Adaptive Stereo Acoustic Echo Cancelation in reverberant environments. Amos Schreibman

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1 Adaptive Stereo Acoustic Echo Cancelation in reverberant environments Amos Schreibman

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3 Adaptive Stereo Acoustic Echo Cancelation in reverberant environments Research Thesis As Partial Fulfillment of the Requirements for the Degree Master of Science in Electrical Engineering Amos Schreibman Submitted to the Senate of the Technion Israel Institute of Technology Sivan 5769 Haifa May 2009

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5 Acknowledgement The research thesis was done under the supervision of Associate Professor Israel Cohen in the Department of Electrical Engineering. I would like to thank him for his dedicated guidance, encouragement to perfection and valuable advices throughout all stages of this research. I express my deep gratitude to my family for their never-ending patience, love and encouragement. Without their truthful support, this work could not have been accomplished and I dedicate it to them. The Generous Financial Help Of The Technion Is Gratefully Acknowledged. i

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7 Contents 1 Introduction Motivation and Background Stereophonic Acoustic Echo Cancelation overview Existing DT resilient algorithms Overview of the Thesis SAEC problem formulation and current solutions Introduction Problem formulation SAEC scheme SAEC normal equations Convergence rate Nonuniqueness problem Current solutions Legacy solutions Half wave rectifier Input slide technique UW-PSP POWER algorithm Conclusions Unified convergence limit suboptimal SA Introduction Variable step size sign based algorithm iii

8 3.2.1 Optimum step size for SA in a noisy environment Suboptimal step size SA Unified convergence limit of the suboptimal SA Effect of non perfect estimation of the error variance in the steady state for the suboptimal SA Experimental results Convergence limit Estimation error effects on the convergence limit Convergence under DT and noise interference Conclusions Appendix: Derivation of (3.15) IR tail effect in SAEC environment Introduction The IR Tail effect IR tail effect in the single channel case IR tail effect in a stereo environment SAEC error as a DT interfearance Proposed solution Stage 1: Preprocessing Stage 2: Projection Stage 3: POWER Experimental results Noiseless Environment convergence and reconvergence Noisy Environment convergence and reconvergence Double Talk convergence Conclusions Conclusions Summary Future research iv

9 Bibliography 78 v

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11 List of Figures 2.1 SAEC scheme Coherence of NLP processed signals Convergence using Benesty s rectifier FRLS tracking Input slide unit Input sliding convergence Input sliding convergence interpretation taken from [1] POWER algorithm taken from [1] POWER and UW-PSP convergence AEC scheme Convergence limit of the proposed algorithm Estimation error affect on the convergence limit Convergence under AWGN and DT interference Convergence under AWGN and continuous DT interference Proposed SAEC algorithm TUC block diagram Near-end room Impulse response. (a) Dominant channel. (b) Weak Channel Noiseless reconvergence. (a) Dominant channel Misadjustment. (b) Weak Channel Misadjustment. (c) Overall Misadjustment (d) Overall ERLE Noisy reconvergence. (a) Dominant channel Misadjustment. (b) Weak Channel Misadjustment. (c) Overall Misadjustment (d) Overall ERLE DT convergence. (a) Dominant channel Misadjustment. (b) Weak Channel Misadjustment. (c) Overall Misadjustment (d) Overall ERLE vii

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13 List of Tables 4.1 Tap update control mechanism ix

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15 Abstract Acoustic Echo cancelation (AEC) is an important tool which is needed to reduce the echo levels heard by the speaker in communication systems such as full-duplex hands free teleconferencing system. This echo is the result from the loudspeaker to microphone coupling in the near-end room. Adaptive filtering techniques are usually used in order to estimate the near-end room loudspeaker to microphone impulse response. The estimated echo signal is subtracted from the received microphone signal, thus reducing the level of the returning echo to the far-end room. Stereophonic communication systems provide a more lifelike conversation quality since the participants can use the added spatial information to distinguish the current speaker location. This work discusses a stereophonic system which consists of two microphones in the far-end room connected to two loudspeakers in the near-end room, and one microphone in the near-end room connected to a single loudspeaker in the far-end room. The analysis for a second set of a near-end room microphone connected to a far-end room loudspeaker is easily deducted due to the problem symmetry. In the discussed stereophonic system, a single speaker signal in the far-end room is received by the far-end room microphones and transmitted to the near-end room loudspeakers. The two loudspeakers signals are then received by the near-end room microphone and transmitted back to the far-end room loudspeaker, thus producing the stereophonic echo. The current available solutions that reduce the stereophonic echo focus mainly on solving problems caused by the high correlation between the two far-end room microphone signals (the near-end room loudspeakers signals). This high correlation is caused by the single origin of the two signals (the far-end room speaker). Therefore the 1

16 two signals differ only by the convolution with the far-end room speaker to microphone impulse responses. These solutions assume the existence of a Double Talk Detector (DTD), a device which freezes the adaptive filters adaptation if a double talk interval in the near-end room is detected. However, in the presence of a constant near-end room interference, the existing solutions are unable to converge and therefore do not function properly. Furthermore, due to the placement of both the near-end room loudspeakers and microphone, it is possible that one of the near-end room loudspeaker to microphone impulse responses will consist of higher energy when compared to the second near-end room loudspeaker to microphone impulse response. In such case, due to the use of Normalized Least Mean Square (NLMS) based solutions in the existing solutions, the residual echo of the dominant channel interferes with the weaker channel convergence. This work suggests the use of a noise and double talk resilient adaptive algorithm in collaboration with the POWER stereophonic echo cancelation algorithm. To produce a complete solution, the noise and double talk resilient algorithm should be able to converge during constant near-end room interference (continuous double talk interval). Additionally, the algorithm must produce good results for a low signal to noise ratio (SNR) environment in order to comply with the problem of a weak channel convergence in the presence of a dominant channel. A new variable step size adaptive algorithm is presented in this work. The algorithm is derived from the Sign Algorithm (SA), and it is noise and double talk resilient. The choice of the SA as the foundation of the noise and double talk resilient algorithm is twofold. Firstly - the SA is a noise robust algorithm due to its bounded gradient; and secondly - the binary decision type of the SA could reduce the correlation between the far-end room microphones signals, thus making the algorithm attractive to stereophonic echo cancelation systems. In order to derive this noise and double talk resilient algorithm, the optimal step size of the SA is found under a noisy environment assumption. This optimal step size is then used to calculate a simple and efficient sub-optimal step size. The convergence limit of the algorithm was analyzed and found to be independent of 2

17 the environment level of interference, i.e. the algorithm converge to the same limit for a noisy or noiseless environment. These findings produce an efficient solution to the demands given above, since the algorithm will continue to converge towards the same limit both during a constant interference in the near-end room and in a low SNR environment. Furthermore, the convergence limit indifference to the environment noise level enables the noise and double talk resilient algorithm to converge during double talk in the near-end room. This feature gives the algorithm a significant advantage which enables the use of the algorithm in an AEC system with a low performance DTD or in an AEC system with no DTD device at all. The main disadvantage of the suggested noise and double talk resilient algorithm is its initial convergence rate. This relatively slow rate is derived from the division of the algorithm step size by the residual error energy level, a variable which is expected to have high values at the beginning of the convergence and at a state of reconvergence. In order to solve this problem, the noise and double talk resilient algorithm was integrated in a new algorithm based on the POWER stereophonic acoustic echo canceler algorithm. At initial convergence or reconvergence, the filters tap values are derived from the NLMS algorithm, while at the remaining time the filters tap values are derived according to the noise and double talk resilient algorithm. The choice between the two filters tap update methods is decided by two ghost filters. Each of the ghost filters converges using a different tap update equation. The ghost filter with the lower error energy level chooses the favorable system filter tap update method. This algorithm was checked under several convergence simulations in which its performance was compared to the POWER and NLMS algorithms under the same convergence conditions. It was found that the noise and double talk resilient algorithm combined with the POWER algorithm produced better performance compared with NLMS and POWER algorithms. The advantages of the noise and double talk resilient algorithm combined with the POWER algorithm were especially noted at low SNR environments, and under constant DT interference. 3

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19 Notation x Vector are noted with underline x[n] The vector x sampled at time stamp n { } T Transpose operator R Matrixes are noted as capital letters <, > Standard euclidian inner product of the l 2 space 2 Standard euclidian norm of the l 2 space Statistical independency E{ } Statistical expectancy operator s[n] Input speech vector g Far-end room speaker to microphone impulse response vector x[n] Input speech vector w ŵ[n] d[n] y[n] R[n] p[n] G ğ[n] D(f, n) Ǵ W X(f, n) α Near-end room loudspeaker to microphone impulse response Adaptive filter Desired signal sample Estimated echo signal sample Reference vector autocorrelation matrix Reference to desired sample cross correlation vector Far-end room speaker to microphone impulse response matrix SAEC input to error impulse response STFT of the desired echo sample Far-end room speaker to microphone IR fourier transform Fourier transform of the desired near-end room loudspeaker to microphone IR STFT of input vector Benesty rectifier nonlinear coefficient 5

20 x[n] c Ĝ 1 G 1 G 2 Preprocessed far-end room microphone signal ISU state control ISU preprocessed c 0 state 1 st channel far-end room speaker to microphone IR matrix ISU preprocessed c 1 state 1 st channel far-end room speaker to microphone IR matrix ISU preprocessed 2 nd channel far-end room speaker to microphone IR matrix µ Adaptive algorithm step size ε[n] η[n] m[n] σ 2 x σε[n] 2 ση[n] 2 σe[n] 2 N Residual echo sample Near-end room interference sample Misalignment vector Input speech vector variance Residual echo variance Near-end room interference variance Adaptation error variance Length of the estimation filter µ OP T Optimal step size for the SA in a noisy environment µ Sub OP T Suboptimal step size for the SA in a noisy environment σ e[n] 2 λ ρ λ [n] σρ 2 σe 2 min I L L Adaptation error estimated variance Error variance estimation regression coefficient Estimated error variance estimation error Estimated error variance estimation error variance Lower bound of the error variance for the suboptimal step size SA L L identity matrix 0 N N N N zeros matrix x[n] R p w ξ 0 σd 2 J ex [ ] δ Cropped input signal Cropped autocorrelation matrix Cropped cross correlation vector Cropped desired IR Minimum achievable residual echo error variance Variance of the desired signal Steady state excess mean squared error of the NLMS adaptive algorithm Room damping constant 6

21 E w con x con [n] R con x con [n] R con p con Squared amplitude of the first echo reflection Concatenated estimated IR vector Full rank concatenated input vector Full rank autocorrelation matrix of the concatenated input vector concatenated cropped input signal Cropped concatenated autocorrelation matrix Cropped concatenated cross correlation vector 7

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23 Abbreviations AEC APA AR DT DTD ERLE FIR FRLS IIR IR ISU LMS LS MAE MSE NLMS NLP PDF POWER PSD SA SAEC SD SNR Acoustic Echo Cancelation Affine Projection Algorithm Auto Regressive Double Talk Double Talk Detector Echo Return Loss Enhancement Finite Impulse Response Fast Recursive Least Squares Infinite Impulse Response Impulse Response Input Slide Unit Least Mean Square Least Squares Mean Absolute Error Mean Square Error Normalized Least Mean Square Non linear Processing Probability Density function Pairwise Optimal Weight Realization Power Spectrum Density Sign Algorithm Stereophonic Acoustic Echo Cancelation Steepest descent Signal to Noise Ratio 9

24 STFT TUC WGN Short Time Fourier Transform Tap Update Control White Gaussian Noise 10

25 Chapter 1 Introduction 1.1 Motivation and Background Stereophonic Acoustic Echo Cancelation overview Acoustic Echo Cancelation (AEC) is necessary for communication systems such as full-duplex hands-free teleconferencing systems, since it reduces the echo caused by the loudspeaker to microphone coupling [2]. Adaptive filtering techniques [3] are usually used to identify the loudspeaker to microphone impulse response. The estimated impulse response is used to synthesize an echo signal estimate which is subtracted from the actual echo signal thus producing lower echo levels. In the single channel case, AEC employed in such a manner manages to simultaneously identify the loudspeaker to microphone impulse response (in the near-end room) and achieve low echo levels, regardless of the speaker to microphone impulse response in the far-end room. Stereophonic communication systems provide a more lifelike conversation quality, since the participants can use the added spatial information to distinguish the current speaker location. However, due to the high correlation between the two loudspeaker signals, the adaptive algorithm employed in a Stereophonic Acoustic Echo Canceler (SAEC) faces two major problems: the first is a very slow convergence rate [4], and the second is convergence to a local minima point dependent of the far-end room speaker to 11

26 CHAPTER 1. INTRODUCTION 12 microphones impulse response (nonuniqueness problem) [5]. This work discusses a stereophonic system which consists of two microphones in the far-end room connected to two loudspeakers in the near-end room, and one microphone in the near-end room connected to a single loudspeaker in the far-end room. It is of note that the analysis for a second set of a near-end room microphone connected to a far-end room loudspeaker is easily deducted due to the problem symmetry. In the stereophonic system discussed, a single speaker signal in the far-end room is received by the far-end room microphones and transmitted to the near-end room loudspeakers. The two loudspeakers signals are then received by the near-end room microphone and transmitted back to the far-end room loudspeaker, thus producing the stereophonic echo. Many solutions to the SAEC problems were suggested during the past years. One of the first solutions suggested using only one adaptive filter per channel [6]. A different approach is to decorrelate the loudspeaker signals by adding a white uncorrelated noise to the far-end room microphones signals [7]. This approach proved to be impaired since the power of the noise needed to be added for acceptable convergence rate degraded the speech quality to unacceptable levels. A partial solution for this is published in [8], [9] which suggests spectral shaping of the noise so its PSD will match the spectrum of the loudspeaker signal. A different approach, which was published in [7], is decorrelation in the frequency domain. This work suggests the use of interleaving non-overlapping comb filters for each reference signal, so that each reference signal has its own frequency domain. Yet another attempted solution, suggested by [10], was the use of first order all-pass filters with unity gain in which the reference signals are filtered. To produce the decorrelation effect, the pole/zero pair value in the filters is randomly changed while the unity gain is kept. A higher order implementation of the filter is suggested by [11]. Another approach for solving the nonuniqueness problem was introduced in [12]. This solution tries to reduce the inter-channel correlation by using an exclusive maximum tap criterion for selecting the adaptive filter taps. This technique requires the use of a

27 CHAPTER 1. INTRODUCTION 13 Non Liner Process (NLP) preprocess to produce satisfactory results. An APA and RLS implementation of this approach can be found at [13]. A similar approach was introduced by [14], which suggested to periodically freeze one of the system s adaptive filter to enable the second filter to adapt independently. One of the first solutions to produce satisfactory results was published in [4]. The technique described in this article suggested preprocessing the far-end microphone signals with a half wave rectifier NLP in order to reduce the correlation between the two signals. This decorrelation technique used with NLMS adaptive algorithm proved to be slow converging, so it was additionally suggested in [4] to employ a robust, fast converging two channel Fast Recursive Least Square (FRLS) algorithm [15] in the SAEC system. [16] proposed a method of estimating the coherence between the two SAEC channels and using this estimation for controlling the levels of the NLP applied to the SAEC input signals. The combination of the NLP with the FRLS did provide good convergence properties, but since the FRLS is a Least Squares (LS) based algorithm, the suggested system had a poor tracking ability which is crucial for a real time, full-duplex communication systems. A NLMS based solution which provides both good convergence speed and a tracking ability is introduced in [5]. This solution suggests to periodically alter one of the far-end room speaker to microphone impulse responses by applying an Input-Slide Unit (ISU) on one of the loudspeaker signals. It is proven by [5] that this technique forces the adaptive process to converge to the true solution of the system. In order to increase the convergence rate of the SAEC system using the input-slide technique a new adaptive algorithm based on the parallel subgradient projection technique is introduced in [17] (Uniform Weight Pairwise Subgradient Projection - UW-PSP), [1] (Pairwise Optimal WEight Realization - POWER). The introduction of the ISU and the POWER algorithm seem to provide a good solution to both the nonuniqueness and the slow convergence problems of the SAEC. Thereby research may be directed to enhancing the convergence quality of the algorithms operating in the SAEC environment.

28 CHAPTER 1. INTRODUCTION 14 The current available solutions to the stereophonic echo problem focus mainly on solving problems caused by the high correlation between the two far-end room microphone signals (the near-end room loudspeakers signals). These solutions assume the existence of a Double Talk Detector (DTD), a device which freezes the adaptive filters adaptation if a double talk interval in the near-end room is detected. However, in the presence of a constant near-end room interference, the existing solutions are unable to converge and therefore do not function properly. Additionally, the adaptive filters operating in the SAEC environment are affected by a more demanding convergence scenarios when compared to the single channel AEC operating under the same conditions. The added convergence difficulty is generated by the addition of the second adaptive filter which results in additional tap mismatch noise, and the addition of a second desired near-end room Impulse Response (IR). The additional near-end room IR impulse response tail introduces additional noise to the SAEC error signal. Due to the asymmetrical nature of speaker and microphone placement in the acoustic rooms, it is possible that one of the SAEC adaptive channels consists of smaller energy IR taps. In this case the weak channel error signal might be negligible when compared to the dominant channel error signal, therefore reducing the weak channel convergence quality [18]. In the search of proper solution to these problems, this work suggests the use of a noise and Double Talk (DT) resilient adaptive algorithm in collaboration with the POWER stereophonic echo cancelation algorithm. To produce a complete solution, the noise and double talk resilient algorithm should be able to converge during constant near-end room interference (continuous double talk interval). Additionally, the algorithm must produce good results for a low SNR environment in order to comply with the problem of a weak channel convergence in the presence of a dominant channel Existing DT resilient algorithms Many solutions to the DT problem have been suggested during the past years. A major class of DT robust algorithms is inspired by the theory of robust statistics [19]. This

29 CHAPTER 1. INTRODUCTION 15 class exhibits robust performance to short bursts or impulsive noise (outliers) present in the near-end desired signal. The basic idea for this class of algorithms is to use a cost function with a bounded gradient for high error values in the error minimization process. Subsequently, the algorithm exhibits lower divergence during near-end disturbance as long as the disturbance is short timed. When a DTD device is employed in the AEC system, this class of outlier robust algorithms can be used as long as the misdetection period of the DTD is relatively short. One of the simplest form of gradient bounded algorithms is the Sign algorithm (SA) [3] (also known as sign LMS), which minimizes the Mean Absolute Error (MAE) cost function. While the SA exhibits good robustness to near-end interference, it possess a slower convergence rate compared to the NLMS algorithm, and does not converge to a zero variance error value even without any interference in the near-end room [20]. More advanced outlier robust algorithms which apply the bounded gradient approach include the Proportion Sign Algorithm (PSA) [21], Adaptive Threshold Nonlinear Algorithm (ATNA) and the algorithms described in: [22], [23], [24], [25] and [26]. Another method for dealing with DT situation is to use a variable step size, which slows the adaptation once a DT situation is declared by the algorithm. Such algorithms include [27] and [28]. Another method is to switch between adaptation algorithms once a DT situation is declared. These algorithm are presented by: [29] [30], [31], [32] and [33]. [34] uses an additional adaptive filter which operates when a DT situation is declared by a Giegel DTD to distinguish between a DT situation and a echo path change. Since the algorithms listed above are dependent of the length of the disturbance or of the existence of a DTD device in the AEC system, they cannot be used in a SAEC system which requires convergence throughout a continuous DT situation. Although DT situation in a typical conversation occurs for around 20% of the time [35] for which the previous listed DT robust algorithms can be applied (with the use

30 CHAPTER 1. INTRODUCTION 16 of a DTD for the algorithm which requires it), there are some cases in which the near-end interference is continuous [36] and therefore a special class of DT robust algorithms should be applied. [37] states that since the DT signal and the echo signal are statistically independent a correlation domain LMS (CLMS) should converge even in the presence of DT. This work is modified in the Expanded CLMS [38] which adds weighted time interval correlation values to the adaptation scheme, and in [39] which uses adaptive lattice filter to accelerate the convergence rate of the algorithm. [40] proposed to inject a Maximum Length (ML) sequence to the far-end microphone signal, and to use this sequence to adapt the algorithm during DT situation. It is also proposed in [40] to decorrelate the input-reference signal in order to accelerate the convergence during single talk. In this work, the author claims that the ML sequence injection cannot be observed by the near-end listener. This work is further researched in [41] and [42]. A different DT robust approach is proposed in [36], which models the near-end interference as the output of an AR system and uses a Prediction Error algorithm to simultaneously estimate both the AR coefficients and the desired Impulse Response coefficients. 1.2 Overview of the Thesis A new variable step size adaptive algorithm is presented in this work. The algorithm is derived from the Sign Algorithm (SA), and it is noise and double talk resilient. The choice of the SA as the foundation of the noise and double talk resilient algorithm is twofold. Firstly, the SA is a noise robust algorithm due to its bounded gradient. Secondly, the binary decision type of the SA could reduce the correlation between the far-end room microphones signals, thus making the algorithm attractive to stereophonic echo cancelation systems. In order to derive this noise and double talk resilient algorithm, the optimal step size of the SA is found under a noisy environment assumption. The optimal step size is than used to calculate a simple and efficient sub-optimal step size. The convergence limit of the algorithm was analyzed and found to be independent of the

31 CHAPTER 1. INTRODUCTION 17 environment level of interference, i.e. the algorithm converges to the same limit for a noisy or noiseless environment. These findings produce an efficient solution to the demands given above, since the algorithm will continue to converge to the same limit both during a constant interference in the near-end room and in a low SNR environment. Furthermore, the convergence limit indifference to the environment noise level enables the noise and double talk resilient algorithm to converge during double talk in the near-end room. This feature gives the algorithm a significant advantage which enables the use of the algorithm in an AEC system with a low performance DTD or in an AEC system with no DTD device at all. The main disadvantage of the suggested noise and double talk resilient algorithm is its initial convergence rate. This relatively slow rate is derived from the division of the algorithm step size by the residual error energy level, a variable which is expected to have high values at the beginning of the convergence and at a state of reconvergence. In order to solve this problem, the noise and double talk resilient algorithm was integrated in a new algorithm based on the POWER stereophonic acoustic echo canceler algorithm. At initial convergence or reconvergence, the filters tap values are derived from the NLMS algorithm, while at the remaining time the filters tap values are derived according to the noise and double talk resilient algorithm. The choice between the two filters tap update methods is decided by two ghost filters. Each of the ghost filters converges using a different tap update equation. The ghost filter with the lower error energy level chooses the favorable system filter tap update method. This algorithm was checked under several convergence simulations in which its performance was compared to the POWER and NLMS algorithms under the same convergence conditions. It was found that the noise and double talk resilient algorithm combined with the POWER algorithm produced better performance compared with NLMS and POWER algorithms. The advantages of the noise and double talk resilient algorithm combined with the POWER algorithm were especially noted at low SNR environments, and under constant DT interference.

32 CHAPTER 1. INTRODUCTION 18 The structure of this work is as follows. Chapter 2 introduces and formulates the problems of the SAEC system: the slow convergence rate and the nonuniqueness problem, chapter 2 also describes and compares the current solutions given to the SAEC problems. In chapter 3 a new suboptimal variable step size SA is derived and shown to be noise and DT robust for a continuous DT situation. Experimental results are given to support the theoretical results of this chapter. Chapter 4 describes the added difficulties caused by the IR to filters length mismatch under which the adaptive filters in the SAEC system work, and a new algorithm which integrates the algorithm derived in chapter 3 in the NLMS based POWER algorithm is introduced. Experimental results which demonstrates the new algorithm advantages are displayed. The thesis is concluded with a brief summary and directions for future research in chapter 5.

33 Chapter 2 SAEC problem formulation and current solutions 2.1 Introduction This chapter describes and formulates the SAEC problems, and discusses the existing SAEC solutions. The chapter outline is as follows: Section 2.2 describes and formulates the problems of SAEC system. Section 2.3 describes and demonstrates the many solutions to the SAEC problems which have been suggested during the past years. Starting from the early attempts of decorrelation by addition of WGN and proceeding in chronological order to the most recent SAEC solutions. This chapter is concluded with a brief summary and conclusions in section Problem formulation In this section the SAEC scheme is described, and the SAEC normal equation and solution are formulated, followed by the explanation of SAEC two major problems: slow convergence rate and the convergence to a local minimum point SAEC scheme Fig. 2.1 describes the SAEC system. It is to be noted that only one microphone channel is analyzed in the near-end room. The analysis for the second channel is easily deducted 19

34 CHAPTER 2. SAEC PROBLEM FORMULATION AND CURRENT SOLUTIONS 20 Figure 2.1: SAEC scheme. from the given description due to the symmetry between the channels SAEC normal equations The input reference vector sample can be defined as the multiplication of the source signal s[n] with each of the far-end room impulse responses g i : x i [n] = g T i s[n], i = 1, 2 (2.1) each input vector x i [n] thus can be defined as [ x i [n] = x i [n] x i [n 1] x i [n N + 1] ] T, i = 1, 2 (2.2) and the two input vectors x 1 [n] and x 2 [n] can be grouped to form the unified input vector x[n]: [ ] T x[n] = x T 1 [n] x T 2 [n] (2.3) The two estimation filters ŵ 1 [n] and ŵ 2 [n] can also be grouped to form the unified estimation vector ŵ[n]: [ ] T ŵ[n] = ŵ T 1 [n] ŵ T 2 [n] (2.4) It can be concluded that the estimated echo sample can be defined as: y[n] = ŵ[n] T x[n] (2.5)

35 CHAPTER 2. SAEC PROBLEM FORMULATION AND CURRENT SOLUTIONS 21 Using (2.5), (2.4) and (2.3), the residual echo sample can be defined as: e[n] = d[n] y[n] = d[n] ŵ[n] T x[n] (2.6) Using the Mean Squared Error (MSE) cost function, the estimation problem can be formulated as: for which the solution is known to be: arg min ŵ R L E{e 2 [n]} (2.7) R[n] ŵ[n] = p[n] (2.8) where R[n] is the reference vector autocorrelation matrix and p[n] is the reference to desired sample cross correlation vector: R[n] = E{x[n] x T [n]} (2.9) p[n] = E{x[n] d[n]} (2.10) Convergence rate A well known fact is that speech is a highly correlated signal and hence its autocorrelation matrix (R[n]) is ill conditioned. The SAEC reference vector as defined by (2.3) adds an additional obstacle. The input reference vectors can be related using the definitions given above as: g T x 2 1[n] = g T x 1 2[n] (2.11) which implies that the reference signals are linearly dependent and hence R[n] is not full rank and not invertible. It is known that the convergence rate of steepest decent (SD) adaptive algorithms is faster as the condition number of R[n] is lower. Since (2.11) implies that the condition number of the matrix goes to infinity, SD adaptive algorithms will not be able to converge. In the practical case though, since the length of the desired impulse response (L) is much longer than the length of the estimation vector (N), R[n] is invertible, but is ill conditioned even for a white signal, and therefore the convergence rate of the stereo case is slow compared to the single channel case [4].

36 CHAPTER 2. SAEC PROBLEM FORMULATION AND CURRENT SOLUTIONS Nonuniqueness problem Using the definitions above, the input reference vector can also be represented as: [ x i [n] = g T i ] T s[n] g T s[n 1] g T s[n N] = Gi s[n] (i = 1, 2) (2.12) i i where G i is the impulse response matrix defined as: g i,0 g i,1 g i,m g i,0 g i,1 g i,m G i = g i,0 g i,1 g i,m g i,0 g i,1 g i,m 1 (i = 1, 2) (2.13) and M is the far-end room speaker to microphone IR length. Using this definition, the residual echo can be expressed as: e[n] = ğ T [n] s[n] (2.14) where ğ[n] = [ (w 1 ŵ 1 [n]) T G 1 + (w 2 ŵ 2 [n]) T G 2 ] T (2.15) (2.14) is used in the MSE cost function to get: E{e 2 [n]} = ğ T [n] E{s[n] s T [n]} ğ[n] (2.16) SD adaptive algorithms search the performance functions expressed in (2.16) heading towards the minima point indicated by the gradient which can be expressed by: = E{e2 [n]} ğ T [n] = E{s[n] s T [n]} ğ[n] (2.17) Usually for SD adaptive algorithms, the algorithm is said to have converged when its gradient is equal to 0. Looking at (2.17) and presuming that the autocorrelation matrix of the input speech signal is non zero and invertible, it is clear that the gradient is equal zero when ğ[n] is zero i.e. (2.15) is zero. Looking at (2.15) it is clear that its homogenic equation has infinite number of solutions depending on G i, which implies that even if the adaptive algorithm applied in the SAEC system is fully converged, it is not guaranteed

37 CHAPTER 2. SAEC PROBLEM FORMULATION AND CURRENT SOLUTIONS 23 that it has converged to the true solution, i.e. w i ŵ i [n], but to any other local minima point. The local minima point is dependent on G i which implies that any change in the far-end room source to microphones impulse responses (e.g. active talker change) causes momentarily high echo levels until the algorithm reconverges to another local minima point. 2.3 Current solutions Many solutions to the SAEC problem were suggested over the past years. Few of them are covered starting from a short historical background which is followed by the current solutions Legacy solutions Single channel One of the first solutions was suggested by [6] which offered using a single adaptive filter and a single reference signal to cancel the echo. The motivation for this solution is that the echo sample can be expressed in the Short Time Fourier Transform (STFT) domain as: D(f, n) = [ W1 Ǵ1 ] + W2 X1 (f, n) (2.18) Ǵ 2 where D(f, n) is the STFT of the desired echo sample, W i (i = 1, 2) is the fourier transform of the desired near-end room loudspeaker to microphone IR, Ǵ i is the far-end room speaker to microphone IR fourier transform and X 1 (f, n) is the STFT of one of the SAEC input vectors Equation (2.18) implies that only one reference signal and one adaptive filter is needed. However, from looking at equation (2.18) it is clear that this solution heavily relies on each of the far end room impulse and therefore cannot assure convergence to the true system solution. Time domain decorrelation A known approach to decorrelate two highly correlated signals is an addition of white noise to both of the signals. The problem with this solution is that to achieve satisfactory

38 CHAPTER 2. SAEC PROBLEM FORMULATION AND CURRENT SOLUTIONS 24 levels of echo attenuation, the power of noise needed to be added causes major degradation of speech quality [7]. A solution for this was introduced by [8], which suggested spectral shaping of the noise so its PSD match the spectrum of the reference speech signal. This solution did manage to hide the noise in each of the reference signals, but when combined together to form the stereophonic channel it was discovered that the spatial identification, which is the essence of stereo communication, was lost due to this addition. Frequency domain decorrelation An approach which has been suggested in the frequency domain is using interleaving nonoverlapping comb filters for each reference signal so that each reference signal has its own frequency domain [7]. Again this solution did not achieve satisfactory results. Another solution which has been suggested by [10] is the use of first order all pass filters with unity gain in which the reference signals are filtered, the pole/zero pair value in the filters is randomly changing while the unity gain is kept. This attempt has proved to degrade both the speech quality and the spatial affect. This is caused by the phase shift due to the change of the pole/zero movement. This algorithm was improved later by [11] which offered a high order implementation of [10] Half wave rectifier The first solution which managed to produce satisfactory results was published in [4], in which it was suggested to preprocess the input reference signals with the following non-linear process (NLP): x 1 [n] = x 1 [n] + α x1[n]+ x 1 [n] 2 x 2 [n] = x 2 [n] + α x2[n]+ x 2 [n] 2 (2.19) equation (2.19) is known as the Benesty rectifier (after the author s name). It is clear that the higher the value of the nonlinear coefficient α is, the higher the decorrelation is. Benesty checked that α can go up to 0.5 without major loss of speech quality. Fig. 2.2 shows the coherence function of two typical input reference signals before (blue) and after (green) the NLP process with α = 0.3. It is clearly seen by observing Fig. 2.2 that the NLP greatly reduces the correlation between the two signals. Benesty found

39 CHAPTER 2. SAEC PROBLEM FORMULATION AND CURRENT SOLUTIONS Coherence α=0 α= Freq[Hz] Figure 2.2: Coherence of NLP processed signals. that the decorrelation was not sufficient to accelerate the convergence speed of normal NLMS algorithm so he suggested a stable FRLS algorithm which he developed for the two channel case [15]. Fig. 2.3 shows the misadjustment of a SAEC system with and without the Benesty rectifier for NLMS and FRLS adaptive process using speech input signals. It is seen that the NLP does bring the misadjustment to a lower level, and while the NLMS solution still gives slow convergence rate, the FRLS solution gives satisfactory results. The use of FRLS algorithm achieves initial fast convergence but has a very poor tacking capability. This causes the Benesty solution to handle any changes in the near end impulse response very poorly. Fig. 2.4 shows the behavior of NLMS and FRLS with and without NLP for near end impulse response change at around samples. While the NLMS based algorithm quickly returns to their pre-change misadjustment levels, the FRLS based algorithms convergence speed is much slower. It is clear that in a full duplex communication system this kind of changes can occur at any time and hence the Benesty solution did not provide a complete solution to the SAEC problem Input slide technique In 2001 Sugiyama [5] suggested a process which enables the use of memoryless SD algorithms while still achieving desirable levels of cancelation and acceptable speed of conver-

40 CHAPTER 2. SAEC PROBLEM FORMULATION AND CURRENT SOLUTIONS NLP RLS NLP NLMS RLS NLMS 1 Misalignment [db] Iteration [n] x 10 4 Figure 2.3: Convergence using Benesty s rectifier. gence. The process relies on an ISU which is shown in Fig The ISU is a 2 weight FIR filter which periodically delays its input signal depending on the period of the input signal c k. c k is a periodical signal whose value equal 0 for half the period and 1 for the rest of the period. Sugiyama suggested attaching the input sliding unit to one of the input reference signals x i [n], thus if x 1 [n] is used, the output from the ISU x 1 [n] can be described as: x 1 [n] c 1 x 1 [n] = (2.20) x 1 [n 1] c 0 by using the impulse response matrix defined in (2.15) we can define an impulse response matrix of channel 1 for the c 0 state as: [ ] Ĝ 1 = 0 M 1 G 1 (2.21) to maintain similar size of impulse response matrix The impulse response matrix of channel 1 for the c 1 state is defined as: [ ] G 1 = G 1 0 M 1 (2.22) and the impulse response matrix of channel 2 for both states (the input sliding unit is processing just one channel): [ ] G 2 = G 2 0 M 1 (2.23)

41 CHAPTER 2. SAEC PROBLEM FORMULATION AND CURRENT SOLUTIONS NLP RLS NLP NLMS RLS NLMS 0 1 Misalignment [db] Iteration [n] x 10 4 Figure 2.4: FRLS tracking. Figure 2.5: Input slide unit. While introducing the nonuniqueness problem it has been proven that the SD algorithm has converged when the equation given in (2.15) is equal zero. Using this fact and equations (2.21), (2.22) and (2.23), It can be said that the SD algorithm is converged for the c 0 state when: [ (w 1 ŵ 1 [n]) T Ĝ1 + (w 2 ŵ 2 [n]) T G 2 ] T = 0 (2.24) and that the SD algorithm is said to be converged for the c 1 state when: [ (w 1 ŵ 1 [n]) T G 1 + (w 2 ŵ 2 [n]) T G 2 ] T = 0 (2.25) by subtracting equations. (2.24) and (2.25) it can be assumed that the SD algorithm is converged for both states when: [ (w 1 ŵ 1 [n]) T (Ĝ1 G 1 )] T = 0 (2.26)

42 CHAPTER 2. SAEC PROBLEM FORMULATION AND CURRENT SOLUTIONS 28 since in the general case Ĝ1 G 1, (2.26) leads to: w 1 = ŵ 1 [n] (2.27) using equation (2.27) with (2.24) results in: w 2 = ŵ 2 [n] (2.28) and hence the algorithm converges to the true solution. It is clear that since the algorithm is based on SD algorithms, it has no tracking problem and thus can handle any change in the near-end room impulse response. Fig. 2.6 describes a SAEC system with WGN as input. For NLMS and FRLS based solution using: no preprocessing, Benesty s rectifier and input sliding technique. The input signals are identical (i.e. x 1 [n] x 2 [n] ) so there is high correlation between them. To represent far end room impulse response change an added delay is introduced to x 2 [n] starting from sample and on samples. The period of c k is set to 10 samples and the change from one state to the other occurs instantly. It is clear that the input sliding technique converges to the true solution faster than the other given solutions. It is also noted that the input sliding technique preformed better when using SD algorithms than when using memory based algorithms. Furthermore, it is seen that the changes in the far end room do not affect the input sliding convergence. When speech signals are used as inputs to the input sliding unit, a quick change between states results in audible clicks. In order to reduce these kinds of artifacts a more subtle change between states should be applied. In order to reduce the aliasing affects after the change between two states, the period time of c k should be sufficiently long. For speech signals sampled at 8 [khz] a period of around 2000 samples and a change time of 200 samples is sufficient to reduce these artifact to acceptable levels UW-PSP Looking at the NLMS weight update equation: ŵ[n + 1] = ŵ[n] + µ d[n] ŵt [n] x[n] x T [n] x[n] x[n] (2.29) it is clear that the next solution of the system ŵ[n+1] is defined exclusively by the present solution of the system ŵ[n], the input reference vector x[n] and the desired sample d[n]

43 CHAPTER 2. SAEC PROBLEM FORMULATION AND CURRENT SOLUTIONS Misalignment [db] slide RLS slide NLMS NLP RLS NLP NLMS RLS NLMS Iteration [n] x 10 4 Figure 2.6: Input sliding convergence.. As shown in equation (2.24)-(2.28), it is clear that the input sliding technique produces two different states of the SAEC system whose simultaneous solution leads to the true solution of the SAEC system. When using simple SD algorithms combined with the input sliding technique the states change serially one after the other and the solution zigzags slowly from one state to the other and towards the true solution. A graphic representation for this can be seen in Fig. 2.7 in the middle image, where ν and ν are the two states of the system, h is the desired solution and h k is the current solution. In the same figure, on the left image, the more traditional solution of decorrelating the input reference signals is presented. As seen in the figure for these solutions there exist only one state ν and the attempt is to bring the possible solution h as close as possible to the true solution h, while trying to bring the current solution h k as fast as possible to the possible solution h. This is done by decorrelating the input reference signals. The UW-PSP algorithm introduced in [43], proposes to simultaneously use the information of the two states produced by the input sliding technique to project the current solution to 2 different new solutions, and then to uniformly average them. Since we know the period Q of the input sliding module, it is clear that two input reference vectors x[n] and desired samples d[n] with Q/2 sampled delay between them are from 2 different states. The algorithm can be made more robust by adding more different states (from different time delays) to the uniform

44 CHAPTER 2. SAEC PROBLEM FORMULATION AND CURRENT SOLUTIONS 30 Figure 2.7: Input sliding convergence interpretation taken from [1]. average. This solution is known to reduce the zigzag loss introduced by the traditional input sliding technique. A graphical representation of this algorithm is shown in Fig. 2.7 on the right image POWER algorithm From the analytical description of the input sliding technique and from the graphical interpretation of Fig. 2.7, it is seen that the true solution is located in the intersection of the two subspaces defined by the two states of the input sliding technique. The POWER algorithm proposed in [1], suggests using the information defined by the two states to project the current solution to 2 (or more) new solutions in a similar way to the UW- PSP algorithm. However, instead of averaging the new solutions, [1] suggests to find the intersections of the two half spaces defined by the vectors between the current and each new solutions (for each 2 states) to find the final new solution, thus causing the current solution to advance in less steps towards the true solution. As in UW-PSP, more states can be added by storing more previous and current state reference vectors and desired samples. A block model for two types of POWER algorithms for 4 stages of each state is shown in Fig The first suggestion (on the left image) projects the solutions, and then finds the intersection of each two states until the final solution is achieved. The second

45 CHAPTER 2. SAEC PROBLEM FORMULATION AND CURRENT SOLUTIONS 31 Figure 2.8: POWER algorithm taken from [1]. and less complex suggestion projects the solutions, then averages the new solutions of the same state, and then finds the intersection of the averages for the final new solution. Fig. 2.8 demonstrates the convergence of the two types of POWER algorithm and UW- PSP algorithms with 8 stored states compared to the simple input sliding technique with NLMS and FRLS algorithms, with speech signals as inputs. It is clear that the POWER and UW-PSP algorithms greatly enhance the convergence rate of the traditional input technique. 2.4 Conclusions In this chapter the slow convergence rate and the nonuniqueness problems which are the major problems of the SAEC system are described and formulated. The solutions suggested over the past years are described and compared with each other. The first feasible solution suggested by [4] used NLP to decorrelate the SAEC input signals with minor degradation to the speech quality in the near-end room [44]. This solution also required RLS based adaptive filtering to achieve sufficient convergence speed, which degraded the tracking capabilities of the solution. In order to use SD

46 CHAPTER 2. SAEC PROBLEM FORMULATION AND CURRENT SOLUTIONS NLMS RLS POWER1 q=8 POWER2 q=8 UW PSP q=8 1.5 Misalignment iteration x 10 4 Figure 2.9: POWER and UW-PSP convergence. based adaptive algorithms in the SAEC system, a new approach had to be applied. An approach which enabled the use of memoryless adaptive algorithms is described in [5]. This approach suggested to periodically delay one of the far-end room input signals thus not allowing the SAEC system to converge to any local minima solution but the true solution of the system with little degradation to the near-end room speech quality. This idea was used afterwards in [17] and [1] to further accelerate the SAEC convergence speed. It is apparent that once the ISU preprocessing technique [5] was introduced and proved to solve the nonuniqueness effect with little degradation to the near-end speech quality, and the pairwise subgradient projections algorithms ( [17], [1]) which accelerated the SAEC system convergence rate, the basic problems of the SAEC system received a good answer. Therefore, further research should be applied in order to improve the SAEC system echo cancelation quality, i.e. the levels of ERLE and misadjustment. The current available solutions assume the existence of a Double Talk Detector (DTD) device, which freezes the adaptive filters adaptation if a double talk interval in the near-end room is detected. However, in the presence of a constant near-end room interference, the existing solutions are unable to converge and therefore do not function properly. Furthermore, due to the placement of both the near-end room loudspeakers and microphone, it is possible

47 CHAPTER 2. SAEC PROBLEM FORMULATION AND CURRENT SOLUTIONS 33 that one of the near-end room loudspeaker to microphone impulse responses will consist of higher energy when compared to the second near-end room loudspeaker to microphone impulse response. In such case, due to the use of NLMS based solutions in the existing solutions, the residual echo of the dominant channel prevents the weaker channel from converging. A possible way to reduce this effect is to employ a noise robust algorithm in the SAEC system. The following chapters deals with the derivation of such algorithm, and its implementation.

48 CHAPTER 2. SAEC PROBLEM FORMULATION AND CURRENT SOLUTIONS 34

49 Chapter 3 A unified convergence limit suboptimal variable step size Sign Algorithm 3.1 Introduction In this chapter a new DT and noise resilient algorithm is presented. The chapter outline is as follows: In section 3.2 a DT and noise robust algorithm is derived from the Sign Algorithm (SA), this algorithm is formulated from the optimal variable step size of the SA under a noisy environment assumption. The algorithm is shown to be noise and DT robust and therefore suitable for use in a SAEC system. The theoretical results of the derived algorithm are supported by experimental results which are presented in section 3.3. This chapter is concluded with a brief summary and conclusions in section Variable step size sign based algorithm The SA is known to be a very good candidate for use in a system where there is a frequent DT situation, due to its bounded gradient [19], and therefore is an attractive starting point for a DT robust algorithm construction. Furthermore, the binary decision type of the SA could reduce the correlation between the far-end room microphones signals thus making the SA an even more appealing algorithm for the SAEC case. The 35

50 CHAPTER 3. UNIFIED CONVERGENCE LIMIT SUBOPTIMAL SA 36 SA has different convergence limits for the clean (noiseless) and noisy cases (as can be deducted from [20]), i.e. a SA employed in an AEC system which does not use DTD diverges to the noisy limit in the presence of DT and reconverges to the clean limit when DT is absent. This results in recurring high echo levels after each DT situation. In this section the optimal step size for the SA in the presence of a constant noise source is derived, and a suboptimal step size is constructed from it. The convergence limit of the suboptimal variable step size SA is found, and the resulting algorithm is shown to solve the DT divergence problem of the SA, thus making it a noise and DT robust algorithm and a good candidate for the SAEC system Optimum step size for SA in a noisy environment The SA tap update equation is given by: ŵ[n + 1] = ŵ[n] + µx[n]sign(e[n]) (3.1) Where ŵ[n] are the estimation filter taps, µ is the algorithm step size, e[n] is the current algorithm error and x[n] is the input reference vector. The error sample e[n] is composed from the current residual echo ε[n] and the near-end interference sample η[n]: e[n] = ε[n] + η[n] (3.2) Fig. 3.1 describes this AEC system. Defining the misalignment vector m[n] as the difference between the desired impulse response w and the filter current solution ŵ[n]: m[n] = w w[n] (3.3) the SA misalignment update equation can be expressed as: m[n + 1] = m[n] µx[n]sign(ε[n] + η[n]) (3.4) (3.3) can be used to expressed the residual error ε[n] as: ε[n] = m T [n] x[n] =< m[n], x[n] > (3.5)

51 CHAPTER 3. UNIFIED CONVERGENCE LIMIT SUBOPTIMAL SA 37 Figure 3.1: AEC scheme. Where <, > is the standard euclidian inner product of the l 2 space, thus the l 2 norm of (3.4) can be expressed as: m[n + 1] 2 = m[n] 2 + µ 2 x[n] 2 2µε[n]sign(ε[n] + η[n]) (3.6) assuming that the input reference vector is an iid white Gaussian vector: x[n] N(0, σ 2 x) (3.7) and that the input reference vector and the misalignment vector are statically independent x[n] m[n] (3.8) (even though this assumption is far from being true as the update equation (3.4) clearly states, it is a common assumption in the analysis of adaptive algorithms [20], and can be regarded as true for small values of the step size µ) the residual echo variance can be calculated using (3.5) and (3.8) to be: σ 2 ε[n] = E{ ε[n] 2 } = E{m T [n] x[n] x T [n] m[n]} = σ 2 xe{ m 2 } (3.9) the statistical expectation of the norm of the input reference vector x[n] can be found using assumption (3.7): E{ x[n] 2 } = E{x T [n] x[n]} = N 1 n=0 E{x[n] 2 } = Nσ 2 x (3.10)

52 CHAPTER 3. UNIFIED CONVERGENCE LIMIT SUBOPTIMAL SA 38 where N is the length of the estimation filter ŵ[n]. Multiplying (3.6) with σx 2 and applying the statistical expectation operator while considering (3.9) and (3.10) yields: σε[n 2 + 1] = σε[n] 2 + µ 2 Nσx 4 2µσxE{ε[n]sign(ε[n] 2 + η[n])} (3.11) modeling the residual echo sample and the near-end room interference as uncorrelated white Gaussian random variables: ε[n] N(0, σ 2 ε[n]) (3.12) η[n] N(0, ση) 2 (3.13) η[n] ε[n] (3.14) The remaining right hand side expectation element of (3.11) can be evaluated using the statistical expectation definition (see Appendix 3.5) to be: E{ε[n]sign(ε[n] + η[n])} = 2 π σε[n] 2 σε[n] 2 + ση 2 = 2 σε[n] 2 π σ e [n] (3.15) combining (3.11) and (3.15) results in: 8 σε[n 2 + 1] = σε[n] 2 + µ 2 Nσx 4 µ π σ 2 xσ 2 ε[n] σ e [n] (3.16) Observing the right hand side of (3.16) it can be noted that the equation is quadratic w.r.t. the step size µ, hence the optimum step size can be found by simple derivation and zero equalization: 8 µ (σ2 ε[n] + µ 2 Nσx 4 µ π σxσ 2 ε[n] 2 ) = 0 (3.17) σ e [n] thus producing the optimal step size for the SA in a noisy environment: µ OP T = 2 σε[n] 2 π Nσxσ 2 e [n] (3.18) Suboptimal step size SA Observing (3.18) it can be noted that the equation nominator includes the element σε[n], 2 which represents the current residual echo energy. The estimation of the residual echo energy in a continuous DT situation is a difficult task since the observable error signal

53 CHAPTER 3. UNIFIED CONVERGENCE LIMIT SUBOPTIMAL SA 39 is a combination of the residual echo and the near-end interference (3.2). A method of estimating the energy of the residual echo for DT situation was suggested in [45] but this method heavily relies on the assumption that the residual echo and the near-end interference are uncorrelated therefore making the estimation of σ 2 ε[n] uncertain in the SAEC case. To overcome this obstacle a different approach should be taken: noticing that σ 2 ε[n] in (3.18) is divided by σ 2 x which results in the inverse of the echo return loss enhancement (ERLE), (3.18) can be rewritten as: 2 µ OP T 1 = N 2 π ERLE[n]σ e [n] (3.19) It is a well known fact that the SA has a constant non zero residual echo energy level convergence limit even for the noiseless case [20]. From this it can be concluded that the steady state ERLE of the SA is a constant value in the noisy and noiseless cases. Hence, if the initial convergence of the SA is overlooked, and since the rest of (3.19) expressions (not including σ e [n]) are constants, a new suboptimal SA step size can be formulated: µ Sub OP T = µ σ e [n] And a new DT robust algorithm can be formulated: w[n + 1] = w[n] + µ sign(e[n])x[n] σ e [n] where σ 2 e[n] is the estimated variance of the error signal given by: and 0 < λ < 1 is the regression coefficient. (3.20) (3.21) σ 2 e[n + 1] = λ σ 2 e[n] + (1 λ) e[n] 2 (3.22) Unified convergence limit of the suboptimal SA Continuing from equation (3.16), while replacing the general algorithm step size µ with the suboptimal step size given by (3.20) produces: σε[n 2 + 1] = σε[n] 2 + µ 2 Nσ4 x 8 σe[n] µ 2 π σ 2 xσ 2 ε[n] σ 2 e[n] (3.23) Notice that by using the suboptimal step size in this proof, it is assumed that the derived algorithm (3.22) is able to perfectly estimate the error variance σ 2 e[n]. Subsection deals with the non-perfect error variance estimation effect.

54 CHAPTER 3. UNIFIED CONVERGENCE LIMIT SUBOPTIMAL SA 40 equation (3.23) can be transformed into a geometric series: σε[n 2 + 1] = σε[n](1 2 + µ 2 Nσx 4 8 σε[n]σ 2 e[n] µσx 2 2 π σe[n] ) (3.24) 2 the geometric series in (3.24) converges iff its step size is smaller than 1: π σε[n] 2 8 µnσ2 x (3.25) Thus producing the lower bound limit for the derived algorithm residual error energy. The independence in (3.25) of the interference energy ση 2 implies that the algorithm converges to the same unified convergence limit both in the noisy and the noiseless cases. This attribute is unique since the NLMS or SA algorithms convergence limit is dependent on the near-end interference energy [20]. This result shows that the derived algorithm is a DT and noise robust algorithm, since, unlike the NLMS or SA algorithms, once a DT situation is encountered, the derived algorithm continues to converge towards the same limit (with slower convergence rate) while the SA and NLMS algorithms diverge toward their noisy limit Effect of non perfect estimation of the error variance in the steady state for the suboptimal SA The analysis of the convergence limit for the derived algorithm given above assumes that the error variance is known for each iteration. Since the variance of the error is unknown, and is estimated in the manner given by (3.22) the effect of non perfect variance estimation should be analyzed. Assume that the algorithm has converged to its steady state. The estimated error variance has an estimation error ρ λ [n]: σ e [n] = σ e [n] + ρ λ [n] (3.26) (the lower index λ is added since the estimation error clearly depends on the regression variable λ). Modeling the estimation error in the steady state as a zero mean gaussian noise: ρ λ [n] N(0, σρ) 2 (3.27)

55 CHAPTER 3. UNIFIED CONVERGENCE LIMIT SUBOPTIMAL SA 41 assuming no correlation between the steady sate variance and the estimation error: and assuming that there is no near end interference: ρ λ [n] σ 2 e[n] (3.28) e[n] = ε[n] (3.29) (3.16) is developed in the same manner given above while using the imperfect error variance estimation σ 2 e[n] in the derived step size which results in the following geometric series: Nσ 4 x σe[n 2 + 1] = σe[n](1 2 + µ 2 ˆσ e[n]σ 2 e[n] 2 8 π µσ 2 x ˆσ e [n]σ e [n] ) (3.30) The series given by (3.30) converges iff the geometric step size is smaller than 1. This results in the following condition: σ e [n]ˆσ e [n] π 8 µnσ2 x = σ 2 e min (3.31) where σ 2 e min is the lower bound of the error variance for the derived algorithm (3.25). applying the estimation error expression given by (3.26) on (3.31) results in the following binomial expression: σ 2 e[n] + ρ λ [n]σ e [n] σ 2 e min 0 (3.32) the expression given by (3.32) is convex, and its solution (while neglecting the pathological case of negative error variance) is given by: σ e [n] 1 2 ( ρ λ[n] + ρ 2 λ [n] + 4σ2 e min ) (3.33) Assuming that in the steady state the error variance estimation error is relatively small compared to the actual variance: ρ 2 λ [n] σ2 e min, (3.33) can be approximated to be: σ e [n] 1 2 (2σ e min ρ λ [n]) (3.34) squaring (3.34) and taking the expectation operator, while considering (3.27) and (3.28) results in: E{σ 2 e[n]} σ 2 e min σ2 ρ (3.35) The result expressed by (3.35) states that the non perfect estimation of the error variance causes a slightly higher error variance lower bound than the bound expected by (3.25). The increase in the bound is dependent on the accuracy of the variance estimator, noted by σ 2 ρ, which depends on the statistics of the input signal and the regression variable λ

56 CHAPTER 3. UNIFIED CONVERGENCE LIMIT SUBOPTIMAL SA σ ε [n]=0 2 σ ε [n]=1 2 σ ε [n]=2 2 σ ε [n]=4 Theoretic Limit 25 σ ε 2 [n] Itterantion [n] x 10 6 Figure 3.2: Convergence limit of the proposed algorithm 3.3 Experimental results In this section experimental results which support the theoretical analysis of the previous section are presented. For all channel scenarios the impulse response of the nearend room are generated by [46]. For this section The near-end room dimensions are [ ] [ ] [m], loudspeaker position is [m], while the microphone [ ] position is [m]. The impulse response length is 256 samples, sampled at 16[KHz]. The adaptive filter length is also 256 samples. The adaptive filter initial value is set to Convergence limit In this section the convergence limit of the derived algorithm is compared to theoretical limit given by (3.25). In this simulation the input is a white gaussian noise with σx 2 = 5, the step size of the proposed algorithm is chosen as µ = 1.25E 6 to achieve a lower bound error variance of σε 2 = 30[dB], the estimation variable is set to λ = To test the derived algorithm convergence limit invariance to the interference energy, additive white gaussian noise (AWGN) is injected to the desired signal with energy levels of ση 2 = 0, 1, 2, 4. The simulation result can be viewed in Fig. 3.2, in which the derived algorithm theoretical

57 CHAPTER 3. UNIFIED CONVERGENCE LIMIT SUBOPTIMAL SA λ=0.7 λ=0.8 λ=0.9 λ=0.99 λ=0.999 Theoretic Limit 35 σ ε 2 [db] itterantion x 10 5 Figure 3.3: Estimation error affect on the convergence limit limit is given in magenta, the derived algorithm error variance for the noiseless case is given in blue and error variance for the noisy cases of ση 2 = 1, 2, 4 is given in green, red and cyan respectively. Looking at Fig. 3.2 it is apparent that the derived algorithm does converge to its given theoretic limit for every interference energy level, it is also noted that the speed of convergence is inverse proportionate to the strength of the interference as anticipated Estimation error effects on the convergence limit As shown in subsection the imperfect estimation of the error variance results in higher levels of error variance for the proposed algorithm than those expected by the theoretical limit. The convergence limit is shown to be higher as the variance of the estimation error is higher. In this section this phenomena is displayed. In this simulation the input is a white gaussian noise with σx 2 = 5, the step size of the derived algorithm is chosen to be µ = 0.4E 8 to achieve a lower bound error variance of σε 2 = 55[dB]. The estimation variable λ is given 5 different values ranging form 0.7 to The simulation result can be viewed in Fig. 3.3, in which the theoretical limit is given in yellow, the derived algorithm error variance is given in blue (λ = 0.7), green (λ = 0.8), red (λ = 0.9),cyan (λ = 0.99) and magenta (λ = 0.999). Looking at Fig. 3.3 it is clear that

58 CHAPTER 3. UNIFIED CONVERGENCE LIMIT SUBOPTIMAL SA Suboptimal SA NLMS Robust NLMS SA Misalignment [db] Iteration [n] x 10 6 Figure 3.4: Convergence under AWGN and DT interference the closer the value of λ to 1 (and hence the smaller is the estimation error variance) the closer is the error variance of the derived algorithm in the steady state to the theoretical limit. However one must keep in mind that the algorithm should be employed in a speech based system, in which the input signal is nonstationary and hence λ should be kept small enough to track the rapid changes in the error variance. By observing Fig. 3.3 it can be concluded that λ value of 0.99 should suffice to achieve the expected convergence limit and still posses good tracking ability Convergence under DT and noise interference In this section the convergence of the suboptimal SA under noise and DT interference is demonstrated. The input signal for the system is speech sampled at 16[KHz], the length of the IR are 1500 samples sampled at 16[KHz] and the length of the adaptive filter is 1250 taps. additive white noise is injected to the near end microphone signal to achieve SNR of 10[dB], an additional speech signal is inserted at 0.5E6, 1E6, 1.5E6 and 2E6 iterations to simulate DT situation. The rest of the scenario settings is given at section 3.3. Fig. 3.4 exhibits the behavior of the Suboptimal SA (blue), NLMS (green), the Robust NLMS algorithm described at [31] (red) and the SA (cyan) algorithms for the stated environment. The step sizes of the algorithms are chosen to achieve the same steady state misalignment

59 CHAPTER 3. UNIFIED CONVERGENCE LIMIT SUBOPTIMAL SA Suboptimal SA NLMS Robust NLMS SA 15 Misalignment [db] Iteration [n] x 10 6 Figure 3.5: Convergence under AWGN and continuous DT interference level of 15[dB] for the noiseless case. Observing Fig. 3.4 it is clear that while the NLMS, robust NLMS and SA are unable to converge or maintain their steady state misalignment levels during the DT phase without the use of a DTD device, the suboptimal SA continues to converge toward its unified convergence limit during those intervals without damaging the echo canceling quality. It can be also noted that the suboptimal SA achieves the noiseless 15[dB] steady state misalignment level while the rest of the algorithms achieve a higher steady state level due to the WGN interference, thereby demonstrating the unified convergence level of the suboptimal SA. Fig. 3.5 exhibits the behavior of the above algorithms for a case of continuous DT interference. Observing Fig. 3.5 it is clear that while the algorithms either fail to converge (NLMS and robust NLMS) or exhibit erratic behavior (SA) which results in high echo levels at the end of each DT phase, the suboptimal SA continues to converge towards its unified convergence limit throughout the entire scenario, thereby emphasizing the robustness of the suboptimal SA to both noise and DT interference.

60 CHAPTER 3. UNIFIED CONVERGENCE LIMIT SUBOPTIMAL SA Conclusions In this chapter sign based variable step size algorithm is derived. The variable step size is derived from the optimal step size of the sign algorithm for the noisy case. It is shown that the optimal variable step size implementation of the sign algorithm is a difficult task for real-life noisy situation due to the need of the residual error energy estimation which is a part of the optimum variable step size expression. The reason of the estimation difficulty is due to the fact that in a noisy situation the observable error signal is composed of both the residual error expression and the system noise. To overcome this difficulty a suboptimal variable step size is suggested which is dependent only on the overall error energy instead of the residual error energy. The suboptimal variable step size sign algorithm is shown to posses a unified convergence limit for both the noisy and noiseless cases, thus an AEC in which the proposed variable step size SA is employed converges to same convergence limit for different levels of ambient noise energy, and also during DT intervals. This in contrast to a NLMS algorithm or SA which converges to a different convergence levels depending on the interference energy level, causing high echo levels after each DT disturbance end. This fact shows that the suboptimal step size SA is a DT and noise robust algorithm. The theoretical results derived in this chapter are demonstrated in various convergence scenarios. The unified convergence limit for both the noisy and noiseless cases of the proposed algorithm is very appealing to the SAEC environment due to the noisy and DT like situations under which each channel is working. The DT and noise robustness characteristic of the proposed algorithm are an additional benefits for an AEC since when employing the derived algorithm in an acoustic environment which suffers from low levels of SNR of the desired signal, the proposed algorithm converges to its predetermined convergence limit regardless of the SNR value. Furthermore due to the DT robustness derived from the unified convergence limit, the derived algorithm can be employed in an AEC system

61 CHAPTER 3. UNIFIED CONVERGENCE LIMIT SUBOPTIMAL SA 47 with low performance or no DTD device present in the system. Looking at (3.21) it is clear that the derived algorithm suffers from slow initial convergence or re-convergence rate due to the division by σ e [n], this division causes the step size to be small for large error energy levels, which is normally the case at the start of the adaptive process or when the near-end room impulse response has suddenly changed (reconvergence situation). Due to this fact the derived sign based algorithm should be integrated in a fast converging algorithms like APA based algorithms to enhance its convergence rate, while maintaining the algorithm DT and noise robustness. For the SAEC case, the APA based POWER algorithm can be used with the derived sign based algorithm as the tap update mechanism to achieve this propose. The next chapter describes the implementation of the proposed algorithm into the POWER algorithm. 3.5 Appendix: Derivation of (3.15) To derive (3.15) the statistical expectation definition is used: E{ε[n]sign(ε[n]+η[n])} using assumption (3.14) the PDF can be separated: ε[n]sign(ε[n] + η[n])p DF (ε[n],η[n]) (ε[n], η[n])dε[n]dη[n] (3.36) P DF (ε[n],η[n]) (ε[n], η[n]) = P DF (ε[n]) (ε[n])p DF (η[n]) (η[n]) (3.37) and by using the normal distribution assumption (3.12) and (3.13), the expectation expression (3.38) can be reformulated into: 1 [ 1 ] ε[n]sign(ε[n] + η[n]) exp{ ε2 [n] 2πση 2πσε [n] 2σε[n] }dε[n] exp{ η2 [n] }dη[n] 2 2ση 2 (3.38) the inward integral sign expression is eliminated by dividing the integration to the intervals ε[n] > η[n] in which the sign expression is 1 and the complementary interval ε[n] < η[n] for which the sign expression is -1, in this way the internal integral expression of (3.38) resolves to: 1 [ ε[n] exp{ ε2 [n] 2πσε [n] η[n] 2σε[n] }dε[n] 2 η[n] ε[n] exp{ ε2 [n] 2σ 2 ε[n] }dε[n]] (3.39)

62 CHAPTER 3. UNIFIED CONVERGENCE LIMIT SUBOPTIMAL SA 48 Due to the anti-symmetric nature of the integrand, the expression given in (3.39) can be minimized to: 1 2ε[n] exp{ ε2 [n] }dε[n] (3.40) 2πσε [n] η[n] 2σε[n] 2 Using a simple variable change technique (x = ε 2 [n]) the integral given in (3.40) can be solved to produce the partial result: 1 ε[n]sign(ε[n] + η[n]) exp{ ε2 [n] 2πσε [n] 2σε[n] }dε[n] = 2σ ε[n] exp{ η2 [n] 2 2π 2σε[n] } (3.41) 2 Combining the partial result given in (3.41) with the expression given in (3.38) results in: σ ε [n] πσ η exp{ (σ2 η + σε[n])η 2 2 [n] }dη[n] (3.42) 2σησ 2 ε[n] 2 which is a standard integral (gaussian integral). Solving (3.42) produces the final result given in (3.15) E{ε[n]sign(ε[n] + η[n])} = 2 π σε[n] 2 σε[n] 2 + ση 2 (3.43)

63 Chapter 4 On the effect of the impulse response tail in a SAEC environment 4.1 Introduction This chapter suggests the use of the noise and Double Talk (DT) resilient adaptive algorithm derived in Chapter 3, in collaboration with the POWER stereophonic echo cancelation algorithm. The chapter outline is as follows: In section 4.2 the effects of the Impulse response tail on the SAEC system adaptive filters is presented. section 4.3 represents the SAEC system as two single channel AEC whose desired signal is dipped with constant reaccuring, highly correlated, DT interfearnce. Section 4.4 presents a new algorithm which integrates the suboptimal step size SA derived in in chapter 3 into the POWER algorithm. Experimental results are presented in section 4.5. This chapter is concluded with a brief summary and conclusions in section The IR Tail effect In this section the effect of the IR tail in the SAEC system is analyzed. Firstly the IR tail effect is analyzed in the single channel case and is shown to be negligible for a large enough adaptive filter. The next section analyzes the IR tail effect in the stereo case and shows that the effect of the IR tail causes major degradation in the misadjustment level of the SAEC system, especially when there exists a dominant (higher energy) IR in the 49

64 CHAPTER 4. IR TAIL EFFECT IN SAEC ENVIRONMENT 50 near-end room, when compared to the same adaptation scenario in the single channel case IR tail effect in the single channel case Let us firstly discuss the single channel case. Assume that an N tap long adaptive filter is employed in a noiseless acoustic environment, whose near-end room loudspeaker to microphone to impulse response has a finite length of L, N < L taps ŵ[n] R N (4.1) w R L (4.2) the assumption of a shorter filter in an AEC is reasonable, since acoustic IR tend to be very long (a few thousands taps) while due to computational powers limits the adaptive filter is limited in length. For the AEC environment given above, the desired signal d[n] is given by: d[n] = w T x[n] (4.3) where x[n] is the full rank input vector. assuming that x[n] is a white gaussian vector: x[n] N(0, σ 2 x) (4.4) the full rank autocorrelation matrix R of the input vector can be defined as: R = E{x[n]x T [n]} = σ 2 xi L L (4.5) where I L L is the L L identity matrix. defining x[n] as the cropped input signal, padded with zeros to maintain system order: where 0 N N [ ] T x[n] = x 0 [n] x 1 [n] x N 1 [n] 0 1 (L N) R L (4.6) is the N N zeros matrix. the cropped autocorrelation matrix R of the cropped input vector can be defined as: R = E{x[n]x T [n]} = σ 2 x I N N 0 (L N) N 0 N (L N) 0 (L N) (L N) (4.7)

65 CHAPTER 4. IR TAIL EFFECT IN SAEC ENVIRONMENT 51 the minimum achievable residual echo error variance expression ξ 0 is given by ( [3]): ξ 0 = σ 2 d p T R 1 p (4.8) σ 2 d is the variance of the desired signal, which by using (4.5) can be expressed by: σ 2 d = E{d 2 [n]} = E{w T x[n]x T [n]w } = σ 2 x w 2 (4.9) where is the standard euclidian norm. p is the crosscorrelation vector between the desired signal and the input vector which by using (4.3),(4.6) and (4.7) can be expressed by: p T = E{d[n]x T [n]} = E{w T x[n]x T [n]} = w T R = σxw 2 T (4.10) where w is the cropped desired IR padded with zeros: [ ] T w = w0 w1 wn (L N) R L (4.11) combining (4.8) with (4.7), (4.9) and (4.10) results in the minimum residual echo variance expression: ξ 0 = σ 2 x w 2 σ 2 xw T w = σ 2 x L 1 n=n w 2 n (4.12) Given the minimum residual echo variance ξ 0, The steady state excess mean squared error of the NLMS adaptive algorithm J ex [ ] can be approximated as ( [3]): J ex [ ] µ 2 µ Nσ2 xξ 0 (4.13) where µ is the NLMS algorithm step size. excess MSE of the given scenario: combining (4.12) into (4.13) results in the J ex [ ] µ 2 µ Nσ4 x L 1 n=n w 2 n (4.14) It is known that under the diffuse sound field assumption, the ensemble average σ 2 w the squared room impulse responses decays exponentially (see [47]); i.e., of σw 2 E{w 2 n } = E 1 e δn (4.15) where δ is the damping constant related to the T 60 room character, and E 1 is the squared amplitude of the first echo reflection. taking the statistical expectation operator on J ex [ ]

66 CHAPTER 4. IR TAIL EFFECT IN SAEC ENVIRONMENT 52 while using (4.15) results in the ensemble average of the excess MSE in the steady state for single channel AEC: J 1ch ex = E{J ex [ ]} µ 2 µ Nσ4 xe 1 e δn e δl 1 e δ (4.16) Observing (4.16) it can be deducted that enlarging the value of N produces two countering effects: the first is enlarging the excess error variance due to the raise in the weight misalignment noise which grows as the filter length grows, this is represented by the multiplication by N, and the second is reduction in the same excess error variance due to the diminished effect of the IR tail noise represented by the multiplication with (e δn e δl ). Due to the exponential decay behavior of the IR tail, it can be concluded that for a large enough N the mean excess error eventually converges to a negligible amount, and therefore its effect on the single channel adaptation is insignificant IR tail effect in a stereo environment Now consider two N tap long adaptive filters which are employed in a stereophonic acoustic environment, whose near-end room loudspeakers to microphone impulse response has a finite length of L, N < L taps ŵ 1 [n], ŵ 2 [n] R N (4.17) w 1, w 2 R L (4.18) To simplify the wiener solution representation, the estimated IR can be concatenated into a single vector: [ w con = T T w 1 w 2 using (4.19) the desired echo sample d[n] can be represented by: ] T (4.19) where x con [n] is the full rank concatenated input vector: d[n] = w T conx con [n] (4.20) [ x con [n] = x T 1 [n] ] T w T 2 [n] (4.21)

67 CHAPTER 4. IR TAIL EFFECT IN SAEC ENVIRONMENT 53 and x i, i = 1, 2, are the near-end room loudspeaker input signals. analysis assume that each input signal is a white gaussian noise: for the rest of the x i [n] N(0, σ 2 x i ), i = 1, 2 (4.22) and also assume that an optimal preprocessing is applied to the input vectors x 1 and x 2 resulting in a completely uncorrelated input vectors: x 1 [n] x 2 [n] (4.23) assumption (4.23) is a very loose assumption since for SAEC the input vectors are generally highly correlated. Nevertheless, as it is described in the following analysis, even for the optimal uncorrelated input assumption, the resulting IR tail effect for the stereo case degrades the system convergence significantly. under assumptions (4.22) and (4.23) the full rank autocorrelation matrix of the concatenated input vector R con can be represented by: 0 L L R con = E{x con [n]x T con[n]} = σ2 x 1 I L L (4.24) 0 L L σx 2 2 I L L Defining x i [n], i = 1, 2 as the cropped input signals according to (4.6) and x con [n] as the concatenated representation of the cropped input signal according to (4.21), the cropped concatenated autocorrelation matrix R con can be represented by: σ x 2 1 I N N 0 N (L N) 0 L L 0 R con = E{x con [n]x T con[n]} = (L N) N 0 (L N) (L N) σ x I N N 0 N (L N) L L 0 (L N) N 0 (L N) (L N) (4.25) proceeding in the same manner as for the single channel case, the desired signal variance σ 2 d for the stereo case can be expressed by: σ 2 d = E{d 2 [n]} = E{w T conx con [n]x T con[n]w con} = σ 2 x 1 w σ 2 x 2 w 2 2 (4.26) p con for the stereo case can be expressed by using (4.20), (4.25) and x con [n] definition as: [ p T = con E{d[n]xT con[n]} = E{w T x con [n]x T con[n]} = w T R con = σx 2 1 w T 1 σx 2 2 w T 2 ] T (4.27)

68 CHAPTER 4. IR TAIL EFFECT IN SAEC ENVIRONMENT 54 where the cropped desired response w i, i = 1, 2 is defined similar to (4.11). using (4.25), (4.26) and (4.27) on (4.8) results in the minimum residual echo variance expression for the stereo case: L 1 ξ 0 = σx 2 1 w σx 2 2 w 2 2 (σx 2 1 w σx 2 2 w 2 2 ) = σx 2 1 n=n L 1 w1,n 2 + σx 2 2 w2,n 2 (4.28) n=n using the result given by (4.28) on (4.13) while keeping in mind that the SAEC system consists of 2N filter taps and that the input vector variance is given by: results in the excess MSE for an NLMS based SAEC: σ 2 x con = σ 2 x 1 + σ 2 x 2 (4.29) J ex [ ] µ L 1 2 µ 2N(σ2 x 1 + σx 2 2 )(σx 2 1 w1,n 2 + σx 2 2 n=n L 1 n=n w 2 2,n) (4.30) using (4.15) as in the single channel case and taking the ensemble average over (4.30) while keeping in mind that δ is a room characteristic and therefore is identical for each cannel, results in the ensemble average of the excess MSE in the steady state for stereo channel AEC: J 2ch ex µ 2 µ 2N(σ2 x 1 + σ 2 x 2 )(σ 2 x 1 E 1 + σ 2 x 2 E 2 ) e δn e δl 1 e δ (4.31) where E i, i = 1, 2 are the squared magnitudes of the first reflections desired IR respectively. Symmetrical case Consider the case of a symmetrical SAEC system, i.e. the case where the near-end room loudspeakers to microphone distance is the same for both channels and the arrangement of the loudspeakers and microphone is symmetrical, resulting in an identical value of the first IR reflection energy: E 1 = E 2 (4.32) furthermore assume that the input signal energy is equal: σx 2 1 = σx 2 2 (4.33) using (4.32) and (4.33) on (4.31) results in the excess MSE for the symmetric stereo channel case: J 2ch sym µ 2 µ 8Nσ4 x 1 E 1 e δn e δl 1 e δ = 8J 1ch ex (4.34)

69 CHAPTER 4. IR TAIL EFFECT IN SAEC ENVIRONMENT 55 the result represented in (4.34) indicates that each adaptive filter employed in the SAEC environment experience an IR tail noise energy 8 times greater than the energy of the noise the filter experience if the same filter is employed in the single channel AEC. In fact to produce the same level of IR tail noise for the symmetrical case the input energy level has to be divided by 2 2, resulting in a reduced speech quality in the near-end room. The amplified excess steady state MSE results in a higher misadjustment and thus a poorer echo cancelation quality. Keep in mind that the convergence rate of a SAEC system is much slower when compared to an AEC system. Therefore both the convergence rate and the steady state error of a SAEC system is degrared when compared to a single channel AEC system. The same analysis can be performed for the n channel case for which the result is expressed by: J nch sym = n 3 J 1ch ex (4.35) For the n channel case the input energy has to be divided by 2 n to achieve the same excess MSE as the single channel case, which is unacceptable for a telecommunication system. Dominant cannel case Stereophonic telecommunication is a real life application. For this reason the symmetrical assumption (4.32) does not represent most SAEC scheme cases. Furthermore due to the varying placements of the near-end microphones and loudspeakers, one dominant channel may exist in the stereo environment whose desired IR energy is larger than the other channel. Consider a SAEC environment in which there exists one dominant channel i.e.: E 1 E 2 (4.36) keeping the assumption of same energy levels of the input signal (4.33) the excess steady state MSE for the dominant channel case SAEC can be represented by: J 2ch dom µ 2 µ 4Nσ4 x 1 E 1 e δn e δl 1 e δ (4.37) Observing (4.37) it can be deducted that while the dominant cannel experiences only 4 times the excess MSE of the single channel case (this in contrast to (4.34) which is 8 time

70 CHAPTER 4. IR TAIL EFFECT IN SAEC ENVIRONMENT 56 the excess MSE of the single channel case). The weak channel filter ŵ 2 [n] experiences an excess MSE of a few orders above the excess MSE the filter experiences in the single channel case due to (4.36). This fact causes the weak channel to produce a poor echo cancelation quality, and even to diverge altogether, causing the overall (both channels) echo cancelation quality to drop significantly. 4.3 SAEC error as a DT interfearance The SAEC system goal is to estimate the system s near-end loudspeaker to microphone impulse response (w 1 and w 2) by using the adaptive filters ŵ 1 [n] and ŵ 2 [n], thus producing the estimated echo sample y[n] which is subtracted from the desired signal d[n] resulting in the residual echo sample e[n]. The residual echo signal expression given by: e[n] = d[n] y[n] (4.38) can be expanded into each channel s contribution both in the near-end room and at the adaptive filters output: e[n] = (w T 1 x 1 [n] + w T 2 x 2 [n]) (ŵ T 1 [n]x 1 [n] + ŵ T 2 [n]x 2 [n]) (4.39) where x i [n], i = 1, 2 are the far-end room microphone signals: x i [n] = g T s[n], i = 1, 2 (4.40) i s[n] is the far-end room source signal, and g i, i = 1, 2 are the far-end source to microphone impulse responses. by manipulating equation (4.39), each input signal contribution can be regrouped: e[n] = w T 1 x 1 [n] ŵ T 1 [n]x 1 [n] + w T 2 x 2 [n] ŵ T 2 [n]x 2 [n] (4.41) to produce the following residual error expression: e[n] = e 1 [n] + e 2 [n] (4.42) Equation (4.42) implies that each of the adaptive filter adaptation process is disturbed constantly by the residual error of the other filter.

71 CHAPTER 4. IR TAIL EFFECT IN SAEC ENVIRONMENT 57 The error expression given by (4.42) is very similar to the error expression of another very common single channel acoustic echo cancelation problem known as double talk situation. In a DT situation a near-end room speaker starts talking simultaneously with the far-end room speaker thus adding uncorrelated noise to the desired signal d[n]. The SAEC problem can therefore be viewed as two single channel AECs whose desired signal is constantly contaminated with a DT interference. Traditional AEC DT solutions utilizes a device called Double Talk Detector (DTD) to overcome the DT problem, the DTD task is to identify a DT situation in order to freeze the adaptation of the system identification process. Reliable DT detection is crucial in order to avoid the divergence of the adaptive system identification algorithm. The detection of DT situation is generally based on a comparison of some attribute between the loudspeaker and the microphone signals. This attribute can be amplitude [48], statistic correlation [49], [50] etc. Observing equation (4.42) again, it is clear that this form of solution cannot be applied to the DT SAEC due to the fact that whenever the residual echo of the first adaptive filter e 1 [n] is present, the interfering DT signal of the second adaptive filter e 2 [n] is also present, and vice versa. This fact is due to the common source of the two channel signals s[n] (4.40) i.e. at no time there can be exclusive filter adaptation. Another problem which differ the DT SAEC system from the single channel AEC DT model, is that the residual echo signal and the DT interference signal in the single channel AEC are uncorrelated since they are originating form two different speakers, on the other hand, in the SAEC system, both e 1 [n] and e 2 [n] are originated from the same source s[n] (4.40) and are therefore highly correlated. To overcome this obstacle it is assumed that the SAEC system employs some sort of preprocessing to the input signals in such a manner that their inter-channel correlation is somewhat reduced.

72 CHAPTER 4. IR TAIL EFFECT IN SAEC ENVIRONMENT Proposed solution Concluding the previous sections it is clear that each channel s adaptive filter in the SAEC system is operating in a more demanding conditions when compared to a single channel AEC working in the same adaptation scenario [18]. The added obstacles are generated either by the constant noise generated by the IR tail effect (4.2) or by the other channel adaptive filter residual error (4.3). Current SAEC solutions are usually based on the NLMS algorithm whose performance is known to degrade significantly when operated in a noisy environment, and therefore the NLMS algorithm does not produce optimal results when operated in the SAEC system. Furthermore, the current SAEC available solutions assumes the existence of a Double Talk Detector (DTD) device which freezes the adaptive filters adaptation if a double talk interval in the near-end room is detected. However, in the presence of a constant nearend room interference, the existing solutions are unable to converge and therefore do not function properly. On the other hand, if the DT and noise robust algorithm which is derived in chapter 3 is implemented in the SAEC system, it is expected to produce better ERLE and misadjustment levels due to a better performance under noisy condition, i.e. the IR tail effect and the DT like characteristic of the SAEC residual error. Furthermore, when implementing a DT and noise robust algorithm in an AEC system, additional advantages can be obtained since the algorithm produces better ERLE and misadjustment level when additional near-end room interference (ambient noise or DT situation) is added to the desired signal, thus enabling the SAEC to be applied with a low performance DTD or even without a DTD device at all, while producing superior ERLE and misadjustment levels even for a very noisy environment. The following section describes a proposed algorithm which combines the fast convergence rate of the POWER algorithm with the noise and DT robustness of the derived suboptimal step size SA algorithm to produce a fast converging high performance SAEC algorithm. The basic outline of the proposed algorithm is the POWER type 1 algorithm described by [1], to improve the system steady state ERLE of each filter (both current and previ-

73 CHAPTER 4. IR TAIL EFFECT IN SAEC ENVIRONMENT 59 Figure 4.1: Proposed SAEC algorithm. ous state filters) the NLMS based projection is replaced by the derived suboptimal SA algorithm whenever each individual filter has reached a converged state. The same action is taken once the system has detected a DT interference in the near-end room thereby continuing the system convergence even for a DT situation. When the system detects a near-end IR change, or at the beginning of the convergence scenario, the NLMS based projection is chosen for the tap update mechanism to achieve the fast convergence rate of the NLMS Algorithm. Fig. 4.1 describes the proposed algorithm (order q = 3) overview, and the following subsections details each of the algorithm stages Stage 1: Preprocessing The first step of the algorithm (which is not shown in Fig. 4.1) is to preprocess the farend microphone signals by the IS unit ( [5]) in order to produce two different input signal states: current and previous. The factor which determines the time period between the two states is the IS unit period parameter Q. since the IS unit shift state every Q/2 iterations, two input samples which are Q/2 iterations apart from each other are from two distinct states. Additional NLP preprocessing of the input signals is recommended to reduce inter channel correlation and improve the projection of the suboptimal SA.

74 CHAPTER 4. IR TAIL EFFECT IN SAEC ENVIRONMENT Stage 2: Projection The inputs to the second stage of the algorithm are the preprocessed concatenated far-end microphone signals defined by: [ x state con [n] = x statet 1 [n] x statet 2 [n] ] T (4.43) where state = curr, prev is derived by the IS unit time period Q, and x state i [n], i = 1, 2 is defined by: x state i [n] = x i [n] x i [n Q/2] state = curr state = prev (4.44) The desired signal d state [n] which is defined similarly to (4.44) and the current concatenated estimation filter ŵ[n] [ ŵ[n] = ŵ T 1 [n] ŵ T 2 [n] ] T (4.45) The order of the proposed algorithm q determines the number of inputs p to this stage by: p = 2 q. The minimum number of inputs for order q = 0 are the two input vectors states: x curr con [n], x prev con [n] and their desired signal samples d curr [n] and d prev [n]. The algorithm order can be increased by using additional inputs which are the delayed form of the input signal needed for the 0 order algorithm: x curr con [n j], x prev con [n j], d curr [n j] and d prev [n j] where j = 0,.., q Fig. 4.1 shows the proposed algorithm of order q = 3, therefore p = 8 inputs are needed, and j = 0, 3. In this stage the current concatenated estimation filter ŵ[n] is projected into the zero state concatenated estimation filter ŵ 0 state,j[n] where state and j are defined above according to the projection block inputs. The projection is performed to each channel of the estimation filter ŵ i [n], i = 1, 2 separately using NLMS or the suboptimal SA algorithms according to the output of the Tap Update Control (TUC) unit. the TUC and Projection units functionality is described in the following sections. Tap update control Algorithm block (TUC) As discussed in the previous chapter, the main disadvantage of the proposed SA algorithm is its slow initial convergence rate. This attribute is caused by the division in the residual

75 CHAPTER 4. IR TAIL EFFECT IN SAEC ENVIRONMENT 61 Current update state Condition Next update state NLMS ERR % < T H1 NLMS NLMS ERR % > T H1 Suboptimal SA Suboptimal SA ERR % < T H2 Suboptimal SA Suboptimal SA ERR % > T H2 Hang delay start Hang delay - Hang delay Hang delay end ERR % < T H2 Suboptimal SA Hang delay end ERR % < T H2 NLMS Table 4.1: Tap update control mechanism error energy level which is expected to be high at the start of the convergence scenario, or due to sudden change in the near-end room loudspeaker to microphone IR (reconvergence scenario). As a result of this attribute, the proposed algorithm cannot be implemented in a straight forward manner as the only tap update mechanism of the adaptive filters. To solve this problem, the tap update equation for each individual filter channel ŵ i [n], i = 1or2 should be defined in the following manner: Initial convergence : NLMS Converged state : Suboptimal SA DT state : Suboptimal SA Reconvergence : NLMS The distinction between each state is a difficult task since both reconvergence and DT states are characterized by the same high residual error energy value, but require two different tap update techniques. A possible solution for this problem is to provide two ghost adaptive filters to each of the algorithm estimation filters ŵ i,algo state,j [n] where state = prev, curr, j = 0,..., q, i = 1, 2 and algo = NLMS, SA SO, where a different tap update equation is applied to each ghost filter according to the algo parameter. The ghost filter which produces a smaller energy level determines the projection manner of the current estimation filter ŵ i [n] into the zero state estimation filter ŵ 0 i,state,j[n]. Since a NLMS based system quickly diverge in a DT situation, the tap update mechanism should

76 CHAPTER 4. IR TAIL EFFECT IN SAEC ENVIRONMENT 62 make the transition from the suboptimal SA tap update to the NLMS tap update a more difficult one. Setting i, state, j constant for a single TUC block, the following hysteresis like tap update scheme is proposed: Define the error energy produced by the ghost filters ŵ algo [n] as ERR algo Define the error energy percentage ERR % defined by ERR % = 100 ERR SASO ERR NLMS ERR NLMS The convergence is initialized with NLMS tap update while the error energy percentage ERR % produced is smaller than a given threshold T H1 the tap update mechanism is kept as NLMS If the system has been in this state for N1 iterations, the NLMS ghost filter taps ŵ NLMS [n] are copied to the suboptimal SA taps ŵ SA SO [n] in order to accelerate the suboptimal SA ghost filter convergence If ERR % is greater than the given threshold T H1 the system changes to the suboptimal SA tap update If ERR % is smaller than the given threshold T H2 the system remains in the suboptimal SA tap update If ERR % is greater than the given threshold T H2 the system changes to a hang period of N 2 iterations during which the suboptimal tap update mechanism is retained If at the end of the hang period ERR % is smaller than the given threshold T H2 the system returns to the suboptimal SA update state If at the end of the hang period ERR % is greater than the given threshold T H2 the system returns to the NLMS update state The system tap update control mechanism is summarized in table 4.1. It is important to emphasize that the tap update control is performed to each channel filter (channel 1

77 CHAPTER 4. IR TAIL EFFECT IN SAEC ENVIRONMENT 63 Figure 4.2: TUC block diagram. and 2) individually at each stage. Since the ghost filters adds additional computation complexity, shorter than system filters can be employed, or a partial transform domain adaptation can be employed to the ghost filters [51]. A block diagram of the TUC unit can be viewed in Fig. 4.2, the σ algo 2 block estimates the residual error energy ERR algo by the following recursive formulae: ERR algo [n + 1] = λerr algo [n] + (1 λ) e algo [n] 2, 0 < λ < 1 (4.46) Projection block The objective of this block is to project the current estimation filters ŵ[n] = [ ] T ŵ T 1 [n] ŵ T 2 [n] into the zero state 0 solution ŵ 0 i,state,j[n], where i indicates the channel number. The projection is performed according to the output from the TUC block: If NLMS update is chosen for the current filter, the tap update is given by: ŵ 0 x state i [n j]e state [n j] i,state,j[n] = ŵ i [n] + µ NLMS [n j] 2 + α x state con (4.47) where µ NLMS is the NLMS step size, α is the regulating factor, and e state [n j] is defined by: e state [n j] = d state [n j] ŵ T [n] x state con [n j] (4.48)