CA2410749A1 - Convergence improvement for oversampled subband adaptive filters - Google Patents

Abstract

A method and system for improving the convergence properties of the adaptive filters is provided. The system includes an oversampled WOLA and a circuit for improving the convergence rate. The circuit may implement whitening by spectral emphasis, whitening by additive noise, whitening by decimation or affine projection algorithm. The system is applicable to echo cancellation.
For echo cancellation, adaptive step size control, adaptation process control using Double-Talk detector may be implemented. The system may further implement a non-adaptive processing for reducing uncorrelated noise and/or cross-talk resistant adaptive noise cancellation.

Description

Convergence Improvement for Oversampled Subband Adaptive Filters Field of the Invention:
The present invention relates to adaptive filter, more specifically to a system and method for improving convergence in subband adaptive filers.
Backaround of the Invention:
It is well known that a noise cancellation system can be implemented with a fullband adaptive filter working on the entire frequency band of interest [4]. The Least Mean-Square (LMS) algorithm and its variants are often used to adapt the fullband filter with relatively low computation complexity and good performance.
However, the fullband LMS solution suffers from significantly degraded performance with colored interfering signals due to large eigenvalue spread and slow convergence [4,5,6]. Moreover, as the length of the LMS filter is increased, the convergence rate of the LMS algorithm decreases and computational requirements increase. This can be a problem in applications, such as acoustic echo cancellation, that demand long adaptive filters to model the return path response and delay. These issues are especially important in portable applications, where processing power must be conserved.
As a result, subband adaptive filters (SAF) become a viable option for many adaptive systems. The SAF approach uses a filterbank to split the fullband signal input into a number of frequency bands, each serving as input to an adaptive filter. The subband decomposition greatly reduces the update rate and the length of the adaptive filters resulting in a much lower computational complexity. Further, subband signals are often decimated in SAF systems. This leads to a whitening of the input signals and an improved convergence behavior [7). If critical sampling is employed, the presence of aliasing distortions requires the use of adaptive cross-filters between adjacent subbands or gap filterbanks [7,8]. However, systems with cross-filters generally converge slower and have higher computational cost, while gap filterbanks produce significant signal distortion.

It is desirable to provide a subband adaptive filter system which can meet a high speed processing, low power consumption and high quality and is applicable to a noise (echo) cancellation.
Summary of the Invention:
It is an object of the present invention to provide a system and method to overcome one or more of the problems cited above.
It is an object of the present invention to provide a system and method for improving convergence of a subband adaptive filter.
The inventors have investigated the convergence properties of an SAF
system based on generalized DFT (GDFT) filterbanks. The filterbank is a highly oversampled one (oversampling by a factor of 2 or 4 or more). Due to the ease of implementation, low-group delay and other application constraints a higher oversampling ratio than those typically proposed in the literature may be used.
The oversampled input signals received by the subband processing blocks are no longer white in spectrum. In fact, for oversampling factors of 2 and 4, their bandwidth will be limited to pl2 and p/4 respectively. As a result, one would expect a slow convergence rate due to eigenvalue spread problem [4,5,6].
On the other hand, while the oversampled subband signals are not white, their spectra are colored in a predicable way and can therefore be modified by further processing to whiten them in order to increase the convergence rate. Thus, the inherent benefit of decreased spectral dynamics resulting from subband decomposition is not lost due to oversampling. Various spectral whitening techniques will be described hereafter. Another method of improving the convergence rate is to employ adaptation strategies that are less sensitive to eigenvalue spread problem. One of these strategies is the Affine Projection (AP) algorithm. Exact and approximate versions of the AP algorithm are proposed to speed up the convergence rate of the SAF system on an oversampled filterbank.
Oversampled SAF systems offer a simplified structure that without employing cross-filters or gap filterbanks, reduce the alias level in subbands. To reduce the computation cost, often a close to one non-integer decimation ratio is used [9].

In accordance with a further aspect of the present invention, there is provided an echo cancellation system which includes a SAF system having functionality of convergence improvement, such as whitening by spectral emphasis, whitening by adding noise, whitening by decimation, Affine projection algorithm. The system may include Double-Talk Detector to control the adaptation process. The system may have functionality of adaptation step size control.
In accordance with a further aspect of the present invention, there is provided an echo cancellation system which includes an adaptive processing and non-adaptive processing.
In accordance with a further aspect of the present invention, there is provided a cross-talk resistant subband adaptive filter.
A further understanding of other features, aspects and advantages of the present invention will be realized by reference to the following description, appended claims, and accompanying drawings, Brief Description of the Drawings:
The invention will be further understood from the foNowing description with reference to the drawings in which:
Figure 1 is a block diagram showing a subband adaptive filters (SAF) system for whitening by spectral emphasis method in accordance with a first embodiment of the invention;
Figure 2 is a block diagram showing a SAF system for whitening by additive noise method in accordance with a secod embodiment of the invention;
Figure 3 is a block diagram showing a SAF system for whitening by decimation method in accordance with a third embodiment of the invention;
Figure 4 is a graph showing signal spectra at various points of Figure 3;
Figure 5 is a graph showing Average Normalized Filter MSE for speech in 0 dB SNR White noise, (a) without whitening, (b) whitening by spectral emphasis, (c) whitening by decimation;

Figure 6 is a graph showing eigenvalues of the autocorrelation matrix of the reference signal for: No whitening, Whitening by spectral emphasis, whitening by decimation, and whitening by decimation and spectral emphasis;
Figure 7 is a graph showing measured mean-squared error for: No whitening, whitening by spectral emphasis, whitening by decimation, and whitening by decimation and spectral emphasis;
Figure 8 is a graph showing measured mean-squared error for Affine Projection Algorithm (APA) with different orders;
Figure 9 is a block diagram showing an adaptive echo cancellation system in accordance with an embodiment of the invention;
Figure 10 is a block diagram showing a task of echo cancellation of Figure 9;
Figure 11 is a block diagram showing a first embodiment of an adaptive processing block (APB) of Figure 10;
Figure 12 is a block diagram showing a second embodiment of an APB of Figure 10;
Figure 13 is a block diagram showing a third embodiment of an APB of Figure 10;
Figure 14 is a block diagram showing a fourth embodiment of an APB of Figure 10;
Figure 15 is a graph showing a coherence function of a diffuse noise;
Figure 16 is a block diagram showing an oversampled SAF system for noise reduction in accordance with a forth embodiment of the invention;
Figure 17 is a block diagram showing one embodiment of an adaptive processing block (APB) and a non-adaptive processing block (NAPB) of Figure 16;
Figure 18 is a block diagram showing a cross-talk resistant APB of Figure 10 and Figure 16.

Detailed Description of the Preferred Embodiment(s):
Whitening by spectral emphasis Figure 1 shows a block diagram of an SAF system that includes the 5 proposed whitening by spectral emphasis method. As shown an unknown plant P(z) is modeled by the adaptive filter, W(z). After WOLA analysis, subband signals are decimated by a factor of M/OS, where M is the number of filters, and OS is the oversampling factor. At this stage, the subband signals are no longer full-band. Rather, as shown in Figure 1 (points 1 and 2), their bandwidth is now p/OS. The emphasis filter (gP~e(z)) then amplifies the high frequency contents of signals at points 1 and 2 to obtain almost white spectra. The filter gain (G) is a design parameter that depends on the analysis filter shape.
Whitening by additive noise Alternatively, high-pass noise can be added to bandpass signals to make them whiter in spectrum. As shown in Figure 2, first the average power (G) of the signal at point 1 is estimated and used to modulate a high-pass noise a(n).
The input to adaptive filter (point 3) is then whitened by adding G.a(n) to the signal at point 1.
Whitening by decimation Figure 3 shows a block diagram of the SAF system with a proposed whitening by decimation method. As shown, the subband signals (for both the reference input x(n) and the primary input d(n)) are further decimated by a factor of DEC<OS. Assume, without loss of generality, that DEC is at its maximum, DEC= OS-1. As demonstrated in Figure 4 (point 3), this increases the bandwidth to p (OS-1)/OS (3p/4 for OS=4) without generating in-band aliasing. Due to the increased bandwidth, the LMS algorithm now converges much faster. To be able to use the adaptive filter (Vlld(z)), it is expanded by OS-1. This may create in-band images (point 4 in Figure 4). However, since the signal at point 1 does not contain considerable energy for w> p/OS, the spectral images will not contribute to any errors.

Affine Projection In order to further increase the convergence rate, a class of adaptive algorithms called Affine Projection have been proposed [12]. Affine Projection Algorithm (APA) forms a link between Normalized LMS (NLMS) and Recursive Least Square (RLS) adaptation algorithms: faster convergence of RLS and low computational requirements of NLMS are compromised in APA.
In NLMS, the new adaptive filter weights have to best fit the last input vector to the corresponding desired signal. In APA, this fitting expands to the P-1 past input vectors (P being the APA order). Adaptation algorithm for the Pt"
order APA can be summarized as follows:
1) update X~ and do 2) en =d~ _ X~W

3) Wn+, = W~ + mX" (X~ X" + a I)-' en where:
X~: an L' P matrix containing P past input vectors _ d~ : a vector of the past P past desired signal samples W~ : adaptive filter weights vector at time n a : regularization factor The convergence of APA is surveyed in [12, 13]. It is shown that as projection order P increases, the convergence rate becomes less dependant on the eigenvalue spread. Increasing the APA order results in faster convergence at the cost of more computational complexity of the adaptation algorithm.
We propose the use of the APA for a SAF system implemented on a highly oversampled WOLA filterbank [1,2,3]. An APA order of P = 2 can be a good choice, compromising fast convergence and low complexity. In this case, the matrixXn X~ can be approximated by R (autocorrelation matrix of the reference signal) [14]. So, for P = 2, it is sufficient to estimate the first two autocorrelation coefficients (r(0) and r(1 )) and then inverse the matrix R , analytically. A first order recursive smoothing filter can be used to estimate r(0) and r(1 ).
Combination of the above techniques It is possible to combine any two or more of the described techniques to achieve a higher performance. For example, whitening by decimation improves the convergence rate by increasing the effective bandwidth of the reference signal. However, it cannot deal with the smallest eigenvalues that are associated with the stop band region of the analysis filter. On the other hand, whitening by spectral emphasis improves the convergence by limiting the stop band loss thereby increasing the smallest eigenvalues. A combination of the two techniques will enable us to take advantage of the merits of both systems.
Performance evaluation Preliminary assessments show that the performance of the whitening by additive noise is very similar to whitening by spectral emphasis. However, the computation cost of whitening by additive noise is less since it does not need emphasis filters. Instead, it needs a very simple filter (per subband) to estimate the signal power.
Figure 5 shows typical convergence behavior of the proposed whitening by decimation compared to no whitening and whitening by emphasis. The application of the SAF system has been 2-microphone adaptive noise cancellation. As shown, whitening by decimation converges mush faster than the other two methods.
Whitening by decimation greatly improves the convergence properties of the SAF system. At the same time, since the adaptive filter operates at a low frequency, the method offers less computation than whitening by emphasis or by adding noise. However, the proposed whitening by decimation is applicable only g to oversampling factors (OS) of more than 2. For detailed mathematical models of SAF systems see [9,15].
Figure 6 shows the theoretical Eigenvalues of the autocorrelation matrix of the reference signal for: No whitening, Whitening by spectral emphasis, Whitening by decimation, and Whitening by decimation and spectral emphasis.
The method employed is described in [6]. As shown, while whitening by spectral emphasis and by decimation both offer improvements (demonstrated by a rise in the eigenvalues), a combination of both method is more promising. This conclusion is confirmed by the mean-squared error (MSE) results shown in Figure 7. Finally, Figure 8 shows the MSE results APA orders of P = 1, 2, 4 and 5 (The APA for P = 1 yields an NLMS system). As shown, increasing the AP
order, improves both the convergence rate and the MSE.
The present invention will be further understood by the additional description A, B and C attached hereto.
IS
Alternate embodiment Echo Cancellation by SAFs using Improved Convergence Techniques As mentioned above, in echo cancellation long filter lengths may be used due to long duration of echo path. As a result, fast adaptation techniques for echo cancellation are now described in detail. Figure 9 shows an application of adaptive systems for echo cancellation. As shown, the Far-End (FE) acoustic input signal is converted to an electric signal x(t) that is sent to the Near-End (NE) speaker. The NE microphone then receives an acoustic echo signal (called FE echo) from the NE speaker. The NE microphone also receives NE input (speech and noise) signal, and converts the total signal (FE echo + NE input) to an electric signal d(t). An adaptive filer minimizes the error signal e(t) to eliminate FE echo. Once converged, the adaptive filter essentially models the transfer functions of the NE speaker and NE microphone, as well as the acoustic transfer function between the NE speaker and the microphone. Echo can also be generated by electrical signals leaking back to the FE side through various (undesired) electrical paths between the FE and the NE sides. The proposed techniques cover both acoustical and electrical echoes, however, in the discussions only acoustical echo is discussed.

Figure 10 demonstrates the task of echo cancellation (Figure 9) implemented in a subband domain. As shown signals x(t) and d(t) are first sampled and then analyzed by two analysis filterbanks to obtain complex frequency-domain subband signals x;(n) and d;(n), i=0,1, ,K-1, K being the number of subbands. Pairs of [x;(n) ,d;(n)] are next used as inputs to Adaptive Processing Blocks (APB in Fig. 10). The outputs of APBs (complex subband signals e;(n)) are then combined in the synthesis filterbank to obtain the time-domain echo-cancelled signal e(n). We now describe a few possible APBs that could efficiently cancel echo. The APB blocks in Figure 10 and its possible examples (described in Figures 11, 12, 13, 14, 16, 17, and 18) can employ any of the convergence improvement techniques introduced above (whitening by spectral emphasis, whitening by adding noise, whitening by decimation, Affine projection algorithm, and a combination of two or more of those techniques) to achieve fast convergence.
Figure 11 shows a possible APB to be employed in Figure 10. As shown a Double-Talk Detector (DTD) is employed to control the adaptation process. The DTD includes two voice-activity detectors (VADs), one (FE VAD) operating on the FE signal and another one (NE VAD) employing the signal d;(n). It also contains a logic that based on the two VAD decisions, specifies when double-talk (both NE and FE sides talking), single-talk (only one of the FE or NE sides talking) or common-pause (none of the two sides talking) situations occur. The DTD allows quick adaptation of the adaptive filter only during FE signal-talk.
In other situations, it stops or slows down the adaptation.
Demonstrated in Figure 12 is another possible APB for Figure 10. As shown, the NE VAD now uses the error signal e;(n). The rational behind using the error signal is as follows. At the start of the adaptation process, the error signal is almost the same as d;(n) since the adaptive filter is identically zeros. As the DTD allows the adaptive filter to adapt, more and more of the echo is cancelled from d;(n). As a result, the DTD detects more instances of FE single-talk and the filter gets more chance to further adapt. This in turn will cancel echo more efficiently. This looping improves the performance of the DTD and as a result the echo cancellation system. This strategy is particularly helpful when there are high levels of echo.

Adaptation Step-size control The NE signal might contain both speech and noise. As a result, the NE
noise might be present even when the DTD detects a FE single-talk situation.
5 This would create problems for the adaptive processor if a large adaptation step-size (rr) were chosen. One solution to this problem is to condition the adaptation step-size on the level of the FE echo (FEE) signal relative to the level of the NE
noise (NEN) signal, i.e. on the ratio of ~FEE~2! ~NEN~2. An estimate of the NEN
energy can be obtained by measuring the energy of d;(n) in common-pause. To 10 estimate energy of the FEE, one can subtract the NEN energy estimate from energy of d;(n) during FE single-talk, i.e.:
~d;(n)~2 in common-pause ~NEN~2 estimate ~d;(n)~2 in FE single talk ~NEN~2 estimate ~FEE~2 estimate Figure 13 shows an APB (to be used in Figure 10) that contains a rr~
Adaptation block. Based on the DTD result and the estimate of ~FEE~21 ~NEN~2 , the block varies the value of the step-size. Various strategies are possible to adapt the step-size. Generally as the ratio of ~FEE~21 ~NEN~2 increases, larger step-sizes are employed.
Alternative method of DTD
The adaptive filters employed for echo cancellation might have high filter orders due to long echo paths. As a result, the adaptive filter may converge slowly. Since the DTD of Figure 12 is relying on the adaptive filter performance, this slow convergence might create a problem for the whole system. Here a possible solution is proposed. Figure 14 shows an APB (to be used in Figure 10) that employs two adaptive filters. Adaptive filter 2 contains is a low-order filter that is basically used for DTD. Adaptive filter 1 works with the rr+Adaptation block and performs similar to the system in Figure 13. The low-order adaptive filter can adapt faster than the adaptive filter 1. Most of the echo would be eliminated quickly at its output (f;(n)), and the NE VAD in the DTD would perform well even before full convergence of the adaptive filter 1.

ll Atte~nate embodiment Combination of adaptive and non-adaptive processing for noise and echo cancellation The general SAF system of Figure 10 performs well for noise cancellation as long as the noises in the two inputs x(n) and d(n) are correlated. It is well-known that the (optimum) adaptive filter is estimated as [5]:
u'~(f) =px~r(f) P~(f) where pxa(f)=a, ra(k)e-~~a' k and rx~(k) is the cross-correlation of input signals x(n) and d(n) at delay k.
So, the cross correlation plays a major role in estimating the transfer function between two inputs. In the case of weak correlation, adaptive filter only removes the correlated portion of the noise and leaves the uncorrelated part intact.
The most valid feature to characterize the correlation of two noise signals x(n) and d(n) (here it assumed that the input signal d(n) contains only noise and there is no speech signal present), is the coherence function[18]:
Gx~ (f) = I P~ U) I2 P~U)~'a~(f) In each frequency f, equation (1 ) characterizes the correlation of two input signals by a value between 0 and 1 and consequently, determines the amount of noise that can be cancelled in that frequency through adaptive filtering. More precisely, the noise reduction factor of adaptive filtering is equal to [18]:
NR( f ) = Input noise power at frequency f __ 1 output noise power at frequency f 1- Gx~ ( f ) Diffuse Noise Field In a diffuse noise field, two microphones receive noise signals from all directions equal in amplitude and random in phase. This results in a squared Sinc (magnitude squared) coherence function for diffuse noise field [19]:
Gxd(l)=sin2(2pldlc)=Sinc2(2~d) (2pfd I c)2 c where d is the microphone spacing and c is the sound velocity (c = 340 m/s).
Figure 15 shows the coherence function of a diffuse noise for d=38 mm.
According to this coherence function, increasing microphone spacing d, will decrease the noise reduction capability of adaptive filter in more subbands.
Although a decrease in distance of two microphones can be proposed as a remedy, but clearly this intensifies the cross-talk problem (described in next Section) and thus, is not an acceptable solution.
Many practical noise fields are diffuse (16,17]. As a result, the noises recorded by the two microphones are only coherent at low frequencies. This implies that the SAF system can partially remove the noise from d(n). There are some other possible scenarios where the two noises present at the two microphones do not have a flat coherence function (of value 1 ) across various frequencies. In such cases, the SAF system can only partially enhance the signal.
The system in Figure 16 can cope with these situations. The system in Figure 16 includes an extra (compared to Figure 10) non-adaptive processing block (NAPB) in each subband. The NAPB can pertorm single-mic or two-mic non-adaptive noise reduction. For example, as shown in Figure 17, the NAPB
can be a single-mic Wiener filter to eliminate the residual uncorrelated noise at the output of the subband adaptive filter. Other single-mic or two-mic noise reduction strategies are also possible. Since the correlated noise is already eliminated by the APB stage, the artifacts and distortions due to the NAPE
would have less degrading effects at the output. For diffuse noises, the important low-frequency region of speech signal will not be distorted since the low-frequency noises at the two mics are correlated and will be eliminated mostly by the APB
without generating artifacts.

Alternate embodiment Cross-Talk Resistant subband adaptive filters for noise cancellation The performance of adaptive noise cancellation systems can be severely limited in cross-talk, i.e. when the speech (or desired) signal leaks into the reference (noise) microphone. To remedy this problem, cross-talk resistant adaptive noise canceller (CTRANC) has been proposed in the literature [20]. However, the proposed systems are implemented in time-domain and not in subband domain.
The use of CTRANC techniques for SAF systems implemented on oversampled filterbanks is now described. Figure 18 shows the block diagram of a cross-talk resistant APB, to be used in systems of Figure 10 or Figure 16. As shown, there are two adaptive filters V;(z) and W;(z) in each subband. After convergence, the signal e;(n) provides the enhanced (subband) speech signal while the signal f;(n) provides the noise signal without speech interference.
The SAF system and the noise cancellation system of the present I S invention may be implemented by any hardware, software or a combination of hardware and software having above described functions.
While particular embodiments of the present invention have been shown and described, changes and modifications may be made to such embodiments without departing from the true scope of the invention.
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Additional Description A
Technical Report Polyphase Analysis of Subband Adaptive Filters Polyphase Analysis of Subband Adaptive Filters Stephan Weissl, Robert W. Stewart2, Moritz Harteneck3, and Alexander Stenger4 1 Dept. Electronics & Computer Science, University of Southampton, UK
2 Dept. Electronic & Electrical Eng., University of Strathclyde, Glasgow, UK
3 Infineon Technologies AG, Munich, Germany 4Telecommunications Institute I, University of Erlangen, Germany s.xeiss~ecs.soton.ac.uk, r.stexart~eee.strath.ac.uk Abstract jest to a number of limitations, which have been inves-tigated, for example, with respect to the required filter Based on a polyphase analysis of a subband adaptive length (3, 14] or to lower bounds for the MMSE and alter (SAF) system, it is possible to calculate the opti- the modelling accuracy (12]. These analyses have been mum aubband impulse responses to which the SAF sys- performed using modulation description (3, ?], time do-tem will converge. In this paper, we gave some insight main (14), or frequency domain approaches (5, 12J.
into how these optimum impulse responses are calcu- Here, we discuss the 5AF
in Fig. l using a polyphase lated, and discuss two applications of our technique. description of the signals and filters therein (2J. This Firstly, the performance limitations of an SAF sys- will provide some new and alternative insight into the tem can be explored with respect to the minimum mean optimality of SAFs. Sec.
2 analyses the subband er-square error performance. Secondly, fullband impulse rora, which leads to the derivation and discussion of an responses can be correctly projected into the subband optimal subband filter structure in Sec. 3. Application domain, which is required for example for translating examples for the proposed techniques are underlined constraints for subband adaptive 6eamforming. Exam- by simulations in Sec. 4.
pies for both applications are presented.
2. Polyphase Analysis of Subband Errors 1. Introduction The aim of this section is to express the subband er-Adaptive filtering in aubbanda is a popular ap- ror signals, Ek{x) ~-o ek(x), in terms of the polyphase proach to a number of problems, where high compu- components of all involved signals and systems. Im-tational cost and slow convergence due to long filters plicitly, this means that we are trying to find a lin-permits the direct implementation of a fullband algo- ear, time-invariant {LTI) description of the error sig-rithm. These problems include acoustic echo cancella- nil. To achieve this task, we first require suitable rep-tion (5, 3), identification of room acoustics (8), equal- resentations for the decimated desired signal in the kth ization of acoustics (10], or beamforming (6, 11]. In subband, Dk (z) ~-o dk (nJ, and for the decimated in-Fig. 1, a subband adaptive filter (SAF) is shown in a put signal in the kth subband, Xk(x) ~--o xk(n], as system identification setup of an unknown system s(n), labelled in Fig. 1. In our notation, superscript {~}d for whereby the input x(n) and the desired signal d(nJ are z-transforms of signals refers to decimated quantities, split into K frequency bands by analysis filter banks while normal variables such as Xk(x) indicate undeci-built of bandpass filters h~(n). Assuming a cross-band mated signals, i.e. in this case the input signal in the free SAF design (3], an adaptive filter wk(nJ is applied kth subband before going into the decimator as shown to each subband decimated by N <_ K. Finally, the in Fig. 1.
fullband error signal e(n] can be reconstructed via a The formulation of the kth decimated desired sig synthesis bank. nil Dk(z) ~--o dk(n] will be the first aim. We define However, subband adaptive filters (SAF) are sub- the expansion of the desired signal D(x) ~-o d(n] into -analysis unknown f lter bank system ; ~]ds[n]
~= dfn]' 'L~,~N, dUn]
s.
N ~ ~dKtLn]
'~~___:~ ' j _ ~ _ _______ (n] of ] e~[n]
;,rv : . ,~~ _O---~-~N--;xx~Ln] extLn]' analysis adaptive synthesis filter bank . filters filter bank Fig. 1. Subband adaptive filter (SAF) in a system identification setup.
type-2-polyphase components (9J Dn(x), X(z) is defined similarly to (3) based on the type-2-polyphase components of the input signal w-i X (z) ~-o x(n~. The matrix An(z) in (6) is a delay D(x) _ ~ x-N+n+1 , Dn{zrr) , (1) matrix defined as n=0 _ 0 IN_"
and a type-1-polyphase expansion (9J of the analysis An{x) - [ x-lIn 0 ] ' (7) filters H~(x), With (5) and {6), the decimated kth desired subband N-1 signal Dk{z) Hk{z) _ ~ x n-Hk,n{zN) . {2) r~T(z)Ao(z) n=0 TzAlx Similarly, for all following polyphase expansions, it Dk(z) = H~(z) ~ ( ) ( ) X(z) = H~(x)S(z)X(x) is assumed for compatibility that systems are rep-resented by a type-1-polyphase expansion, and sig- ST(z)Aw-i{x) nals by type-2-polyphase expansions. Bringing these (8) polyphase components of (1) and (2) into vector form, can be assembled. For brevity, the substituted matrix D(x) - (Da(x) Dl (z) . . . DN i(z)]T (3) S(x) holds differently delayed polyphase components of the unknown system.
Hk(x) - (H,~~o(z) Hk~l . . . Hk~N 1 {x)~ {4) With the type-2-polyphase components of X (x) arid the polyphase representation of the analysis filter bank Dk(x) can now be expressed as in (2) it is comparably simple to derive the kth deci d s mated input signal Xk (x) as Dk(z) _ ~ (x) ' D(x) - (5) Xk (x) - Hk (x) . X (x) . (9) To trace the desired signal back to the input signal -X (x) ~-o x(n~, the expression D(x) = S(z) . X (x) can Finally, with (8), (9), and the transfer function be appropriately expanded such that the nth polyphase of the kth adaptive filter W~ (x) ~-o wk (n) it is pos component in (3) may be written as sible to formulate the kth subband error signal, Dn(z) _ ~T(z) . An(x) . X{z) . (6) Ek{z) ~-o ex(n):
Ek {x) = D~(z) - Wk (z) . X~ (x) {10) The vector S(x) contains the type-1-polyphase com- ( l ponents of the unknown system S(z) - -~-o s(n), while - S Hk (z)~S(x) - H~
(x).Wk(x) 5X(z)(11) Fig. 3. SAF standard solution in the kth subband.
Flg. 2. SAF optimal polyphase solutions in the kth subband.
3.2. Interpretation Note, that for the description of E~ (z), the time-varying decimators have been swapped with all system Alternatively, the nth optimum solution can be writ-elements in the SAF structure of Fig. 1, and (11) only ten as contains LTI terms.

Wk Pt (x) _ ~ A~~n(z) ' ,S"(z) . (14) 3. Subband Error Minimization "-o and interpreted as a superposition of polyphase com This section discusses the optimum subband filters ponents S"(x) of the unknown system S(z), "weighted"
to solve the identification problem outlined in Sec. 1, by transfer functions based on the polyphase analysis of the subba,nd errors in the previous Sec. 2. " H z A~I~(,z) = z-~(n+v)/1Vl . k~~n+v) modm( ) ' (15) Hk~n(x) 3.1. Optimum Subband Filters From this, we can observe, that the length of the opti As no external disturbance of the SAF system in mum subband responses is obviously given by 1/N of Fig. 1 by observation noise is present, ideally the at- the order of S(z), but extended by the transfer func tainable minimum mean square error (MMSE) should dons (15). These extending transients are causal for be zero. This is identical to setting Ek(x) in (11) equal poles of Ak~n(z) within the unit circle, and aca,usal for to zero. As independence of the optimum solution from stabilized poles outside the unit-circle (13], motivating the input signal's polyphase components in X (x) is de- a non-causal optimum response.
sirable, the requirement for optimality (in every sense) Further, for an ideal, alias-free filter bank, the is given by polyphase components Hyn(z) in (15) should not differ in magnitude but only in phase, which is compensated Hk (x) ~ S(x) ' XT ~ Wk,opt (z) . (12) for by the delay element in (15). Hence all N solutions become identical, an the N optimum polyphase filters Hence, we obtain N)cancellation conditions indicated can be replaced by a single filter Wk,opt(z) as shown by superscripts { } , which have to be fulfilled: in Fig. 3, which is equivalent to the original standard (n) Hk (x) ~ AT (x) ~ S_(z) setup in Fig. 1. In general, and particularly if a,liasing Wit,oPt(x) = H~~"(x) do E {0;N-1} . is present, the optimum polyphase solutions Wk opt(z) (13) wdl differ. In this case the optimum standard SAr' so lution according to Fig. 3 gives the closest l2 match to Therefore, ideally W,t(x) in (11) and (12) should be ~1 N polyphase solutions:
replaced by an N x N diagonal matrix with entries N_1 Wk")(z), n = 0(1)N-1. For the kth subband, this Wk,oPt(x) = N ~ Wk pt (x) .
(lfi) solution with N polyphase filters is depicted in Fig. 2. "=o ._ the structure of the standard SAF system in Fig. 3, the desired signal PSD analytlCal solution (16) calculated from (18) is given by . ,,r'" _ ',, ,, - - - simulated the mean of the two optimum polyphase solutions, enalylical prod.
s .~,.". , Wo,opt (x) = 1.5 + 0.5x-1 error signal PSD: ~ , , n "~' , This result obviously very closely agrees with the sim-s anaycal prea~,;on ulation result in {17).
Based on the above analytical solutions, it is now ' possible to predict the subband error signal as due to °o o., o.Z o.3 0., o.s o.8 o.~ o.0 °.9 , the mismatch of {18) and (4.1). The PSD of the 0th adapted subband error signal, Seo(e~n), can be anal Flg. 4. Comparison between simulated and ana- lytically predicted by inserting the optimum standard lytically predicted PSDs in the 0th subband. solution {16) into (11), seo(e~~) = I~%'b(e~o)Ia = 1- cosSt , (19) The error made in this approximation explains error which can be used to determine the minimum mean and modelling limitations of the SAF approach and squared error of the SAF
system alternative to spec-represents an alternative coefficient / time-domain de- tral methods (12J.
Fig. 4 demonstrates the excellent fit scription as opposed to spectrally motivated SAF error between the analytically calculated PSD in (19), and explanations in the literature (3, 12J. Interestingly, in the measured results from the RLS simulation. Also (7J the same polyphase structure as in Fig. 2 is obtained shown is the analytically predicted and measured PSD
using modulation description (2J 5 , although only for of the 0th desired subband signal Sdo (esn ) = 6+2 cos St the critically sampled case. (hence the uncancelled error signal) calculated via (5).
4. Applications and Simulations 4.2. Subband Projection We now want to explore two applications for the polyphase analysis presented in Secs. 2 and 3. A second application example is concerned with sub stituting subband adaptive system identification with 4.1. Error Limits the proposed analysis. If a digital impulse response is given in the fullband, but should be projected into A very basic example given in the following will the subband domain, an SAF
identification is mostly demonstrate the ability of the proposed analysis to pre- required. This could be to produce computationally ef dict optimal subband responses and error terms in the fi~ent sound processing from a given (fullband) room context of SAF systems. For this example, a 2-channel transfer function (8J, or the projection of constraints critically decimated standard SAF system as in Fig. 1 into the subband domain when performing subband based on a Haar filter bank (2J should adaptive iden- captive beamforming (11J.
tify an unknown system S(x) = 1 + z1, using unit vari- We assume an SAF system with K = 8 channels ance Gaussian white noise excitation. Looking at the decimated by N = 6, and wide analysis filters to im-channel k = 0 produced by the Haar lowpass filter prove spectral whitening in the subbands (11. Analysis Ho(x) = 1 + x-1, an RLS adaptive algorithm (4J con- and synthesis banks are derived from the two different verges to the solution prototype filters shown in Fig. 5. With a lowpass full-band response s(nJ given, an RLS adaptive identifica WO,adapt {x) = 1.4873 + 0.5067x-1 . (17) tion yields in the subband k = 0 the coefficients shown Analytical evaluation via (14) and (15) yields the m Fig. 6, along with the analytic solution according to N = 2 optimum polyphase solutions for the band k = 0 (14) and (16). For the analytic solution, the roots of the denominator polynomial in (15) have been substi Wo o~pt(z~ = 2 , Wolopt(x) = 1 + z-1 , (18) tuted by appropriate causal and a causal FIR filters.
Obviously, the match between adaptive and analyti-which refers to the optimal subband adaptive filter cal solution is very close; and themore direct analytical structure shown in Fig. 2. If this setup is simplified to approach can replace an adaptive projection.

., °s 6. Acknowledgements o.,s _ _ a°e~~ w°,°,rae $yre is o,o,o,ype ;c °' The authors gratefully acknowledge Dr. Ian r °°~ ' K. Proudler, of DERA, Malvern, UK, who partially o ~~ _ supported this work. S. Weiss would like to thank the ' ' Royal Academy of Engineering for providing a travel o ,o zo °o ,o s° so ~o e° eo grant.
, ° ! ~ . - _ ~n~ss ~ N~ References --ZO , .
-'° ~ r~ ; (1) P.L. de LeGn II and D.M. Etter. "Experimental -oo -~ ~j!I ' ~, "y''Y r ~7 ,11I I~~~Ip ~" ~ , ,.. ........ Results with Increased Bandwidth Analysis Filters ' ~y Srs ~I~r4suar4,'y"n;i,,u~,4~i'.,y~ ~ r, ' , Wr s: ! a y y ' 4 , f ,~ .~ ,~ "'~, in Oversampled Subband Acoustic Echo Cancelers .
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~~~~~~~~~~'~ (2J N.J. Fliege. Multirute Digital Signal Processing: Mul Flg. 5. Prototype filters. tirate Systems, Falter Banks, Waveleta. Wiley, 1994.
[3J A. Gilloire and M. Vetterli. "Adaptive Filtering z5 ~ ~a~~~ in Subbands with Critical Sampling: Analysis, Ex-periments and Applications to Acoustic Echo Can-celation". IEEE ?fans Signal Processing, Vo1.40 (No.B):pp.1862-1875, Aug. 1992.
o.s (4J S. Haykin. Adaptive Filter Theory. Prentice Hall, 2nd -o.so ,o zo ao ~o so ao ~o eo 00 ,oo ed> 1991.
[5j W. Kellermann. "Analysis and Design of Multirate Systems for Cancellation of Acoustical Echoes" . In aae n8 Proc. ICASSP, vol 5, pp.2570-2573, New York, 1988.
--~-- ena~ytic [6] W. Kellermann. "Strategies for Combining Acoustic ' Echo Cancellation and Adaptive Beamforming Micro-° phone Arrays" . In Proc. IEEE ICASSP, vol I, pp.219--' 222, Munich, April 1997.
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Fig. 6. Adaptive and analytic subband response [8] M. Schonle, N.J. Fliege, and U. Zolzer. "Parametric for k = 0. Approximation of Room Impulse Responses by Multi rate Systems". In Proc. IEEE ICASSP, vol I, pp.153 5. ~iOriClilSiOriS 156, Minneapolis, May 1993.
(9] P.P. Vaidyanathan. Multirnte Systems and Filter Banks. Prentice Hall, 1993.
We have introduced an analysis of an SAF system, [10] S. WeiB, S.R. Dooley, R.W. Stewart, and A.K. Nandi.
which formulates the subba,nd errors in dependency "Adaptive Equalization in Oversampled Subbands".
of LTI polyphase components only. The main result IEE Elec. Let., Vo1.34(No.lS):pp.1452-1453, Jvly was a structural difference between what the optimum 1998' [11J S. Weiss, R.W. Stewart, M. Schabert, LK. Proudler, SAF requires and what the standard SAF structure and M.W. Hoffman. "An Efficient Scheme for Broad-provides. As a qualitative measure, this difference in band Adaptive Beamforming". In Aeilomnr Conf structure gives alternative insight into the inaccuracies Sig. Sys. Comp., Monterey, CA, Nov. 1999.
and limitations of the standard SAF approach. But [12j S. WeiB, R.W. Stewart, A. Stenger, and R. Raben-as demonstrated, the approach can also be utilized stein. "Performance Limitations of Subband Adaptive to quantify errors. Different from alias measurement Filters". In Proc.
EUSIPCO, vol. III, pp. 1245-1248, methods for error prediction ~12), the analysis also of Rodos, Sept. 1998.
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(14) R.J. Wilson, P.A. Naylor, and D. Brookes. "Perfor-cation for the latter, an example was given that allows mice Limitations of Subband Acoustic Echo Con-us to substitute the subband projection by SAF system trollers". In Proc.
IWAENC, pp.176-179, London, identification with the proposed analytical polyphase Sept. 1997.
approach.

Additional Description B
Technical Report Highly Oversampled Subband Adaptive Filters for Noise Cancellation on a Low-Resource DSP System

Claims (18)

1. A method of implementing adaptive echo cancellation, the method comprising steps of:
analysing a primary signal and an echo signal corresponding to the primary signal to produce frequency domain signals and a frequency domain echo signals in a plurality of frequency bands;
adaptively processing the frequency domain signals and the frequency domain echo signals using improving convergence technique in each frequency band; and synthesizing the outputs of the adaptive processing blocks to output a time domain echo cancelled signal.

2. A method as defined in claim 1, wherein the processing step includes the step of whitening the input of an adaptive filter by spectral emphasis.

3. A method as defined in claim 1, wherein the processing step includes the step of whitening the input of an adaptive filter by adding noise.

4. A method as defined in claim 1, wherein the processing step includes the step of whitening the input of an adaptive filter by decimating the frequency domain signals and frequency domain echo signals.

5. A method as defined in claim 1, wherein the processing step includes the step of implementing affine projection algorithm.

6. A method as defined in claim 1, wherein the processing step includes the step of employing a double-talk detector to control the adaptation process.

7. A method as defined in claim 1, wherein the processing step includes the step of controlling the adaptation step size.

8. A method as defined in claim 1, wherein the processing step includes the step of performing non-adaptive noise reduction for eliminating uncorrelated noise.

9. A method as defined in claim 1, wherein the processing step includes the step of performing a cross talk resistant adaptive processing using two adaptive filters in each frequency band.

10. A system for echo cancellation, the system comprising:
analysis filterbank for analyzing a primary signal and an echo signal corresponding to the primary signal to produce frequency domain signals and a frequency domain echo signals in a plurality of frequency bands;
processing module for adaptively processing the frequency domain signals and the frequency domain echo signals using improving convergence technique in each frequency band; and synthesis filterbank for synthesizing the outputs of the adaptive processing blocks to output a time domain echo cancelled signal.

11. The system as defined in claim 10, wherein the processing module includes an adaptive processing block having an adaptive filter and a module for whitening the input of the adaptive filter by spectral emphasis.

12. The system as defined in claim 10, wherein the processing module includes an adaptive processing block having an adaptive filter and a module for whitening the input of the adaptive filter by adding noise.

13. The system as defined in claim 10, wherein the processing module includes an adaptive processing block having an adaptive filter and a module for whitening the input of the adaptive filter by decimating the frequency domain signals and frequency domain echo signals.

14. The system as defined in claim 10, the processing module includes a module for implementing affine projection algorithm.

15. The system as defined in claim 10, wherein the processing module includes a double-talk detector to control the adaptation process.

16. The system as defined in claim 10, wherein the processing module includes a module for controlling the adaptation step size.

17. The system as defined in claim 10, further comprising a non-adaptive noise reduction module for eliminating uncorrelated noise.

18. The system as defined in claim 10, wherein the processing module includes a cross talk resistant adaptive processing module having a pair of adaptive filters in each frequency band.