Sparsity Promoting LMS for Adaptive Feedback Cancellation

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1 7 5th Euroean Signal Processing Conference (EUSIPCO) Sarsity Promoting MS for Adative Feedback Cancellation Ching-Hua ee, Bhaskar D. Rao, and Harinath Garudadri Deartment of Electrical and Comuter Engineering University of California, San Diego {chl438, brao, Abstract In hearing aids (HAs), the acoustic couling between the microhone and the receiver results in the system becoming unstable under certain conditions and causes artifacts commonly referred to as whistling or howling. The least mean square (MS) class of algorithms is commonly used to mitigate this by roviding adative feedback cancellation (AFC). The seech quality after AFC and the amount of added stable gain (ASG) with AFC are used to assess these algorithms. In this aer, we introduce a variant of the MS that romotes sarsity in estimating the acoustic feedback ath. By using the l norm as a diversity measure, the aroach does not enforce, but takes advantage of sarsity when it exists. The erformance in terms of seech quality, misalignment, and ASG of the roosed algorithm is comared with other roortionate-tye MS algorithms which also leverage sarsity in the feedback ath. We demonstrate faster convergence comared with those algorithms, quality imrovement of about.5 (on a objective scale of the hearing-aid seech quality index ()), and about 5 db ASG imrovement comared with the normalized MS (). Index Terms Adative feedback cancellation, MS, roortionate adatation, sarsity, hearing aids I. INTRODUCTION In hearing aids (HAs), acoustic feedback is a well-known roblem that causes howling and whistling effects that are annoying to the users. Under certain conditions, the receiver signal feeding back to the microhone will make the system become unstable. This not only degrades the audio quality but also limits the amount of amlification that can be rovided by the HA. To overcome this many adative feedback cancellation (AFC) techniques have been roosed for modern HAs ]. In AFC, an adative filter is continuously adjusting to aroximate the imulse resonse (IR) of the acoustic feedback ath. In the adatation stage, least mean square (MS) algorithms ] are the most widely used techniques due to comutational simlicity and their effectiveness. However, the estimate given by the MS is inevitably biased due to the correlation between the incoming signal and the feedback signal 3]. Several methods have been roosed to address this bias issue such as the filtered-x MS (FXMS) 4], 5], the rediction-error-method (PEM) based AFC 6], insertion of robe noise 7], hase modulation 8], PEM with frequency shifting 9], the dual-microhone aroach ], etc. Assuming that the bias roblem can be well handled by the above techniques, a question of interest is: Can we further imrove the convergence behavior of the AFC from other asects? Observing that tyical feedback ath IRs are (quasi) sarse as shown in Fig., one might think of taking advantage of this sarseness for imrovements. This can actually be carried out by the concet of roortionate adatation that originated from the roortionate normalized MS (P) algorithm ]. The main idea behind roortionate adatation is to udate each filter coefficient indeendently of the others by assigning to the corresonding ste size a weight in roortion to the magnitude of the estimated coefficient. In other words, it redistributes the adatation gains among all coefficients and emhasizes the large ones in order to seed u their convergence. However, the original P has the roblem that it is more beneficial for systems with very sarse structures. For AFC alication where the feedback ath IRs are usually quasi-sarse, other roortionate-tye MS algorithms can be more suitable. For examle, the imroved P (IP) ] and the IP-l 3] have the flexibility for identifying systems of different levels of sarsity. Attemts have been made to incororate these roortionate algorithms into AFC 4] 6] and imrovements have been reorted. However, these roortionate algorithms were not formally derived by minimizing any underlying objective functions so that their usage can be further otimized. Moreover, the arameters within these algorithms do not have direct connections to the sarsity degree of the underlying system they aim to identify. In this aer, we roose a new AFC aroach that incororates sarsity in the feedback ath estimate based on the MS algorithm of 7], 8]. This method, called Sarsity romoting MS (SMS), takes advantage of sarseness in the feedback ath when such sarsity is resent while not enforcing it. By adding the l norm as a diversity measure to the objective function of the ordinary MS, the algorithm is derived using the affine scaling method 9] in the minimization rocedure. The benefit of using the l norm is brought by its direct connection to the system sarseness which rovides a ractical way of arameter selection. Our algorithm has the advantages of enjoying theoretical suort, simler arameter otimization, and more straightforward leverage of (quasi) sarsity in acoustic feedback aths. Simulation results using feedback aths measured with real HAs show that the SMS outerforms other roortionate-tye MS algorithms in terms of audio quality, misalignment, and added stable gain (ASG). ISBN EURASIP 7 36

2 7 5th Euroean Signal Processing Conference (EUSIPCO) II. AFC SYSTEM Fig. shows the tyical AFC framework. The AFC filter W (z) is an FIR filter laced in arallel with the HA rocessing G(z) that continuously adjusts its coefficients to emulate the IR of the feedback ath F (z). x(n) is the desired inut signal and d(n) is the actual inut to the microhone, which contains x(n) and the feedback signal y(n) generated by the HA outut s(n) assing through F (z). ŷ(n) is the estimate of y(n) given by the outut of W (z). e(n) = d(n) ŷ(n) is the feedbackcomensated signal which, ideally, should be identical to x(n). In ractice, however, the AFC is not erfect and therefore ŷ(n) y(n), resulting in distortion between e(n) and x(n). x(n) d f (n) d(n) A(z) y(n) e(n) ŷ f (n) e f (n) ŷ(n) Feedback Path F (z) Hearing Aid Processing G(z) Coy of W (z) AFC Filter W (z) Coefficient Adatation u(n) u f (n) Fig.. Block diagram of the AFC framework. s(n) H(z) A(z) In our system we adot the PEM based AFC framework 6] where a time-varying re-filter A(z) is resent and adated using linear rediction of e(n) ]. We also emloy a bandlimited filer H(z) from the FXMS aroach to concentrate on the frequency regions where oscillation is likely to occur 5]. This filter can also be viewed as a very rough aroximation of the feedback ath in the frequency domain 4]. In this framework, the MS for coefficient adatation is carried out in the re-filtered signal domain, where the refiltered signal u f (n) and e f (n) are used to udate the -ta AFC filter w(n) = w (n), w (n),..., w (n)] T as: w(n ) = w(n) µ(n)u f (n)e f (n), () where u f (n) = u f (n), u f (n ),..., u f (n )] T and a time-varying ste size µ(n) is usually emloyed to imrove the convergence rate: µ µ(n) = ˆσ (n) ɛ, () with µ > the ste size arameter, ɛ a small ositive constant to revent division by zero, and ˆσ (n) = ρˆσ (n ) ( ρ)(u f (n) e f (n)) the ower estimate term with a forgetting factor < ρ. This is actually the (modified) normalized MS () ] and has been widely used in seech rocessing esecially for AFC in HAs 5], 6], ]. III. PROPORTIONATE AGORITHMS FOR AFC To incororate the roortionate adatation idea into the AFC, the udate rule () becomes: w(n ) = w(n) µ(n)p(n)u f (n)e f (n), (3) where P(n) = diag{ (n), (n),..., (n)} (4) is an -by- diagonal matrix assigning different weights to the ste sizes for different filter tas. We refer to it as the roortionate matrix. Alying the P ], at the n-th iteration the diagonal entries of P(n) are comuted as: l (n) = r l (n)/( i= r i(n)), (5) for l =,,..., where r l (n) = max{κ max{ζ, w (n),..., w (n) }, w l (n) }, (6) with ositive constants κ and ζ. When using the IP ], we have: r l (n) = ( α) w(n) w l (n). (7) Substituting (7) into (5) gives: l (n) = ( α) w l (n) i= w i(n) δ, (8) where a small constant δ > is added to revent division by zero and α is a arameter that can be chosen for different sarsity levels: When α = it behaves like the P while for α = it reduces to the. When using the IP-l 3], the l norm aroximation w(n) = lim β i= e β w i(n) ] r l (n) = ( α) w(n) which results in: i= e β w i(n) ] (9) is used to relace the l norm in the IP. This gives: ] e β w l(n), () l (n) = ( α) i= e β w l(n) ] e β w i(n) δ, () with another arameter β that rovides control for identifying systems with different degrees of sarsity. The P is more suitable for very sarse systems. The IP and IP-l have arameters (α and β) for fitting different sarsity degrees but without direct connections to the system sarsity levels. Moreover, the formulations of these algorithms do not have theoretical foundation. IV. PROPOSED SPARSITY PROMOTING MS AGORITHM In this section we resent the derivation of the roosed AFC algorithm that romotes sarsity. et E ] denote the exectation oerator. In the re-filtered signal domain of AFC, the ordinary MS considers the following mean squared error (MSE) minimization roblem: min J(w) = E e f (n) ] = E d f (n) ŷ f (n) ] w = E d f (n) w T u f (n) ] () = E d f (n) ] w T b w T Rw, ISBN EURASIP 7 37

3 7 5th Euroean Signal Processing Conference (EUSIPCO) ] where R E u f (n)u T f (n) and b E u f (n)d f (n)]. To enforce sarsity on the solution w, we modify () as: J(w) = E d f (n) ] w T b w T Rw γ w, (3) where the l norm diversity measure w = l= w l, (, ] is added with a regularization arameter γ > to romote sarsity in the solution w 7], 8]. From 9], the gradient of the l norm term w.r.t. w is given by: w w = Π(w)w, (4) where Π(w) = diag( w l ). Alying this to the gradient of (3) and setting to zero, we obtain the following condition for the otimal solution w o : w o = Π (wo )(Π (wo )RΠ (wo ) γ I) Π (wo )b, (5) where γ γ and I denotes the identity matrix. This suggests the following iterative rocedure for comuting w o : w(n ) = M(n)(M(n)RM(n) γ I) M(n)b where we have defined: = M(n)(R(n) γ I) b(n), (6) M(n) Π (w(n)) = diag( wl (n) ) (7) and auxiliary variables: R(n) M(n)RM(n) and b(n) M(n)b. At the n-th iteration, for w R, we define the corresonding affinely scaled variable as 9]: q(w) Π (w(n))w = M (n)w. (8) With this transformation into the affine scaling domain, we have at the n-th iteration: q(n) q(w(n)) = M (n)w(n). (9) q(n ) q(w(n )) = M (n)w(n ). () Using (), the udate rule (6) can be equivalent to: q(n ) = (R(n) γ I) b(n). () Observing that () is actually the minimizer of the following quadratic objective function: J(q) = E d f (n) q T M(n)u f (n) ] γ q, () we can derive the steeest descent algorithm as: q(n ) = q(n) µ qj(q) = ( µ γ)q(n) µ(b(n) R(n)q(n)), (3) where µ > is the ste size arameter. Transforming (3) back to the original coefficient domain using (9) and () we obtain: w(n ) = ( µ γ)w(n) µm (n)(b Rw(n)). (4) Finally, to derive the adative algorithm, we relace R and b in (4) by their corresonding instantaneous values u f (n)u T f (n) and u f (n)d f (n), resectively ]. This gives: w(n ) = ( µ γ)w(n) µm (n)u f (n)e f (n), (5) where M (n) = diag( w l (n) ) from (7). This is actually the MS algorithm roosed in 7]. Note that M (n) has a similar role as the roortionate matrix P(n) in (3). In fact, it indicates that the weight assigned to each ste size is in roortion to w l (n). Based on this and the revious discussion on roortionate adatation for AFC, we roose the following udate rule: w(n ) = w(n) µ(n)p(n)u f (n)e f (n), (6) where µ(n) is the normalized ste size as (), P(n) = diag{ (n), (n),..., (n)} as (4) with l (n) = r l (n)/( i= r i(n)), for l =,,..., as (5), and: r l (n) = w l (n) c, (7) where (, ] and c > is a small ositive constant. Several observations can be made here. First, comaring (6) to (5), we have set γ to zero as it has negligible effect as long as we kee it small 8]. In such case, due to the resence of P(n), the algorithm will still exloit, though not enforce, the sarsity of the solution if it already exists. We then refer to it as Sarsity romoting MS (SMS). However, a ractical issue arises when γ = for (5): Once any of the AFC filter tas becomes, it will not get udated anymore. We therefore suggest adding a small constant c > to all the tas as in (7) before they are used to comute P(n). The arameter is much more influential: When is smaller (close to or less than ), the algorithm becomes more sarsity romoting due to the nature of the l norm; while when =, it reduces to the. This indicates that a sarse system would benefit more from a smaller while for a disersive system, close to would be more referable. From this it becomes clearer the advantage of incororating the l diversity measure. For the quasi-sarse feedback IRs in AFC, we exect that the otimal value would lie between and. V. SIMUATION RESUTS The above algorithms are evaluated using comuter simulations in MATAB at a samling rate of 6 khz. The exerimental set u was as follows: The HA rocessing G(z) = gz d with g = and d corresonding to a delay of 8 ms. The feedback ath IRs were measured using a behindthe-ear HA with oen fitting on a dummy head and truncated to a length of 63 samles as shown in Fig.. The AFC filter length was = to cover the significant art of the IRs. The forgetting factor ρ =.985. The ste size arameter µ =.5. Small ositive constants ɛ = δ = c = 6. The band-limited filter H(z) =.8z.8z as used in 4]. The re-filter A(z) was an FIR filter of order udated every ms via linear rediction of e(n) ]. For the urose of evaluation, the following metrics were used. First, to measure the distortion between e(n) and x(n), ISBN EURASIP 7 38

4 7 5th Euroean Signal Processing Conference (EUSIPCO) Fig.. Measured acoustic feedback ath IRs of f : no obstruction, f : with a cellhone close to the ear, and f 3 : with a cellhone right on the ear. the hearing-aid seech quality index () was used 3]. The score ranges from to, where the higher the score, the better the quality. For evaluating convergence erformance the normalized misalignment was considered: π Misalignment = log F (ejω ) ˆF (e jω ) dω π F, (8) (ejω ) dω where F (e jω ) and ˆF (e jω ) are the frequency resonses of the measured and estimated feedback IRs, resectively. Note that the estimated feedback resonse is given by the band-limited filter and the AFC filter together as ˆF (e jω ) = H(e jω )W (e jω ). For estimating the ASG we used 4]: ) ASG = log (min ω log (min ω F (e jω ) F (e jω ) ˆF (e jω ) ). (9) In our first exeriment we examine the effect of in the SMS on channels with different sarsity levels. In addition to the measured feedback ath f we considered two other artificial channels as lotted in Fig. 3. The inut was a stationary seech-shaed noise. Fig. 4 shows the convergence behavior in terms of misalignment. We can see that for the feedback ath case, =.5 outerforms other values. For the sarser system, a smaller around. is more referable; while around.8 gives the best erformance for the disersive one. These results show that the SMS is exloiting the underlying system structure in the way we exect. Furthermore, in Fig. 4 we can see that the SMS (with a good choice of ) can imrove the convergence rate over the and still maintains low steady-state error, which is not achievable by only using a larger ste size arameter µ for the. In the next exeriment, we ran the AFC system with the SMS on 5 male and 5 female seech signals from TIMIT database and measured the corresonding of the feedback-comensated signal e(n). Average scores over the 5 seech files for different values of are shown in Fig. 5. We can see that the otimal almost lies in the same range even as the feedback IR differs. This means, for a given HA device, if we have some rough knowledge about the Fig. 3. Artificial IRs of a sarse channel and a disersive channel =. =. =.5 =.8 = =. =. =.5 =.8 = =. =. =.5 =.8 = Fig. 4. Misalignment comarison of SMS with different on the measured feedback ath f, the sarse channel, and the disersive channel with seech-shaed noise inut. sarsity degree of its feedback channel, the SMS is robust since is not very sensitive near the otimal oint. From the results we found around.5 to be a good choice, which also corresonds to the result we had in Fig Fig. 5. Effect of on seech quality of SMS for f, f, and f 3. We also comare the SMS with other roortionate algorithms for the AFC. The arameter settings were P: κ =. and ζ =.; IP: α = ; IP-l : α = and β = 5. For the SMS we used =.5. The (which is equivalent to IP and IP-l with α = and SMS with = ) is also comared. Fig. 6 comares the tracking erformance in terms of misalignment and ASG with seech-shaed noise inut. To model a highly time-varying feedback back environment, the feedback ath was initially f, switching to f then f 3 at the /3 and /3 of the inut sequence, resectively. We can see that the roortionate algorithms ISBN EURASIP 7 39

5 7 5th Euroean Signal Processing Conference (EUSIPCO) basically have faster convergence seed comared to the. Among all, the SMS shows the best convergence behavior and can rovide u to about 5 db additional ASG comared to the. Finally, for further verification, we ran the algorithms on the seech dataset and measured the average under 4 different feedback scenarios as shown in Fig. 7. We see that the SMS outerforms all the other ones, esecially obvious under an adverse feedback situation such as the last two cases (about.5 imrovement comared to the in the last case). - - P IP IP-l ASG (db) P IP IP-l Fig. 6. Misalignment and ASG with seech-shaed noise inut. f f f 3 f 3 P IP IP-l Fig. 7. Comarison of seech quality for different feedback environments. The first three cases were fixed environments with f, f, and f 3. The last case f 3 was the feedback ath changing from f to f then f 3 at /3 and /3 of the inut sequence, resectively. VI. CONCUSION In this aer we introduce the SMS for AFC to exloit sarsity in estimating the feedback ath. This aroach extends the MS algorithm by incororating the l diversity measure in the objective function. We derive udate rules and discuss guidelines for choosing the system arameters. We resent simulation results with seech-shaed stimulus and seech segments with real-world feedback aths, including conditions where the feedback ath changes over time. The results show that for the SMS (i) choice of is not very sensitive around the otimal oint; (ii) imrovement of about.5 over the ; (iii) ASG imrovement by 5 db comared to the ; and (iv) better erformances comared with other roortionate-tye MS algorithms. ACKNOWEDGMENT This work was suorted by NIH/NIDCD grants #RDC546 and #RDC5436. REFERENCES ] T. van Waterschoot and M. Moonen, Fifty years of acoustic feedback control: State of the art and future challenges, Proc. IEEE, vol. 99, no., ,. ] S. 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Gazor and T. iu, Adative filtering with decorrelation for coloured AR environments, IEE Proc. Vision, Image, Signal Process., vol. 5, no. 6, , 5. ] J. E. Greenberg, Modified MS algorithms for seech rocessing with an adative noise canceller, IEEE Trans. Seech Audio Process., vol. 6, no. 4, , 998. ] H. Scheker,. T. T. Tran, S. Nordholm, and S. Doclo, Imroving adative feedback cancellation in hearing aids using an affine combination of filters, in Proc. IEEE Int. Conf. Acoust., Seech, Signal Process. (ICASSP), 6, ] J. M. Kates and K. H. Arehart, The hearing-aid seech quality index () version, J. Audio Eng. Soc., vol. 6, no. 3,. 99 7, 4. 4] J. M. Kates, Room reverberation effects in hearing aid feedback cancellation, J. Acoust. Soc. Am., vol. 9, no., ,. ISBN EURASIP 7 4

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