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Many situations are either not well suited to linear estimation procedures, or the parameter is not well described as a randomvariable. For example, signal delay is observed nonlinearly and usually no a priori density can be assigned. In such cases, maximum likelihood estimators are more frequentlyused. Because of the Cramr-Rao bound , fundamental limits on parameter estimation performance can be derived for any signal parameter estimation problem where the parameter is not random.

Assume that the data are expressed as a signal observed in the presence of additive Gaussian noise.

l l 0 L 1 r l s l n l
The vector of observations r is formed from the data in the obvious way. Evaluating the logarithm of the observationvector's joint density, p r r -1 2 2 K n 1 2 r s K n r s where s is the signal vector having P unknown parameters, and K n is the covariance matrix of the noise. The partial derivative of this likelihood function with respect to the i th parameter i , for real-valued signals, i p r r r s K n i s and, for complex-valued ones, i p r r r s K n i s If the maximum of the likelihood function can be found by setting its gradient to 0 , the maximum likelihood estimate of the parameter vector is the solution of the set of equations
i i 1 P ML r s K n i s 0

The Cramr-Rao bound depends on the evaluation of the Fisher information matrix F . The elements of this matrix are found to be

i j i j 1 P F i j i s K n j s
Further computation of the Cramr-Rao bound's components is problem dependent if more than one parameter is involved, andthe off-diagonal terms of F are nonzero. If only one parameter is unknown, the Cramr-Rao bound is given by
2 b 2 1 b 2 s K n s
When the signal depends on the parameter nonlinearly (which constitute the interesting cases), the maximum likelihoodestimate is usually biased. Thus, the numerator of the expression for the bound cannot be ignored. One interestingspecial case occurs when the noise is white. The Cramr-Rao bound becomes 2 b 2 n 2 1 b 2 l 0 L 1 s l 2 The derivative of the signal with respect to the parameter can be interpreted as the sensitivity of the signal to theparameter. The mean-squared estimation error depends on the"integrated" squared sensitivity: The greater this sensitivity, the smaller the bound.

For an efficient estimate of a signal parameter to exist, the estimate must satisfy the condition we derived earlier . s K n r s I b s K n s r b Because of the complexity of this requirement, we quite rightly question the existence of any efficient estimator,especially when the signal depends nonlinearly on the parameter (see this problem ).

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 ML s K n s 0 The form of this equation suggests that the maximum likelihood estimate is efficient. The amplitude estimate isgiven by ML r K n s s K n 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 s K n s The condition for an efficient estimate becomes s K n r s s K n s ML whose veracity we can easily verify.

In the special case where the noise is white, the estimator has the form ML r s , 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 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 LIN 2 r K n s 1 2 s K n s which we found in the white-noise case becomes a weighted version of the maximum likelihood estimate (see example ). LIN 2 2 n 2 r s 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.

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Source:  OpenStax, Statistical signal processing. OpenStax CNX. Dec 05, 2011 Download for free at http://cnx.org/content/col11382/1.1
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