Modified Kalman Based NLMS Algorithm For Noise Cancellation

This paper deals with the problem of Adaptive Noise Cancellation (ANC) for the speech signal corrupted with an additive white Gaussian noise. After explaining the least Mean Square (LMS)-based adaptive filter and Kalman filter, it examine the hybrid Kalman-based LMS (KNLMS) technique for adaptation of the ANC. The proposed technique suggests a way to normalize LMS algorithm using Kalman filter. This simulation shows that the KNLMS method converges faster and is more stable compared to the LMS and its Normalized version, NLMS. . INTRODUCTION As we know, noise is an important factor that affects speech recognition performance and many other applications .here , background noise can be significantly suppressed using the proposed adaptive noise cancellation scheme. The Least Mean Squares (LMS) algorithm for adaptive filters has been extensively studied and tested in a broad range of applications [1–4]. In [1] and in [2] a relation between the Recursive Least Squares (RLS) and the Kalman filter [6] algorithm is determined, and in [1] the tracking convergence of the LMS, RLS and extended RLS algorithms, based on the Kalman filter, are compared. However, there is no link established between the Kalman filter and the LMS algorithm. Speech enhancement is widely becoming an important topic of research due to the use of speech-enabled systems in a variety of real world applications. The aim is to minimize the effect of noise and to improve the performance of voice communication systems when input signals are corrupted by background noise. Several methods have been reported so far in the literature to enhance the performance of speech processing systems; some of the most important ones are: Wiener filtering [1], spectral subtraction [2]-[3], thresholding [4]-[5], Kalman filtering, and Least Mean Square (LMS) or Recursive Least Squares (RLS)-based adaptive filters [6]. Kalman filtering is known as an effective speech enhancement technique that models the speech signal as an autoregressive (AR) process and represents it in the state-space domain. Many approaches using Kalman filtering have been reported in the literature. They usually operate in two steps: first, noise and driving process variances and speech model parameters are estimated; then, the speech signal is estimated by Kalman filtering. In fact these approaches differ only by the choice of the algorithm used to estimate model parameters and the choice of the models adopted for the speech signal and the additive noise [7]. On the other International journal of Systems and Technologies ISSN 0974 2107 48 Double Blind Peer Reviewed Journal IJST side, LMS-based adaptive filters have been widely used for speech enhancement . In practice, LMS is replaced with its Normalized version, NLMS. By combining the ideas of two above-mentioned techniques, a new Kalman-based normalized LMS (or briefly called KNLMS) algorithm have been derived that has some advantages to the classical LMS and Kalman filters. Firstly, since it is based on the wellknown Kalman filter algorithm which has been shown and justified to be stable, the stability of the algorithm is sufficiently guaranteed. Secondly, it suggests a new way to modify the step size that results in optimum convergence for a large range of input signal powers. Finally, KNLMS prevents the amplification of measurement noise that occurs in the NLMS algorithm for low order filters. This paper examine the application of KNLMS in adaptive noise cancellation from noisy speech. The performance of the KNLMS will be compared to the classical LMS and NLMS algorithms. The rest of this paper is organized as follows: The basic concepts of LMS technique and Kalman filtering are respectively explained in sections I and II. Section III introduces KNLMS algorithm. Simulations,Implimentation and results are presented in section IV, V and VII . Finally, section VIII consists of some concluding remarks I LMS Algorithm Given an input signal, u(n) and a desired signal, d(n) determine the coefficient filter, w(n) that minimizes the mean square of the error signal, e(n) . The error signal is defined as the difference between the output of the filter, y(n) and the desired signal. An iterative algorithm for finding w(n) is the well-known LMS for the case of Finite Impulse Response (FIR) filters this algorithm is given by (1) This equation updates the vector of the filter coefficients . The output of the filter is where and N is the filter length. Also as mentioned above, It is known that the LMS algorithm is stable for a limited range of step size; actually, step size is inversely proportional to the power of the reference signal [1]. As a robust version of LMS, normalized LMS (NLMS) is widely used. For NLMS, equation (1) is changed to: (2) It has been shown [1] that the NLMS is stable as long as Of course, this is conditioned to . To prevent the instability of the algorithm in weak portions of input signal, equation (2) is practically modified as (3) Where q is selected to be small enough compared to International journal of Systems and Technologies ISSN 0974 2107 49 Double Blind Peer Reviewed Journal IJST II Kalman Filter For linear but time variant systems the Kalman filter is based on a state space formulation of a continuous or discrete time system. Here let the system should be considered as discrete time only. The Kalman filter gives an estimate of the state of the system given a set of outputs. For the case of Gaussian signals, and given the assumed linear model, the state estimate is optimum in the sense that it minimizes the norm of the difference between the estimate and the actual state. The system is described by the equations, (4) (5) Where we have, system state vector , measured signal vector , state transition matrix , state noise , observation matrix , measurement noise . The state noise and measurement noise assumed Gaussian random variables with known autocorrelation functions. The autocorrelation of the state noise is and that of the measurement noise is i.e.