Time distributed update of the NLMS algorithm coefficients for Acoustic Echo Cancellers

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1 Time distributed update o the NLMS algorithm coeicients or Acoustic Echo Cancellers Fotis E. Andritsopoulos, Yannis M. Mitsos, Christos N. Charopoulos, Gregory A. Doumenis, Constantin N. Papaodysseus Abstract Hands ree operation is a standard eature o many ixed and mobile telephone sets. In hands ree telephony, echoes appear at the ar end, because o the open-air acoustic path between the loudspeaker and microphone at the near end. This paper presents an eicient realization o an Acoustic Echo Cancellation (AEC) algorithm, as well as its implementation structure and perormance aspects. The algorithm is based on the Least Mean Square (LMS) amily o algorithms [2]. The innovative eatures o this algorithm provide signiicant gains in terms o computational complexity and signal quality. The proposed method is based on the segmentary update o the coeicients o the N-LMS ilter. It is proved that the convergence behavior o the introduced algorithm is similar to the N-LMS; it is also proved that segmentary update provides similar quality to the original N-LMS, while signiicantly reducing complexity. The algorithm can be easily implemented and it is particularly suited or simultaneous echo cancellation on multiple voice channels. It is hence appropriate or echo cancellation implementation in access devices as well as in central oice equipment. The results presented in this paper have been collected rom the algorithm s implementation on a standard commercial DSP. KEY WORDS AEC, Acoustic Echo Cancellation, LMS, digital ilters, hands ree telephony, signal processing.. Introduction One o the most important issues in interactive voice communication is the perceived quality o the communication. In videoconerencing systems, even i the video quality is comparatively good, poor audio will negatively impact the perceived quality o the communication. The term near end speaker reers to the person using the hands ree device. Alternative the term local is used. In our case ar end speaker is a person using a normal handset communicating with the near end speaker. In hands ree telephony, the acoustic signal leaving the ar end speaker travels through an electric path, which terminates at a loudspeaker. The purpose o the loudspeaker is to reproduce and ampliy the audio signal, so that it can reach the near end speaker at audible level. As the audio signal leaves the speakerphone, it is emmited into the room and The authors are with the Department o Telecommunications and Electronic Engineering o the National Technical University o Athens, Heroon Polytechniou 9, Zograou 5780, GREECE. it is relected rom the room s solid suraces, (walls, urniture, the loor or the ceiling). Thus, the signal propagates through many paths, each o dierent acoustic length, beore it can reach the persons ears. The ar end signal also reaches the microphone (which is installed on the hands ree device). The paths rom the near-end loudspeaker to the near-end microphone are called collectively as echo paths. Because o the dierent acoustic length o the paths, there is a inite time dierence between the shortest and longest echo path. When the echo path time delay is larger than a threshold, it is detectable by the human Mic Echo Path Speaker D C Near End Signal x(n)+r(n) + r(n) LMS Adaptive Filter u(n)=x(n)+r(n) r(n) Far End Signal y(n) Figure. Echo Canceller Model ear ath the ar end; this annoying phenomenon is called acoustic echo. Two common approaches to solve this problem make use either o hal-duplex devices or ull-duplex acoustic echo cancellers. In contrast to hal-duplex, the goal o ull-duplex telecommunication systems is to provide echo-ree communications by using an acoustic echo canceller to electronically cancel echoes beore they are transmitted back over the network. This paper proposes an LMS based implementation scheme or AEC on the TMS320C5402 DSP, which is compliant with ITU-T s speciications (G.65 [0]). This implementation introduces some modiications on the existing LMS algorithm, in order to save a signiicant amount o processing power. 2. Echo Canceller Model To solve the aorementioned problem, developers are using the digital signal processing technique o acoustic echo cancellation (AEC) to stop the eedback and allow or ull-duplex communication. To cancel echoes, the AEC system must model the open-air path between the loudspeaker and microphone. This path is a unction o the loudspeaker and microphone, as well as o their placement within the room and the room s acoustics, including its B A

2 construction materials, dimensions, urnishings and their locations, and movement o people in the room. Figure shows the principle o the echo canceller or one direction o transmission. Let y(n) represent the ar-end signal, r(n) is the undesired echo, and x(n) is the near-end talking person s signal. The near-end signal is superimposed with the undesired echo on port D. The received ar-end signal is available as a reerence signal or the echo canceller, and is used by the canceller to generate a modelled replica o the echo called ˆr(n). This replica is subtracted rom the near-end signal (plus the echo) to yield the transmitted near-end signal u(n) where u(n) = x(n) + r(n) ˆr(n). Ideally, the residual echo error e(n) = r(n) ˆr(n) () will be very mall ater echo cancellation, thus u(n) x(n). 3. LMS And Cancellation Theory Cancellation theory is based on adaptive modelling o the open air path using adaptive inite impulse response (FIR) ilters. The adaptive iltering algorithm uses the previous voice sample values and the calculated errors to update the FIR ilter s coeicients, based on a least mean square (LMS) algorithm. Then, it uses the updated new coeicients and the latest sample values to calculate the FIR ilter s output. An adaptive ilter discrets samples o an input signal y(n), a set o ilter coeicients, a k (n), and produces an output signal ˆr(n), called the desired response. The desired response signal provides a rame o reerence or adjusting the ilter coeicients. In the case o the AEC, the input signal y(n) is the signal sent to the local loudspeaker and the desired response signal ˆr(n) is an approximation o the undesired echo signal that enters the local microphone. When the near end person is quiet and the ar end person is talking, the near-end signal contains only the echoes to be removed. When only the remote person is talking, the goal o the adaptation algorithm is to drive the error signal e(n) to zero so that no echo is heard at the remote end. An important actor or a robust implementation o an acoustic echo canceller, is the accurate identiication o the speech state that may be present on the line(no talk, near end talk, ar end talk, double talk). This detection is achieved by computing the energy o the appropriate signal (ar-end, near-end, etc), and then comparing this energy to an adaptive threshold. An FIR ilter is presented in Figure 2. Equation 2 describes the transer unction o an FIR ilter. The ilter is used to approximate the actual transer unctions o the echo paths. N ˆr(n) = k=0 a(k) y(n k) (2) where N is the number o ilter s taps, y is the signal that needs iltering and a stands or ilter s coeicients. Note that y and a are vectors. a = [a(0),a(),...,a(n )] and y(n) = [y(n),y(n ),...,y(n N + )] y(n) FIR Filter H(z) a(k) Figure 2. FIR ilter r(n) The process o re-calculating the vector a in order to driver ˆr(n) = 0, is called ilter adaptation. I the ilter is adapted at the wrong time, such when the microphone is picking up near-end speech, the ilter can misconverge, creating audible artiacts that are transmitted to the ar-end. The problem is compounded by the act that the acoustic signature o the room and microphone s position are not constant. Changes in the echo path caused by motions o participants must be compensated or in a timely manner. Even more signiicant, echo path changes are caused by participants moving the microphones or adjusting the loudspeaker volume control. I the ilter is adapting poorly, moderately high levels o loudspeaker volume will oten result in positive acoustic gains in the communication loop, a prime condition or singing and howling. The recursive least mean square algorithm provides a low cost way (in terms o processing requirements) to determine the optimum (Wiener) ilter coeicients without explicitly computing the cross correlation and auto correlation sequences. The goal is to minimize the sum o mean squared error E: E = E { S e 2 (n) n=0 } = E { S n=0( r(n) N k=0 a(k)y(n k) ) 2 } (3) where E stands or the mean value operator, e(n) is the residual error signal and S is the stepsize that determines the speed o converge. Ater minimizing the above ormula, we ind a recursive gradient (steepest descent) algorithm that determines the minimum value o E. The process starts with any arbitrary choice or the initial values o the ilter coeicients (usually all equally to 0).Ater each new sample input y(n) enters the adaptive FIR ilter the corresponding output ˆr(n) and the error signal e(n) are computed and the ilter coeicients are updated according to: a k (n + ) = a k (n) + e(n)y(n k) (4) where k =0,,...,N and n =,2,..., is the step size parameter and y(n) is the input sample (located at the k-th tap o the ilter at time n). This ormula is used in the simple LMS algorithm; in the N-LMS [2] algorithm (where a simple normalization o the step size is perormed), the ormula is made Normalized LMS. a k (n + ) = a k (n) + µ e(n)y(n) (5) P(n)

3 where P(n) is an estimate o the power o the input signal y(n), and µ is the step size. Notice that, or a given time sample n, the quantity µ P(n) e(n) is constant and need only be calculated once. The convergence properties o the N-LMS algorithm are largely determined by the step size parameter and the power o the signal P(n). The AEC algorithm may update the coeicients ater every new sample (i.e. up to 8000 times per second) 2. The N-LMS algorithm requires one addition and one multiplication per sample and tap, so an n-tap ilter has an O(n) complexity while the aster stabilized version o the Kalman [3] algorithm has an O(4n) complexity and would require 4 times more additions and multiplications. In addition, at each sample it will be required the calculation o one Kalman gain, with a more complex ormula [4] than the µ P(n) e(n) quantity that the LMS needs. Formulating the problem as the minimization o the error, e(n) = r(n) a T (n)y(n) (6) results in the mean square error being a quadratic unction o the ilter coeicients. A normalized LMS (N-LMS) algorithm is used in the LMS adaptive ilter unction to update the FIR ilter s coeicients. N-LMS is almost the same as LMS, except that equation 5 is used to normalize the step size. The equation 5, is also used to update the FIR ilter s coeicients. In general, making the step size larger speeds the convergence, while a smaller step size reduces the asymptotic cancellation error. The convergence time constant is inversely proportional to the power o y(n), so the algorithms converge very slowly or low-power signals. The generic recursive equation or estimating the average power is: P y (n) = L 2 y(n) (7) where L y (n) is presented in equation 8.The term ρ is a constant with range values between zero and one. L y (n) = ( ρ)l y (n ) + ρ y(n ) (8) 4. Innovation Introduced As described our implementation introduces a modiied N-LMS algorithm, ocusing on the update o the ilter s coeicients, in order to acquire more space in processing power domain. This is perormed by updating segmentary the coeicients o the FIR ilter. This method is depicted in Figure 3. The previous Figure shows that in a speciic timeslot,we update a part o the available coeicients () instead o updating all o them. With the sequential update o the coeicient s vector, we can acquire the whole updated vector in a ew sequential cycles. Using this technique we can run longer ilters than a speciic DSP does without aecting the quality o the AEC. A undamental assumption or this concept is that or a given processing power, it is easible to run the whole (ull update) 2 Normally, or 8KHz PCM coded voice samples the coeicients are calculated irregularly, when the residual error e(n) exceeds a threshold, or when the talk state changes FIR 64 2 End processing Timeslots FIR Figure 3. Segmentary update o the FIR s coeicients FIR ilter during one available timeslot. The experiments proved that or an FIR ilter o N = 52, i we update its coeicients in 8 dierent timeslots, 3 the perormance o the AEC is excellent when comparing with the ull update o the N-LMS algorithm or 52 taps. A mathematical model o this concept is provided below in order to prove theoretically the aorementioned concept. In order to study the problem, we use the ollowing notation a 64 i (n), i =, or the coeicients when the system order is M = 64, while the notation i (n),i =, is used when M = 256. Moreover, let and ˆr 64 (n) = a 64 i (n) y 64 (n) (9) ˆr 256 (n) = i (n) y 256 (n) (0) or the residual errors or system order M = 64 and M = 256 respectively. Now we observe something ully conirmed in practice, namely that during LMS algorithm convergence and ater ew iterations [2], quantities r 64 (n) and r 256 (n) are o the same order and, even more, o quite close values. This is also depicted in Figure 9 where the mean values o r 64 (n) and r 256 (n) or a speciic experiment are presented. Moreover, since the value o power P(n) is independent o the employed echo path, it ollows that the irst sixty our (64) components o (n) will be identical to the components o a 64 (n). In addition, the second subset o 64 components o (n) namely 64 (n),a (n),...,a (n) are quite close to a64 (n) too, since they correspond to an input sequence each time shited by one sample per iteration. We emphasize that speech is a relatively slowly varying signal [6], so or a sampling requency greater than a proper threshold, y(n) and y(n ) will have a relatively small dierence. This eature is exploited also in PCM coding [7]. The quantizer used in classical PCM codes instantaneous values o speech samples. Adjacent speech samples, however, are oten highly correlated. This means that the variance o the dierence between successive samples x(n) and x(n ) is smaller than the variance o x(n). Consider the dierence signal d(n) = x(n) x(n ). The variance o d, assuming x is a zero mean stationary process, is: 3 Updateable segment size = 64 σ 2 d = σ2 x{2( r )} ()

4 where r = E[x(n)x(n )] is the normalized correlation E(x 2 ) between adjacent samples. I r < 0.5, then σ 2 d < σ2 x, and it would be better to quantize d instead o x. This is because or a given number o bits in the coder, the variance o the quantization noise is proportional to the variance o the signal at the input to the quantizer when the signal x(n) is speech. Next, i we partition as ollows: (n) 2 (n) 3 (n) (n) =. = (n) (n)] 2 (n)] 3 (n)] (2) 4 (n)] one can similarly deduce that a 64 3 and a 64 4 are close enough to a 64 too. Notice that the brackets on the terms a 64,a64 2,... stand to indicate the set o coeicients or the irst 64 coeicients, the second and so on. Now, suppose that we execute the LMS algorithm with M = 256 number o taps, but at a certain iteration we update only the irst 64 that orm the vector, while keeping the other coeicients unchanged. Then, obtain a signal approximation ˆr (n) = (n+) y + 2 (n) y (n) y (n) y 4 (3) but, the actual LMS approximation corresponding to a 64 (n) is ˆr(n) = (n + ) y + 2 (n + ) y (n + ) y (n + ) y 4 Now, according to the previous analysis, term (n + ) y is quite close to the desired value r(n + ), while at the same time the other three terms dierence is the ollowing: δr = ( 2 (n + ) 2 (n)) x 2 + ( 3 (n + ) 3 (n)) x 3 + ( 4 (n + ) 4 (n)) x 4 But, as described previously, that quantity i (n + ) i (n),i =,...,4 is o relatively very small value, providing that ˆr (n+) and ˆr(n+) are pretty close and close to r(n + ) too, in this case. With very similar arguments one can prove that ˆr(n+ ) and ˆr (n + ) are pretty close when only 2 (n) or 3 (n) or 4 (n) are each time updated. So the error that can be inserted due to the segmentary update is very small when applied on speech signals. The N-LMS algorithm itsel consists o the three equations reerenced above (2, 5 and 6). The analysis o the algorithm when it is employed in a single-channel eedorward adaptive control task is presented by []. An L-coeicient FIR echo canceller, requires 2L + 2 multiplications, 2L additions and 2L + storing operations. I we deine a divider actor 4 then we can perorm the N-LMS 4 While the actor is always even and greater than zero or the modiied algorithm, we have that >. in a time distributed way in terms o the coeicients update, according with what proposed in the previous sections o this paper. The processing power saving can be easily expressed as a unction o. Notice that the actor stands or the number o steps that the updating algorithm will be taking place. Especially or = the algorithm degenerates to the N-LMS. The multiplications that are required or each iteration, (which are described by the equations 2 and 5), are L and 2 + L respectively. Thus, the multiplications that are needed by the modiied algorithm are less than the N- LMS does. I MUL are the number o multiplications that are required by the N-LMS and MUL 2 are the number o multiplications or the modiied algorithm then we have: MUL = 2L + 2 (4) MUL 2 = L L = L( + ) + 2 (5) MUL 2 is always less than MUL because + < 2. One can determine the gain in terms o processing power when using the proposed method by calculating the ratio MUL 2 /MUL. The actual gain should be + G mul = MUL 2 = L MUL 2L + 2 (6) I we want to acquire a quantitative analysis o the MUL in contrast with the MUL 2, we can rewrite the previous unction as ollowing: G mul = L + 2L = + 2 (7) Similarly, or the addition operations we declare SUM the number o addition operations that are required by the N-LMS and SUM 2 the addition operations or the proposed algorithm. SUM = 2L (8) SUM 2 = L + L F = L + (9) This quantity is always smaller when comparing with the SUM = 2L. The gain can be calculated by the ollowing equation: G sum = L + 2L = + 2 (20) The ollowing Figure, represents the behaviour o the gain 5 when using the proposed algorithm or acoustic echo cancellation. According to this graph, the gain that is acquired by using partitioning on coeicient update procedure, is signiican when the actor takes short values. For urther partitioning the gain remains the same while the penalty on the quality o the echo canceller is increasing. Our experiments proved that using the value o 4 or the actor, the processing power gain that derived rom 5 Expressed as + 2

5 other ways to detect the type o the speech with less computational complexity [5]; however just a ew aults in the speech detection routine, are able to drive the FIR ilter to an arduous misconvergence. This has as eect the whole system to become completely unstable. The speech detection unction uses data rom the power estimate unction to determine the sequence o operations to be perormed or the corresponding mode. According to Figure 6, i a near-end signal or example Figure 4. Behaviour o the quantity + 2 or various values o operations such as multiplications and additions (MAC), was about to 40% when running the algorithm with 256-tap ilters. This gain was acquired without aecting the quality o the received signal. The trade-o between the processing gain and the quality, seems that the above value is the optimum one. Moreover, Figure 4 shows that the partitioning into many blocks (i.e. above 0) does not provide any additional gain. The vertical axis o Figure 4 denotes the quantity + 2, while the horizontal axis denotes the valus o the actor. 5. Implementation Description The method was used or speech detection in our implementation (which is presented in Figure 5), has the advantage o accuracy and avoids alse speech state detection. The cost or this algorithm is the high complexity and the signiicant requirements regarding processing power. Three dierent windows are used or power estimation (very short, short and long), hence more operations are needed to detect the type o speech. This, however, ensures that the detection will be accurate. There are many Figure 6. Sotware structure is detected by the near-end detector, the NEAR SPEECH lag is set to, and the NES HANG hangover counter is reset to the highest value (600) to avoid toggling o the speech modes. Hangover counters are used to indicate i there is necessity or ilter s coeicient updating or not. In the case that the value o the NES HANG hangover counter reaches the zero that means that there is no near end signal on the line or a long time. The highest value o these hangover counters has been selected to be 600 (assuming the sampling requency or voice channels in telephony is 8KHz). Next, the program perorms double-talk Table. LMS Mode Bit settings according to hangover counters Counter Counter 2 Filtering Updating DT HANG 0 NES HANG 0 YES NO DT HANG 0 NES HANG<0 YES NO DT HANG<0 NES HANG 0 NO NO DT HANG 0 NES HANG<0 YES YES Figure 5. Complex Speech Detector detection even i no ar-end speech is detected. This avoids alse detection which would lead to a misconvergence o the ilter. I double-talk is not detected and there is no arend speech, the program decreases the DT HANG hangover counter by one and goes to the control lms routine. I neither ar-end speech nor double-talk is detected, the program goes to the near-end speech detector. I near-end

6 speech is detected, the NES HANG hangover counter is set to the highest value, and the program jumps to the routine that will control the unctionality o the LMS algorithm. This routine checks all hangover counters beore setting the AEC UPDATE and AEC FILTERING mode bits. Figure 6, presents the structure o the designed sotware in terms o speech detection and signal iltering. 6. Simulation Results The ollowing Figures represent the estimated error ˆr(n) and the residual error e(n) respectively as produced by the sotware module described in this paper. While time passes, the modiied LMS converges with an acceptable behavior. The residual error reaches a minimum value, as the estimated desired response ˆr(n) is increasing. That indicates that the residual error has been eliminated. These results have been taken rom simulated samples on the TMS320C5402 DSP using as tool the Code Composer [8]. The behavior o the system is similar to the one presented in Figures 7 and 8, when using real audio signals or input. The vertical axis denotes the amplitude o the signal in Volts, while the horizontal one provides the number o the iterations that perormed. Implementing the algorithm in C language exactly as described in previous sections on the aorementioned DSP, we achieved a perormance o 3 MIPS/Voice Channel. The length o the ilter matches the length o the echo path. The above results acquired using an echo path o 64ms. For a sampling requency o 8 khz, the number o taps (ilter coeicients) is, thereore, equal to 52. A study on the whole algorithm proves that most o processing power is spent to the updating o the ilter s coeicients. This is the reason that segmentary updating o the ilter s coeicients provides a signiicant gain in terms o processing power. One common approach [9] to reduce the computational overhead o the normalized LMS (NLMS) algorithm is to update only a subset o the adaptor the segemntary updatingive ilter coeicients using a selectively algorithm. It is known that the mean square error is not equally sensitive to the variations o the coeicients. This technique used as a basis or the modiied sotware structure o the acoustic echo canceller in order to support long ilters (long echo paths), or more channels. The implementation o this technique, had as eect to support 32 simultaneous 6 (and real time) acoustic echo cancellers or echo paths that corresponds to 64ms delay using the same DSP 7. In Fig Figure 9. Mean square values o ˆr 64 (n) and ˆr 256 (n) Figure 7. Estimated error ˆr(n) Figure 8. Residual error e(n) ure 9, we can study the mean square values o the errors or a speech signal on an echo path o 256 taps. As presented, the values that come rom both methods are quite similar. The circles stand or the updating using the ull update algorithm o the N-LMS, and the dots or the segmentary updating. Each value o the circle and dot has been calculated per 000 iterations o the algorithm, in order the graph to be more distinct. The error that inserted by this way, does not aect the perormance o the system, according to subjective and objective tests o G-65 when the partial update is perormed in a ew steps. That means that the aspect ratio o the length o the coeicients vector, to the length o the partial updating, must be above a lower bound. For example, or ilters o 52 taps, the segmentary updating took place on 8 steps (update blocks o 64 samples). 7. Conclusions This paper presented an innovative method or acoustic echo cancellation on long echo paths which is based on the N-LMS algorithm. Even the N-LMS is not the most accurate algorithm or error minimization, it is widely used or echo cancellation implementations. The modiication that inserted in the existing algorithm is based on the segmentary update o the FIR s ilters coeicients providing 6 In worst case where adaptation is in use 7 TMS320C5402

7 as eect a signiicant gain in terms o long ilters execution due to processing power saving. The theoretical model proves that this modiication can work with high accuracy comparing with the ull update N-LMS algorithm, when speech signals are used as input. The gain o the processing power due to this algorithm, can be used either or cancellation on longer ilters or on supporting more channels than the N-LMS does or a ixed echo path. As uture work on the segmentary update o the coeicients, could be the tradeo between the number o segments and the perormance or quality. There is a golden section on the maximum number o the segments and the minimum acceptable quality. The study o the variance on voice signals, and the outputs o the above algorithm could provide us with such results or a maximum perormance o this method. [] Scott C. Douglas, Adaptive Filters Employing Partial Updates, Trans. Circ. and Syst. II, vol 44 pp209-26, Mar 997 [2] Haykin S., Adaptive Filter Theory, Prentice-Hall, Englewood Clis, New-Jersy, 986 [3] Greg Welch, Gary Bishop An introduction to the Kalman ilter Department o Computer Science, University o North Carolina at Chapel Hill Chapel Hill, NC [4] Grewal Mohinder S., Andrews Augus P. Kalman Filtering Theory and Practice Using MATLAB, John Willey & Sons, Inc., 200 [5] Pavel Sovka, Petr Pollak, Jan Kybic Extended Spectral Subtraction Czech Technical University, Faculty o Electrical Engineering, Czech Republic [6] T. J. Moir Automatic Variance Control and Variance Estimation Loops, I.I.M.S., Massey University Albany Campus, Auckland, New Zealand [7] M. J. Narasimha, Dierential Coding Techniques dierential co [8] Texas Instruments Code Composer Studio User s Guide February 2000 [9] T. Aboulnasr, K. Mayyas Complexity Reduction o the NLMS Algorithm via Selective Coeicient Update University o Ottawa [0] ITU-T Recommendation G.65 Echo Cancellers 993