Maximum Likelihood Estimators of Signal Parameters 1.3 2003/05/18 19:00:00 GMT-5 2003/08/18 17:04:20 GMT-5 Don Johnson dhj@rice.edu Jeffrey Silverman jsilv@rice.edu Kevin Duh kevinduh@rice.edu Elizabeth Gregory lizzardg@rice.edu Don Johnson dhj@rice.edu Eileen Krause erkrause@rice.edu Mariyah Poonawala mariyah@rice.edu Matthew Jeanes mjeanes@rice.edu Kyle Clarkson kclarks@rice.edu Many situations are either not well suited to linear estimation procedures, or the parameter is not well described as a random variable. For example, signal delay is observed nonlinearly and usually no a priori density can be assigned. In such cases, maximum likelihood estimators are more frequently used. Because of the Cramér-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 θ n l The vector of observations r is formed from the data in the obvious way. Evaluating the logarithm of the observation vector's joint density, p r θ r -12 2 K n 12 r s θ K n r s θ where s θ 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 θ i , for real-valued signals, θ i p r θ r r s θ K n θ i s θ and, for complex-valued ones, θ i p r θ r r s θ K n θ i s θ 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 θ θ ML r s θ K n θ i s θ 0 The Cramér-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 θ i s θ K n θ j s θ Further computation of the Cramér-Rao bound's components is problem dependent if more than one parameter is involved, and the off-diagonal terms of F are nonzero. If only one parameter is unknown, the Cramér-Rao bound is given by ε 2 b θ 2 1 θ b θ 2 θ s θ K n θ s θ When the signal depends on the parameter nonlinearly (which constitute the interesting cases), the maximum likelihood estimate is usually biased. Thus, the numerator of the expression for the bound cannot be ignored. One interesting special case occurs when the noise is white. The Cramér-Rao bound becomes ε 2 b θ 2 σ n 2 1 θ b θ 2 l 0 L 1 θ s l θ 2 The derivative of the signal with respect to the parameter can be interpreted as the sensitivity of the signal to the parameter. 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. ≟ θ s θ K n r s θ I θ b θ s θ K n θ s θ θ r θ 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 θ s l and is observed in an array's output in the presence of additive noise. The maximum likelihood estimate of the amplitude is the solution of the equation r θ ML s K n s 0 The form of this equation suggests that the maximum likelihood estimate is efficient. The amplitude estimate is given by θ 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 Cramér-Rao bound is given by ε 2 s K n s The condition for an efficient estimate becomes ≟ s K n r θ s s K n s θ ML θ whose veracity we can easily verify. In the special case where the noise is white, the estimator has the form θ ML r s , and the Cramér-Rao bound equals σ 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-noise ratio for the estimate, defined to be θ 2 ε 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 θ LIN σ θ 2 r K n s 1 σ θ 2 s K n s which we found in the white-noise case becomes a weighted version of the maximum likelihood estimate (see example). θ LIN σ θ 2 σ θ 2 σ n 2 r s Seemingly, these two estimators are being used to solve the same problem: Estimating the amplitude of a signal whose waveform is known. They make very different assumptions, however, about the nature of the unknown parameter; in one it is a random variable (and thus it has a variance), whereas in the other it is not (and variance makes no sense). Despite this fundamental difference, the computations for each estimator are equivalent. It is reassuring that different approaches to solving similar problems yield similar procedures.