3.7 Maximum likelihood estimators of signal parameters

Let the unknown parameter be the signal's amplitude; the signal is expressed as $s(l)$ and is observed in an array's output in the presence of additive noise. The maximum likelihood estimate of theamplitude is the solution of the equation $$(r-({}_{\mathrm{ML}})s)^T{K}_n^{(-1)}s=0$$ The form of this equation suggests that the maximum likelihood estimate is efficient. The amplitude estimate isgiven by $$({}_{\mathrm{ML}})=\frac{r^T{K}_n^{(-1)}s}{s^T{K}_n^{(-1)}s}$$ The form of this estimator is precisely that of the matched filter derived in the colored-noise situation (see equation ). The expected value of the estimate equals the actual amplitude.Thus the bias is zero and the Cramr-Rao bound is given by $$(^2)\ge s^T{K}_n^{(-1)}s^{(-1)}$$ The condition for an efficient estimate becomes $$(s^T{K}_n^{(-1)}(r-s), s^T{K}_n^{(-1)}s(({}_{\mathrm{ML}})-))$$ whose veracity we can easily verify.

In the special case where the noise is white, the estimator has the form $({}_{\mathrm{ML}})=r^Ts$ , and the Cramr-Rao bound equals ${}_n^2$ (the nominal signal is assumed to have unit energy). The maximum likelihood estimate of the amplitude has fixed error characteristics that do not depend on the actual signal amplitude. A signal-to-noiseratio for the estimate, defined to be $\frac{^2}{(^2)}$ , equals the signal-to-noise ratio of the observed signal.

When the amplitude is well described as a random variable, its linear minimum mean-squared error estimator has the form $$({}_{\mathrm{LIN}})=\frac{{}_{}^{2}r^T{K}_n^{(-1)}s}{1+{}_{}^{2}s^T{K}_n^{(-1)}s}$$ which we found in the white-noise case becomes a weighted version of the maximum likelihood estimate (see example ). $$({}_{\mathrm{LIN}})=\frac{{}_{}^2}{{}_{}^2+{}_n^2}r^Ts$$ Seemingly, these two estimators are being used to solve the same problem: Estimating the amplitude of a signalwhose waveform is known. They make very different assumptions, however, about the nature of the unknownparameter; in one it is a random variable (and thus it has a variance), whereas in the other it is not (and variance makesno sense). Despite this fundamental difference, the computations for each estimator are equivalent. It isreassuring that different approaches to solving similar problems yield similar procedures.