Hirano Akihiro

Nakayama Kenji

Tokui N.

Kanazawa University Repository for Academic Resources

A synchronized learning algorithm forreflection coefficients and tap weights in ajoint lattice predictor and transversal filter著者 Tokui N., Nakayama Kenji, Hirano Akihirojournal orpublication titleIEEE Proc. ICASSP'2001, Salt Lake Citypage range 1472-1475year 2001-05-01URL http://hdl.handle.net/2297/6824A SYNCHRONIZED LEARNING ALGORITHM FOR REFLECTION COEFFICIENTS ANDTAP WEIGHTS IN A JOINT LATTICE PREDICTOR AND TRANSVERSAL FILTERNaoki Tokui, Kenji Nakayama and y Akihiro HiranoGraduate School of Natural Science and Technology Kanazawa Universityy Faculty of Engineering, Kanazawa University2-40-20, Kodatsuno, Kanazawa, 920-8667, Japane-mail:tokui@leo.ec.t.kanazawa-u.ac.jpABSTRACTIn order to achieve fast convergence and less computation foradaptive filters, a joint method combining a whitening process andthe NLMS algorithm is a hopeful approach. One of them is tocombine a lattice predictor and a transversal filter supervised bythe NLMS algorithm. However, the filter coefficient adaptationis very sensitive to the reflection coefficient fluctuation. In thispaper, the reason of this instability is analyzed. The filter coef-ficients are updated one sample behind the reflection coefficientupdate. This causes large error, in other words, sensitivity of theirmismatch is very high on filter characteristics. An improved learn-ing method is proposed in order to compensate for this mismatch.The convergence property is close to that of the RLS algorithm.Computational complexity can be well reduced from that of theRLS algorithm. Simulation results using real voices demonstrateusefulness of the proposed method.1. INTRODUCTIONAs VLSI technology has been developed, adaptive filters have beenapplied to audio acoustic processing, control systems, telecommu-nication systems, and others. Among them, acoustic echo cancel-lation and noise cancelation are very important.When very high-order adaptive filters are required, fast con-vergence and less computation for real signals are very important.The normalized LMS (NLMS) algorithm can be implemented withless computation. However, a very long time is required for con-vergence. On the contrary, the recursive least squires (RLS) algo-rithm can converge fast, at the expense of computational complex-ity.One method to overcome this problem is to join a whiten-ing process and the NLMS algorithm. The whitening process in-cludes orthogonal transform and linear prediction [1–6]. The for-mer method requires many frequency bands in order to realizegood orthogonalization [1–3]. A lattice predictor is used in thelatter method [1,4–5]. Order of the predictor is determined by thatof an AR model generating the input signal, which is not so high.Therefore, this method is hopeful. However, filter coefficient adap-tation is unstable due to the reflection coefficient fluctuation. Eventhough the lattice predictor works well, the output error cannot bewell reduced. Any methods to stabilize convergence for reflectioncoefficient fluctuation have not been proposed.In this paper, the learning process of the lattice predictor basedNLMS algorithm is analyzed based on a relation between the re-flection and filter coefficient update. A reason of the unstable adap-                                                     	  	    	   	   Fig. 1. Lattice-based structure for joint-process filter of order Mtation is clarified. Based on this analysis, a synchronized algo-rithm for reflection coefficients and tap weights is proposed. Thismethod can achieve fast convergence like the RLS algorithm witha moderate number of computations. Computer simulation usingreal voice will be demonstrated to confirm usefulness of the pro-posed learning method.2. A JOINT LATTICE PREDICTOR AND NLMSTRANSVERSAL ADAPTIVE FILTERFigure 1 shows a block diagram. The 1st-stage is the lattice pre-dictor and the 2nd-stage is the transversal filter.fm(n) and bm(n) are the forward and the backward predictionerrors, respectively, at themth-stage and the nth-sample. They arecalculated by the following recursive formulas.fm(n) = fm 1(n) + m(n)bm 1(n  1) (1)bm(n) = bm 1(n  1) + m(n)fm 1(n) (2)m = 1; 2; : : : ;M   1f0(n) = b0(n) = u(n): (3)m(n) is the reflection coefficient at the mth-stage and the nth-sample.The input signal for the transversal filter is the backward pre-diction error bm(n). Letting b(n), w(n) and y(n) be the back-ward prediction error, the filter coefficients and the output, respec-tively, they are related byb(n) = [b0(n); : : : ; bM 1(n)]T (4)w(n) = [w0(n); : : : ; wM 1(n)]T (5)y(n) = wH(n)b(n): (6)3. ANALYSIS OF LEARNING PROCESS3.1. Update of Reflection CoefficientsThe reflection coefficient m(n) is determined so as to minimizethe following prediction error.Jm= Ejfm(n)j2+Ejbm(n)j2: (7)From the condition,@Jm@m(n)= 0 (8)m(n) is given bym(n) =  2E[bm 1(n  1)fm 1(n)]E[jfm 1(n)j2+ jbm 1(n  1)j2]: (9)Furthermore, letting the numerator and the denominator be 2N;m(n) and D;m(n), respectively, they are approximatelyupdated byN;m(n) = (1  ) (N;m(n   1)+ bm 1(n  1)fm 1(n)) (10)D;m(n) = (1  ) (D;m(n  1)+ jfm 1(n)j2+ jbm 1(n  1)j2(11)1 >  > 0 (12)3.2. Update of Filter CoefficientsThe filter coefficients are updated by the NLMS algorithm as shownine(n) = d(n)  y(n) (13)w(n+ 1) = w(n) +kb(n)k2+ Æb(n)e(n): (14) is a step size and Æ is a small positive number. The other algo-rithms can be also employed.3.3. Relation between Reflection and Filter Coefficient Updatem(n) is updated at the nth-sample, and will be used at the (n +1)th-sample. b(n) is obtained by using m(n  1), that is the pre-vious values. The filter coefficients are updated at the nth-sampleusing b(n) resultingw(n+1), which will be used at the (n+1)th-sample.In Eq.(14), e(n) and b(n) are obtained using m(n   1), notm(n). This means the filter coefficientsw(n+1) can reduce thecost function in collaboration with m(n   1), not with m(n).However, at the (n + 1)th-sample, w(n + 1) is combined withm(n) to generate the output y(n+1). This means the filter coeffi-cient update is always one sample behind the reflection coefficientupdate.This relation is held when m(n) is used to generate b(n) ande(n). The index on the time axis is only shifted. An essential pointis the fact that the reflection coefficients used in updatingw(n+1)and in generating the filter output y(n+1) together withw(n+1)are different.These relations are shown in Fig.2.                                                               Fig. 2. Flow diagram of updating reflection coefficients and tapcoefficients.4. AN IMPROVED LEARNING METHOD4.1. Transfer Function RepresentationThe transfer function of the joint adaptive filter shown in Fig.1consists of the reflection coefficients and the filter coefficients. Inthis section, an equivalent transfer function in the time domain isobtained.First, b(n) is expressed byb(n) =KH(n)u(n) (15)u(n) = [u(n); : : : ; u(n M + 1)]T: (16)Here,K(n) has the following structure.K(n) =26666641 K1;2(n) K1;3(n)    K1;M(n)0 1 K2;3(n)    K2;M(n)......1......0 0......KM 1;M(n)0 0 0    13777775(17)K1;2(n) = 1(n) (18)K1;3(n) = 2(n) (19)K2;3(n) = 2(n)1(n) + 1(n  1) (20)K1;4(n) = 3(n) (21)K2;4(n) = 3(n)(1(n) + 2(n)1(n  1)) + 2(n  1) (22)K3;4(n) = 3(n)2(n)+2(n 1)1(n 1)+1(n 2): (23)Using the above expression, the filter output is given byy(n) = wH(n)KH(n)u(n): (24)In this expression,wH(n)KH(n) represents the equivalent trans-fer function in the time domain.  and H indicate complex conju-gate and Hermitian transposition, respectively.                                                             Fig. 3. Flow diagram of the modified filter coefficients4.2. Compensation of Filter CoefficientsFrom the discussions in the previous section, w(n+1) is updatedusingK(n), therefore, the following output at the next sample canreduce the cost function.y^(n+ 1) = wH(n + 1)KH(n)u(n+ 1): (25)However,K(n) is updated at the (n+1)th-sample, then the actualoutput becomesy(n + 1) = wH(n+ 1)KH(n+ 1)u(n+ 1): (26)y(n + 1) cannot reduce the error well. In order to overcome thismismatch, the filter coefficients are modified so that the equivalenttransfer function satisfies Eg.(25). That is,K(n+ 1)^w(n+ 1) =K(n)w(n+ 1): (27)From this condition, we obtain^w(n+ 1) =K 1(n+ 1)K(n)w(n+ 1): (28)This modified filter coefficients will be used at the (n + 1)th-sample to generate b(n+1) and y(n+1). The coefficients updateand modification processes are shown in Fig.3.4.3. Computational ComplexityEquation (28) requires inverse matrix operation. However, the ma-trix K(n) is an upper triangle matrix, in which the diagonal ele-ments are unity. In addition, the order of the lattice predictor de-pends on the input signal model, not on the unknown system. Forexample, in the case of voice, the signal generation process canbe modeled by using approximately a 20th-order AR model. ThismeansK(n) is also a band matrix. Taking these properties into ac-count, the computational complexity can be reduced intoO(LM),where L is the order of the AR model for the input signal. In thecase of acoustic echo canceller, usually M = 1000  4000 andL = 20, then the computational load is well reduced comparedwith that of the RLS algorithm. Table 1 lists the number of com-putations of the proposed and the conventional algorithms. “Con-ventional Lattice” indicates the joint adaptive filter, which has thesame structure as the proposed method. However, the filter coeffi-cients are updated one-sample behind the reflection coefficients.Table 1. Comparison of LMS Algorithm with Preceding LatticePredictorMultiplier AdderProposed 2ML + 5M + 9L 2ML+ 4M + 5LCon. Lattice 5M + 9L 4M + 5LNLMS 5M 4MRLS 3M2 + 4M 2M2 + 3M                                                ! " #   ! $  % " & " ! ' "   (    (a)                      ) *  +   " # (      ! " #   ! $  % " & " ! ' "   (    !   , (b)Fig. 4. (a)Impulse response of unknown system (b)Frequency re-sponse of unknown system.5. SIMULATION AND DISCUSSIONS5.1. Simulation ProblemSimulation was carried out based on system identification. Anunknown system is the 12th-order IIR lowpass filter. The ampli-tude response and the impulse response are shown in Figs.4(a) and4(b), respectively. The impulse response spreads over 50 samples.Therefore, the adaptive filter lengthM is set to 50 taps. The NLMSalgorithm and the RLS algorithm are used for comparison.5.2. Colored Input SignalColored signal is generated through a 2nd-order AR model withthe white noise input. The amplitude response of the AR modelis shown in Fig.5. In the joint adaptive filter and the NLMS algo-rithm,  = 1, Æ = 0:001,  = 0:999, and  = 0:95 in the RLSalgorithm.The learning curves are shown in Fig.6. The “ConventionalLattice” cannot converge. The error is saturated at  40dB due tothe one-sample delay mismatch. The learning curve of the pro-posed method is close to that of the RLS algorithm. This meansthe one-sample delay can be compensated for, at the same time,the input signal whitening by the lattice predictor is successful.                  !   ,  "   ! *   *      !   Fig. 5. Amplitude response of 2nd-order AR model.- !  ! *     .  /   0  *  ! $     *    ! "1"02*,+***!*345                                                       / 6   6 - ! " 2  "   ! "    6     # 7 * !  !   Fig. 6. Learning curves for colored noise.5.3. Voice Input SignalThe voice signal used in the simulation is shown in Fig.7. A sam-pling frequency is 8kHz, then 20; 000 samples mean 2:5 seconds.Figure 8 shows the learning curves. The proposed method cancatch up with the RLS at 2000 iterations. The number of itera-tions is the same as that of the signal samples. On the contrary, theNLMS algorithm requires 10000 samples until converge.From these simulation results, the proposed method is usefulfor both stationary and nonstationary processes. Its convergencespeed is close to that of the RLS.6. CONCLUSIONSJoint adaptive filters combining a whitening process and the NLMStransversal filter is useful to make convergence speed fast withless computation. However, the lattice predictor based adaptivefilter has the instability problem. In this paper, the reason of theinstability has been clarified. In the conventional method, the fil-ter coefficients are updated one-sample behind the reflection co-efficients. Sensitivity of this one-sample delay mismatch is veryhigh. The synchronized learning algorithm has been proposed tocompensate for this mismatch. The computer simulation using thecolored signal and the voice signal demonstrate that the proposedmethod can converge fast like the RLS with less computations.7. REFERENCES[1] S. Haykin, “Adaptive filter theory,” PRENTICE-HALL, 3rdEdition, 1996.      8 !  # ,"9 ! , ! *        !  &  *       "   "  *       # :   ,      * Fig. 7. Input signal of voice.+***!*3458 !  #    .  /   0  *  ! $     *    ! "                                            / 6   6 7 * !  !   - ! " 2  "   ! "    6     # Fig. 8. Learning curves for voice.[2] J.J. Shynk, “Frequency–domain and multirate adaptive filter-ing,” IEEE SP MAGAZINE, pp.14–37, Jan., 1992.[3] F. Beaufays, “Transform–domain adaptive filters: an analyt-ical approach, ” IEEE Trans. Signal Process., Vol. 43, No. 2,pp.422–431, Feb., 1995.[4] J.H. Yoo, S.H. Cho and D.H. Youn, “A lattice/transversaljoint(LTJ) structure for an acoustic echo canceller, ” 1995IEEE Symposium on Circuits and Systems, Vol. 2, pp.1090–1093, 1995.[5] S.H. Leung and C.C. Chu, “Adaptive LMS filter with latticeprefilter, ” Electron. Lett., Vol. 33, Iss. 1, pp.34–35, 2nd Jan.,1997.[6] G. Mandyam and N. Ahmed, “The discrete laguerre trans-form: derivation and applications,” IEEE Trans. Signal Pro-cess., Vol. 44, No. 12, pp.2925–2931, Dec., 1996.[7] A. Fertner, “Frequency–domain echo canceller with phaseadjustment,” IEEE Trans. Circuits and Systems — II, Vol. 44,No. 10, pp.835–841, Oct., 1997.[8] V.N. Parikh and A.Z. Baraniecki, “The use of the mod-ified escalator algorithm to improve the performance oftransform-domain LMS adaptive filters,” IEEE Trans. SignalProcess., Vol. 46, No. 3, pp.625–635, Mar., 1998.

A synchronized learning algorithm for reflection coefficients and tap weights in a joint lattice predictor and transversal filter

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A synchronized learning algorithm for reflection coefficients and tap weights in a joint lattice predictor and transversal filter

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