];


2. AP naturally includes the time-domain dilation described by (5);

3. The ‘Bulk’ computation of multiple values is more efficient than computing each valueindependently. (Details of ‘Bulk’ sampling computation are given in Annex A)

2.4.3 Spectral translation through permutation and shift

For a given signal 𝑥(𝑡) and its Fourier transform 𝑋(𝜔), the algorithm defines a transformation 𝑃𝜔′,𝜎 as follows.

𝑃𝜔′,𝜎𝑥(𝑡) = 𝑒−𝑗2𝜋𝜔′𝑡𝜎∗

𝑁 𝑥(𝑡𝜎∗), (7)

where 𝜎 is invertible modulo 𝑁, i.e., 𝜎𝜎∗ ≡ 1 𝑚𝑜𝑑 𝑁. Using the properties of the Fouriertransform, the transform of 𝑃𝜔′,𝜎𝑥(𝑡) is 𝑋(𝜔′ + 𝜎𝜔).

2.4.4 Spectral-band separation using filter bank

To build a filter-bank, the algorithm selects the Boxcar (rectangular) function as the filter time-domain response, i.e.:

ℎ𝐾,𝑁(𝑡) = √𝑁

𝐾𝑡 ∈ [0, 𝐾]

0 𝑡 ∉ [0, 𝐾]

. (8)

It is well known that this filter frequency-domain response is a sinc-function, and is not a good filter for the following reasons:

it does not have sharp transition from pass to reject bands, and

it has poor frequency rejection because of its side lobes; the peak of the first side-lobe isonly 13 dB lower than the peak of the pass-band.

In spite of the poor performance, this filter is used as it is easy to implement.

2.4.5 Fast ‘significant’ frequency detection using group-testing or bit-testing

If we have a time-domain signal with just one “significant” frequency, the group testing or bit-testing method will help us “discover” the frequency bin without using the Fourier transform or FFT. Group-testing or bit-testing thus involves a projection operation where the “significant” frequencies are projected either into the space of odd or even frequencies, followed by a norm estimation of each projection.


In other words, the method finds the (𝑏 + 1)𝑡ℎ bit of 𝜔0, provided we know the 𝑏 least significant bits, which is assumed to be known by trying all possibilities exhaustively. Once we know the value of any bit positions of 𝜔0, we can modulate the relevant signal, i.e., translate the spectrum, so that all the known bit positions become zeros. Bit testing for the general filter pair and the special filter pair when 𝑁 is a power of two are presented below.

2.4.5.1 The special case when 𝑵 is equal to power of two

Define a filter pair in (9), called even (+) and odd (−) filters, and let 𝑁 equal power of two,

gb±(t) =

√N

2𝛿(𝑡) ± 𝛿 ((𝑡 +

𝑁

2𝑏+1) 𝑚𝑜𝑑 𝑁), (9)

where 𝛿(𝑡) is the usual delta function.

It is easy to see that

Gb±(ω) =

1 𝜔 𝑒𝑣𝑒𝑛 𝑛𝑢𝑚𝑏𝑒𝑟0 𝜔 𝑜𝑑𝑑 𝑛𝑢𝑚𝑏𝑒𝑟

, (10)

and Gb+(ω) = 1, if 𝜔 = 0 𝑚𝑜𝑑 2𝑏+1 and Gb

+(ω) = 0, if 𝜔 = 2𝑏 𝑚𝑜𝑑 2𝑏+1. Gb−(ω) = 1 −Gb+(ω), if 2𝑏divides ω. In Annex B a Matlab code on how to apply these filters to identify

the frequency in a given signal is presented. In the code, 𝑥𝑒 and 𝑥𝑜 are output time-domain signals from different orders of even and odd filters. If the amplitude of 𝑥𝑜 is bigger or equal to the amplitude of 𝑥𝑒, i.e.,

|𝑥𝑜| ≥ |𝑥𝑒|, (11)

it is evident that the current least significant bit is one, otherwise, the current bit is zero.

2.4.5.2 General case

Define a general filter pair (12), in which 𝑁 may not equal a power of two,

gb±(t) =

√N

2𝛿(𝑡) ± 𝛿 ((𝑡 + ⌊

𝑁

2𝑏+1⌋) 𝑚𝑜𝑑 𝑁), (12)

where ⌊∙⌋ means to take the integer that is smaller and closest to or equal to the real number. Applying DFT to gb

±(t), we have


𝐺b±(𝜔) =

1 ± 𝑒𝑥𝑝𝑗2𝜋𝜔 ⌊

𝑁2𝑏+1

⌋

𝑁

2,

(13)

and consider the case where 𝑏 = 0

G0+(𝜔) =

1 + 𝑒𝑥𝑝 𝑗2𝜋𝜔 ⌊

𝑁2⌋

𝑁

2.

Define ∆= 𝑁

2− ⌊

𝑁

2⌋, so ∆= 0, when 𝑁 is power of two, or ∆= 1

2 , when 𝑁 is not power of

two. Thus ⌊𝑁2⌋ =

𝑁

2− ∆, then G0+(𝜔) can be written

G0+(𝜔) =

1+(−1)𝜔𝑒−𝑗2𝜋𝜔∆/𝑁

2.

Now, suppose there is an ω0 that satisfies 0 ≤ ω0/N ≤ γ/π, then πω0∆N≤ γ/2.

If ω0 is even, then

G0+(𝜔0) =

1 + 𝑒−𝑗2𝜋𝜔0∆/𝑁

2,

using the first two terms after the Taylor expansion, one has

G0+(𝜔0) ≈ 1 −

𝑗𝜋𝜔0∆

𝑁.

Hence,

|G0+(𝜔0) − 1| ≈

𝜋𝜔0∆

𝑁≤γ

2≤ γ.

Similarly for G0−(𝜔0),

G0−(𝜔0) ≈

𝑗𝜋𝜔0∆

𝑁.

Hence,

|G0−(𝜔0)| ≤ γ.

Similarly, if ω0 is odd, then |G0+(𝜔0)| ≤ γ,


and

|G0−(𝜔0) − 1| ≤ γ .

Thus, for sufficiently small γ, the filters G0± behave similarly to when 𝑁 is a power of

two, provided 0 ≤ ω0 ≤ Nγ/π. For general Gb±(𝜔), we will perturb 𝑁/2b+1 by ∆ to get

an integer ⌊𝑁/2b⌋, when |𝜔| ≤ 𝑁𝛾/𝜋, the perturbation 2𝜋𝜔∆/𝑁 ≤ 𝛾, so that |e2𝜋𝜔∆

𝑁 −

1| /2 ≈ γ/2 ≤ γ.

The following table shows that for the given γ and 𝑁, which part of frequency spectrum can be found using the filter pair when 𝑁 is not equal to a power of two. From the table, one can find that only a small portion of the spectrum bins can be identified using the bit-testing filter pair given in (13) - when 𝑁 is not a power of two. For example, for a complex time-domain signal with 1000 points, the signal frequency bin has to be less than 95 in order to use the bit-testing method when 𝛾 = 0.3. Annex C presents the Matlab code and demonstrates the use of the bit-testing method for signal lengths that may not be a power of two.

From the plot in Annex C, one can find that 0.78 is the optimum value for γ. Once 𝛾 >0.78, the bit-testing method for the general signal length cases starts having frequency-bin prediction error. In practical applications, usually it is easy to make the signal length equal to a power of two.

Table 1: The frequency can be found by the bit-testing filter pair once N and 𝛾 are given, and N

does not equal power of two.

𝛄 0.01 0.1 0.2 0.3 0.5 𝛑/𝛄 314.159 31.4159 15.7079 10.4719 6.2831

𝐍 𝛚𝟎 ≤ 𝐍𝛄/𝛑

1000 3.18309 31.8309 63.6619 95.4929 159.1549 5000 15.91549 159.1549 318.3098 477.4648 795.7747 10000 31.83099 318.3098 636.6197 954.9296 1591.5494 50000 159.1549 1591.549 3183.0988 4774.6482 7957.7471 100000 318.3098 3183.098 6366.1977 9549.2965 15915.4943 500000 1591.549 15915.494 31830.988 47746.4829 79577.4715 1000000 3183.099 31830.988 63661.977 95492.9658 159154.9430


2.5 Identification of “significant” frequency bins

Having the information given in the last section, we can discuss more details of the greedy pursuit algorithm presented in section 2.3. Since the first step in the method is to identify frequency bins, in this subsection, we will explore two ways to identify ‘significant’ frequency bins. To begin with, let us assume that:

there are 𝑀 ‘significant’ frequency bins in signal 𝑥;

signal length is a power of two;

frequencies in the signal are located on frequency bins, and

the number of signal samples, 𝐾, is at least at 𝑂(𝑀).

2.5.1 Isolation of “significant” frequencies: Method 1

The following methodology is adopted for isolation of ‘significant’ frequencies:

Permute the spectrum of 𝑥 by a random dilation σ, getting 𝑃0,𝜎𝑥(𝑡);

Filter 𝑃0,𝜎𝑥(𝑡) by a filter-bank of approximately 𝐾 equally-spaced frequency-domaintranslations of the LPF given in (8). In other words,

o time-domain shape is a Boxcar function with approximately 𝐾 taps, and

o frequency bandwidth is approximately 𝑁/𝐾.

After filtering using the filter-bank, there are 𝐾 new signals, 𝑓𝑘. Some have a singleoverwhelming Fourier mode, 𝜎𝜔, that corresponds to a “significant” mode, 𝜔 in 𝑥, andundoing the permutation would bring back 𝜔 from 𝜎𝜔;

Then by applying bit-testing the “significant” bins (and hence frequencies) are identified.

This method can be tersely summarized as “permute original signal – filtering – undo-permutation” to obtain the 𝑀 “significant” frequencies.

The Matlab code given in Annex D demonstrates the method to identify frequency bins using the “permute original signal – filtering – undo-permutation” approach. The code also enables the reader:

to permute the original signal, to form the LPF using a Boxcar function in time-domain, to filter the permuted signal using the 𝑘𝑡ℎ translated filter in the filter-bank, to undo permutation on the output of the 𝑘𝑡ℎ filter, and to read information given in each figure at the end of annex.


2.5.2 Isolation of “significant” frequencies: Method II

The 𝐾-shattering of a signal is the second method to isolate the signals that only contain one or no ‘significant’ frequencies. The spectrum of the signal is randomly permuted (or dilated) and decomposed into 𝐾 signals using sub-band decomposition techniques with the hope that each signal in 𝐾 signals would contain one ‘significant’ bin. Then the dilation or permutation is undone to restore the frequencies to their original places in the spectrum.

For given parameter 𝐾, that is a power of two, the 𝐾-shattering of signal 𝑥 with length of 𝑁 is a collection of

𝑓𝑘(t) = [1

√𝑁(𝑃𝑘𝑁/𝐾,𝜎ℎ𝐾,𝑁(𝑡)) ∗ 𝑥(𝑡)]

𝑘

, (14)

where 𝑘 = 0, 1, 2, … , 𝐾 − 1, and 𝜎 is a random number invertible 𝑚𝑜𝑑 𝑁.

From (7), we have:

𝑃𝑘𝑁/𝐾,𝜎ℎ𝐾,𝑁(𝑡) = 𝑒−𝑗2𝜋𝑘𝑡𝜎∗

𝐾 ℎ𝐾,𝑁(𝑡𝜎∗) (15)

and (15) gives a shifted LPF with its frequency center translated to 𝑘𝑁/𝐾. Instead of using “permute original signal – filtering – undo-permutation” approach, the FFS algorithm translates the permuted LPF and forms the filter-bank as given in (15), and then applies these filters to the original signal. This forms the K-shattering of the signal. Equation (14) gives the 𝑡𝑡ℎ sample of the output signal. The output signal is the original signal 𝑥 filtered by the 𝑘𝑡ℎ permuted filter in the filter-bank.

Annex E gives the Matlab code. The advantage of the K-shattering method is that there is no need to undo permutation on the output signals of permuted filters in the filter-bank.

2.5.3 Using FFT in K-shattering

Using (15) in (14) and considering time-domain convolution, one has:

𝑓𝑘(t) = 1

√𝑁∑ 𝑒

−𝑗2𝜋𝑘𝑢𝜎∗

𝐾 ℎ𝐾,𝑁(𝑢𝜎∗) ∙𝑢 𝑥(𝑡 − 𝑢),

Letting 𝑢𝜎∗ = 𝑣, and using 𝜎𝜎∗ = 1 𝑚𝑜𝑑 𝑁,

𝑓𝑘(t) = 1

√𝑁∑ 𝑒

−𝑗2𝜋𝑘𝑣

𝐾 ℎ𝐾,𝑁(𝑣) ∙𝑣 𝑥(𝑡 − 𝑣𝜎).

Considering the LPF given in (8), the 𝑓𝑘(t) can be written as:


𝑓𝑘(t) = 1

𝐾∑ 𝑥(𝑡 − 𝑣𝜎) ∙ 𝑒−

𝑗2𝜋𝑘𝑣𝐾

𝐾−1

𝑣=0

(16)

The following about the above equation can be inferred from the previous discussions:

a. 𝑥(𝑡 − 𝑣𝜎) (𝑣 = 0, 1, 2, … , 𝐾 − 1) is the AP used to pick data from the original signalas given in (6);

b. (16) can be recognized as the FFT of 𝑥(𝑡 − 𝑣𝜎).

A few interesting observations of equation (16) are in order.

1. There are two independent variables 𝑘 and 𝑡 in the left hand side. One can fix 𝑘 andvary 𝑡 or do the reverse. From the last figure of Annex E, one can recognize thatinstead of fixing a 𝑘 to get the entire set of the time-domain signal of 𝑓𝑘 (asprogrammed in Annex E), the 𝐾-shattering of signal gives an alternate method to get𝐾 samples of 𝑓𝑘 using FFT as given in (16) by fixing 𝑡𝑓𝑖𝑥, for example 𝑡𝑓𝑖𝑥 = 250.

2. Computational time can be significantly reduced using (16) when an application onlyneeds 𝑓𝑘(𝑡)(𝑘 = 0, 1, … , 𝐾 − 1) at a few time samples, for example, in bit-testingtheoretically only one time-domain sample is needed for condition (11);

3. When 𝐾 is a small number, say 8, 16 or 32, the FFT can produce 𝐾 points of 𝑓𝑘(𝑡𝑓𝑖𝑥)from just one FFT.

Even though using a Boxcar function as the LPF time-domain response gives the fastest way to identify the ‘significant’ frequency bins, the leakage of the filter in the frequency-domain is an issue. As an improvement, a Dolph-Chebyshev window will be investigated and reported in a subsequent report.

2.6 The Fourier sampling algorithm

In this section, we summarize the implementation steps of the algorithm [6] and give the Matlab code of the FFS algorithm in Annex F.

The algorithm:

identifies a set of “significant” frequencies in the spectrum of the residual signal 𝑥 − 𝑦through:

o isolation of “significant” frequencies using K-shattering of the signal;

o bit testing of each signal 𝑓𝑘(𝑡) to find one “significant” 𝜔 by learning the bits of 𝜔 oneat a time, from the least to the most significant bit through frequency translations.

estimates the energy of each “significant” frequency , where ‘Bulk’ sampling is used tospeed up the computation;

adds the contributions of these newly found “significant” frequencies to the representationsignal 𝑦, and


iterates in a greedy pursuit as described earlier in this report.

Note that, although the ‘Bulk’ sampling can speed up (A.1) and (A.10) calculations, in the Annex F we use the Matlab function ‘bsxfun’ to calculate them. Tests show that ‘bsxfun’ is much faster than ‘Bulk’ sampling. However, ‘bsxfun’ is a Matlab specialized function that cannot be used in hardware based DSP implementations. Since the ‘Bulk’ sampling is a smart way of using FFT, it has the potential for hardware based DSP implementations.


This page intentionally blank


3 Application of FFS in wideband digital receiver

In the last section, we outlined the key concepts of the FFS algorithm and described the algorithm for Matlab implementation. In this section, we will discuss the signal detection and spectrum estimation results for UWB digital receiver applications.

3.1 The UWB receiver model and assumptions for the study

For the convenience of discussion and study, here we make some assumptions for our digital receiver model. The wideband digital receiver can be viewed as having three parts [7]:

a general amplifier that includes all the RF/Microwave front-end, and is specified by its amplification, noise figure, intermediate frequency (IF) and corresponding bandwidth;

an analog-to-digital converter (ADC) that is specified by its sampling speed and number of bits. Note that the only impairment considered in this report is the quantization error [9];

the algorithm that encompasses signal processing methods, such as FFT, MUSIC and eigenvalue decomposition.

The report focuses on the application of FFS for UWB digital receiver applications. We assume:

The ADC sampling speed (𝑓𝑠) is 2.56 GHz;

A system that is perfectly terminated with 50Ω;

The data input to the algorithm is complex (representation-wise);

The receiver is not working in a saturated state;

The length of sampled signal (𝑁) is a power of two;

Noise generated from all portions of the digital receiver is already accounted for, and

Assuming that frequencies in the original signal are all centered on different frequency bins. The deviants from these assumptions will be discussed in a subsequent report.

3.2 Signal detection using FFS

3.2.1 Probability of detection of signal having one frequency

This section shows that with the different sets of parameters in the FFS algorithm how well the method can detect the presence of the signal. The following are the parameters used:

‘𝐾’ determines the number of filters in the filter-bank, and the number of taps in the Boxcar function used as the LPF time-domain response, 𝐾 = 𝐾8 ∗ 𝑚 where m is the initial estimated number of frequencies in the original signal;

‘Jreps’ represents the number of iterations in the greed pursuit process of the algorithm;


‘reps’ defines the number of time samples to be obtained from K-shattering the signal,and bit-testing by the even and odd filters;

‘ns’ is the number of frequencies in the original signal, and in this case ns = 1.

Table 2 lists all the combinations of parameters that were used in the study.

Table 2: Parameters used in FFS for single frequency detections

K8 Jreps reps SNR

8, 16, 32 15, 10, 5 20, 15 10, 5 20, 15, 10, 5, 0, -5, -6, -7, -8, -9, -10, -11, -12, -13, -14, -15

In this report, the probability of detection (POD) is defined as the probability that the FFS method correctly estimates the frequency with different SNR. Each POD is estimated from 1000 runs, and the original signal length (N) is fixed as 212.

Note that, 𝜎 and 𝑡0 are randomly picked from [1 3 5 … 𝑁 − 1] and [0 1 2 3 …𝑁 − 1],respectively. The original signal power and single frequency in the signal are randomly selected within [−30 0] dBm and [0 𝑓𝑠] GHz, respectively.

Figure 1 shows the POD results obtained for different values of K and under different set of parameters. The results are grouped in three plots by different values of K. In each plot, there are four groups of data with different colors that represent different ‘reps’. In each color, there are three line-types that reflect different ‘Jreps’. From the figure, we have the following observations:

1. Increasing ‘reps’ can increase POD for any given K;2. Changing ‘Jreps’ does not alter the POD significantly, if other parameters are kept the

same.3. When the number of filters (K) increases in the filter-bank, the algorithm can detect the

frequency in the signal with lower SNR.

Based on the above observations, one can choose parameters to handle different detection scenarios. For example, 100% POD, K = 16, reps = 20 and Jreps = 5 is a good combination when SNR reaches the -10 dB level.


Figure 1: POD of signal frequency vs. SNR for FFS using different sets of parameters.

-15 -10 -5 0 5 10 15 200

20

40

60

80

100

black: reps = 5

magenta: reps = 10

blue: reps = 15

red: reps = 20

solid line: Jreps = 15

dashed line: Jreps = 10

dotted line: Jreps = 5

PO

D (%

)K = K8*m = 8 * 1

-15 -10 -5 0 5 10 15 200

20

40

60

80

100

black: reps = 5

magenta: reps = 10

blue: reps = 15

red: reps = 20




PO

D (%

)

K = K8*m = 16 * 1

-15 -10 -5 0 5 10 15 200

20

40

60

80

100

black: reps = 5

magenta: reps = 10

blue: reps = 15

red: reps = 20




PO

D (%

)

K = K8*m = 32 * 1

SNR (dB)


While obtaining the results given in Figure 1, the program tracked the usage of (amount) time-domain sampled data in the original signal. Table 3 shows examples that for 100% POD the algorithm did not have to use all the 4096 data points from the original signal. For example, for some randomly selected 𝜎 and 𝑡0, FFS estimated the signal frequency with less than 45% data,when (K, Jreps, reps) = (8, 5, 5) and SNR is 5 dB. The following observation in the table provides an idea about the sparse usage of the data during the computations.

Table 3: Examples of a portion of time samples used in the estimations with 100% POD

K Jreps reps SNR (dB)

Number of samples in the original signal that are used during the computation.

1 8 5 5 5 1839

2 8 10 5 5 2740

3 8 10 5 10 2780

4 8 5 10 5 2833

5 16 5 5 5 2876

6 8 5 10 0 2878

7 16 5 5 0 2927

8 16 5 5 10 2978

9 8 5 15 5 3290

10 8 15 5 10 3364

11 8 5 15 0 3367

12 8 15 5 5 3379

13 8 5 15 -5 3379

14 8 5 20 -7 3621

15 8 10 10 10 3645

16 8 10 10 0 3655

17 8 10 10 5 3673

18 16 10 5 0 3677

19 8 5 20 -6 3679

20 16 5 10 0 3688

3.2.2 POD of single frequency using sparse sampling

We further explored the effect on POD when only fewer samples from the original sampled signal are utilized. Instead of randomly selecting 𝜎 and 𝑡0 for (6), we

set 𝜎 to one of the small odd numbers from the following set 1, 3, 5, 7, …, 31 and set 𝑡0 = 𝑡𝑖𝑛𝑖𝑡𝑖𝑎𝑙 + 𝜎 ∙ (𝑖 − 1), where i = 1, 2, …, reps, here the initial point ( 𝑡𝑖𝑛𝑖𝑡𝑖𝑎𝑙) is

randomly selected with [0 N),


With these settings, PODs were obtained with 𝐾 at 2, 4, 8 and 16 and are given in four figures from Figure 2 through Figure 5. The following observations are in order:

1. Figure 2 and Figure 3 provide the same observations as with Figure 1;

2. Interestingly, as 𝐾 increases, the POD reduces dramatically even at high SNR in Figure 4 and Figure 5. We conjecture that the randomness requirement for compressive sampling for the FFS algorithm may be violated. A separate study may be conducted to verify this conjecture.

3. Using K equals to 2 or 4, and large number of reps, for example 20, this method provides 100% POD when the SNR is at 0 dB, and

4. From this study, we have observed that a very small amount of original time-domain samples can be used for estimations. For example: K=4, 𝜎=1, only 193 time samples were used. Even though we have studied only limited cases, the following conclusions may be generally valid:

the FFS algorithm may provide 100% POD for detection of a single frequency in an UWB signal (signal has only one frequency) using less than two hundred time-domain samples, when the SNR is higher than 0 dB.

As in FFT, the frequency resolution of the estimation is still determined by the sampling frequency (𝑓𝑠) and total number of samples in the original time-domain data (N), and not determined by the how many time samples are used in the estimation.

The significance of this study is that the FFS algorithm can detect a signal using a few time-domain samples and achieve high frequency resolution in an ultra-wide frequency band, and at the same time, can take advantage of the short-length FFT implemented on high speed DSP hardware. Further studies on implementation issues will be reported in a future report.


Figure 2: POD with 𝐾 = 2 and fixed 𝜎

-20 -10 0 10 200

20

40

60

80

100sigma=1, Samples=179

PO

D %

-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


PO

D %

-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


PO

D %

-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


SNR (dB)

PO

D %

-20 -10 0 10 200

20

40

60

80


SNR (dB)-20 -10 0 10 200

20

40

60

80


SNR (dB)-20 -10 0 10 200

20

40

60

80


SNR (dB)

red: reps = 20, blue: reps = 15, magenta : reps = 10 and black: reps = 5 solid: Jreps = 15, dashed: Jreps = 10 and dotted: Jreps = 5



-20 -10 0 10 200

20

40

60

80


PO

D %

-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


PO

D %

-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


PO

D %

-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


SNR (dB)

PO

D %

-20 -10 0 10 200

20

40

60

80


SNR (dB)-20 -10 0 10 200

20

40

60

80


SNR (dB)-20 -10 0 10 200

20

40

60

80


SNR (dB)


PO

D %



-20 -10 0 10 200

20

40

60

80


PO

D %

-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


PO

D %

-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


PO

D %

-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


SNR (dB)

PO

D %

-20 -10 0 10 200

20

40

60

80


SNR (dB)-20 -10 0 10 200

20

40

60

80


SNR (dB)-20 -10 0 10 200

20

40

60

80


SNR (dB)



Figure 6: POD with 𝐾 = 2 and 𝑡𝑖𝑛𝑖𝑡𝑖𝑎𝑙 = 1 with fixed 𝜎

-20 -10 0 10 200

20

40

60

80


PO

D %

-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


PO

D %

-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


PO

D %

-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


-20 -10 0 10 200

20

40

60

80


SNR (dB)

PO

D %

-20 -10 0 10 200

20

40

60

80


SNR (dB)-20 -10 0 10 200

20

40

60

80


SNR (dB)-20 -10 0 10 200

20

40

60

80


SNR (dB)

reps = 20, Jreps = 15


3.3 Double tone instantaneous dynamic range (IDR) study

In this study, there are two tones in the original signal. The idea is to understand the impact on detection due to the presence of an additional tone in the estimation. Two additional parameters, namely the signal strength, and the relative location of the second tone with respect to the first tone is varied to understand the impact. The first tone (the strong one) is 100 dB above the noise and its frequency bin is randomly picked from [0 fs], where fs is the sampling frequency. The frequency of the second tone was varied as follows:

randomly selected (results are given in Figure 7);

one bin lower than the first frequency (Figure 8) (here frequency difference between two adjacent bins is 0.625 MHz);

five bins lower than the first frequency (Figure 9), and

ten bins lower than the first frequency (Figure 10).

The second tone amplitude is set at 0, 5, 10, 20, 30, 40, 50 and 60 dB lower than the first one. The goal is to study the POD of two frequencies under different settings of the FFS parameters.

Comparing the results from Figure 7 to Figure 10, we note that

the FFS algorithm can effectively detect two tones;

the IDR is increased when number of filters (𝐾) in the filter-bank is increased;

increasing number of iteration (Jreps) in the isolation step of the algorithm can increase the IDR;

the number of samples for estimation after K-shattering the signal has less influence on the IDR;

A 40 dB IDR at more than 90% POD can be achieved when 32 filters in the filter-bank and 15 iterations are used in the isolation stage.

Note IDR is the ratio of the stronger signal power to the weaker signal minimum power that is needed for 90% POD of the weaker signal [7].


Figure 7: IDR study: frequencies of two tones are randomly selected.

-60 -50 -40 -30 -20 -10 00

20

40

60

80

100P

OD

%

K = K8 * m = 8 * 2

-60 -50 -40 -30 -20 -10 00

20

40

60

80

100

PO

D %

K = K8 * m = 16 * 2

-60 -50 -40 -30 -20 -10 00

20

40

60

80

100

black: reps = 5

magenta: reps = 10

blue: reps = 15

red: reps = 20

solid: Jreps = 15

dashed: Jreps = 10

dotted: Jreps = 5

PO

D %

Power difference between two tones (dB)

K = K8 * m = 32 * 2


Figure 8: IDR study: Randomly positioned strong first tone frequency and the second tone

frequency is positioned one bin less than the first one.

-60 -50 -40 -30 -20 -10 00

20

40

60

80

100P

OD

%

K = K8 * m = 8 * 2

-60 -50 -40 -30 -20 -10 00

20

40

60

80

100

PO

D %

K = K8 * m = 16 * 2

-60 -50 -40 -30 -20 -10 00

20

40

60

80

100

black: reps = 5

magenta: reps = 10

blue: reps = 15

red: reps = 20

solid: Jreps = 15

dashed: Jreps = 10

dotted: Jreps = 5

PO

D %


K = K8 * m = 32 * 2


Figure 9: IDR study :Randomly positioned strong first tone frequency and the second tone frequency is positioned five bins less than the first one.

-60 -50 -40 -30 -20 -10 00

20

40

60

80

100

PO

D %

K = K8 * m = 8 * 2

-60 -50 -40 -30 -20 -10 00

20

40

60

80

100

PO

D %

K = K8 * m = 16 * 2

-60 -50 -40 -30 -20 -10 00

20

40

60

80

100

black: reps = 5

magenta: reps = 10

blue: reps = 15

red: reps = 20

solid: Jreps = 15

dashed: Jreps = 10

dotted: Jreps = 5

PO

D %


K = K8 * m = 32 * 2


Figure 10: IDR study: Randomly positioned strong first tone frequency and the second tone frequency is ten bins less than the first one.

-60 -50 -40 -30 -20 -10 00

20

40

60

80

100

PO

D %

K = K8 * m = 8 * 2

-60 -50 -40 -30 -20 -10 00

20

40

60

80

100

PO

D %

K = K8 * m = 16 * 2

-60 -50 -40 -30 -20 -10 00

20

40

60

80

100

black: reps = 5

magenta: reps = 10

blue: reps = 15

red: reps = 20

solid: Jreps = 15

dashed: Jreps = 10

dotted: Jreps = 5

PO

D %


K = K8 * m = 32 * 2


This page intentionally blank


4 Conclusions

This report introduced the fast Fourier sampling method developed in [4] and provided a description of the algorithm and the program implementation. The report focussed on applying the Fourier sampling method to ultra-wideband digital receiver applications. The following conclusions were made from this study:

A single tone signal above -10 dB SNR can be easily detected using FFS.

It was observed that the algorithm used about 4.36% of time-domain sampled data to detect a single tone signal at 0 dB SNR.

The frequency resolution of the estimation is determined by the length of original time-domain sampled data and sampling frequency, and not determined by how many samples are used in the estimation.

For two-tone signals, a 90% probability of detection has up to 40 dB instantaneous dynamic range

Future work includes 1) using a sharper-edge LPF in FFS, for instance, a rectangular window convolved with low side-lobe Dolph-Cheybshev window [10]; 2) using a window function on time-domain data to increase detection dynamic range, and 3) using digitized data coming from a CS ADC [11]. New algorithms proposed in [12] should be studied as this would be relevant for communication signals as the spectrum at different times represents the message transmission itself.

For the ultra-wideband digital receiver design, such as those receivers used in electronic warfare, the FFS method can help to quickly scan the band of interest and approximately locate the frequencies of the signals of interest. This will enable fast tuning of narrow-band receivers to the sub-bands that contain the signals of interest. This method can be useful in areas which use wideband signal acquisition where speed is paramount such as in radar countermeasure and radar warning systems.

Since the algorithm is probabilistic in nature, several detection processes need to be computed in a parallel fashion (which is possible with current Graphical Processor Units (GPU)) and the results of this parallel bank of detectors needs to be combined using a pre-defined strategy (like majority voting). Since the receivers used in countermeasures may always be collecting the signals, the proposed method may be amenable for implementation. Future studies will explore such implementations.


References …..

[1] E. J. Candès, “Compressive Sampling,” Int. Congress of Mathematics, 3, pp. 1433-1452, Madrid, Spain, 2006.

[2] R. Baraniuk, “Compressive sensing,” IEEE Signal Processing Magazine, 24(4), pp. 118-121, July 2007.

[3] E. J. Candès and T. Tao, “Near optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inform. Theory, vol. 52, no. 12, pp. 5406–5425, Dec. 2006.

[4] A. C. Gilbert, S. Guha, P. Indyk, S. Muthukrishnan, and M.J. Strauss, “Near-optimal sparse Fourier representations via sampling,” in ACM Symp. Theoretical Comput. Sci., pp. 152-161, 2002.

[5] A. C. Gilbert, S. Muthukrishnan, and M.J. Strauss, “Improved time bounds for near-optimal sparse Fourier representations,” in Proc. SPIE Wavelets XI, San Diego, CA, 2005, pp. 59141A.1–15.

[6] A. C. Gilbert, M. J. Strauss and J. A. Tropp, “A tutorial on fast Fourier sampling,” IEEE Signal Processing Magazine, March 2008, pp. 57-66.

[7] J. Tsui, Special design topics in digital wideband receivers, Artech House, 2009.

[8] J. Tsui, Digital techniques for wideband receivers, second edition, Artech House, 2001.

[9] W. Keter, Mixed-signal and DSP design techniques, Newnes, 2003.

[10] H. Hassanieh, P. Indyk, D. Katabi, and E. Price. Nearly optimal sparse Fourier transform. In STOC, 2012.

[11] P. K. Yenduri, A. C. Gilbert, M. P. Flynn and S. Naraghi, “Rand PPM: A lowpower compressive sampling analog to digital converter,” Acoustics, Speech and Signal Processing (ICASSP), 2011, IEEE, May 22-27, 2011, pp.5980-5983.

[12] P.K. Yenduri and A.C. Gilbert, “Continuous Fast Fourier Sampling,” SAMPTA 09, Marseille, France 2009.

http://www-stat.stanford.edu/~candes/papers/CompressiveSampling.pdf

http://dsp.rice.edu/files/cs/baraniukCSlecture07.pdf


The concept of ‘Bulk’ sampling Annex A

In this annex, we closely follow [6] and present the concepts. The mathematical equations are taken from [6].

A.1 Reformulation of the original model

Let a signal 𝑦 can be represented as a linear combination of complex sinusoids, exp (2𝜋𝜔𝑙𝑡/𝑁) with frequencies and their Fourier coefficients 𝜔𝑙, 𝑐𝑙𝑙∈𝐿, and let 𝑡𝑘 = 𝑡0 + 𝜎𝑘 for 𝑘 ∈ [0, 𝐾) be an arithmetic progression. One can compute 𝑦(𝑡𝑘) using the ‘Bulk’ sampling [4] method. The ‘Bulk’ sampling method is given as follows.

As given by (3), 𝑦(𝑡𝑘) can be written using the original model as:

𝑦(𝑡𝑘) =1


𝑗2𝜋𝜔𝑙(𝑡0+𝜎𝑘 )/N𝑙 , (A.1)

and let 𝑢𝑙 =1

√𝑁𝑐𝑙𝑒

𝑗2𝜋𝜔𝑙𝑡0/N and 𝑣𝑘(𝜔) = 𝑒𝑗2𝜋𝜔𝑘/N, then one has:

𝑦(𝑡𝑘) = ∑ 𝑢𝑙𝑣𝑘(𝜔)𝑙 , (A.2)

where 𝜔 = 𝜎𝜔𝑙.

Fix 𝑅 to be a power of two at least ⌊2𝜋𝐾⌋, and using Taylor series around 𝑟𝑁/𝑅 for some 𝑟 that is equal to 0, 1, …𝑅 − 1, to approximate 𝑣𝑘(𝜎𝜔𝑙), we have ∆𝑙,𝑟= 𝜎𝜔𝑙 − 𝑟𝑁/𝑅, and the 𝑛𝑡ℎ order derivative of 𝑣𝑘(𝜔) can be shown to be

𝑣𝑘(𝑛)(𝜔) = (

2𝜋𝑗𝑘

𝑁)𝑛𝑒𝑗2𝜋𝜔𝑘/𝑁 . (A.3)

So 𝑣𝑘(𝜎𝜔𝑙) can be written as:

𝑣𝑘(𝜎𝜔𝑙) = ∑1

𝑛!𝑣𝑘(𝑛)(𝑟𝑁/𝑅) ∆𝑙,𝑟

𝑛

∞

𝑛=0

= ∑1

𝑛!(2𝜋𝑗𝑘

𝑁)𝑛

𝑒𝑗2𝜋𝑘𝑟/𝑅∆𝑙,𝑟𝑛

∞

𝑛=0

. (A.4)

Substitute (A.3) to (A.2), and change the order of the summations, one gets:

𝑦(𝑡𝑘) = ∑1

𝑛!

∞

𝑛=0

(2𝜋𝑗𝑘

𝑁)𝑛

∑𝑢𝑙∆𝑙,𝑟𝑛

𝑙

𝑒𝑗2𝜋𝑘𝑟/𝑅 , (A.5)


or written in detailed fashion as

𝑦(𝑡𝑘) =1

√𝑁∑

1

𝑛!∞𝑛=0 (

2𝜋𝑗𝑘

𝑁)𝑛∑ (𝜎𝜔𝑙 − 𝑟𝑁/𝑅)

𝑛𝑢𝑙𝑙 𝑒𝑗2𝜋𝑘𝑟/𝑅. (A.6)

A.2 Fast computation

Define a vector with length 𝑅 for any given 𝑛:

𝑓𝑟,𝑛 = 𝑠(𝑟) ∙ 𝑢𝑙 ∙ (𝜎𝜔𝑙 − 𝑟𝑁/𝑅)𝑛, (A.7)

𝑟 = 0, 1, 2, … , 𝑅 − 1, 𝑙 = 1, 2, 3, … , 𝐿 and the way to obtain the elements in 𝑓𝑟,𝑛 is as follows:

1. For each 𝜔𝑙 there is 𝑟𝑙 that satisfies

min𝑟∈[0,𝑅]|(𝜎𝜔𝑙 𝑚𝑜𝑑 𝑁) − 𝑟𝑁/𝑅|. (A.8)

2. Once 𝑟𝑙 is determined, one can set 𝑠(𝑟 ∈ 𝑟𝑙) = 1 and 𝑠(𝑟 ∉ 𝑟𝑙) = 0.

Now to calculate 𝑦 = [𝑦(𝑡0), 𝑦(𝑡1), 𝑦(𝑡2),… , 𝑦(𝑡𝑘−1)] 𝑇, here 𝑇 is transpose, using the equationgiven below

𝑦(𝑡𝑘) =1

√𝑁∑

1

𝑛!∞𝑛=0 (

2𝜋𝑗𝑘

𝑁)𝑛[∑ 𝑓𝑟,𝑛𝑒

𝑗2𝜋𝑘𝑟/𝑅𝑅−1𝑟=0 ]. (A.9)

The significance of (A.9) is that for any given 𝑛 one can

use FFT to calculate term ∑ 𝑓𝑟,𝑛𝑒𝑗2𝜋𝑘𝑟/𝑅𝑅−1

𝑟=0 in (A.9) to simultaneously obtain results for all 𝑘 (0 ≤ 𝑘 < 𝐾 < 𝑅), and then

multiply by 1

√𝑁

1

𝑛!(2𝜋𝑗𝑘

𝑁)𝑛

and sum over the first few 𝑛.

The value of the 𝑛𝑡ℎ term decays exponentially as 𝑛 increases, and only log2 (𝑁

𝛼) terms are

needed to make the error less than 𝛼‖𝑦‖ [5].

A.3 ‘Bulk’ sampling for coefficient (energy) estimation in section 2.6

In the previous part of this annex, we discussed how to use the ‘Bulk’ sampling concept to obtain time-domain samples that are defined based on AP given in (6), when we know frequencies and their coefficients. In this section, for a given signal 𝑥, we discuss how to estimate the coefficients to within precision ±𝛽‖𝑥‖, (‖𝑥‖22 = ∑ |𝑥(𝑡)|2𝑡 ), using the ‘Bulk’ sampling, when we know


frequency and corresponding time-domain data given by 𝑡𝑘 = 𝑡0 + 𝜎𝑘 for 𝑘 ∈ [0, 𝐾). Instead of computing coefficients one-by-one, our goal is to compute

𝑐(𝜔𝑙) =√𝑁

𝐾∑ 𝑥(𝑡0 + 𝜎𝑘)𝑒

−𝑗2𝜋𝜔𝑙(𝑡0+𝜎𝑘)/𝑁

𝐾−1

𝑘=0

, (A.10)

by using ‘Bulk’ sampling where 𝜔𝑙 (𝑙 = 0, 1, 2, … , 𝐿 − 1).

From (A.10), we have

𝑐(𝜔𝑙) = ∑ 𝑎𝑘𝑙 𝑒−𝑗2𝜋𝜔𝑙𝜎𝑘/𝑁𝐾−1

𝑘=0 , (A.11)

where

𝑎𝑘𝑙 =

√𝑁

𝐾𝑥(𝑡0 + 𝜎𝑘)𝑒

−𝑗2𝜋𝜔𝑙𝑡0/𝑁. (A.12)

Now let’s consider a function

𝑓(𝜔) = ∑ 𝑎𝑘𝑙 𝑒−𝑗2𝜋𝜔𝑘/𝑁𝐾−1

𝑘=0 . (A.13)

Expand the function in the Taylor series about 𝑟𝑁/𝑅 for some 𝑟, and ∆𝑟= 𝜔 − 𝑟𝑁/𝑅, where 𝑅 is a power of two and 𝑅 > 𝐾, then we have

𝑓(𝜔) = ∑ ∑ (1

𝑛!(−𝑗2𝜋𝑘

𝑁)𝑛

𝑎𝑘𝑙 )

𝐾−1

𝑘=0

𝑒−𝑗2𝜋𝑟𝑘/𝑅

∞

𝑛=0

∆𝑟𝑛 . (A.14)

For each 𝑛, the expression ∑ (1

𝑛!(−𝑗2𝜋𝑘

𝑁)𝑛𝑎𝑘𝑙 )𝐾−1

𝑘=0 𝑒−𝑗2𝜋𝑟𝑘/𝑅 can be calculated simultaneously for

all the 𝑟. We can compute the DFT of the sequence 1𝑛!(−𝑗2𝜋𝑘

𝑁)𝑛𝑎𝑘𝑙 𝑘𝑛 of length 𝑅 using the FFT

by letting 𝑎𝑘𝑙 = 0 for 𝑘 ≤ 𝐾 < 𝑅.

Once we have DFT data for 𝑛 = 1,2,3,… for each 𝜔𝑙, we find 𝑟𝑙 that minimizes |(𝜎𝜔𝑙 𝑚𝑜𝑑 𝑁) − 𝑟𝑙𝑁/𝑅|. We then approximate 𝑐(𝜔𝑙) by a Taylor series. Only the first few terms of the Taylor series are needed.


A.4 Summary

The ‘Bulk’ sampling method uses the FFT to speed the computations of (A.1) and (A.10). However, in our Matlab code, we did not use the ‘Bulk’ sampling method, instead we used the Matlab function ‘bsxfun’, which is faster than the ‘Bulk’ sampling method. However, the ‘Bulk’ sampling method may be the appropriate algorithm when implemented in DSP hardware to compute (A.1) and (A.10), as FFT plays an important role.


Even and odd filters for bit-testing when the Annex Bsignal length equals power of two

Parameters

omegak = 0; % set predicted frequency to zero.

power=4;

N = 2^power; % the length of the time-domian signal

dt = 1/N; % time step

B=zeros(1,power); % set all the bits equal to zeros

freq = floor(rand*N), % randomly pick a frequency bin within [0,N].

freq =

14

The time-domain signal

x_signal = exp(1i*2*pi*freq*dt*(0:N-1)); % A complex time-domain signal.

Set figure size

h=figure;

set(h,’Position’,[1 1 770 888])

Bit testing to find frequency

t0 = floor(rand*N); % % randomly pick a time as t in Equation (9)

for b=0:log2(N/2) % loop for bits

ge=zeros(1,N); %set even filter to zero

go=zeros(1,N); %set odd filter to zero

t1=mod(t0+N/2^(b+1),N);

ge(t0+1)=+0.5; % even filter in time-domain

ge(t1+1)=+0.5;

go(t0+1)=+0.5; % odd filter in time-domain

go(t1+1)=-0.5;

% Translate signal frequency, so that all the known bit positions

% become zeros.

bit_move=exp(-1i*2*pi*omegak/N*(0:N-1));


x = x_signal.*bit_move;

% Find the translated signal frequency

X=fft(x);

[~,I]=max(X); % the translated signal frequency bin is I-1

% then filtered by even and odd filters in time-domain

xe=cconv(ge,x,N); %even filter output

xo=cconv(go,x,N); %odd filter output

subplot(5,1,1),stem(0:N-1,abs(X ),’r’), grid,axis([0 N-1 0 1])

title([‘Signal freqyency bin = ‘ int2str(I-1) ‘ before filtering’])

xlabel(‘ Frequency bin ‘)

subplot(5,1,2),stem(0:N-1,abs(fft(ge)),’r’),grid,axis([0 N-1 0 1])

title([‘ Even filter spectrum (b= ‘ int2str(b) ‘ )’])


subplot(5,1,3),stem(0:N-1,abs(fft(go)),’b’),grid,axis([0 N-1 0 1])

title([‘ Odd filter spectrum (b= ‘ int2str(b) ‘ )’])


subplot(5,1,4),stem(0:N-1,abs(xe),’b’), grid,axis([0 N-1 0 1])

title([‘The absolute value of time-domain signal from even filter (b= ‘ int2str(b)

‘ )’])

xlabel(‘ Time step ‘)

subplot(5,1,5),stem(0:N-1,abs(xo),’b’), grid,axis([0 N-1 0 1])

title([‘The absolute value of time-domain signal from odd filter (b= ‘ int2str(b)

‘ )’])

xlabel(‘ Time step ‘)

snapnow;

% Randomly pick a time for signal norm comparsion

test_point=floor(rand*N)+1;

if abs(xo(test_point)) > abs(xe(test_point))

omegak = omegak+2^b;

B(b+1)=1; % record nonzero bits

end

end

omegak

B

Results from above Matlab code are given in next figure.


The input signal has a frequency, 𝜔 = 14. 𝐺0+(14) = 1, and 𝐺0−(14) = 1, so the amplitude of the output time-domain signal (𝑥𝑒) from 𝐺0+ is one, and the amplitude of the output time-domain signal (𝑥𝑜) from 𝐺0− is zero, as one can see in the fourth and fifth plots in the figure. Since |𝑥𝑜| <|𝑥𝑒|, the current least significant bit is zero.


From the last figure, we know that the frequency of the input signal to 𝐺1± does not change, i.e.,

𝜔 = 14. Since 𝜔 = 2𝑏 𝑚𝑜𝑑 2𝑏+1, here 𝑏 = 1, we have 𝐺1+(14) = 0 and 𝐺1−(14) = 1 thatresults in |𝑥𝑜| > |𝑥𝑒|, so that the current least-significant bit is one.


Since the previous least significant bit is one, there is a frequency translation to the signal before the input to 𝐺2

±. Now 𝜔 = 12, that satisfies 𝜔 = 4 𝑚𝑜𝑑 8. We have 𝐺2+(12) = 0 and 𝐺2−(12) =1 that results in |𝑥𝑜| > |𝑥𝑒|, so that the current least-significant bit is one, again.


,

Following the same reasoning, one can find this least significant bit is one, as well. Eventually, the predicted frequency bin (𝜔𝑘) = 𝐵(1) ∗ 20 + 𝐵(2) ∗ 21 + 𝐵(3) ∗ 22 + 𝐵(4) ∗ 23 = 14,where 𝐵 = [0 1 1 1].


Bit-testing frequency if signal length does not Annex Cequal power of two

The program shows how to use the filter pair 𝐺b±(𝜔), when the original signal length is not

equal to a power of two, and also finds the optimum number for γ that is discussed in section 2.4.5.2.

clear

format long

Parameters

eps=1e-10;

MC=1000000; %nNumber of simulation times in the Monte Carlo method

gamma=0.5:0.01:1.5; % the range of gamma

mc=zeros(length(gamma),1);

Loop for Monte Carlo method and using for cores for parallel computing

matlabpool 4

parfor ig=1:length(gamma) % loop for Gamma

for i=1:MC

N=1024+floor(rand*1000000); % randomly pike

OMEGA=floor(N*gamma(ig)/pi);% given a Gamma and N determine the upper limit of

frequency

B=floor(log2(OMEGA))+1; % total number of bits needs for OMEGA

omega=floor(rand*OMEGA); % randomly pick frequency in [0 OMEGA]

Bit=zeros(1,B);

omega0=omega;

for b=0:B-1

Gp=abs(1+exp(1i*2*pi*omega0*floor(N/2^(b+1))/N))/2;

Gm=abs(1-exp(1i*2*pi*omega0*floor(N/2^(b+1))/N))/2;

if Gm>=Gp

Bit(b+1)=1;

omega0=omega0-2^b;

end

end

I=find(Bit>0);

O=sum(2.^I)/2;

if abs(O – omega) < eps % If the prediction successful

mc(ig)=mc(ig)+1;

end


end

end

matlabpool close

Starting matlabpool using the ‘local’ profile ... connected to 4 workers.

Sending a stop signal to all the workers ... stopped.

Plot result

results = mc/MC*100; %probability of successfully predicting frequency for a given gamma

plot(gamma,results),grid

xlabel(‘gamma’)

ylabel(‘Probability (%)’)

I=find(results<100);

gamma(I(1)-1)

ans =

0.78

Probability to correctly calculate frequency vs. different γ

0.5 1 1.560

65

70

75

80

85

90

95

100

gamma

Pro

babi

lity

(%)

0.78


Matlab code to identify frequency bin in Annex Doriginal signal

This program shows the details on how to isolate a ‘significant’ frequency bin in the original signal. That is:

to permute original signal; to filter the permuted signal, here assuming each filtered-permuted signal

has one or no ‘significant’ frequency, and to undo permutation on filtered permuted signal

At the end of the annex, the figures and discussions can help the reader to better understand the isolation process.

Clear

rng(‘shuffle’)

h=figure;

set(h, ‘Position’, [1 1 700 400])

Parameters

C = ‘r’,’b’,’m’,’y’; % Cell array of colros.

freq =[232 260]; % frequency bins

freq=sort(freq)’;

M=length(freq); % number of frequencies in thesignal

K=4*M; % Number of filters in the filter bank

n=floor(log2(K)); % insure N = 2^power;

if 2^n < K, K = 2^(n+1); end % make sure K is power of two

N=2^10; % Signal length

dt=1/N; % time step

t=dt*(0:N-1);

To generate signal

x_signal=zeros(1,N);

for i=1:M

x_signal = x_signal+exp(1i*2*pi*freq(i)*t);

end

% Plot the original signal spectrum

X_signal=fft(x_signal)/N;


[~,I]=find(abs(X_signal)>0.1);

XX=zeros(N,M);

for i=1:M

XX(I(i),i)=abs(X_signal(I(i)));

if i==1

subplot(5,1,1), stem(XX(:,i),’color’, Ci),hold on

grid,axis([0 N-1 0 1.1])

title(‘The spectrum of the original signal’)

end

subplot(5,1,1), stem(XX(:,i),’color’, Ci)

end

To permute the original signal

% random pick sigma and t0

sigma = 0;

t0 = floor(random(‘unif’, 0,N-1,1,1));

while gcd(sigma,N) ~= 1

sigma = random(‘unid’, N,1,1);

end

sigmai = mod_inversion(sigma, N); % sigma* sigmai = 1 mod N

if mod(sigma*sigmai,N) ~= 1,

display( ‘sigmna*sigmai=1 mod N’)

break

end

sigma_i_tau = mod(sigma * (0:N-1)+t0 , N);

sigmai_i_tau = mod(sigmai*((0:N-1)-t0), N);

x_p=x_signal(sigma_i_tau + 1); % the permuted signal

X_p=fft(x_p)/N; % the spectrum of the permuted signal

% Plot the spectrum of the permuted signal

% in order to match the bin colors in the original signal and the permuted signal

% here we plot one permuted bin at a time.

for i=1:M

x = exp(1i*2*pi*freq(i)*t);

X = abs(fft(x(sigma_i_tau + 1))/N);

if i==1

subplot(5,1,2), stem(X,’color’, Ci),hold on

grid,axis([0 N-1 0 1.1])

title(‘The spectrum of the permuted signal’)

end

subplot(5,1,2), stem(X,’color’, Ci),

end


Low pass filter Boxcar function used as the LPF time-domain response

g(1:K/2)=sqrt(N)/K;g(K/2+1:N-K/2-1)=0;g(N-K/2:N)=sqrt(N)/K;

The permuted signal filtered by the filter-bank

Kcenter=[N/K/2:N/K:N]; % the center frequencies of the filters in the filter-bank

for k=1:K

%k-th translated filter, center freqruency is Kcenter(k)

gm=g.*exp(1i*2*pi*Kcenter(k)/N*(0:N-1)); % the translated filter

% - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

% Following program uses the translted filters to filter permuted signal

% - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

% the spectrum of the k-th translated filter

GM=fft(gm)/(sqrt(N)/K)/K;

subplot(5,1,3), plot(0:N-1, abs(GM)),grid,axis([0 N-1 0 1.2]) % plot current filter

title([‘The spectrum of the ‘ int2str(k) ‘-th filter in the filter-bank’])

% the spectrum of the permuted signal filtered by the k-th translated filter

X_p_f=GM.*X_p;

subplot(5,1,4), stem(0:N-1, abs(X_p_f)), % plot spectrum

grid ,axis([0 N-1 0 1.2])

title([‘The spectrum of the permuted signal filtered by the ‘ int2str(k) ‘-th

filter’])

% the k-th filtered-permuted time domain signal

x_p_f =ifft(X_p_f)*N;

% undo permutation on the k-th filtered-permuted signal

x_f= x_p_f(sigmai_i_tau+1);

% the spectrum of filtered signal after undo-permutation

X_f=fft(x_f)/N;

subplot(5,1,5), stem(0:N-1, abs(X_f)),grid, axis([0 N-1 0 1.2])

title([‘The un-permuted spectrum of the permuted-signal filtered by the ‘ int2str(k)

‘-th filter’])

xlabel(‘Frequency bin’)

pause(1)

snapnow

end

In the following calculations: 𝜎 = 739, 𝜎∗ = 715, and 𝑡0 = 235


The first plot in the figure reveals that there are two frequency bins in the original signal. After permutation of the original signal, the permuted signal has frequency bins shown in the second plot, in which the red bin is the permuted bin that is related to the first bin (red) in the first plot, and the blue bin (in the second plot) is the permuted spectrum of the blue bin in the first plot.

The spectrum of the 1st filter of the filter-bank is shown in the third plot, since the LPF time-domain response is a Boxcar function, the spectrum of the filter has a sinc-function shape.

The other two plots show, after the filter is applied to the permuted signal, there is no strong signal output from the 1st filter, since the two frequency bins in the second plot are not inside the filter pass-band. However, there are some spectrum leakages because the sinc-function filter does not perfectly supress signals outside its pass-band.


From the figure, one can find that the 2nd filter output is exactly the same as the 1st filter.


The permuted signal spectrum has no bins inside the 3rd filter pass band.


The red bin in the permuted signal (the second plot) is inside the 4th filter pass band, so the output signal (the fourth plot in the figure) has a bin that aligns with the red bin in the second plot. After undoing the permutation of the signal obtained from the 4th filter, one gets the correct frequency bin shown in the last plot of the figure.

Note that, the red bin of the permuted signal is located at the peak of the 4th filter pass band. It may be not always the case.


It can be found that the blue bin of the permuted signal is inside the 5th filter pass band, however, it is not aligned at the peak portion of the filter pass band.


One can see the outputs from both the 5th and 6th filters have the blue bin of the permuted signal.


In the outputs of the 7th and 8th filters, there are no strong signals. However, the signal leakages always happen.


This page intentionally left blank


𝑲 -shattering method to isolate frequncy bins Annex Ein a signal

clear,clf

rng(‘shuffle’)

h1=figure(1);

set(h1, ‘Position’, [1 1 700 350])

h2=figure(2);

set(h2, ‘Position’, [1 1 700 880])

Parameters

C = ‘r’,’b’,’m’,’y’; % Cell array of colros.

freq =[232 260]; % frequency bins

freq=sort(freq)’;

M=length(freq); % number of frequencies in thesignal

K=4*M; % Number of filters in the filter bank

n=floor(log2(K)); % insure N = 2^power;

if 2^n < K, K = 2^(n+1); end % make sure K is power of two

N=2^10; % Signal length

dt=1/N; % time step

t=dt*(0:N-1);

To Generate signal

x_signal=zeros(1,N);

for i=1:M

x_signal = x_signal+exp(1i*2*pi*freq(i)*t);

end

% Plot the original signal spectrum

X_signal=fft(x_signal)/N;

[~,I]=find(abs(X_signal)>0.1);

XX=zeros(N,M);

for i=1:M

XX(I(i),i)=abs(X_signal(I(i)));

if i==1

figure (1)

subplot(3,1,1), stem(XX(:,i),’color’, Ci),hold on

grid,axis([0 N-1 0 1.1])


title(‘The spectrum of the original signal’)

end

figure (1)

subplot(3,1,1), stem(XX(:,i),’color’, Ci)

end

To get permutation parameters Random pick sigma and t0

sigma = 0;

t0 = floor(random(‘unif’, 0,N-1,1,1));



end

sigmai = mod_inversion(sigma, N); % sigma* sigmai = 1 mod N



break

end

sigma_i_tau = mod(sigma * (0:N-1)+t0 , N);

sigmai_i_tau = mod(sigmai*((0:N-1)-t0), N);

Low pass filter Boxcar function used as the LPF time-domain response

g(1:K/2)=sqrt(N)/K;g(K/2+1:N-K/2-1)=0;g(N-K/2:N)=sqrt(N)/K;

To filter original signal using permuted filters in the filter-bank

Kcenter=[N/K/2:N/K:N]; % the center frequencies of the filters in the filter-bank

for k=1:K

%k-th treanslated filter, center freqruency is Kcenter(k)

gm=g.*exp(1i*2*pi*Kcenter(k)/N*(0:N-1)); % the translated filter

% to permute the k-th translated filter

gm_p=gm(sigma_i_tau+1);

% spectrum of the permuted k-th translated filter

GM_p=fft(gm_p)/(sqrt(N)/K)/K;

figure (1)

subplot(3,1,2), plot(0:N-1, abs(GM_p)),grid,axis([0 N-1 0 1.2]) % plot current filter

title([‘The spectrum of the ‘ int2str(k) ‘-th permuted filter in the filter-bank’])

% the spectrum of the original signal filtered by the k-th translated permuted filter

X_f=GM_p.*X_signal;


figure (1)

subplot(3,1,3), stem(0:N-1, abs(X_f)), % plot spectrum

grid ,axis([0 N-1 0 1.2])

title([‘The spectrum of the signal from the ‘ int2str(k) ‘-th permuted filter in the

filter bank’])

xlabel( ‘Frequency bins’)

snapnow

% the absolute value of the time-domain signal output from the the k-th translated

permuted filter

x_f=abs(ifft(X_f)*N);

figure(2)

subplot(K,1,k), stem(0:N-1, x_f)

grid ,axis([0 N-1 0 1.2])

title([‘Absolute value of the time-domain signal output from the the ‘ int2str(k) ‘-

th permuted filter’])

pause(1)

end

figure(2)

xlabel( ‘Time samples’)

In following calculations: 𝜎 = 635, 𝜎∗ = 179, and 𝑡0 = 311

There is no strong signal output from the 1st permuted filter in the filter-bank. However, there is signal leakage, since the filter does not have a good suppression outside its pass band.


The filter output is the same as that of the 1st filter.


The output from the 4th filter contains the second bin of the original signal.

The output from the 5th filter contains the first bin of the original signal, and the following outputs from the rest of the three filters do not have “significant” bins.


The figure shows the absolute value or the amplitude of the output time-domain signals from permuted filters in the filter-bank. It is obvious that the outputs from the 4th and 5th filters have “significant” frequencies. The reason that other outputs do not equal zero is because the LPF does not perfectly supress signals outside its pass-band.

tfix=250

𝑓1(𝑡)

𝑓2(𝑡)

𝑓3(𝑡)

𝑓4(𝑡)

𝑓5(𝑡)

𝑓6(𝑡)

𝑓7(𝑡)

𝑓8(𝑡)


This page intentionally left blank


The Matlab code of the Algorithm-1 in [6] Annex F

A format long

clf,clear

rng(‘shuffle’,’multFibonacci’)

Parameters

fs= 2.56e9; % Hz,sampling frequency

N = 2^12; % length of signal x(t)

dt=1/fs; % time sampling step

time=dt*(0:N-1); % total sampling time

ns=2; % number of frequencies in the signal

%--------------------------------------------------

K8 = 16; % K will be K8 times of the number of frequency bins (ns) in x

reps = 15; % number of time samples randomly picked from the K-shattering of signal

Jreps =15; % repeat Jreps times of indetification and estimation in FFS

B = log2(N);% number of bits of the length of the x_signal

m = ns; % one guesses there are m frequencies in the x_signal

K = K8*m; % the number of filters in the filter-bank in FFS

%-----------------------------------------------------------

% Randomly pick signal frequencies and each freq is on a FFT bin.

freq = abs(floor(rand(1,ns)*(N-1)));

freq = sort(freq);

freq = freq*fs/N;

% Amplitude of the strongest signal

p1 = -40+rand*40; %dBm, ramdomly set the strongest signal power from -40 dBm to 0 dBm

P = [p1 p1-10 ]; %dBm, popwer of frequency bins, the length of P must be n

amp = sqrt((10.^(P/10)/1000*50*2)/2)’; %in volts, power of sinewave = Amp^2/2/(50 Ohm)

SNR = 14; %signal to noise ratio between noise floor and the weakest signal

Noise = min(P)-SNR; %Noise power dBm, SNR(dB) lower than the weakest signal;

%if SNR is 14dB, then one has 90% detection with 10^-7

%probability of false alarm.

Generate signal

% Signal to be studied is the sum of complex sine-wave signals that have

% random inital phases.

phase=zeros(ns,N);

for i=1:ns

phase(i,1:N)=(freq(i)*time+rand*ones(1,N))*2*pi;

end


if ns==1

x_signal = bsxfun(@times,amp,exp(1i*phase));

else

x_signal = sum(bsxfun(@times,amp,exp(1i*phase)));

end

% Add noise

% Help from wgn in Matlab: If output type is ‘complex’, then the real and

% imaginary parts of y each have a noise power of p/2.

% Since x is a comlex signal, and at each freq-bin wave has amp*(real+j*imag)

x_signal = x_signal + wgn(1,N,Noise,50,’dBm’,’complex’); %50 ohm system

% Plot spectrum of original signal

figure(1)

plot((0:N-1)*fs/N/1e9,10*log10((abs(fft(x_signal)*dt)*fs/N).^2/50*1000)),grid,axis([0

2.56 -100 0])

xlabel(‘ Frequency (GHz) ‘)

ylabel(‘ Power (dBm)’)

title(‘ Original signal spectrum’)


Parameters for greedy-pursuit method

% Lambda holds identified frequencies and their corresponding coefficients

Lambda = zeros(K,2); %(:,1)=omega, (:,2)=coefficients

% Randomly pick sigma and it must be a odd integer since N equals power of

% two, and t0

a = zeros(1,Jreps);

for J = 1:Jreps

sigma = 0;



end

sigmai = mod_inversion(sigma, N);



break

end

a(J)=sigma;

end

sigma=a;

for j=1:reps

a(j)=floor(rand*N)+1;

end

t0=a; % t0 can be viewed as time samples for K-shattering of signal

clear a J j

%some constants

pi2N=2*pi*1i/N;

for J=1:Jreps % repeat Jreps times

% Identification

omegak=zeros(1,K);

for b=0:B-1 % loop for bit-testing

vote=zeros(K,1);

for j=1:reps % apply bit-testing on ‘reps’ time sample points

tk = t0(j);

% Arithmetic Progression way to pick data for original signal

t_sigma_l=mod( (tk + sigma(J)*(0:K-1)), N);

% Bit-testing on selective time sample in t0 from K-shattering of

% signal

vk=sum(bsxfun(@times,Lambda(:,2),exp(pi2N*Lambda(:,1)*t_sigma_l)));

u=fft(x_signal(t_sigma_l+1)-vk); % the residual signal

tk = tk + N/2^(b+1); % for bit-testing

t_sigma_l=mod( (tk + sigma(J)*(0:K-1)), N);

vk=sum(bsxfun(@times,Lambda(:,2),exp(pi2N*Lambda(:,1)*t_sigma_l)));


v=fft(x_signal(t_sigma_l+1)-vk); % the residual signal

bit_move=exp(-1i*pi*omegak/2^b); % move known bit to zero

% Norm of the signal output at time sample (tk) from even and odd filters

even = abs(u+bit_move.*v);

odd = abs(u-bit_move.*v);

I=find(odd >= even);

vote(I) = vote(I)+1;

end

% Since there is noise, if there are more than half of

% sampled data have |odd|>=|even|, the current bit is one.

I = find(vote > reps/2);

omegak(I)=omegak(I)+2^b;

end

omega=unique(omegak)’; % that contains the newly find ‘siginificant’ bins from the

x-y

length_omega=length(omega);

% Estimation the energy related to the

% the newly find ‘siginificant’ bins in the x-y

c=zeros(length_omega,1); %the coefficient of just identified freq

C=zeros(length_omega,reps); %the coefficient of just identified freq found at each

time sample point

for j=1:reps % do a number of times and take median

% Random pick t

t = t0(j);

% Time sample points from AP

t_sigma_l = mod( (t + sigma(J)*(0:K-1)), N);

% Generate the current representation signal

y = sum(bsxfun(@times,Lambda(:,2),exp(pi2N*Lambda(:,1)*t_sigma_l)));

% The current residual siganl sampled at t_sigma_l

u = x_signal(t_sigma_l+1)-y;

% The coefficients related to the current iteration identified

% ‘significant’ bins

C(:,j) = sum(bsxfun(@times,u,exp(-pi2N*omega*t_sigma_l)),2 )/K;

end

for l=1:length_omega

c(l)=median(real(C(l,:))+1i*median(imag(C(l,:));

end

clear C vk

% If omega is in Lambda(:,1), then Lambda(omega,2) =

% Lambda(omega,2) + c_omega, else add a new pair (omega,c_omega) to

% Lambda

clear O Ia b ia ib a

if J == 1,

[O,I]=sort(omega,’descend’);

Lambda=[O c(I)];

else

[~,ia,ib]=intersect(Lambda(:,1),omega);


if length(ia) >=1

Lambda(ia,2) = Lambda(ia,2) + c(ib);

end

omega(ib)=0;

[I,~] = find(omega>0); %index of new freq-bins not in Lambda(:,1)

a = [ Lambda(:,1);omega(I)];%add newly found bins in Lambda

b = [ Lambda(:,2);c(I)];

Lambda=[a b];

end

clear O Ia b ia ib a

%Retain K pairs from Lambda whose coefficients have greatest magnitude

[b,I]=sort(Lambda(:,2),’descend’);

Lambda=[Lambda(I,1) b];

if length(Lambda(:,1)) > K

a=Lambda(1:K,;

Lambda = a;

end

end

Retain m pairs from Lambda

clear O Ia b ia ib

if length(Lambda(:,1)) > 1

a=Lambda(1:m);

[b,I]=sort(a(:,1));

Lambda=[b Lambda(I,2)];

end

Compare the estimated results to input bins and their amplitudes

[~,ia,ib]=intersect(freq/fs*N,Lambda(:,1));

[freq(ia)/fs*N

Lambda(ib,1)’] % frequency bin

% [amp’

% abs(Lambda(ib,2)’)]

[P(ia)

10*log10(abs(Lambda(ib,2)’).^2/50*1000)] %in dBm

ans =

744 987

744 987


ans =

-25.033172788324336 -35.033172788324336

-25.075645467103719 -35.055706804470553

Check time-domain results Randomly pick 𝐾 time-domain samples from the original signal

t = t0(1);

% Time sample points using AP

t_sigma_l = mod( (t + sigma(1)*(0:K-1)), N);

x = x_signal(t_sigma_l+1);

% Calculate these data using estimated frequencies and their coefficients

if length(Lambda(:,1))==1

y = bsxfun(@times,Lambda(:,2),exp(pi2N*Lambda(:,1)*t_sigma_l));

else

y = sum(bsxfun(@times,Lambda(:,2),exp(pi2N*Lambda(:,1)*t_sigma_l)));

end

figure(2)

subplot(2,1,1),plot(real(x),’r*’), grid,hold on

subplot(2,1,1),plot(real(y)),hold off

title(‘Comparison between randomly sampled original (red) and estimated (blue) signals’)

ylabel(‘ Real part of signal (volts)’)

subplot(2,1,2),plot(imag(x),’r*’), grid,hold on

subplot(2,1,2),plot(imag(y)),hold off

ylabel(‘ Imaginary part of signal (volts)’)

xlabel(‘Time-domain index’)


List of symbols/abbreviations/acronyms/initialisms

ADC Analog –to-digital converter

AP Arithmetic Progression

CS Compressive Sensing

DFT Discrete Fourier Transform

DND Department of National Defence

DR Dynamic Range

DRDC Defence Research & Development Canada

DRDKIM Director Research and Development Knowledge and Information Management

DSP Digital Signal Processing

EW Electronic Warfare

FFS Fast Fourier Sampling

FFT Fast Fourier Transform

GPU Graphical Processing Unit

IDR Instantaneous Dynamic Range

IF Intermediate Frequency

LPF Low Pass Filter

MUSIC Multiple Signal Classification

O Order of

poly Polynomial function

POD Probability of detection

R&D Research & Development

REW Radar Electronic Warfare

RF Radio Frequency

SFFT Sparse Fast Fourier Transform

SNR Signal to Noise Ratio

TIF Technology Investment Fund

UWB Ultra-wideband

DOCUMENT CONTROL DATA (Security classification of title, body of abstract and indexing annotation must be entered when the overall document is classified)

1. ORIGINATOR (The name and address of the organization preparing the document.Organizations for whom the document was prepared, e.g. Centre sponsoring a contractor's report, or tasking agency, are entered in section 8.)

Defence R&D Canada – Ottawa3701 Carling AvenueOttawa, Ontario K1A 0Z4

2. SECURITY CLASSIFICATION (Overall security classification of the document including special warning terms if applicable.)

UNCLASSIFIED(NON-CONTROLLED GOODS)DMC: AREVIEW: GCEC April 2011

3. TITLE (The complete document title as indicated on the title page. Its classification should be indicated by the appropriate abbreviation (S, C or U) in parentheses after the title.)

Study of fast Fourier sampling algorithm for ultra wideband digital receiver applications

4. AUTHORS (last name, followed by initials – ranks, titles, etc. not to be used)

Wu, C.; Rajan, S.

5. DATE OF PUBLICATION(Month and year of publication of document.)

November 2013

6a. NO. OF PAGES (Total containing information, including Annexes, Appendices, etc.)

90

6b. NO. OF REFS (Total cited in document.)

12 7. DESCRIPTIVE NOTES (The category of the document, e.g. technical report, technical note or memorandum. If appropriate, enter the type of report,

e.g. interim, progress, summary, annual or final. Give the inclusive dates when a specific reporting period is covered.)

Technical Report

8. SPONSORING ACTIVITY (The name of the department project office or laboratory sponsoring the research and development – include address.)

Defence R&D Canada – Ottawa3701 Carling AvenueOttawa, Ontario K1A 0Z4

9a. PROJECT OR GRANT NO. (If appropriate, the applicable research and development project or grant number under which the document was written. Please specify whether project or grant.)

05dz16, 01aa02

9b. CONTRACT NO. (If appropriate, the applicable number under which the document was written.)

10a. ORIGINATOR'S DOCUMENT NUMBER (The official document number by which the document is identified by the originating activity. This number must be unique to this document.)

DRDC Ottawa TR 2013-139

10b. OTHER DOCUMENT NO(s). (Any other numbers which may be assigned this document either by the originator or by the sponsor.)

11. DOCUMENT AVAILABILITY (Any limitations on further dissemination of the document, other than those imposed by security classification.)

Unlimited

12. DOCUMENT ANNOUNCEMENT (Any limitation to the bibliographic announcement of this document. This will normally correspond to theDocument Availability (11). However, where further distribution (beyond the audience specified in (11) is possible, a wider announcement audience may be selected.))

Unlimited

13. ABSTRACT (A brief and factual summary of the document. It may also appear elsewhere in the body of the document itself. It is highly desirablethat the abstract of classified documents be unclassified. Each paragraph of the abstract shall begin with an indication of the security classification of the information in the paragraph (unless the document itself is unclassified) represented as (S), (C), (R), or (U). It is not necessary to include here abstracts in both official languages unless the text is bilingual.)

The compressive sensing (CS) concept is a new paradigm in sampling of signals. The key concept of CS is to sample and obtain as much information about the signal with limited observations. The Fast Fourier Sampling (FFS) algorithm is a novel technique that exploits the idea of sparsity like CS and effectively provides a spectral representation using minimal sampled observations. This report provides a Matlab-based algorithm implementation and explores the applicability of the FFS to signal interception and detection in ultra-wide instantaneous bandwidth radio frequency/microwave digital receivers. The aim of the study is to quickly detect sparsely distributed carrier frequencies in an ultra-wide frequency band.

L’acquisition comprimée constitue un nouveau modèle d’échantillonnage de signaux. Cette technique consiste essentiellement à échantillonner un signal et à obtenir le plus d’information possible à partir d’un nombre limité de valeurs observées. L’algorithme de transformation de Fourier rapide (TFR) est une nouvelle technique fondée sur le concept de dispersion, comme l’acquisition comprimée, qui fournit une représentation spectrale à l’aide d’un minimum de valeurs échantillonnées. Le présent rapport porte sur l’utilisation d’un algorithme MATLAB et étudie l’applicabilité de la TFR dans l’interception et la détection de signaux à l’aide de récepteurs numériques à bande ultralarge de radiofréquences ou de faisceaux hertziens instantanés. L’étude vise à permettre la détection rapide de fréquences porteuses dispersées sur une bande ultralarge.

14. KEYWORDS, DESCRIPTORS or IDENTIFIERS (Technically meaningful terms or short phrases that characterize a document and could behelpful in cataloguing the document. They should be selected so that no security classification is required. Identifiers, such as equipment model designation, trade name, military project code name, geographic location may also be included. If possible keywords should be selected from a published thesaurus, e.g. Thesaurus of Engineering and Scientific Terms (TEST) and that thesaurus identified. If it is not possible to select indexing terms which are Unclassified, the classification of each should be indicated as with the title.)

Fast Fourier Sampling; FFT; Ultra wideband digital receiver; Compressive sensing

study of fast fourier sampling algorithm for ultra...

Documents