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(2) FEEDBACK CANCELLATION IN HEARING AIDS: AN UNBIASED MODELLING APPROACH . Ann Spriet , Marc Moonen , Ian Proudler. . . Katholieke Universiteit Leuven - ESAT, Kasteelpark Arenberg 10, B-3001 Leuven-Heverlee, Belgium e-mail: {spriet,moonen}@esat.kuleuven.ac.be Katholieke Universiteit Leuven - Lab. Exp ORL, Kapucijnenvoer 33, B-3000 Leuven, Belgium QinetiQ Ltd., Malvern Technology Centre, St Andrews Road, Malvern, Worcestershire, WR14 3PS, UK e-mail: i.proudler@signal.QinetiQ.com. ABSTRACT In this paper, we present an unbiased adaptive modelling approach to feedback cancellation in hearing aids. The approach is based on a closed loop identification of the feedback path as well as the (linear prediction) model of the near-end input signal. In general, both models are not simultaneously identifiable in the closed loop system at hand. We show that -under certain conditions e.g. if a delay is inserted in the forward path- identification of both models is indeed possible. Simulation results demonstrate that -under these conditions- the unbiased modelling approach outperforms the biased continuous adaptation algorithm. 1 INTRODUCTION Acoustic feedback, which is caused by leakage from the loudspeaker to the microphone, limits the maximum amplification that can be used in a hearing aid without instability. To increase the maximum gain, a feedback cancellation algorithm is used that estimates the feedback signal and subtracts it from the microphone signal. Since the acoustic path between the loudspeaker and the microphone can vary significantly depending on the acoustical environment, the feedback canceller must be adaptive. Currently available adaptive feedback cancellers can be divided in to two classes: algorithms with a continuous adaptation and algorithms with a noncontinuous adaptation [1],[2]. The latter only adapt the filter when instability is detected or when the input signal level is low. Due to the reactive, rather than proactive, adaptation, these systems may be objectionable. A continuous adaptation scheme
(3) continuously adapts the filter coefficients of the filter . This is depicted in Figure 1. Since the input signal to
(4) the microphone is non-white and due to the forward path , and the in
(5) put to the adaptive filter are correlated, generally . Ann Spriet is a Research Assistant supported by the Fonds voor Wetenschappelijk Onderzoek (FWO) - Vlaanderen. This research work was carried out at the ESAT laboratory and Lab. Exp. ORL of the Katholieke Universiteit Leuven, in the framework of the Concerted Research Action GOA-MEFISTO-666 (Mathematical Engineering for Information and Communication Systems Technology) of the Flemish Government, IUAP P4-02 (1997-2001) ‘Modelling, Identification, Simulation and Control of Complex Systems’ and FWO Research Project nr. G.0233.1 (’Signal processing and automatic patient fitting for advanced auditory prostheses’). The scientific responsibility is assumed by its authors.. +. (R(z)) U(z). G. F. F F0 −. +. − Y(z). X(z). +. Figure 1: Concept of a (biased) adaptive feedback canceller.
(6) .
(7) . of the feedback path causing a biased estimate [3]. To reduce the correlation, delays are included in the for
(8) ward path or in the cancellation path (i.e. at the input of
(9) the adaptive filter ). The correlation can also be reduced by inserting a noise signal at the input of the loudspeaker that is uncorrelated with or by adding nonlinearities in.
(10) the forward path [4]. #%
(11) .
(12) !"
(13) $#%
(14) , with white noise Suppose that "
(15) and monic, inversely stable and known. In [5], it is shown that the bias of the adaptive filter can be avoided by means of a filtered-X algorithm minimizes the filtered er
(16) /.0
(17) $*that '&)(
(18) * +,
(19) )- 1 The concept of the filteredror 2&)(
(20) X algorithm is illustrated in Figure 2. In practice, is unknown and time varying. In addition, the performance of the filtered-X algorithm strongly depends on the quality 3&4(5
(21) of the estimate of so that it is desirable to estimate '&)(
(22)
(23) 2&)(
(24) adaptively. In general though, and.
(25) 7 98.
(26) are not identifiable in closed loop if 6 , is lin :
(27) ear and the filter is fixed [5]. In this paper, we show "&)(
(28) that -under certain conditions- identification of both
(29) and is indeed possible. In Section 2, the identification method is described. Section 3 derives the conditions under which the identification scheme has a unique optimal solution. In Section 4, the theory is verified through simulation. 2 CONCEPT Consider the two-channel identification scheme depicted in.
(30) .
(31) Figure 3 with adaptive FIR filters ; and < , with coefficient vectors = and > and filter lengths ?A@ and ?CB , respectively. The two-channel adaptive filter minimizes.
(32) H D | L w yz is replaced by a previous estimate of e ~ w yz}vxw yz during adaptation, W J and ] J depend on previous values of. (R(z)) +. forward G path. F0 +. ~nw yz and v,wyz . In this case, the optimisation criterion is nonlinear in T and [ . Assume that the system in Figure 3 is sufficiently linear and stationary so that we can use the -transform theory. Then, according to Parseval’s theorem,. U(z) feedback path. F. − Y(z). W(z). H(z). X(z). f. F l. − −1. Figure 2: Filtered-X algorithm. (R(z)) +. G. B. − Y(z) X(z). H(z). W(z). £,wyz. R. w yz. z. y. . y Q. (3). E(z). w yz*w£,w yz e | L w yz¢0wyzz Y2ª wyz Q | w yz¢0wyz Y w yz Q. (4) (5). (1). R _qp)a bcrp)a bCeGf*chgigjgsp)a bCetlCuNeGf*cSo U . and ] J We H would like the filter vxw yz to identify the H D product D eZ{ w yz}| w yz and the filter ~nwyz to identify { wyz such that w yz equals %w yz . To avoid the trivial solution ~Cw yz R R f v,wyz H R D , the first tap of ~Cw yz is set to : ~nw yz f Y y ~Aw yz . In general, a bc is speech-like and a segment of a bc can be modelled by an all-pole model, so we assume f. f Y y. H D. wyz. %wyz Q. ª w yz Y2© wyz f e © w yz)w | wyz e. w yz | L w yz$z¬«. (6). vxw yz Y ~nw yz}| wyz ª w yz f e © w yz*w | wyz e | L wyzz R. ©. w yz*wv,w yz Y ~nwyz$| w yz$z fZe © w yz*w| w yz e | L wyzz¯. {. w yz$%w yz (7) «. Section 3 studies under which conditions Hminimization of D R { Q v,wyz R (3), H has the unique solution C ~. w y z. w y z D eZ{ wyz$| w yz .. _7`a bcd`Va bCeGf*chgigjgk`Va bCe'lnm3eGf*cSo U. wyz%w yz R. ©. Y® ~nwyz Y. M JSR!TVUXWVJZY2[\U4]XJ Q. {. H D. The output wyz of the two-channel adaptive filter thus equals. EIH D JK)LNMPOJQ with. w yz R. R. ¢ w yz R. Figure 3: Two-channel identification scheme.. . ¢0wyz. F. A. +. where WVJ^R. JK)L. 5 l,0 . w yz}wy. where ª w yz is the noise signal injected at the input of the loudspeaker. Substitution of (5) in (4) results in. U(z). F0 +. D EGF. f. M OJ R. with ¡ the unit circle and wyz R v,w yz¢ w yz Y ~nw yz}£xw yz the -transform of the sequence ¤ M J¥5JK)Li¦¨§¨§¨§¨¦ EqH D . The inputs ¢0w yz and £w yz of the two-channel adaptive filter are given as. +. H. EIH D . (2). with %wyz a white noise signal (in case of unvoiced sounds) or H an H D (in case of voiced sounds). Hence, D impulse train { w yz R f Y y wyz is an FIR filter. L w yz is an initial estimate of | w yz with The filter | D D HVH assumed to be stable. It may be replaced during identificationH of obD ~nw yz and v,wyz by aH previously D w yz}vxw yz . The filter ~ w yz should be tained estimate e ~ constrained to be stable. D If | L EIwyH z D is kept fixed during adaptation, the cost function EGF JK)LNMPOJ is linear in T and [ . If. 3 UNIQUE SOLUTION/IDENTIFIABILITY To analyse (7), we distinguish between two cases: ª w yzIR± ° ª ! R (noise injection) and w yz (no noise injection). 3.1 Case 1: ª wyzIR± ° (noise injection) ª ² R a bc are uncorrelated, miniIf wyz±° and if ³ a bc H and D wyz$w y zµ Q results in minimization of mization of ´ H H D D Y ´ a D wyz$ D wy z O w yz} O w y z c µ , where D w yz and O wyz equal D wyz R. v,wyz Y ~Cw yz}| w yz ª w yz f¶e © w | wyz e | L wyzz. O wyz R. ~nw yz Y. ©. w yz*w vxw yz Y n ~ w yz}| wyzz f e © w| w yz e | L wyzz ¯ . (8) wyz (9) «. i.e. sufficiently Assume lnm and l u are adequately H D chosen large. Minimizing ´ D wyz$ D w y zµ results in vxw yz R H D e ~nw yz}| wyz leading to ´ D w yz} D w y z·µ R® . Plugging R ~nwyz this into (9), weH obtain w H yD z Minimization of O wyz D « ´ O w yz} O w y z µ with ~Cw yz R f Y y ~Aw yz corresponds to linear prediction of H D w yz . Since wyz R { w yz$%wyz , this results in ~Cw yz R { wyz . Hence the optimal solution is found to be unique and to equal the desired solution..
(33) 3.2 Case 2: ¸0¹ º»V¼¾½ (no noise injection) If ¸0¹º»¶¼¿½ , minimization of ÀÂÁxùº»$ù º\Ä)Å»ÆÇ reduces to Ç minimization of À ÁxÃZÈ·¹º»$ÃZÈ·¹º¬Ä4Å*» ÆÇ . Ç. 3.2.1 Delay É in the forward path Suppose Ê ¹º»t¼Ëº\Ä Æ¶Ê Ì ¹ º» with É9ͲΠand Ê Ì ¹ º» , Ï ¹º»*Ð ÏXѹ º» are causal. For causal FIR filters Òn¹º» and Ó,¹ º» , Ô. ¹ º» ¼. Ê Ì ¹º»j¹ Ó,¹º»Õ2ÒC¹ º»}Ï ¹ º»$». (10). Î Ö'º Ä Æ Ê Ì ¹º»j¹ Ï ¹º»Ö'Ï Ñ ¹ º»$» Ô. is a causal IIR filter, which may be specified as ¹ ºÔ »×¼ Ø Ø ÑnÕÙº¬Ä4Å Õ%ÚÚÚ . Since the first tap of Òn¹ º» ÕÛºÜÄ Æ ¹ º» Å Î of ÒC¹ º» , minimizain (9) equals the first tap ÝÑÞ¼ tion of À5Á Ã È ¹ º»}Ã È ¹º¬Ä4Å»·ÆÇ corresponds to linear prediction of ß"¹ º» , such that theÇ optimal solution corresponds to à Ô ÒC¹ º»4Õ2º\Ä Æ ¹º»âáß×¹ º»V¼¾ã'Ä4Å5¹ º»âß×¹ º» or Òn¹º»PÕ0º Ä Æ. Ê Ì ¹ º»*¹Ó,¹ º»4Õ"Òn¹ º»}Ï ¹º»» Î Ötº Ä Æ Ê Ì ¹ º»*¹Ï ¹ º»VÖtÏ Ñ ¹º»». In general, Òn¹ º» and Ó,¹ º» (11). Equating powers of º änæ are large enough such ã'Ä)Ź º» and ÖZã3Ä)Ź º»}Ï ¹ º» and ä æ ÉÍGäCå. Ä)Å ¹º»*Ú (11). are not uniquely determined by in (11) we see that -if ä0å and that Òn¹º» and Ó,¹º» can model respectively i.e. äCå"Í×äCçZèé. with ä ç èé. ÍGä ç èé Õ"änê. ¼±ã. ¼ëänì±ÕíÎ , and if. - the solution is unique and equals ÒC¹ º» ¼±ã. Ä)Å ¹º»*îZÓ,¹ º» ¼ÙÖZã. Ä)Å ¹º»$Ï ¹ º»Ú. (12). If É¿¼dä å but ä å is smaller than the length of ã3Ä4Å5¹ º» the solution of (11) is unique but biased because ã2Ä4Å5¹ º» is under modelled. Note that the biased, continuous adaptation algorithm depicted in Figure 1 can be interpreted as a special case of the two-channel adaptive filtering scheme in which Òn¹º»ï¼ Î . For significantly large É , the correlation between ß"¹º» and º\Ä Æ ß×¹ º» will be negligible such that the minimization of À ÁxÃ È ¹º»$Ã È ¹º¬Ä4Å*»ÆÇ decouples Ç in to minimization of ÀPÁ ÒC¹ º»âß"¹º»$Òn¹ º\Ä)Å*»âß"¹º¬Ä4Å»·ÆÇ Õ Ô Ô Ç ÀPÁ ¹º»}ß"¹ º» ¹ º\Ä)Å»}ß"¹ º\Ä)Å*» ÆÇ . Since Òn¹º»A¼sÎ , only the Ç second term can be minimized and hence, Ó,¹ º» converges to Ï ¹ º» . Also note that in (11) ò the error Óx¹ º»ðÕdÒn¹º»$Ï ¹ º» ó is weighted by . The larger èõ òó ñ ó ê Çô ó ê·ö ó ÷ ÷ ÷. Å Ä Ç òó èõ òó ñ ó ê Çô ó ê·ö ó ÅÄ Ç Ç ô Ä /Ç ô ô ñ Ç/ô. ÷. Çô Ä Çôô ñ Ç/ô ÷ , the smaller the bias of the feedback ÷. path will be in a biased approach. 3.2.2 Delay É È in the cancellation path Suppose a delay ÉÈ is added to the cancellation path i.e. Ó,¹º»0¼øº\Ä ÆùúÓx Ì ¹ º» with Ó, Ì ¹ º» causal and suppose Ï ¹ º»A¼ º¬Ä ÆùûÏ Ì ¹ º» with Ï Ì ¹ º» causal. If ÉÈúÕ×Éð;ä ç èé with ÉüÍÙÎ and if ä å Í!äCçZèé and ä æ ÍýäCçZèéþÕ"äCê , the solution of (11) is unique and equals the desired solution. If the first É È taps of the feedback path Ï ¹º» differ from zero, the solution will be biased.. 3.2.3 Time varying Ïѹ º»Ð Ê Ì ¹º» or nonlinear Ê Ì ¹ º» In general, (11) implies that there are several solutions for Òn¹ º» and Ó,¹º» . If Ê Ì ¹º» or Ï Ñ ¹º» are time varying, the positions of the spurious solutions will change with time such that it is likely that -with sufficient averaging- the algorithm will converge to the desired solution. Hence, if Ï Ñ ¹ º» is at each time instant replaced by the most recent estimate of ÖZÒÿÄ)Å5¹º»$Ó,¹º» , the adaptive algorithm may converge to the desired solution, even without adding a delay in the forward path. ¹ º» reduces the correlation between ß"¹º» A nonlinear Ê Ì ò ó ò ó and ó ñ ó ê Çô ó ß×¹ º» such that it decouèõ ê·ö ó ÅÄ Ç. ñ Çô. Ç/ô Ä. Çôôô. ples the minimization of À ÁÃ È ¹ º»}Ã È ¹º¬Ä4Å»·ÆÇ into Ç À ÁÒn¹º»}ß"¹ º»}ÒC¹ º\Ä)Å»}ß"¹ º\Ä)Å*» ÆÇ minimization of Õ Ô Ô Ç À Á and thus also makes ¹º»}ß"¹ º» ¹ º\Ä)Å»}ß"¹ º\Ä)Å» ÆÇ Ç Òn¹ º»ÐÜÓ,¹ º» identifiable. 4 SIMULATION RESULTS Section 3 shows that under certain conditions the filters Òn¹º» and Ó,¹º» are identifiable even if no additional noise ¸0¹º» is injected in the system. Inserting e.g. a large enough delay É in the forward path Ê ¹º» renders the system identifiable. Inserting a delay ÉÈ in the cancellation path only results in an unbiased solution if the first ÉÈ taps of Ï ¹ º» equal ½ . Making Ê Ì ¹ º» nonlinear or inserting a noise signal ¸0¹º» also helps to make the system identifiable but may degrade the sound quality of the microphone signal. Hence, inserting a delay É in the forward path is the preferred option. This Section illustrates the performance of the two-channel identification method through simulation for this scenario. The two cases, adaptive and fixed Ïѹº» , are considered. For comparison, the results obtained with the continous adaptation algorithm of Figure 1 are given too. 4.1 Recursive Algorithm In the simulations, Recursive Least Squares (RLS) is used to update the two-channel adaptive filter. If however at each time instant the filter Ï Ñ ¹ º» is replaced by the most recent estimate of ÖZÒÿÄ)Ź º»}Ó,¹º» during adaptation, Ð depend on previous estimates of and such that the optimisation problem becomes nonlinear. This dependency is effectively ignored in our implementation, which corresponds to neglecting the second term in the gradient of the cost function.
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(35) ¼ Õ . "!#. Ú. (13). This algorithm resembles a pseudo-linear regression algorithm (cfr. the pseudo-linear regression algorithm used in output error IIR adaptive filters [6]). 4.2 Simulation Results In the simulations, the acoustic feedback path model Ï ¹à º' » is a $ % th order FIR filter. The hearing aid input signal & á is a speech-shaped noise signal created by passing Gaussian.
(36) 5. F0(z)=0 F0(z)=A−1(z)B(z). 0. Misalignment ζ [dB]. −5. continuous adaptation algorithm −10. −15. two−channel adaptive algorithm −20. −25. filtered X (optimal speech model) 0. 1. 2. 3. 4. 5 samples. 6. 7. 8. 9. 10 4. x 10. Figure 4: Frequency domain misalignment (*),+-/+1 . 0 of the feedback path estimate 2/354768):9 0<;=),9>0 for ?A@CBDB . 0. 5 CONCLUSIONS. F0(z)=0 F0(z)=A−1(z)B(z). −5. In this paper, we have presented an unbiased adaptive modelling approach to feedback cancellation in hearing aids. The approach performs a closed loop identification of the feedback path and the (linear prediction) model of the near-end input signal. In general, both models are not simultaneously identifiable in closed loop. We show that -under certain conditions e.g. if a delay is inserted in the forward path- identification of both models is possible. Simulation results demonstrate that -under these conditions- the unbiased technique outperforms the biased continuous adaptation algorithm.. −10. Misalignment ζ [dB]. −15. −20. −25. −30. −35. −40. −45. model, which we consider in some sense an optimal solution, is depicted too. The solid lines correspond to a fixed filter + n with + n ),9>0~@L , the dotted lines are the ones obtained for a continuously adapted + n ):9 0A@2/34j6),9>0|;}),9>0 . In this simulation, these two lines nearly coincide. Other simulations have shown that for + n ):9 0@C2/354j68):9 0<;=):9 0 the convergence of the misalignment of the feedback path estimate strongly depends on the initialisation of the covariance matrix, but always outperforms the biased continuous adaptation algorithm. The two-channel adaptive filter performs nearly as well as the optimal filtered-X algorithm and clearly outperforms the biased continuous adaptation algorithm. Figure 5 shows the misalignment of the speech model estimate obtained with the two-channel adaptive algorithm. The misalignment of the speech model drops significantly. This indicates that also the speech model estimate converges to the true model.. 0. 1. 2. 3. 4. 5 samples. 6. 7. Figure 5: Frequency domain misalignment speech model estimate 3J):9 0 for ?1@CBKB .. 8. 9. 10 4. x 10. (E),FG-HFI. 0. of the. noise through a BL th order Q all-pole Q filter F#):9 0 . The forward path model equals 9 4NMPO , with O @SR . Figure 4 shows the misalignment (*),+-+ . ) (in dB) of the estimated feedback path + . as a function of the number of samples for the continuous adaptation algorithm and for the two-channel adaptive filter for ?T@UJVHW XY@BDB . The filter lengths of the adaptive filters are set to the true model orders i.e. UAZ#@[U V W X , UJ\]@^U1V W X_ U1` in the two-channel adaptive filter technique and Ub` a @cUJ` in the biased continuous adaptation algorithm. The misalignment (*),+-P+J. 0 is computed in the frequency domain as. 4j6 oo +q),rstIv u 02+q. ):rstwv u 0 oo mk ljnp (*),+- +1. 0d@fehgi o i i o x - (14) 4j6 oo +q),r stwv u 0 oo kmljnp e g i o i ox where UAyJ@bzD{ equals the number of frequency points used. Here +q . ):9 0 is the obtained estimate of the feedback path. . ):9 0 equals 2/354768),9>0|;}),9>0 . In the two-channel approach +q For comparison, the misaligment of the feedback path estimate obtained with Filtered-X RLS using the correct speech. References [1] J. A. Maxwell and P. M. Zurek, “Reducing Acoustic Feedback in Hearing Aids,” IEEE Trans. SAP, vol. 3, no. 4, pp. 304–313, July 1995. [2] J. E. Greenberg, P. M. Zurek, and M. Brantley, “Evaluation of feedback-reduction algorithms for hearing aids,” J. Acoust. Soc. Amer., vol. 108, no. 5, pp. 2366–2376, Nov. 2000. [3] M. G. Siqueira and A. Alwan, “Steady-state analysis of continuous adaptation in acoustic feedback reduction systems for hearing-aids,” IEEE Trans. SAP, vol. 8, no. 4, pp. 443–453, July 2000. [4] H. A. L. Joson, F. Asano, Y. Suzuki, and S. Toshio, “Adaptive feedback cancellation with frequency compression for hearing aids,” J. Acoust. Soc. Amer., vol. 94, no. 6, pp. 3248–3254, Dec. 1993. [5] J. Hellgren, Compensation for hearing loss and cancellation of acoustic feedback in digital hearing aids, Ph.D. thesis, Dep. of Neuroscience and Locomotion, Division of Technical Audiology, Linköpings universitet, SE-581 Linköping, Sweden, Apr. 2000. [6] J. J. Shynk, “Adaptive IIR Filtering,” IEEE ASSP Magazine, vol. 6, no. 2, pp. 4–21, Apr. 1989..
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