Title Sinusoidal Signals. Author(s) Sakai, Hideaki; Fukuzono, Hayato. Conference: Issue Date DOI

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1 Tite Anaysis of Adaptive Fiters in Fee Sinusoida Signas Authors) Sakai, Hideaki; Fukuzono, Hayato Proceedings : APSIPA ASC 2009 : Asi Citation Information Processing Association, Conference: Issue Date DOI Doc URLhttp://hd.hande.net/2115/39734 Right Type proceedings Additiona Information Fie Information TA-P1-7.pdf Instructions for use Hokkaido University Coection of Schoary and

2 Anaysis of Adaptive Fiters in Feedback Canceation for Sinusoida Signas Hideaki Sakai, Hayato Fukuzono Department of Systems Science, Kyoto , Japan E-mai: Abstract Anaysis of couping effects in adaptive fiters is presented for feedback canceation systems in hearing aids. As the incoming signa, a sum of sinusoids in white noise is assumed in this paper. In the proposed scheme two LMS-type adaptive fiters are used. The one is set for identifying the impuse response of the feedback path. The other is associated with frequency estimation. We propose a scheme using the constrained LMS agorithm for the adaptive fiter for frequency estimation. The stationary point of the proposed agorithm is shown to be the desired ones. We aso show that the simutaneous stabiity near the stationary point is guaranteed. Simuation resuts show the vaidity of the theoretica findings. Fig. 1. Bock diagram of the proposed agorithm for feedback canceation I. INTRODUCTION In hearing aids an adaptive fiter is used to mode the feedback path from the receiver speaker) to the microphone and cances its effect. The conventiona LMS agorithm is biased, since the input signa and the incoming signa which acts as observation noise in the adaptive fiter are correated due to the feedback path. In 1, the PEM-AFC Prediction Error Method-based Adaptive Feedback Canceer) agorithm has been proposed for continuous unbiased canceation for the case where the incoming signa is an AR autoregressive) process. But its convergence mechanism to the desired point was not discussed. In 2, under an assumption that the incoming signa is a sum of sinusoids and white noise, a simiar agorithm with two LMS-type adaptive fiters was proposed where the one is to identify the feedback path and the other is associated with estimation of frequencies. For the atter the LMS agorithm for inear prediction is used. The stabiity anaysis of the former was conducted under the assumption that the atter converges to have the desired notch characteristic. In 3, the stabiity anaysis for simutaneous adaptation of both adaptive fiters was taken into account where the incoming signa is an AR process. In this paper, as another method of frequency estimation we use the constrained LMS agorithm in 4 where the order of the fiter is equa to the number of sinusoids. Aso, anaysis of the couping effects of the two adaptive fiters is conducted for the proposed agorithm. It is shown that the couping effect disappears near the stationary point. Finay, simuation resuts show the vaidity of the theoretica findings. II. THE STATIONARY POINT OF THE PROPOSED SCHEME Fig.1 shows a bock diagram of the proposed scheme. An input signa to the hearing aids is denoted by dn) and an ampified output signa from the receiver is denoted by xn). The transfer functions of the forward path ampifying circuit) and the feedback path are Gz) and Hz), respectivey. The forward path transfer function Gz) is the desired characteristic of the hearing aids and is fixed and known, but Hz), the transfer function of feedback path, is unknown and may be sowy time-varying. So, the adaptive fiter which is denoted by W z) is used to identify Hz). The signas xn) and en) in Fig.1 in the steady state are written by xn) = Gz)en) 1) en) = yn) W z)xn) = dn) + Gz)Hz) W z))en), 2) and W z) and Hz) are given by W z) = w 0 + w 1 z w L1 z L+1 3) Hz) = h 0 + h 1 z h L1 z L+1 4) where z 1 denotes the unit deay operator and for exampe, Gz)en) means the fitering operation to the signa en) by the transfer function Gz). We aso see that 1 en) = Qz)dn), Qz) = 5) 1 + W z) Hz))Gz) where we assume that Qz) is stabe. As in 2, we use the foowing modified LMS agorithm for W z). The tap weight vector wn) is updated by wn + 1) = wn) + µx n)e n) 6)

3 where µ is the positive step size, wn), x n) are L dimensiona vectors defined by wn) = w 0 n), w 1 n),, w L1 n) T 7) x n) = x n), x n 1),, x n L + 1) T 8) and x n), e n) are defined by x n) = ˆP z)xn), e n) = ˆP z)en) 9) where these signas are the fitered version of xn), en) by ˆP z), respectivey. The operation ) T denotes transpose of matrices or vectors. This ˆP z) needs to be appropriatey seected. The stationary point of the adaptive fiter was derived by the standard averaging method. The averaged system corresponding to 6) is given by wn + 1) = wn) + µex n)e n). 10) The stationary point of wn) is obtained by soving Ex n)e n) = 0 11) with respect to wn) = w. Denoting the PSD Power Spectra Density) of the incoming signa dn) as Se jω ), from 1), 5) and 9) the -th eement of Ex n)e n) is written as Ex n)e n)) = e jω Ge jω ) Qe jω ) ˆP e jω ) 2 Se jω )dω. 12) We assume that the incoming signa dn) is a sum of K sinusoids and white noise and express it as dn) = A p e jωpn + νn) 13) where νn) is white noise with mean 0 and variance σ 2. From 13), we have Se jω ) = A p 2 δω ω p ) + σ 2. 14) It has been shown in 2 that for sufficienty arge L the ony soution of 11) is W opt z) = Hz), if ˆP z) has the notch characteristic ˆP e jω 1 ) = ˆP e jω 2 ) = = ˆP e jω K ) = 0 15) and the order q 1 of ˆP z) satisfies q 1 K 16) where the forward path transfer function Gz) is set to contain q time unit deays such that Gz) = z q G c z) 17) where G c z) is taken to be of minimum phase. We introduce an adaptive fiter W p z) for frequency estimation into the system, and update ˆP z). In 2, the LMS agorithm for inear prediction for en) is used since at the desired stationary point en) = dn). It has been stated in 2 that if the order of W p z) is sufficienty arge compared to the number of sinusoids, then desirabe unbiased feedback canceation can be done. Let us consider what agorithm shoud be used for W p z) in order that at the stationary point it has the notch characteristic with the order q 1 = K. Expressing the autocorreation matrix of en) by R e where en) is defined by en) = en), en 1),, en q + 1) T. 18) When en) = dn), we find that the fiter whose coefficients are eements of the eigenvector corresponding to the minimum eigenvaue of R e has the notch characteristic to the sinusoids. For detais see 4.5 of 5.) If w p is the constrained unit-norm weight vector, then minimization of the quadratic form w T p R e w p eads to the eigenvector corresponding to the minimum eigenvaue of R e by the method of Lagrange mutipiers. The adaptive agorithm is obtained by cacuating the gradient of the squared output with respect to the unnormaized weight vector w r, where w p = w r w r, w p = 1. 19) The resuting constrained LMS agorithm was derived in 4 as en)ˆγn) w r n + 1) = w r n) µ p w r n) w rn)ˆγn) 2 w r n) 2 w p n) = w rn) w r n) 20) where µ p is the positive step size. The tap vectors w p n), w r n) are the estimates of w p, w r with w p n) = w p0 n), w p1 n),, w pq1) n) T 21) w r n) = w r0 n), w r1 n),, w rq1) n) T, 22) and the output error ˆγn) is ˆγn) = w p0 n)en) + + w pq1) n)en q). 23) We set as W r z) = w r0 + w r1 z w rq1) z q+1, 24) and W p z) can be expressed as W p z) = W r z)/ w r. Since at the stationary point, the coefficients of W r z) correspond to the eignvector associated with the minimum eigenvaue of R e, we update ˆP z) as ˆP z) = W r z). 25) Thus, by using the constrained LMS agorithm for the adaptive fiter for frequency estimation, we can make ˆP z) to have the notch characteristic with the order q 1 K at the stationary state.

4 III. STABILITY OF THE FEEDBACK CANCELLATION ALGORITHM Here stabiity anaysis near the desired stationary point is presented when both W z) and W p z) are operating. For 6) and 20) the averaged system is given by wn + 1) w rn + 1) = wn) w rn) µe x n)e n) en)ˆγn) w wrn)ˆγn)2 r n) w r n) 2 26) where for simpicity we set µ = µ p. The above discrete-time system can be approximated by the continuous-time inear system near the stationary point as ) d wt) wt) wopt = Φ 27) dt w r t) w r t) w ropt where Φ is w Ex n)e n) en)ˆγn) wr ˆγn)2 w E w r w r 2 def = A B C D w r Ex n)e n) w r E en)ˆγn) wr ˆγn)2 w r w r 2. 28) From Lyapunov stabiity theory, if Φ + Φ T is positive definite then the adaptive fiters are stabe near their stationary points. The -th eement of Ex n)e n) is expressed as 12), and the -th eement of E en)ˆγn) w r w r ˆγn) 2 w r is 2 1 Qz) 2 Sz 1 )W rz) z w w ) rw r z 1 ) dω. r 2 w r 4 29) At the stationary point, W z) is W opt z) = Hz) and from 5) we have Qz) = 1, 30) and W r z), W p z) are W r,opt z) = ˆP opt z), W p,opt z) = W r,opt z)/ w ropt respectivey where W p,opt z), W r,opt z) and ˆP opt z) have the notch characteristic in 15). We first cacuate the bock A in the foowing. The, k)-th eement of A is A k = w k z Gz 1 ) Qz) ˆP z) 2 Sz 1 )dω 31) where = 0,, L 1, and k = 0,, L 1. Partia differentiating and using 14), 17) and 30), we have A k = σ2 z +k+2q G 2 cz 1 ) ˆP opt z) 2 dω z k G c z) 2 ˆP opt z) 2 dω. 32) The function z +k+2q G 2 cz 1 ) ˆP opt z) 2 can be expanded in positive powers of z. So, the first term of RHS of 32) is 0. Hence, A k is A k = σ2 z k G c z) 2 ˆP opt z) 2 dω. 33) We next cacuate the bock B in the foowing. The, k)-th eement of B is B k = z Gz 1 ) Qz) 2 Sz 1 ) W r z) 2 dω w rk 34) where 0 L1 and 0 k q 1. Partia differentiating and using 14), 17) and 30), we have B k = σ2 σ2 z +q+k G c z 1 )W r,opt z)dω z +qk G c z 1 )W r,opt z 1 )dω. 35) Noting that the order of W r,opt z) is q 1, the function z +q+k G c z 1 )W r,opt z) can be expanded in positive powers of z, and so the first term of RHS is 0. Noting that k q 1, the function z +qk G c z 1 )W r,opt z 1 ) can be expanded in positive powers of z, and so the second term of RHS is 0. Hence, B k = 0. 36) We then cacuate the bock C in the foowing. The, k)-th eement of C is C k = 1 π Qz) 2 Sz 1 )W r z) w k z w r w ) rw r z 1 ) dω 2 w r 4 37) where 0 q 1 and 0 k L1. Partia differentiating and using 14), 17) and 30), we have C k = σ2 σ2 z kq+ G c z)w r,opt z) w ropt 2 dω w r,opt G c z) W r,opt z) 2 z kq w ropt 4 dω z k+q+ G c z 1 )W r,opt z) w ropt 2 dω w r,opt G c z 1 ) W r,opt z) 2 z k+q w ropt 4 dω. 38) Noting that q 1 and the order of W r,opt z) is q 1, the functions z kq+ G c z)w r,opt z) and G c z) W r,opt z) 2 z kq can be expanded in negative powers of z, and so the first and second terms of RHS are 0. Noting that the order of W r,opt z) is q 1, the functions z k+q+ G c z 1 )W r,opt z) and G c z 1 ) W r,opt z) 2 z k+q can be expanded in positive powers of z, and so the third and fourth terms of RHS are 0. Hence, C k = 0. 39) From 36) and 39) it is seen that two adaptive fiters are decouping near the stationary point.

5 We finay cacuate the bock D in the foowing. The, k)- th eement of D is D k = 1 π Qz) 2 Sz 1 )W r z) w rk z w r 2 w rw r z 1 ) ) w r 4 dω 40) wopt,w r opt where 0 q 1 and 0 k q 1. Partia differentiating and using 14), 17) and 30), we have D k = A p 2 e jω pk) w ropt 2 ) z 2wrk,opt W p,opt z) + zk dω w ropt 3 w ropt 2 { π σ2 δ k) w ropt 4w ) r,optw rk,opt W p,optz) w ropt 4 } w r,opt Wp,optz 1 )z k + W k) p,optz)z dω. 41) w ropt 3 By cacuating the integras in 41) and using 19), we see that the second and third terms cance. Hence D k becomes D k = A p 2 ejω pk) w ropt 2. 42) Hence, from 33), 36), 39) and 42) the Hermitian form of Φ is ξ H η π ξ H Φ = σ2 L1 2 ξ η i z i G c z) 2 i=0 ˆP opt z) )dω 2 A p 2 q1 2 + w ropt 2 η i e jωpi. 43) i=0 The bock A for W z) is positive definite, so W z) is stabe as in 2. But, the bock D for W r z) is positive semidefinite. From the second term of RHS of 43) we see that a vectors η 0 such that η H Dη = 0 correspond to the coefficient vector satisfying 15). This means that the adaptive fiter W p z) has the desired notch characteristic whenever it stops. Hence, the proposed agorithm is stabe for the desirabe characteristic athough Φ is positive semidefinite. IV. SIMULATION RESULTS To see the vaidities of the theoretica findings in previous sections, some simuation resuts are presented. First, we present resuts for sinusoida signas. We use the foowing misaignment as the performance index ζ h = L1 i=0 h i w i ) 2 / L1 i=0 h i 2. The feedback and forward path transfer functions are set to Hz) = 0.2z 1 and Gz) = 2.0z q respectivey. For the adaptive fiter W z) we set the tap weight ength to L = 2 and the step size to µ = The adaptive fiter W p z) is q 1)-th order and its step size is µ p = Referring to 13), νn) is Gaussian i.i.d, with zero mean and variance σ 2 = In Fig.2 it is a sum misaignment db the LMS for inear prediction the Constrained LMS number of iteratons x 10 5 Fig. 2. Pots of misaignment ζ h in the case where K = 4 and q = 5. misaignment db the LMS for inear prediction the Constrained LMS number of iterations x 10 4 Fig. 3. Pots of misaignment ζ h in the case where the input is actua music signa and q = 100. of two sinusoids with K = 4, A 1 = A 2 = A 3 = A 4 = 1/2, ω 1, ω 2 = ±π/4, ω 3, ω 4 = ±π/7, and q is set to 5. When the LMS agorithm for inear prediction is used, the misaignment does not converge at a. As for the constrained LMS agorithm, the desired convergence is obtained. Second, we present resuts for actua music signa. Fig.3 shows the resuts where L = 5, G c z) = 3, and q = 100 with µ p = 10 2 and µ = It is seen that the agorithm with the constrained LMS performs a itte bit better than the LMS for inear prediction. V. CONCLUSION In this paper we have presented convergence anaysis of the modified LMS-type agorithm for hearing aids with sinusoida input signa. It has been shown that the adaptive fiters have the desired stationary points and the stabiities near the points. REFERENCES 1 A. Spriet, I. Prouder, M. Moonen, and J. Wouters: Adaptive Feedback Canceation in Hearing Aids with Linear Prediction of the Desired Signa. IEEE Trans. Signa Processing, vo.53, no.10, pp , October, H. Sakai: Anaysis of an Adaptive Agorithm for Feedback Canceation in Hearing Aids for Sinusoida Signas. Proc. ECCTD 2007, pp H. Sakai: Recent Topics in Adaptive Fitering. Proc. IEEE TENCON 2008, Hyderabad, India, Nov invited paper). 4 V. U. Reddy, B. Egardt, and T. Kaiath : Least Squares Type Agorithm for Adaptive Impementation of Pisarenko s Harmonic Retrieva Method. IEEE Transactions on Acoustics, Speech, and Signa Processing, vo. ASSP-30, No. 3, pp , P. Stoica and R.Moses : Introduction to Spectra Anaysis., Prentice Ha, 1997.