Paper ID 1569422051

Approximation of Real Impulse Response

Using IIR StructuresA. Primavera1, S. Cecchi1, L. Romoli1, P. Peretti1 and F. Piazza1

1A3Lab - DIBET - Universita Politecnica delle MarcheVia Brecce Bianche 1, 60131 Ancona Italy

www.a3lab.dibet.univpm.it

Abstract

In this paper, we propose a new approach to the approximation and simulation of areal impulse response. Starting from a preliminary analysis of the mixing time, theimpulse response is decomposed in the time domain considering the early and latereflections. Therefore, an IIR structure composed of a cascade of second-order sec-tions and four all-pass filters is employed to synthesize the first part of the impulseresponse, using a parametric optimization process in the frequency domain. Then,a recursive structure composed of comb and all-pass filters is used to synthesize thelate reflections, exploiting a minimization criterion in the cepstral domain. Severalresults are reported taking into consideration a real impulse response, confirmingthe validity of the proposed approach.

Introduction

FIR Filtering is probably one of the most recurrent operations in DSP. It is an expensive task espe-cially for long impulse responses (IRs).

FIR IIRProblem

In the last 30 years, the problem of simulating real IRs by means of recursive structures has beendeeply investigate:• Multirate filter banks are used in combination with comb filters [1];• Short Time Fourier Transform and Steigliz-McBride algorithm [2];• Moorer Structure(FIR+IIR) and autotuning procedure [3, 4].

State of the Art

We propose a new algorithm exclusively based on IIR filters and an autotuning procedure usefulto obtain the value of the filters parameters.PROS: Create a realistic reverberation with a low computational cost.CONS: None.

Proposed Solution

Proposed Structure

The proposed structure is mainly composed of two parts as the Moorer’s reverberator [5].

Fig.1 Hybrid reverberator block diagram for the single channel case.

Based on convolution with a real IR for theexact reproduction of the early echoes.

Early Reflections Device (classic)

Based on Second Order Structure (i.e.peak filters) and all-pass filter to approxi-mate the early echoes and reduce the com-putational complexity.

Early Reflections Device (proposed)

Based on IIR filters network (e.g. comband/or all-pass) for the simulation of the re-verberation tail.

Late Reflections Device

Early Reflection Device

Starting form a preliminary analysis of the mixing time [3] it is possible to extract the early reflectionsfrom the given impulse response.

Fig.2 Early reflections device block diagram.

Her(z) =N∏p=1

Hp(z)M∏l=1

HAl(z)

Hp(z) =a2z

−2 + a1z−1 + a0

b2z−2 + b1z−1 + 1HA(z) =

−g + z−D

1− gz−D

The Second Order Structures (SOSs) are used to reproduce the magnitude.SOS

The all-pass filters are used to adjust the phase response.All-pass

Two different automatic procedures (e.g., for magnitude and phase) are employed to determine thevalue of the parameters of the used IIR filters.

Autotuning Procedures

Early Reflection Approximations (Magnitude)

The presented approach derives from a technique applied to the loudspeaker equalization [6].

Fig.3 Search of the biggest area and design of first SOS.

• The filter varies around the 0 dB line;• A frequency warping is applied to respect the

logarithmic behavior of the human ear;• The SOSs are obtained evaluating the areas

enclosed by the magnitude of the early reflec-tions and the 0 dB line exploiting a parametricoptimization process.

An iterative procedure is used to deter-mine the parameters of the SOSs (fp, Gpand Qp):

1. Set p = 0;2. Search for the biggest area (error);3. fp is the mean of the zerocrossing point

of the selected area;4.Gp is the value of the error at fp;5.Qp is randomly choose between 1.5

and 3;6. An optimization algorithm is performed

in order to adjust fp, Gp and Qp;7. Update the error function, set p = p+ 1

and return to point 2.

SOS Approximation Procedure

Early Reflection Approximations (Phase)

All-pass filters are used to adjust the phase response obtaining a better approximation of the earlyreflections.

An offline adaptation procedure, based on Simultaneous Perturbation Stochastic Approximation(SPSA) [3, 4], has been employed to iteratively find the value of the parameters of the all-passfilters.

Fig.4 General Scheme of the adaption procedure:F (z, θ) is the transfer function of the artificial system while

HTarget represents the real IR.

It is based on the evaluation of the error be-tween the unwrapped phase:

L1 =1

N

N−1∑n=0

(φr(n)− φa(n))

where:• φr is the phase response of the real IR;• φa is the phase response of the structure

described in Fig.2.

Loss function

Late Reflection Approximations

The Schroeder’s reverberator has been used to reproduce the late reflections.An offline adaptation procedure, based on SPSA, has been employed to iteratively find the value ofthe parameters of the Schroeder’s structure.

Fig.5 Late reflections device block diagram.

HLBCF (z) =1

1− gcLp(z)z−n=

1

1− gc1−d

1−dz−1(z)z−n

HAP (z) =−1 + (1 + ga) z

−n

1− gaz−n

It is computed in cepstral domain [7]:

L2 = max

max√√√√√ K∑

i=1

M∑j=1

[Tr(i,j) − Ta(i,j)]2

where:• Tr is a matrix representing the Mel-Frequency

Cepstral Coefficients (MFCC) derived fromthe real IR.

• Ta is the MFCC obtained by the artificial IR.

Loss function

Results (1)

The proposed procedure has been tested using a real IR with a T60 of 683 ms.

Fig.6 Real IR. Fig.8 Magnitude response of the real IR and the artificial one.

Fig.7 Artificial IR derived with the proposed approach. Fig.9 Phase evolution of the real IR and of the artificial responsewithout (Artificial 2) and with all-pass filters (Artificial 1).

Results (2)

Fig.10 Energy Decay Relief of the real IR.

Fig.11 Energy Decay Relief of the approximated IR.

IIR FIR

Proposed approach implementation

Additions 140 4686

Multiplications 116 4696

Memory Required 10000 300000[samples]

Tab.1 Computational complexity comparison betweenthe proposed method and the FIR filter approach

(Uniform Partitioned Overlap and Save) [8].

With N = 20, Na = 4, Nl = 8, Nap = 4, Sample Rate =

48000, and I/O Latency = 64 samples.

Computational Cost

Tests proved that 20 SOSs are enoughto guarantee a good approximation ofthe real IR.

Listening Test

Conclusions

• A novel approach for the approximation and simulation of a real impulse response has been presented.

• The IR is decomposed in early and late reflections exploiting the mixing time evaluation.

• The early reflection are approximated using a cascade of SOSs and all-pass filters, minimization criteria are em-ployed to obtain the value of the parameters of the used IIR filters.

• The late reflections approximation is based on the Schroeder’s structure and a minimization criterion in cepstraldomain.

• Different tests have been carried out in order to evaluate reverberation quality, in terms of subjective evaluation andobjective measures.

• As confirmed by the listening tests the artificial effect generated sounds really similar to the natural one validatingthe proposed approach.

• Future works will be oriented to the real time implementation of the approach considering an embedded platform.

References

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[2] J.C. Sarris and G.E. Cambourakis, “Time frequency analysis and parametric approximation of room impulse responses,” in Proc. IEEE International Conference on Acoustics, Speech and Signal Processing(ICASSP’03), Hong Kong, apr. 2003, vol. 6, pp. 297–300.

[3] A. Primavera, L. Palestini, S. Cecchi, F. Piazza, and M. Moschetti, “A Hybrid Approach for Real-Time Room Acoustic Response Simulation,” in Proc. 128th Audio Engineering Society Convention (AES’10),London, UK, May 2010.

[4] G. Constantini and A. Uncini, “Real-Time Room Acoustic Response Simulation by an IIR Adaptive Filter,” Electronics Letters, vol. 39, pp. 330–332, 2003.

[5] J.A. Moorer, “About This Reverberation Business,” Computer Music Journal, vol. 3, no. 2, pp. 13–28, 1979.

[6] G. Ramos and J.J. Lopez, “Filter Design Method for Loudspeaker Equalization Based on IIR Parametric Filters,” J. Audio Eng. Soc., vol. 54, no. 12, pp. 1162–1178, Dec. 2006.

[7] S. Heise, M. Hlatky, and J. Loviscach, “Automatic Adjustment of Off-the-Shelf Reverberation Effects,” in Proc. 126th Audio Engineering Society Convention (AES’09), Munich, Germany, May 2009.

[8] W. G. Gardner, “Efficient Convolution without Input-Output Delay,” J. Audio Eng. Soc., vol. 43, no. 3, pp. 127–136, Mar. 1995.