TECHNICAL MEMORANDUM

Transcription

1 Bell Laboratories subject: Proposals for Dynamic Resource Allocation for Network Echo Cancellation Work Project No File Case date: March 9, 1999 from: Tomas Gaensler Lund University Jacob Benesty Dept. BL11332 MH 2D-557 (98) M. Mohan Sondhi Dept. BL11332 MH 2D-536 (98) Steven L. Gay Dept. BL11332 MH 2D-519 (98) BL TM TECHNICAL MEMORANDUM 1 Introduction Current adaptation algorithms for network echo cancelers are designed without regard to the fact that, invariably, a single canceler chip handles many conversations simultaneously. This implies that for Nc channels, the processor must handle Nc times the peak computational load of a single channel. If the number of channels is large, however, it should be possible to reduce the demands on the processor to something close to Nc times the average load. Some additional computational capacity would, of course, be necessary to take care of statistical uctuation in the requirements, but the required safety margin becomes smaller as Nc becomes larger. (With the speed and memory now available on a chip, the number of channels can be several hundred, so the safety margin might not have to be large.) Once the problem is looked upon as that of dealing with a large number of channels, it is also possible to take advantage of other knowledge about speech patterns and characteristics of long distance circuits to further reduce the computational load. In this memo we show how the computational requirement can, in principle, be reduced by a very large factor { perhaps as large as thirty. Basically we capitalize on three facts. First, during a telephone conversation, there are many Lucent Technologies { PROPRIETARY Use pursuant to Company Instructions

2 -2- pauses in each speech signal. These pauses have been exploited to decrease the idle time in telephone connections, so called TASI (Time Assignment Speech Interpolation) networks, since the 196s [3]. An echo canceler too can take advantage of these pauses. Second, network echo paths are sparse, i.e. only a few coe±cients are nonzero. By utilizing this sparseness property of the responses, it is possible to increase the convergence rate and decrease the complexity of adaptive lters [4],[5],[6],[7]. Finally, echo paths do not change much during a conversation. Hence the adaptive lter need not be updated continuously. It is estimated that it needs to be updated perhaps only 1% of the time. These three features can be exploited to design an e±cient algorithm. The rest of the memo is organized as follows: In Section 2 the PNLMS algorithm used in Lucent's next generation echo canceler is described. Section 3 gives some proposals on how to simplify the PNLMS algorithm, and shows the reduction in complexity that may be achieved. Performance simulations are presented in Section 4, and Section 5 gives a discussion of the results and problems associated with this approach. 2 The PNLMS and PNLMS++ Algorithms Figure 1 shows the various signals of interest. In derivations and descriptions, the following notations are used for these quantities: xn = Far-end signal, vn = Background noise and possibly near-end signals, yn = Echo and background noise possibly including near-end signal, x n = [x n ;:::;x n L+1] T ; Excitation vector; h ep = [h ;:::;h L 1] T ; True echo path; h n = [h ;n ;:::;h L 1;n] T ; Estimated echo path: Here L is the length of the adaptive lter, and n isthetimeindex. The PNLMS algorithm was proposed in [1]. In this algorithm, an adaptive individual step-size is assigned to each lter coe±cient. The step-sizes are calculated from the last estimate of the lter coe±cients in such a way that a larger coe±cient receives a larger increment, thus increasing the convergence rate of that coe±cient. This has the e ect that active coe±cients are adjusted faster than non-active coe±cients (i.e. small or zero coe±cients). Hence, PNLMS converges much faster than NLMS for sparse impulse responses (i.e., responses in which only a small percentage of coe±cients is signi cant). Most impulse responses in the telephone network have this characteristic. The PNLMS algorithm is described by the following equations: e n = y n h T n x n ; (1) h n = h n 1 + ¹ x T G n nx n + ± G n x n e n ; (2) G n = diagfg;n;:::;g L 1;ng: (3) G n is a diagonal matrix which adjusts the step-sizes of the individual taps of the lter, ¹ is the overall step-size parameter, and ± is a regularization parameter which prevents division by zero and stabilizes the solution when speech is used as the input (far-end) signal. The diagonal elements of G n are calculated as follows [1]: X l;n = maxf½ maxf± p ; jh;n 1j; ::: ; jh L 1;n 1jg; jh l;n 1jg; l L 1; (4) L 1 g l;n = l;n = i= i;n; l L 1: (5) Lucent Technologies { PROPRIETARY Use pursuant to Company Instructions

3 -3- Parameters ± p and ½ are positive numbers with typical values ± p =:1; ½ =5=L. ½ prevents coe±cients from stalling when they are much smaller than the largest coe±cient and ± p regularizes the updating when all coe±cients are zero at initialization. A variant of this algorithm is the PNLMS++ [2]. In this algorithm, for odd-numbered time steps the matrix G n is calculated as above, while for even-numbered steps it is chosen to be the identity matrix G n = I; (6) which results in an NLMS iteration. Alternating between NLMS and PNLMS iterations has several advantages over using PNLMS alone. For example, it makes the PNLMS++ algorithm much less sensitive to the assumption of a sparse impulse response, without sacri cing performance. 3 Proposals In this section we outline the principles of a simpli ed algorithm that exploits the sparseness of network echo paths and the fact that the adaptive lter does not need to be updated when the residual error is small. We also present the theoretical complexity gains one can achieve. Any overhead that depends on choice of implementation platform is not considered in this memorandum. It is assumed that one or more computation engines serve the channels and some logic has been designed to control and distribute the resources. Decisions, e.g., which channels should be updated, are based on results from voice activity and double-talk detection. The PNLMS algorithm takes advantage of the sparseness of the impulse response to improve convergence rate, but its complexity is greater by a factor of 2 compared to that of the NLMS algorithm. In the following we show that we can do much better than that. 3.1 A Simple Algorithm to Update only Active Channels and Coe±cients In a two-way conversation each talker is active only about half of the time; additionally, there are pauses between sentences and syllables. During these inactive time slots no coe±cient updating is needed. Furthermore, since network echo path responses are sparse, we can focus computations on only the active (non-zero) coe±cients. The following algorithm, which may be seen as a simpli ed PNLMS++ algorithm, saves a large number of multiplications at the expense of some additional overhead. The key di erences compared to the current implementations of PNLMS++ are: 1. No coe±cient is updated if the channel is inactive or double-talk has been detected. 2. No coe±cient is updated if the residual error is su±ciently small. 3. Step-sizes for the active taps can all be made equal instead of the step-sizes speci ed by the matrix G n. 4. All coe±cients are updated (i.e., an NLMS iteration is made), every Mth iteration. Only the active coe±cients are updated at all other iterations. 5. The index of active coe±cients is updated every Mth iteration. Periods of inactivity are easily identi ed with a look-ahead of one, or a few, samples at the outputs of the voice activity and double-talk detectors. Hence the rst item is easily implemented. The implementation of item 2 is explained in the next subsection. Items 3{5 are implemented as follows: Let us rst de ne an \active set", i.e., the set of active tap weights. To this end, de ne a threshold T, and sort the tap weights in descending order of absolute value. Then de ne the active set A s Lucent Technologies { PROPRIETARY Use pursuant to Company Instructions

4 -4- as the rst L a weights in this list, such that their cumulative magnitude just exceeds T times the cumulative magnitude of all the L taps of the lter. In symbols L 1 A s = fl : minfjh l;n jg > maxfjh l;n X L 1 jg; T jh l;n j X jh l;n X j T jh l;n j +maxfjh k;n jgg: (7) l2as l62as k62as l=1 l=1 The threshold T is selected in the range T =[:9; 1). With T =1theactivesetcoversallthe taps, and we get the standard NLMS algorithm. From an implementation point of view it may be appropriate to limit the maximum size of the set to L max thus L a L max <L. The maximum load of a channel can thereby be limited. Appendix A shows a simple way to identify the set A s. For M 1 consecutive iterations the coe±cients X of the active set are updated as follows: e n = y n l2as h l;n = h l;n 1 + Every Mth iteration a full NLMS iteration is made, i.e., l2as h l;n x n l; (8) ¹ Pl2As x2 n l + ± x n le n ; l2 A s : (9) e n = y n h T n x n; (1) h n = ¹ h n 1 + x T x n n + x ne n : ± (11) Increasing M will reduce the average complexity but also worsen tracking performance. In the simulations described in Section 4 we used M = Stopping Adaptation when Small Residual Error is Detected Echo paths on the network vary slowly, in general. Hence adaptation is needed only at a small percentage of iterations, perhaps no more than 1%. This would yield a huge reduction in computations, since on a vast majority of iterations we need to compute only the convolution with the small number (L a )ofcoe±cientsintheseta s. Therefore we propose that when the error signal is su±ciently small, we do not update or sort the tap weights. Asymptotically, for network echo cancelers, the complexity of this algorithm would thus be reduced essentially to the computation of a convolution on the active taps only. A good decision variable, to decide if the residual error is small enough, is the normalized mean square error (in db) de ned as follows: <e 2 (n) > N»(n) =1log 1 ( <y 2 ); (12) (n) > N +± y where is the mean square error, <e 2 (n) > N = 1 N <y 2 (n) > N = 1 N nx nx n N+1 n N+1 e 2 (n) (13) y 2 (n) (14) is the mean square output, ± y is a regularization parameter which prevents division by zero during silences between words, and N is the length of the window used to estimate energy. N should preferably be chosen small in order not to degrade the tracking performance of the adaptive algorithm when the echo path changes. The adaptive algorithm proceeds as follows: At each iteration n, (12) is computed and»(n) is compared to a threshold Th (a typical range is Th =[ 4; 3]). The decision rule is simple: if»(n) Th, then the residual error is not considered small enough and the algorithm continues to update; If»(n) <Th, then the residual error is considered negligible and the algorithm neither updates nor sorts the taps of the lter. However, the convolution on the active taps is still performed. Lucent Technologies { PROPRIETARY Use pursuant to Company Instructions

5 Theoretical Reduction in Complexity The complexity of the proposed algorithm is compared to a PNLMS implementation with respect to multiplications and other required computations. Computations required for the various steps of the proposed algorithm are: Eq. (7): k L log 2 flg +2L + L a (sorting instructions) 1, Eq. (8): L a (multiplications), Eq. (9): 2L a (multiplications), Eq. (1): L (multiplications), Eq. (11): 2L (multiplications), where k is a proportionality constant for the sorting algorithm. Let the probability of active speech be denoted by p s and the probability of active adaptation by p a. Then, assuming equal weight for multiplications and sorting instructions, the average required number of computations is C = p s p a 3L M ; full NLMS update; (15) (M 1)(1 + 2p a )L a C1 = p s ; Update of active coe±cients; (16) M L + L a + kl log C2 = 2 flg p s p a ; Update of the active set; (17) M and the average total number of computations is Cp:alg = C + C1 + C2: (18) For comparison, note that the implementation of PNLMS++ via equations (1-3) requires: Eq. (1): L (multiplications), Eq. (2): 3L (multiplications), Eq. (4): 2L (comparisons). Again, with equal weight for multiplications and comparisons, the total number of computations 2 is As an illustration, assuming the expected typical values CPNLMS++ = 6L: (19) p s ¼ :5; (2) p a ¼ :1; (21) L a = 1; (22) k ¼ 1; (23) M = 1; (24) L = 768; (25) 1 This assumes that an algorithm like quicksort is used. 2 Strictly speaking, this is the complexity of PNLMS. For PNLMS++ some of the computations can be eliminated at all even-numbered iterations. However, the next generation of Lucent's echo canceler does not take advantage of this. Lucent Technologies { PROPRIETARY Use pursuant to Company Instructions

6 -6- we nd that Cp:alg ¼ 7; (26) CPNLMS++ = 468; (27) which shows that the average complexity of the proposed algorithm could be drastically less than that of PNLMS. However, for various reasons, the estimate in eq (26) should not be taken literally. First, it does not allow for a safety margin, and it is not yet clear how large that needs to be made. Second, at present there is no hard evidence to justify the estimate pa ¼ :1. Third, the estimate of complexity of the sorting algorithm is not rigorous. Finally, the choice of M = 1, that controls the initial convergence rate has not yet been optimized. Nevertheless, a reduction in complexity by a factor as large as 25 or 3 appears to be possible. 3.4 Overload Management As mentioned earlier, a safety margin must be provided to allow for statistical uctuations in the computational load. There is, however, a nite probability (which depends on the safety margin allowed) of temporarily running out of resources. Some possibilities to handle such an overload situation are to ² Not perform coe±cient updates on some of the active channels (or increase the level of the threshold Th). ² Neither perform coe±cient updates nor calculate residual echoes of some of the active channels. Just apply echo suppression. 4 Simulations of the Proposed Algorithm ITU-T G.168 [8] recommends certain test signals to be used for evaluating the performance of echo cancelers. The signals used are the so called Composite Source Signals (CSS) that have properties similar to those of speech with both voiced and unvoiced sequences as well as pauses. In the rst part of our simulations, we use these as excitation signals and show the convergence of the proposed algorithm without halting adaptation. In these simulations, results from three kinds of echo paths are shown: a sparse, a dispersive, and a multire ection path. These three paths represent a variety of responses that can be expected in practice. Figure 2 shows these impulse responses and their magnitude functions. The far-end signal (CSS) used is shown in Fig. 3a. The parameter settings chosen for the following simulations are: ² ¹ =:2, L = 512 (64 ms), ± =2 1 5, ±p =:1, ½ =:1. ² T =:98, L max = 2. ² ¾x = 21, SNR ¼ 37 db (echo-to-pcm quantization noise ratio). ² Hybrid attenuation: 6, 11 db. ² h 1 =. The performance is evaluated by using the mean square error (MSE) which is given by, MSE = P e ; (28) Py Pe = LPFfe 2 ng: (29) Lucent Technologies { PROPRIETARY Use pursuant to Company Instructions

7 -7- Here LPF denotes a lowpass lter, chosen here to be a leaky integrator with a single pole at :999. This choice does not signi cantly a ect the convergence curves shown in the gures. Py is calculated analogously. Figures 3b{d show the convergence behavior of the proposed algorithm compared to PNLMS++ and NLMS. The proposed algorithm has somewhat poorer initial convergence rate than PNLMS++ but is considerably faster than NLMS. With longer echo paths the performance improvement compared to NLMS will be even greater. In the second part of the simulations, we use a speech signal as the excitation, and show the convergence and tracking of the proposed algorithm when adaptation is halted. The following parameters are used: ² ¹ =:2, L = 512 (64 ms), ± = ² Th = 38:5 db,n = 4, ± y = ² T =:98, L max = 2. ² ¾ x = 19, SNR ¼ 39 db. ² Hybrid attenuation: 6, 11 db. ² h 1 =. Figures 4, 5, and 6 show the mean squared error and the misalignment for three di erent algorithms when the input is speech and the echo path is the one shown in Fig. 1a. In Fig. 4 the algorithm used is NLMS. In Fig. 5 we used the algorithm proposed above, but without stopping adaptation during periods of small residual error (i.e. with Th set to 1). In Fig. 6 the same algorithm was used but with Th selected such that the adaptation was halted more that 55% of the time. We can see that the proposed algorithm (whether or not adaptation is stopped) outperforms the NLMS algorithm. It is also of interest to see what happens in a tracking situation, i.e., when the echo path changes. Figures 7, 8, and 9 show the results of repeating the experiments of Figs. 4-6, except that the echo path is that of Fig. 1a for the rst 1 seconds and that of Fig. 1c for the next ten seconds. Again, we can see that the proposed algorithm (whether or not adaptation is stopped) tracks better than the NLMS algorithm. Lucent Technologies { PROPRIETARY Use pursuant to Company Instructions

8 -8-5 Discussion In this memorandum a number of proposals have been made for decreasing the complexity of the adaptive algorithm in a multi echo canceler system. Emphasis has been placed on nding simple procedures for choosing active regions of the impulse response and halting adaptation when the residual error is small. Though more careful analysis and development of the algorithms are needed, these proposals give some idea of what can be done from an algorithm point of view in order to improve the e±ciency of the implementations. An important aspect of the problem that we have not discussed in the memorandum is the possibility of reducing the requirement of storage capacity. Reduction in storage requirements is necessary if the reduction of computational complexity is to be fully exploited. One possibility for reducing storage requirements is to store the L a coe±cients in the set A s with full precision, and the rest with reduced precision. Another possibility is to store L max coe±cients with full precision and the rest with reduced precision. Since the inactive coe±cients are, in general, much larger in number, this procedure can signi cantly reduce the memory requirement. What remains to be seen is how few bits can be used for the inactive taps without degrading performance. Also, whether this scheme can be implemented without a large increase in overhead. Tomas Gaensler Jacob Benesty M. Mohan Sondhi {TG/JB/MMS/SLG-jb Steven L. Gay Atts. References Figures 1-9 Appendix Lucent Technologies { PROPRIETARY Use pursuant to Company Instructions

9 References [1] D. L. Duttweiler, \Proportionate normalized least mean square adaptation in echo cancelers," Lucent Bell Laboratories technical memorandum BL1771N TMS, [2] S. L. Gay, \PNLMS++ for network echo cancellation," Lucent Bell Laboratories technical memorandum BL TM, [3] K. Bullington and J. M. Fraser, \Engineering aspects of TASI," Bell System Tech. J., pp , March [4] S. Kawamura and M. Hatori, \A tap selection algorithm for adaptive lters," IEEE proc. of ICASSP, pp. 2979{2982, [5] V. Madisetti, D. Messerschmitt, and N. NordstrÄom, \Dynamically reduced complexity implementation of echo cancelers," IEEEproc.ofICASSP, pp. 1313{1316, [6] A. Sugiyama et al., \A fast convergence algorithm for adaptive FIR lters under computational constraint for adaptive tap-position control," IEEE Trans. Circ. Syst. II, Vol. 43, pp , Sept [7] J. Homer et. al., \LMS estimation via structural detection," IEEE Trans. Sign. Proc., Vol. 46, no. 1, Oct [8] International Telecommunication Union, ITU-T G.168, Section 3.4, 1997.

10 xn DTD Adaptive algorithm h n h ep vn + wn en yn Figure 1: Block diagram of the echo canceler and double-talk detector.

11 .5 (a) (b) 1 db (c) (d) 1 db (e) (f) db ms khz Figure 2: The three hybrids. (a, c, e) Impulse responses. (b, d, f) Corresponding magnitudes of the frequency responses.

12 (a) (b) 1 db (c) 1 db (d) 1 db Figure 3: Mean square error performance for composite source signal. Far-end CSS signal (a), Sparse hybrid (b), Disperse hybrid (c), Hybrid with three re ections (d). Solid line: Proposed algorithm, Dashed line: NLMS, Dotted line: PNLMS++.

13 7 (a) (b) MSE (db) Misalignment (db) Figure 4: Behavior of the (a) MSE ({) as compared to original echo level ({ {) and (b) misalignment of the NLMS algorithm with speech signal as input and hybrid of Fig. 1a.

14 7 (a) (b) MSE (db) Misalignment (db) Figure 5: Behavior of the (a) MSE ({) as compared to original echo level ({ {) and (b) misalignment of the proposed algorithm, without stopping adaptation, with speech signal as input and hybrid of Fig. 1a.

15 7 (a) (b) MSE (db) Misalignment (db) Figure 6: Behavior of the (a) MSE ({) as compared to original echo level ({ {) and (b) misalignment of the proposed algorithm, where adaptation is stopped more than 55% of the time, with speech signal as input and hybrid of Fig. 1a.

16 7 (a) (b) MSE (db) Misalignment (db) Figure 7: Behavior of the (a) MSE ({) as compared to original echo level ({ {) and (b) misalignment of the NLMS algorithm with speech signal as input and when the echo path changes at time 1 s from the one in Fig. 1a to the one in Fig. 1c.

17 7 (a) (b) MSE (db) Misalignment (db) Figure 8: Behavior of the (a) MSE ({) as compared to original echo level ({ {) and (b) misalignment of the proposed algorithm, without stopping adaptation, with speech signal as input and when the echo path changes at time 1 s from the one in Fig. 1a to the one in Fig. 1c.

18 7 (a) (b) MSE (db) Misalignment (db) Figure 9: Behavior of the (a) MSE ({) as compared to original echo level ({ {) and (b) misalignment of the proposed algorithm, where adaptation is stopped more than 45% of the time, with speech signal as input and when the echo path changes at time 1 s from the one in Fig. 1a to the one in Fig. 1c.

19 A Matlab Code for Calculation of the Active Set The following Matlab code shows an algorithm for calculation of the active set A s. First the magnitude of the estimated response is sorted in ascending order and the sorted taps' original index are stored. The cumulative sum of the sorted vector is calculated and the index (II) is found where the cumulative sum is larger than a part (T ) of the total sum. The active set is the L II +1 values of the sorted index vector (I). In a real implementation simpli cations can be made since the magnitude of the impulse response vector only needs to be partially sorted. [h sort ; I] = sort(abs(h)); (3) h cum = cumsum(h sort ); (31) II = min( nd(h cum T h cum (L))); (32) A s = I(max(L L max +1;II):L); (33)

20 Bell Laboratories Document Cover Sheet for Technical Memorandum Title: Proposals for Dynamic Resource Allocation for Network Echo Cancellation Authors Tomas Gaensler Jacob Benesty M. Mohan Sondhi Steven L. Gay Electronic Address Location MH 2D-557 MH 2D-536 MH 2D-519 Phone (98) (98) (98) Company (if other than Lucent{BL) Lund University Document No. BL TM Filing Case No Work Project No Keywords: Echo Cancellation, NLMS, PNLMS, Adaptive lters MERCURY Announcement Bulletin Sections CMM{ Communications MAS{ Math and Statistics Abstract Network echo canceler chips are designed to handle several channels simultaneously. With the processing speeds now available, a single chip might handle several hundred channels. In current implementations, however, the adaptation algorithm is designed for a single channel, and the computations are replicated N c times, where N c is the number of channels. With such an implementation, the computational requirement is N c times the peak load for a single channel. The number of computations required in each channel, however, varies widely over time. Therefore, a considerable reduction in computational load can be achieved by designing the system for the average load plus a margin to account for load variations. The reduction in complexity is achieved by exploiting three features: (a) the inherent pauses in conversations, (b) the sparseness of network echo paths, and (c) the fact that an adaptive lter does not need to be updated when the error signal is small. In this memorandum it is shown that, in principle, such a design can reduce the computational load by a very large factor { perhaps as large as thirty. It remains to be seen whether a customized hardware architecture can be implemented to fully take advantage of the proposed algorithm. Pages of Text 8 Other Pages 13 Total 21 No. Figs. 9 No. Tables No. Refs. 8 Mailing Label Lucent Technologies { PROPRIETARY Use pursuant to Company Instructions tm.sty (1996/2/4) BELL LABORATORIES

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