Convergence of the SMI and the diagonally loaded SMI algorithms with weak interference (adaptive array)

Abstract

The sample matrix inversion (SMI) algorithm is commonly used in adaptive arrays since it offers rapid convergence to the maximum signal-to-interference-plus-noise ratio (SINR) solution. However, in some applications, such as digital communications or satellite television communications, other measures of performance such as the signalto-interference ratio (SIR) may be equally important. In this paper approximations are derived for the power levels at the output of an adaptive array that uses the diagonally loaded SMI algorithm. Diagonal loading is a technique where the diagonal of the covariance matrix is augmented with a positive or negative constant prior to inversion. We examine how SINR and SIR at the array output vary with the number of samples taken when the input signals are continuous wave. It is shown that positive loading produces more rapid convergence with a reduction in output SIR. Negative loading provides an improved SIR level, but it is shown that the output power levels are erratic and slow to converge. Simulation results are given which verify the theoretical predictions.

Analysis of Modified SMI Method For Adaptive Array Weight Control

“…Ganz, Moses, and Wilson [7] have provided a statistical analysis of the modified SMI weight and power estimators assuming that the true noise In summary, the estimated weights and resultant output powers are asymptotically unbiased and consistent. The biases of the weight and power estimators decrease at a rate proportional to ( 1 / K ) while the asymptotic standard deviations decrease at a rate proportional to ( l / f l ) .…”

Adaptive minimum bit error rate beamforming assisted receiver for QPSK wireless communication

“…Classically, this has been achieved by minimizing the mean square error (MSE) between the desired output and the actual array output, and this principle is rooted in the traditional beamforming employed in sonar and radar systems. Adaptive implementation of the theoretical minimum MSE (MMSE) beamforming solution can readily be realized using temporal reference techniques [2][3][4][13][14][15][16][17]. Specifically, block-data based beamformer weight adaptation can be achieved using the so-called sample matrix inversion (SMI) algorithm [13,14], while sample-by-sample adaptation can be carried out using the least mean square (LMS) algorithm [15][16][17].…”

Adaptive minimum bit-error rate beamforming

“…Classically, the beamformer's weight vector is determined by minimizing the MSE term of , which leads to the following MMSE solution: (8) with being the first column of . Although the system matrix is generally unknown, the MMSE solution can be readily realized using the block-data based adaptive SMI algorithm [11], [12]. The MMSE solution can also be implemented using the stochastic gradient algorithm known also as the LMS algorithm.…”

“…The array input signal takes values from the signal set defined as (10) This set can be partitioned into two subsets depending on the specific value of as follows: (11) Similarly, the beamformer's output takes values from the scalar set . Thus, the real part of the beamformer's output can only take values from the set (12) which can be divided into two subsets conditioned on as follows: (13) Note that the term beamforming here in fact refers to linear beamforming. An implicit assumption is that and are linearly separable, that is, there exists a weight vector such that the two scalar sets and are completely separable by a linear decision boundary.…”

“…Adaptive implementation of the minimum MSE (MMSE) beamforming solution can be realized using temporal reference techniques [2]- [4], [11]- [14]. Specifically, block-based beamformer weight adaptation can be achieved, for example, using the sample matrix inversion (SMI) algorithm [11], [12], while sample-by-sample adaptation can be carried out using the least mean square (LMS) algorithm [13], [14].…”