An Overlap-Save Frequency-Domain Implementation of the Delayless Subband ANC Algorithm

1706 IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 16, NO. 8, NOVEMBER 2008

Correspondence

An Overlap-Save Frequency-Domain Implementation ofthe Delayless Subband ANC Algorithm

Ming Wu, Xiaojun Qiu, and Guoyue Chen

Abstract—An efficient active noise control algorithm based on thedelayless subband adaptive filter was previously proposed by Park et al.(“A delayless subband active noise control system for wideband noisecontrol, IEEE Transactions on Speech and Audio Processing, vol. 9, no. 8,pp. 892–897, November 2001), whereby the computational complexity isreduced significantly by using a short impulse response filter to model thesecondary path transfer function in a subband-decomposed form. In thiscorrespondence, an overlap-save frequency-domain implementation of thealgorithm is proposed. Simulations show that the proposed implementa-tion has better performance than the original.

Index Terms—Active noise control (ANC), delayless subband adaptivefilter, frequency-domain adaptive filter.

I. INTRODUCTION

Active noise control (ANC) is a method for attenuating unwanteddisturbances by the introduction of controllable secondary sources,whose outputs are arranged to interfere destructively with the originalprimary source [1]–[3]. Recently, the subband adaptive filteringtechnique is often adopted in ANC systems [4]–[10], which has twoadvantages: 1) faster convergence is possible because the dynamicrange is greatly reduced in each subband and each subband can haveits own step size; and 2) the computational complexity for the controlfilter update is reduced by approximately the number of the subbands,since both the number of taps and weight update rate can be decimatedin each subband. However, the conventional subband approach intro-duces a delay into the signal path which limits their performance inANC. To avoid the signal path delay while retaining the advantages ofsubband processing, a delayless subband adaptive filter architecturewas proposed by Morgan and Thi, in which the adaptive weights arecomputed in subbands, but collectively transformed into an equivalentset of wideband filter coefficients [5]. Park et al. improved this delay-less subband adaptive filter architecture [8], where the computationalcomplexity is further reduced by decomposing the secondary pathtransfer function into a set of subband functions. In the delaylesssubband adaptive filter architecture, before the subband/widebandtransformation, the time-domain adjusted subband adaptive weightsof each subband should be transformed into the frequency-domain.Thus, the computational complexity could be further reduced if thefrequency-domain adaptive weights are obtained directly by meansof a frequency-domain adaptive algorithm. In this correspondence, an

Manuscript received August 24, 2007; revised August 04, 2008. Current ver-sion published October 15, 2008. This work was supported by the NSFC andNCET under Project 10674068. The associate editor coordinating the review ofthis manuscript and approving it for publication was Prof. Sen M. Kuo.

M. Wu and X. Qiu are with Key Laboratory of Modern Acoustics andInstitute of Acoustics, Nanjing University, Nanjing 210093, China (e-mail:[email protected]; [email protected]).

Guoyue Chen is with Department of Electronics and Information Systems,Akita Prefectural University Faculty of Systems Science and Technology, Akita015-0055 Japan (e-mail: [email protected]).

Digital Object Identifier 10.1109/TASL.2008.2005030

Fig. 1. Delayless subband ANC system proposed by Park et al.

overlap-save frequency-domain implementation of the Park algorithmis proposed. Simulations show that the proposed implementation hasbetter performance than the original.

II. DELAYLESS SUBBAND ANC SYSTEM PROPOSED BY PARK et al.

Fig. 1 shows a block diagram of the delayless subband ANC systemproposed by Park et al. [8], which can be subdivided into five parts.

1) Subband Filtering: The input signal �� and error signal ��are fed to� contiguous single-sideband bandpass filters and thendownsampled by a factor � �� to produce � complex sub-band input signals �� and � complex subband error signals��, where the � contiguous single-sideband bandpass filtersare usually realized by means of the polyphase FFT technique[11].

2) Filtered Reference Signal Generation: In each subband, the inputsignal is filtered to obtain the filtered reference signal

��

�� (1)

where �� is the subband-decomposed secondary path responsevector, and�� (an integer multiple of�) is the length of the wideband secondarypath.

3) Adaptive Weight Update: In the th subband, the adaptiveweights are updated according to

�� (2)

where �� is the th subband weight, � is the step size,�� is the fil-tered reference input vector, � (an integer multiple of �) is thelength of the wideband controller, and superscript * denotes com-plex conjugate.

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IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 16, NO. 8, NOVEMBER 2008 1707

Fig. 2. Frequency-domain implementation of delayless subband ANC systemproposed by Park et al.

4) Subband/Wideband Weight Transformation: The adjusted sub-band adaptive weights �� are then transformed by an��-point (complex) fast Fourier transform (FFT) to obtain thesubband frequency-domain coefficients

�� (3)

where � denotes the Fourier matrix of the proper size. By meansof the frequency stacking technique [5], the subband frequency-domain coefficients are transformed to the wideband frequency-domain coefficients ��. Finally, the inverse FFT is used toobtain the time-domain coefficients of the wideband control filter

�� (4)

5) Signal Path Convolution with Wideband Filter: The control signalfor the secondary source is the output of the wideband control filter

��

��

��

�� (5)

where �� is the th element of vector ��.The computational load of parts (2)–(4) in Fig. 1 might be reduced

if the adaptive weights are updated in the frequency domain. Park et al.[8] also proposed a frequency-domain implementation of the delaylesssubband ANC system as shown in Fig. 2. (Note that in this implemen-tation the FFTs in Fig. 1 are unnecessary because we are already inthe frequency domain.) Multiplying both sides of (2) by the Fouriertransform matrix, the time-domain algorithm for the subband adaptiveweights update is changed into a frequency-domain algorithm

��

��

�

�� (6)

where

�� (7)

As given in (1), the filtered reference input �� is the linear con-volution of �� with �. Therefore, �� can be calculated ap-proximately by the FFT technique,

��

�� (8)

where �� is a diagonal matrix whose elements are given by the trans-formed vector

�� (9)

and �� represents the transformed vector of the�th subband inputsignal defined as

�� (10)

To further reduce the computational load, Park et al. suggest that thesubband weights should be adjusted every �� subband samples bymeans of a block update scheme [8]

��

��

�

�� (11)

However, it can be seen from (11) that only decimated samples of thesubband error signal are used for � � �� to update the subbandweights, which will result in large variance at steady-state or reduce theconvergence speed.

III. PROPOSED FREQUENCY-DOMAIN IMPLEMENTATION

In this section, a new frequency-domain implementation of the de-layless subband ANC system is proposed, as shown in Fig. 3. It canbe seen that three parts of the proposed method are different from thatproposed by Park et al. [8], which are discussed as follows.

1) Adaptive Weight Update: By means of the block update tech-nique, (2) can be rewritten as

��

�� (12)

where

��

(13)

and

��

(14)

It is well known that a Toeplitz matrix�� can be transformed to acirculant matrix by doubling its size

��

�

��

��

��(15)

where��

�� is also a Toeplitz Matrix. By means of (15), (12) can beexpanded as

��

��

��

��

��

��

��

�

(16)

where��

�� are adaptive weights that are ultimately discarded. Mul-tiplying both sides of (16) by the Fourier transform matrix�, the adap-tive weight update of the proposed method is obtained

��

��

�� (17)


Fig. 3. Proposed frequency-domain implementation.

where

��

��

��

��(18)

��

�

��

�

(19)

�� (20)

For the circulant matrix�� is a diagonal matrix. The ele-

ments of��

�

�� are the discrete Fourier transform of the first columnof ��, defined as

��

(21)

2) Filtered Reference Signal Generation: In a similar way to thatof Park et al. [8], �� can also be calculated approximately by theFFT technique

�� (22)

where �� is a diagonal matrix whose elements are given by the ele-ments of the transformed vector

�� (23)

and

��

(24)

3) Subband/Wideband Weight Transformation: In this part, thewideband weight �� is recovered from the frequency-domain

subband adaptive weights ��. After frequency stacking, ��

��is converted to a �� -length wideband weight vector ��. Since

��

��

��

��

� ��

��

��

��

�(25)

it is obvious that the first� components of �� correspond to��after the subband/wideband weight transformation, and the last� com-ponents of �� correspond to the ��

�� after subband/widebandweight transformation. So the � -length adaptive weight �� caneasily be obtained by discarding the last � components of ��.

IV. COMPUTATIONAL COMPLEXITY

In this section, the computational complexity of the proposed imple-mentation is measured by the number of real multiplications per inputsample. The computational complexity of the subband filters and wide-band signal path filter are the same as that proposed by Park et al. [8],requiring, respectively,

� ��

��

�(26)

and

� � � (27)

real multiplications per input sample, where is the order of thepolyphase prototype filter. Equations (22) and (24) are used forgenerating the reference signals in the frequency domain. For (22),� � �� multiplications are required for each subband. For (24),

IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 16, NO. 8, NOVEMBER 2008 1709

Fig. 4. Normalized computational complexity based on the number of real mul-tiplications. � � � � �� .

one ��-point FFT is required for each subband. Thus, generatingthe reference signal requires a total of

��

��

�(28)

real multiplications per sample.Equations (17) and (19) are used for the adaptive weight update. For

(17), � �� multiplications are required for each subband. For(19), one ��-point FFT is needed for each subband. The adaptiveweight update then requires a total of

��

��

�(29)

real multiplications per input sample.For the subband/wideband weight transformation, since the adaptive

weight is already in the frequency domain, only a �� -order IFFT isneeded, requiring

��

��

��

�� (30)

real multiplications per input sample.A plot of normalized computational complexity based on the number

of real multiplications is presented in Fig. 4. The normalized compu-tational complexity is defined as the ratio of the total number of realmultiplications of each system to that of the fullband structured ANCalgorithm. It can be seen from Fig. 4 that the computational benefit ofthe proposed system is significant in comparison to the implementa-tions proposed in Fig. 1 and Fig. 2 with weight update (6) when thenumber of subbands is small. The computational load of the proposedimplementation is a little larger than that of the implementation in Fig. 2with weight update (11).

V. SIMULATIONS

Computer simulations are carried out to validate the proposed imple-mentation. The primary path and the secondary path used for the sim-ulation were measured in the listening room of Nanjing University andwere both modeled by 512-tap FIR filters, as shown in Fig. 5. A 256-tapFIR low-pass filter was designed for the polyphase prototype using the

Fig. 5. Measured impulse response used in the simulations. (a) Primary path.(b) Secondary path.

Fig. 6. White noise input learning curves of the implementations in (a) Fig. 1,(b) Fig. 2 with weight update (6), (c) Fig. 2 with weight update (11), and (d)Fig. 3.

method proposed by M. Harteneck [12]. All signals were sampled at4000 Hz. The subband-decomposed secondary path response vector�� was obtained by the offline modeling technique [4] in the cases areconsidered next.

1) White Noise Input Signal: In this simulation, the step size for the�th subband is normalized as

��

��(31)

where � � �� is adopted and � � � indicates the Euclidean normoperator of the vector. Fig. 6 shows the learning curves of the sub-band ANC implementations in Figs. 1–3 when the number of subbands� � ��. It can be seen that the proposed implementation has betterconvergence characteristics than the implementations proposed by Parket al. [8]. The computational load of the ANC implementation in Fig. 2with (11) for weight update is less than that of the proposed imple-mentation; however, it converges much slower than the others. This isbecause only decimated samples of the subband error signal for ��are used to update the subband weights.

The filtered reference signal �� is the linear convolution of�� with �, and the frequency-domain filtered reference signal�� should be calculated by (1) and (7). However, to reduce the

computational load, the filtered reference signal in Figs. 2 and 3, istaken as the circular convolution of �� with � and calculatedusing the FFT technique. Fig. 7(a) shows the learning curve of the im-plementation in Fig. 2 when �

�� is calculated by (8)–(10). Fig. 7(b)

shows the learning curve of the implementation in Fig. 2 when ��

is calculated by (1) and (7). It can be seen that the performance ofthe implementation in Fig. 2 degrades when the frequency-domainfiltered reference signal is calculated by the FFT technique. Fig. 7(c)


Fig. 7. White noise input learning curves of the implementations in (a) Fig. 2with reference signal calculated by (8)–(10), (b) Fig. 2 with reference signalcalculated by (1) and (7), (c) Fig. 3 with reference signal calculated by (22)–(24),and (d) Fig. 3 with reference signal calculated by (1) and (7).

shows the learning curve of the implementation in Fig. 3 when thefrequency-domain filtered reference signal is calculated by (22)–(24).Fig. 7(d) shows the learning curve of the implementation in Fig. 3when the frequency-domain filtered reference signal is calculatedby (1) and (7). It can be seen that for the proposed implementationin Fig. 3 in which the frequency-domain filtered reference signal iscalculated by the FFT technique, the performance is almost unaffected.This is because �� has been padded by �� zeros before the FFTis applied.

2) Band-Limited Random Noise Input Signal: Here, we use band-limited random noise from 400 to 1400 Hz with three harmonics (480,960, and 1440 Hz) added. For the implementations shown in Figs. 2and 3, the convergence rate can be improved if an individual step sizeis adopted for each frequency bin. In this simulation, the step size forthe �th frequency bin is given by [13]

��

��(32)

where

��

(33)

� is smoothing factor, and �� is the �th element of vector ��.In the simulation of the implementation shown in Fig. 1, the normal-

ized step size for each subband is calculated by (31). For the imple-mentations shown in Fig. 2 and Fig. 3, the individual step size for eachfrequency bin is calculated by (32), (33), and the parameter � � �.The step size � for each implementation is set to the largest value pos-sible while still assuring that the adaptive filters converge.

For the implementation shown in Fig. 1, it can be seen from Fig. 8(a)and (b) that the convergence rate is improved as the number of subbandsincreases. This is due to the decrease of the dynamic range of inputsignal in each subband. However, a too large number of subbands hasno benefit for improving the convergence rate, as shown in Fig. 8(b) and(c). This is because as the number of subbands increases, the aliasingbetween subbands increases. As shown in Fig. 8(e), the proposed im-plementation has better performance than the others. This is becausean individual step-size is adopted for a different frequency bin.

VI. CONCLUSION

For an input signal with large dynamic range, one way to improve al-gorithm performance is to increase the number of subbands. However,

Fig. 8. Band-limited random noise input learning curves of the implementa-tions in (a) Fig. 1 with � � ��, (b) Fig. 1 with � � ��, (c) Fig. 1 with� � ��, (d) Fig. 2 with� � ��, and (e) Fig. 3 with� � ��.

when the number of subbands increases, the aliasing between subbandsalso increases, which degrades performance. In this correspondence, anovel frequency-domain implementation of the delayless active noisecontrol system is proposed using a frequency-domain algorithm for thesubband weight update, whereby the performance can be improved ifan individual step size is adopted for each frequency bin. Simulationsshow that the proposed implementation has better performance thanprevious implementations proposed by Park et al.

ACKNOWLEDGMENT

M. Wu would like to thank W. W. Gao for encouragement and sup-port.

REFERENCES

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[3] S. J. Elliott, Signal Processing for Active Control. London, U.K.:Academic, 2001.

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[5] D. R. Morgan and J. C. Thi, “A delayless subband adaptive filter archi-tecture,” IEEE Trans. Signal Process., vol. 43, no. 8, pp. 1819–1830,Aug. 1995.

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[9] L. Larson, J. M. de Haan, and I. Claesson, “A new subband weighttransform for delayless subband adaptive filtering structures,” in Proc.IEEE Digital Signal Process. Workshop, 2002, pp. 201–206.

[10] X. Qiu, N. Li, G. Chen, and C. H. Hansen, “The implementation of de-layless subband active noise control algorithms,” in Proc. ACTIVE’06,Adelaide, Australia, Sep. 18–20, 2006, pp. 17–26.

[11] P. P. Vaidyanathan, “Multirate digital filters, filter banks, polyphasenetwork, and applicaton: A tutorial,” Proc. IEEE, vol. 78, no. 1, pp.56–93, Jan. 1990.

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An Overlap-Save Frequency-Domain Implementation of the Delayless Subband ANC Algorithm

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