Wavelet Domain Processing of Generalized Sidelobe Canceller for Wideband Smart Arrays

In this paper we discuss the application of wavelet transform domain processing in smart antenna arrays, specifically, in the design of a wideband generalized sidelobe canceller (GSC) beamformer. The wavelet transform domain least mean squares (LMS) processing is shown to have the effect of decreasing the spreading in the eigenvalues of the correlation matrix of the output data. This results in faster convergence of the beamformer and thus a better interference rejection capability. Introduction: One of the main applications of smart and adaptive antenna arrays is the reduction of the vulnerability of the reception of desired signals to the presence of interference signals in radar, sonar and communication systems [1]. Smart array systems have the ability to automatically sense and suppress interference sources while simultaneously enhancing desired signal reception [2]. This is preferably done without prior knowledge of the signal/interference environment. The wideband GSC beamformer has been introduced by Griffths and Jim [3], as an implementation of the linearly constrained minimum variance adaptive beamforming algorithm introduced by Frost [4]. The output signals from an array of N sensors are fed to delay elements needed to steer the array in the desired look direction. Two different processing paths are then taken; the first main path consists of a conventional beamformer followed by a fixed filter used to control the frequency response of the beamformer in the look direction. The second path consists of a blocking matrix followed by a set of tapped delay lines. The function of the blocking matrix is to prevent any signal coming from the look direction from passing into the this path such that the adaptive processor following the blocking matrix will provide an estimate of the interference signal which is then subtracted from the received signal in the main path. In [5] the use of additional directional derivative constraints to ensure the insusceptibility of the array to slight deviation in the direction of the desired signal was discussed. Recently, the use of the wavelet transform in adaptive processing has received much attention due to its capability of decorrelating a wide class of signals [6], [7], [8]. In this paper we consider the use of the wavelet transform as an orthogonalizing transform in smart antenna arrays, specifically in the wideband GSC beamformer. Let Rx(m,n) be the correlation between the m th and n samples of the input process x. furthermore, denote by Ru (m,n), the cross correlation between the m component of the process x in the j wavelet domain band and the n component of the process x in the k wavelet domain band. It has been shown [6] that for a wide class of processes, Ru (m,n) decays at a rate much faster than Rx(m,n) as m tends to infinity and n is fixed or as n tends to infinity and m is fixed. Thus, the multi resolution components of x(n) will be effectively partially uncorrelated. The Wavelet Based GSC Beamformer: Fig. 1 shows the block diagram of the wavelet domain based GSC. In order to speed up the adaptation process, wavelet transform domain LMS is implemented in each of the N ‘ adaptive filters consisting the blocking matrix. Let ri(n) denote the received signal vector from the N sensors at time n after the pre-steering delays, xi(n) denote the output of the i th row of the blocking matrix B at time n, where i =1, 2, ......N’, and the vector Xi(n) = [xi(n), xi(n-1), ...... xi(n-L+1)] T denotes the tap input vector of the i th tapped delay line. The vector Xi(n) is transformed to the wavelet domain vector Ui(n) by multiplying it with the wavelet transform matrix Q, i.e. Ui(n) = Q Xi(n), where, Ui(n) = [u i,0 (n), u i,1 (n),......... u i,L-1(n)] , Q = {QM QM-1.........Q2 Q1}, where,