A Practical Solution for the Regularization of the Affine Projection Algorithm Constantin Paleologu 1,∗ , Jacob Benesty 2 , and Silviu Ciochin˘ a 1 1 University Politehnica of Bucharest, Telecommunications Department, Bucharest, Romania 2 University of Quebec, INRS-EMT, Montreal, Canada ∗ Corresponding author (E-mail: pale@comm.pub.ro) Abstract—The regularization of the affine projection algorithm (APA) is of great importance in echo cancellation applications. The regular- ization parameter, which depends on the level of the near-end signal, is added to the main diagonal of the input signal correlation matrix to ensure the stability of the APA. In this paper, we propose a practical way for evaluating the power of the near-end signal or, equivalently, the signal-to-noise ratio that is explicitly related to the regularization parameter. Simulation results obtained in the context of acoustic echo cancellation support the appealing performance of the proposed solution. Keywords - echo cancellation; adaptive filters; regularization; affine projection algorithm (APA). I. I NTRODUCTION The affine projection algorithm (APA) [1]–[4] is an attractive and popular choice for echo cancellation [5], [6]. As compared to the normalized least-mean-square (NLMS) algorithm [7], the APA achieves a superior convergence rate, especially for highly-correlated input signals (like speech). For numerical reasons, a regularization parameter is needed within the APA [1]. This positive constant is added to the main diagonal of the input signal correlation matrix to be inverted to ensure a stable behavior of the APA. It can be shown that the value of the regularization parameter depends on the level of the system noise. In echo cancellation, the problem is even more complicated, since the system “noise” is in fact the near-end signal (e.g., near-end speech and/or different types of background noise). Different solutions have been proposed to control the value of this important parameter, e.g., see [8]–[10] and the references therein. In many of these works, the main goal was to find a variable regularization parameter, which acts like a variable step-size, thus controlling the entire adaptation of the algorithm. More recently, the problem of regularization in the adaptive filtering context was addressed in [11], where the main issue was to attenuate the effects of the system noise in the adaptive filter estimate. Following this work, several solutions for selecting the regularization parameter of the APA were presented in [12], [13]. In this paper, we present another practical solution for the regular- ization problem in the context of the APA. In Section II, the general framework developed in [12] is summarized. Next, in Section III, we propose a practical way for evaluating the key term needed in the formula of the regularization parameter, i.e., the signal-to- noise ratio (SNR), in terms of the estimation of the near-end signal power. Simulation results presented in Section IV (considering an acoustic echo cancellation scenario) support the theoretical issues and indicate the good performance of the developed solution. Finally, the conclusions are provided in Section V. II. REGULARIZATION OF THE APA We consider the echo cancellation application, which is basically a system identification problem. The reference (microphone) signal at the discrete-time index is ()= x ()h + ()= ()+ (), (1) where x()= [ () (− 1) ⋅⋅⋅ (− + 1) ] is a vector containing the most recent time samples of the far-end (loudspeaker) signal (), superscript denotes transpose of a vector or a matrix, h = [ ℎ0 ℎ1 ⋅⋅⋅ ℎ−1 ] is the impulse response of the echo path (the system we need to identify, i.e., from the loudspeaker to the microphone), and () is the near-end signal (i.e., the system “noise”). The objective is to estimate or identify h with an adaptive filter: ˆ h()= [ ˆ ℎ0() ˆ ℎ1() ⋅⋅⋅ ˆ ℎ−1() ] . The APA is one of the most popular algorithms used for echo cancellation. The basic version of this algorithm is defined by the following equations [1]: e() = d() − X () ˆ h(− 1), (2) R() = I+ X ()X(), (3) ˆ h() = ˆ h(− 1) + X()R −1 ()e(), (4) where e() is the error signal vector of length (with denoting the projection order), d() = [ () (− 1) ⋅⋅⋅ (− + 1) ] is a vector containing the most recent time samples of the desired signal, X()= [ x() x(− 1) ⋅⋅⋅ x(− + 1) ] is the input data matrix, R() is the correlation matrix to be inverted, is the regularization parameter, Iis the × identity matrix, and is the step-size parameter. For =1, the NLMS algorithm is derived. There are three main parameters that controls the performance of the APA. The first one is the projection order, ; the convergence rate of the APA increases when increases, but also the compu- tational complexity. The second one is the step-size parameter, (theoretically, 0 << 2); its main role is to compromise between two important performance criteria, i.e., the convergence rate and misadjustment [7]. Last but not least, there is the regularization parameter, . At a first glance, the regularization matrix Iis used to pre- vent a “bad” matrix inversion, since X ()X() can be very ill- conditioned, especially for speech inputs. On the other hand, in practice, the selection of is of great importance in terms of the adaptive filter performance. The value of this parameter is related to the level of the system noise, i.e., () in (1). Based on (1), we can define the SNR as [5]: SNR = 2 2 , (5) where 2 = [ 2 () ] and 2 = [ 2 () ] are the variances of () and (), respectively [(⋅) denotes the expectation]. A rule of thumb is that the regularization parameter requires large values 978-1-4799-2385-4/14/$31.00 ©2014 IEEE