# Angle Arrival Estimation in Non Gaussian Noise.

December 2000

## Date of Award

Winter 12-12-2001

## Institute Name (Publisher)

Indian Statistical Institute

## Document Type

Master's Dissertation

## Degree Name

Master of Technology

Computer Science

## Department

Applied Statistics Unit (ASU-Kolkata)

## Supervisor

Sengupta, Debasis (ASU-Kolkata; ISI)

## Abstract (Summary of the Work)

Several important problems in signal processing field,among then Direction finding with narrow-band sensor arrays,estimation of th Parameters of multiple superimposed exponential signals in noise And resolution of overlapping echoes can be reduced to parameters in the following model:Y(t)=A(Î¸)x(t) + e(t) t=1,2,...NY(t) is a Cm*1 is the noisy data vectorX(t) is a Cn*1 is the vector of signal amplitudesE(t) is a Cm*1 is an additive noise vectorA(t) is a Cm*n and has the following special structureA(Î¸)=[a(w1),a(w2),...a(wn)]Where {w1} are real parameters and a(w1) which is Cm*1 is a so Called transfer vector between ith signal and y(t) and Î¸=[W1,....Wn]TThere are three main problems associated with fitting models of the form as discussed to the data {y(1),y(2)....y(N)}1.Estimation of the number of signals nMethod s for estimating n are well documented in the Literature and won't be discussed here.In this discussion We will assume that the number of signals is given.2.Estimation of the signal amplitudesOnce an estimate of Î¸ is available, the estimation of x(t) Reduces to a simple least square fit.We will omit any Explicit discussion on the problem of estimating x(t)3.Estimation of the parameter vector Î¸Methods for accomplishing this task and their performance Are the main topics of discussion.A class of methods for estimating Î¸ which has received significant Attention is based on the eigendecomposition of the sample Covariance matrix of y(t).A representative member of this class Is the MUSIC(Multiple Signal Characterization) algorithm. There has been considerable interest recently in the analyzing The statistical performance of the MUSIC.Some interesting and- related studies of the resolvability of MUSIC have been reported However an expression for the covariance matrix of the MUSIC estimate of Î¸ has not been derived in these papers.Here we discuss An explicit expression for the covariance matrix that holds for sufficiently large values of N.Comparison with the performance corresponding to CRAMER RAO LOWER BOUND is of interest.An expression for the CRLB On the covariance matrix of any unbiased estimator of the parameters Î¸ in the general model does not appear to be avaliable in the literature.Here we derive the CRLB on the covariance matrixThe classical maximum-likelihood(ML) method can also be used Under appropriate assumptions to estimate the parameter vector 0 Next, we introduce some basic assumptions on the model. The MUSIC and the ML methods are based on different sets of assumptions. However some assumptions are common to both methods. The common assumptions are listed first:Al:m>n and the vectors a(w) corresponding to (n+1) different values of w are linearly independent.A2:E{e(t)}=0 E{e(t)e'(t)}=o2I and E{e(t)eT(t)}=0 This is a more restrictive assumption that is essential for the MUSIC algorithm; for the ML method, relaxation of A2 is possible in principle, but would lead to considerable complications.

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843156

## Control Number

ISI-DISS-2000-140