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Derived PDF of Maximum
Likelihood (ML) Signal
Estimator which Employs
an Estimated Noise

Christ D. Richmond
Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
77 Massachusetts Avenue
Cambridge, MA 02139
email: cdrichmo@athena.mit.edu

A classical problem in many radar and sonar applications is the adaptive detection/estimation of a given signal(s) in the presence of zero mean Gaussian noise. Reed, Mallett, and Brennan derived and analyzed an adaptive detection scheme where the noise adaptation and non-trivial nature of their analysis resulted from the use of a noise sample covariance matrix (SCM). The case now considered is that of adaptive signal estimation. Specifically, the exact probability density function (pdf) for the ML signal estimator, alias the minimum variance distortionless response (MVDR) and the linearly constrained minimum variance beamformer output, is derived when the estimator relies on a SCM for evaluation. By using a complex Wishart probabilistic model for the distribution of the SCM it is shown that the pdf of the adaptive ML (AML) signal estimator, i.e., the SCM based ML signal estimator, is the confluent hypergeometric function known as Kummer's function. The AML signal estimator remains unbiased, only asymptotically efficient, and converges in distribution to the Gaussian non-adaptive beamformer output (known noise covariance). Steinhardt showed that the marginal pdfs for the MVDR weight vector are proportional to a complex univariate Student's t statistic. I show that the exact joint pdf for the distinct elements of the MVDR weight vector is proportional to a complex multivariate extension of the t-distribution. When the sample size of the SCM is fixed, there exists a dynamic tradeoff between signal-to-noise ratio and noise adaptivity as the dimensionality of array data is varied suggesting the existence of an optimal array data dimension which will yield the best performance. I show that all these results generalize to the multiple signal case which includes the well-known SCM based generalized sidelobe cancellor weight matrix.



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