Sparsity-Aware Pseudo Affine Projection Algorithm for Active Noise Control

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1 Sparsity-Aware Pseudo Aine Projection Algorithm or Active Noise Control Felix Albu *, Amelia Gully and Rodrigo de Lamare * Department o Electronics, Valahia University o argoviste, argoviste, Romania elix.albu@valahia.ro; el: Department o Electronics, University o York, UK ajg540@york.ac.uk, rcdl500@york.ac.uk Abstract his paper describes new algorithms promoting sparsity in modiied iltered-x algorithms or active noise control. he proposed algorithms are based on recent techniques incorporating approximations to the l 0 -norm in the cost unction used. An approximation to the aine projection algorithm leads to new zero-attracting and reweighted zero-attracting modiied iltered-x pseudo aine projection algorithms. he results o simulations indicate that the proposed techniques demonstrate good perormance and reduce numerical complexity compared to the original aine projection based algorithms. I. INRODUCION Active noise control (ANC) is a technique or removing noise rom a system by subtracting the eect o a noise-generating plant rom a signal [1]. ANC algorithms are based on classical adaptive algorithms, and take into account the additional electro-acoustic path between the ilter output and measured error signal, known as the secondary path []. A common technique to account or this path is known as the iltered-x (FX) scheme, originally developed or least-mean square (LMS) algorithms [3] and since incorporated into other adaptive algorithms, including the aine projection (AP) algorithm [4]-[6]. he FX scheme eliminates potential algorithm instability caused by the additional delay in the secondary path [7], but it generally exhibits slow convergence speed [8]. he modiied iltered-x (MFX) scheme was introduced in [9], and improves upon the convergence o FX algorithms by introducing additional iltering steps to approximate the instantaneous error signal. Recent developments in the ield o compressive sensing have led to the introduction o sparsity-promoting penalties in adaptive iltering algorithms [10]-[11], producing zeroattracting (ZA) and reweighted zero-attracting (RZA) algorithms. hese have been shown to provide aster convergence when the system in question has a degree o sparsity, which is oten the case in the systems encountered in ANC. In [1], these sparsity constraints were incorporated into MFX algorithms and shown to improve perormance or systems with a degree o sparsity. However, these algorithms suer rom high computational complexity. In this paper, two approximations are proposed that lead to signiicant complexity reduction in the zero-attracting and reweighted zero-attracting MFxAP algorithms o [1]. Experimental results demonstrate that the proposed algorithms do not exhibit signiicant perormance loss or systems with a moderate degree o sparsity. In particular, there is almost no perormance loss when the step size is close to one, or in case o non-sparse echo paths. he rest o this paper is structured as ollows. In Section II the sparsity-inducing modiied iltered-x algorithms rom [1] are presented. In Section III the proposed algorithms are developed with reerence to the ZA-MFxAP and RZA- MFxAP algorithms rom which they derive. In Section IV, the results o simulation trials perormed with these algorithms are presented and urther directions o research are described. Section V presents the conclusions. Notation: hroughout this paper, uppercase boldace letters will be used to denote matrices, and lowercase boldace letters to denote vectors. Sgn{ i is the signum unction, I is an identity matrix o appropriate dimensions, and i denotes p the pth norm o a vector. All vectors are column vectors. Following convention, the term l 0 -norm and i notation 0 denotes the number o non-zero elements in a vector. II. SPARSIY-INDUCING FILERED-X ALGORIHMS In broadband eedorward ANC, an error signal is used to update an adaptive ilter such that its output can be subtracted rom the acoustic signal at the listener through intererence cancellation, reducing disturbing noise. However, the nature o an ANC system introduces a secondary path, comprising the transer unctions o each electro-acoustic component. It was shown in [7] that this path can be split into two parts, one estimated as part o the plant, and one occurring ater the acoustic summing junction, denoted here s n. In FX algorithms, an estimate o the secondary path, by s ˆ, is used to ilter the input signal algorithm input is the iltered signal x x n, so that the n. his accounts or the secondary path delay, but introduces strict limits on step size that negatively impact upon convergence speed [8]. he MFX algorithms improve the convergence speed by estimating the instantaneous error signal ê( n ) [9] APSIPA APSIPA 014

2 II.1. Zero-attracting MFxAP (ZA-MFxAP) algorithm he ZA-MFxAP incorporates a sparsity-inducing penalty to attract coeicients towards zero. Zero-attracting algorithms use an l 1 -norm penalty as an approximation and the weight update recursion is [1]: ( 1) = μu eˆ w n w n n n { w ρu U sgn w -α sgn (3) where ρ = μα is known as the zero-attraction strength. II.. Reweighted zero-attracting MFxAP (RZA-MFxAP) Algorithm Fig. 1. Modiied iltered-x AP structure [1] he MFxAP structure is illustrated in Fig. 1. It is well suited to ANC as it does not require that d = U w( n 1) be available, where U = [ χ χ( n 1 )... χ( n L) ] is the regressor matrix, χ = [ x, x( n 1 ),... x( n K 1 )], K is the projection order and L is the ilter length. Instead, the ollowing condition is set: = ( n ) d ˆ U w 1 (1) We deine the iltered regressor matrix as U n = [ χ n χ n 1... χ n L ], where = x x ( n ) x ( n K ) χ [, 1,... 1 ]. hereore, condition (1) states that the estimated output or the next iteration should equal the estimated desired signal or the current iteration. We have yˆ = U w, ˆ = H e n U n U n λ, and 1 λ = U H U e ˆ ( n ) [1]. hereore the MFxAP recursion is given by: μ ˆ 1 wn = wn U ne n () U 1 n = U n U n U n δ I and μ is where H H the step size. Next, the sparsity-inducing iltered-x aine projection algorithms proposed in [1] are presented. hese are obtained by incorporating the zero-attracting and reweighted zeroattracting strategies considered in [10] into the modiied iltered-x recursion (). he RZA-MFxAP algorithm uses a log-sum penalty in place o the l 1 -norm, as this provides a closer approximation to the behavior o the l 0 -norm. he RZA-MFxAP recursion is [1]: where Ψ ( 1) = μu ˆ e ρ' U U Ψ γεψ wn wn n n sgn w = 1 ε sgn { w In this case the strength o the zero-attraction is controlled by ρ ' = μγε where ε is known as the shrinkage magnitude. III. HE PROPOSED ALGORIHMS We ound by simulations that or all investigated sparseness levels, the usual step size values and typical ANC situations the common term o MFxAP recursion has the greatest contribution to the perormance or both ZA-MFxAP and RZA-MFxAP algorithms. For the ZA-MFxAP, we ind: ˆ. sgn n n n n n μu e ρu U α w (5) For the RZA-MFxAP, we ind: ˆ ' (4) (6) μu e ρ U U Ψ γεψ his is illustrated by a simulation example shown in Fig. or a non-sparse plant. Also, we note that or a step size close or equal to 1 we have: e e ˆ,0...0 (7)

3 hese observations allow us to reduce both the computational complexity and the memory requirements o the ZA-MFxAP and RZA-MFxAP algorithms in a similar way to that o [13]. I we denote the irst column o U as U 1, and use (7), we obtain: ˆ = e U e U 1 (8) his step reduces the numerical complexity rom KL to L. hereore the complexity reduction is higher or high projection orders. By including (8) in (3) we obtain the weight recursion o a new algorithm called the Zero-Attracting Modiied Filtered-X Pseudo Aine Projection (ZA-MFxPAP): ( 1) = μu w n w n 1 n e n { w ρu U sgn w -α sgn Similarly, using (8) in (4) we obtain the weight recursion o the Reweighted Zero-Attracting Modiied Filtered-X Pseudo Aine Projection (RZA-MFxPAP) as ollows: ( 1) = μu wn wn nen 1 γε ρ' U U Ψ Ψ (9) (10) Additional computational savings can be obtained rom an eicient computation o R = H δ as ollows [13]: where R r R n r 0 n n = r ( 1) = ( n 1) x x( n L) ( n L) r r χ χ = r 0, r and R ( n 1) is the top let ( K 1) ( K 1) R ( n 1). hereore, the update o (11) (1) submatrix o R needs only K multiplications instead o K L multiplications. he approximation in (7) leads to other computational and memory savings due to use o scalars instead o K length vectors. he complexity o each algorithm in terms o multiplications is given in able 1. In these complexity calculations we ignore the cost o inversing a K K matrix, 3 O( K ) [14]. hese results show that the complexity reduction rom ZA-MFxAP to ZA-MFxPAP is the same as that between RZA-MFxAP and RZA-MFxPAP, and is given by: C = KL 6L K 6 K (13) REDUCION It can be seen that there is a small complexity dierence between the original algorithms and the proposed ones. Algorithm ZA-MFxAP ZA-MFxPAP RZA-MFxAP RZA-MFxPAP Number o multiplications per iteration 3K L L (K1) 7KL 3L K L L (K1) K(L) 9L 3K L L (K1) 7KL 4L K L L (K1) K(L) 10L able 1. Algorithm complexity in terms o multiplications per iteration IV. SIMULAION RESULS his section compares the results o simulations o the proposed algorithms outlined in (9) and (10) with those o the conventional ZA-MFxAP and RZA-MFxAP algorithms o (3) and (4). he results are averaged over 10 simulation trials. hree types o plant were used to perorm simulations: a nonsparse path (density 785/800), a partially-sparse path (density 73/800) and a sparse path in which the ourth coeicient is set to one and all remaining coeicients to zero (density 1/800), all o which are identical to those used and described in [1]. hree types o secondary path, also identical to those in [1], were also generated. In Fig. the non-sparse plant was used and the ZA-MFxAP and RZA-MFxAP algorithms were run or 1,000 iterations with the secondary path set as sparse at the start o the experiment, changed to partially-sparse at iteration 4,000 and to non-sparse at iteration 8,000. It can be seen that the norms o the same terms o weight update or MFxAP are much greater than those o the speciic terms o the zero-attracting versions rom [1]. Figs. 3-5 show mean-square deviation (MSD) convergence curves or each o these experiments. For all the algorithms, the parameters were tuned to the same values as in [1], with the exception o step size, μ, or which a value o 1 was used throughout. For each primary path ormulation, the algorithms were run or 0,000 iterations with the secondary path set as sparse at the start o the experiment, changed to partiallysparse at iteration 10,000 and to non-sparse at iteration 70,000, giving suicient time or algorithm convergence in each case. he irst 10,000 iterations in each igure illustrate algorithm perormance when the secondary path is sparse. he advantages and disadvantages o ZA-MFxAP and RZA- MFxAP algorithms over FxAP and MFxAP were shown in [1] and as such are not considered here. It is seen in Fig. 3a that as the secondary path density increases, dierences between the convergence curves ZA- MFxAP and ZA-MFxPAP algorithms start to appear. he proposed algorithms exhibit aster convergence, but also a

4 higher steady state MSD. he same conclusions can be obtained rom Fig. 3b regarding the RZA-MFxAP and RZA- MFxPAP when the projection order is 4. However, when the projection order is increased to 16, the original algorithms perorm better than the proposed algorithms. he results o running the algorithms or a non-sparse plant can be seen in Fig. 5. In this case, the perormance o the proposed algorithms is almost identical to that o the original algorithms. his recommends these numerically less complex algorithms as an alternative to FxAP, MFxAP, ZA-MFxAP and RZA-MFxAP algorithms, particularly or high projection orders, when the plant has a low degree o sparsity. Also, it can be noted that when the plant is non-sparse, the approximations used to derive ZA-MFxPAP and RZA- MFxPAP are valid. For all simulations considered here, when the secondary path is sparse, the proposed algorithms have almost identical perormance to the original algorithms. Fig.. a) he norm o the let side o (5) or ZA-MFxAP algorithm; b) he norm o the let size o (6) or RZA-MFxAP algorithm; c) he norm o the right side o (5) or ZA-MFxAP algorithm; d) he norm o the right side o (6) or RZA-MFxAP algorithm. It can be deduced that or a sparse plant, the simpliying approximations used in the derivation o the proposed algorithms are not eective. Fig. 4. MSD results or partially-sparse plant. Fig. 3. MSD results or sparse plant. he perormance o the algorithms when the plant is partially sparse can be seen in Fig. 4. When the secondary path is semisparse, the ZA-MFxPAP algorithm exhibits similar behavior in terms o both MSD convergence speed and steady-state perormance to the ZA-MFxAP algorithm (Fig. 4a). In the case o the RZA-MFxPAP algorithm, as the secondary path density increases in Fig. 4b and 4c, steady-state dierences are clearly visible between RZA-MFxAP RZA- MFxPAP. Fig. 5. MSD results or non-sparse plant. Fig. 6 shows the multiplication saving o the proposed algorithms over the original ones. In Fig. 6a the path length was varied and the projection order was 16. In Fig. 6b the echo path length is 800 and the projection order is varied rom 1 to 0. As expected rom (13), the complexity saving is seen to have a linear relationship with respect to L, and a quadratic relationship with respect to K. It can be seen that the computational complexity saving increases with the increase o the path lengths or projection order. Since there is only a dierence o L multiplications

5 between the zero-attracting versions and re-weighted zeroattracting versions the multiplication saving o the RZA- MFxPAP over the RZA-MFxAP is the same as those o ZA- MFxPAP over the ZA-MFxAP. However, the reweighed algorithms require L divisions per iteration, making them more complex to implement than the zero-attracting algorithms. Fig. 6. a) he multiplication savings o ZA-MFxPAP over ZA-MFxPAP or a variable L; b) he multiplication savings o ZA-MFxPAP over ZA-MFxPAP or a variable K Further work might incorporate recent improvements to sparse techniques [15]. It would thereore be worthwhile to consider algorithms incorporating a variable shrinkage magnitude parameter as an improvement to the algorithms proposed here, allowing steady-state perormance to be improved without negatively aecting convergence rate. In addition, variable step-size versions or variable projection versions can be designed as in [5], and [16]. Also, a practical implementation is envisaged in order to veriy the eectiveness o the proposed method. V. CONCLUSIONS his paper has proposed new adaptive algorithms that exploit sparsity and approximations to the error and weight update o the aine projection algorithm, based on modiied iltered-x algorithms or active noise control. he simulation results demonstrate that with any degree o sparsity in the primary or secondary path, a good compromise between convergence speed, steady-state error and numerical complexity is obtained by the proposed ZA-MFxPAP and RZA-MFxPAP algorithms, particularly at a relatively high projection orders, such as 16, and or non-sparse echo paths. [] S. M. Kuo and D. R. Morgan, Active noise control: a tutorial review, Proc. IEEE, vol. 87, no. 6, pp , Jun [3] B. Widrow, D. Shur, and S. Shaer, On adaptive inverse control, in Proc. 15th Asilomar Con. Circuits Systems Computers, Paciic Grove, CA, 1981, pp [4] S. M. Kuo and D. R. Morgan, Active Noise Control Systems: Algorithms and DSP Implementations, John Wiley and Sons Inc., New York, NY, [5] A. Gonzalez, F. Albu, M. Ferrer, and M. de Diego, Evolutionary and variable step size strategies or multichannel iltered-x aine projection algorithms, IE Signal Process., vol. 7, no. 6, pp , Aug [6] M. Bouchard and F. Albu, he Gauss-Seidel ast aine projection algorithm or multichannel active noise control and sound reproduction systems, Special Issue on Adaptive Control o Sound and Vibration, International Journal o Adaptive Control and Signal Processing, vol. 19, nr. -3, pp , March-April 005. [7] E. Bjarnason, Analysis o the iltered-x LMS algorithm, IEEE rans. Speech Audio Process., vol. 3, no. 6, pp , Nov [8] M. Rupp and A. H. Sayed, Modiied FxLMS algorithms with improved convergence perormance, in Proc. 9th Asilomar Con. Signals Systems Computers, Paciic Grove, CA, 1995, pp [9] E. Bjarnason, Active noise cancellation using a modiied orm o the iltered-x LMS algorithm, in Proc. 6 th Eur. Signal Process. Con., Brussels, Belgium, 199, pp [10] Y. Chen, Y. Gu, and A. O. Hero, Sparse LMS or system identiication, in Proc. IEEE Int. Con. Acoust. Speech Signal Process., aipei, aiwan, 009, pp [11] R. Meng, R. C. de Lamare, and Nascimento V. H., Sparsityaware aine projection adaptive algorithms or system identiication, in Proc. Sensor Signal Process. Deence, London, UK, 011, MOD, pp [1] A. Gully, R. C. de Lamare, Sparsity aware iltered-x aine projection algorithms or active noise control, in Proc. IEEE Int. Con. Acoust. Speech Signal Process., Florence, Italy, 014, pp [13] F. Albu, M. Bouchard, Y. Zakharov, Pseudo Aine Projection Algorithms or Multichannel Active Noise Control, IEEE rans. Audio, Speech Language Process., Vol. 15, Issue 3, March 007, pp [14] G. H. Golub and C. F. Van Loan, Matrix computation, 3rd edition. Baltimore, MD: he John Hopkins Univ. Press, [15] R. C. de Lamare and R. Sampaio-Neto, Sparsity-aware adaptive algorithms based on alternating optimization and shrinkage, IEEE Signal Process. Lett., vol. 1, no., pp. 5 9, Feb [16] F. Albu, C. Paleologu and J. Benesty, A Variable Step Size Evolutionary Aine Projection Algorithm, in Proc. o ICASSP 011, Prague, Czech Republic, May 011, pp REFERENCES [1] B. Widrow, J. R. Glover, J. M. McCool, J. Kaunitz, C. S. Williams, R. H. Hearn, J. R. Zeidler, E. Dong, and R. C. Goodlin, Active noice cancelling: principles and applications, Proc. IEEE, vol. 63, no. 1, pp , Dec