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One extension of parametric estimation theory necessary for its
application to array processing is the estimation of signalparameters. We assume that we observe a signal
$s(l, )$ , whose characteristics are known save a few parameters
$$ , in the presence of noise.
Signal parameters, such as amplitude, time origin, and frequencyif the signal is sinusoidal, must be determined in some way. In
many cases of interest, we would find it difficult to justify aparticular form for the unknown parameters'
*cannot* be
used in many cases. The minimum mean-squared error
*linear* estimator does not require this
density, but it is most fruitfully used when the unknownparameter appears in the problem in a linear fashion (such as
signal amplitude as we shall see).

The only parameter that is linearly related to a signal is the amplitude. Consider, therefore, the problem where theobservations at an array's output are modeled as

$$\forall l, l\in \{0, , L-1\}\colon r(l)=s(l)+n(l)$$

The signal waveform
$s(l)$ is known and its energy normalized to be unity (
$\sum s(l)^{2}=1$ ). The linear estimate of the signal's amplitude is
assumed to be of the form
$()=\sum h(l)r(l)$ , where
$h(l)$ minimizes the mean-squared error. To use the
Orthogonality Principle expressed by
this equation , an inner product must be
defined for scalars. Little choice avails itself butmultiplication as the inner product of two scalars. The
Orthogonality Principle states that the estimation error mustbe orthogonal to all linear transformations defining the kind
of estimator being sought.
$$\forall h()\colon ((\sum_{l=0}^{L-1} {h}_{\mathrm{LIN}}(l)r(l)-)\sum_{k=0}^{L-1} h(k)r(k))=0$$ Manipulating this equation to make the universality constraint
more transparent results in
$$\forall h()\colon \sum_{k=0}^{L-1} h(k)((\sum_{l=0}^{L-1} {h}_{\mathrm{LIN}}(l)r(l)-)r(k))=0$$ Written in this way, the expected value must be 0
for each value of
$k$ to satisfy the constraint. Thus, the quantity
${h}_{\mathrm{LIN}}()$ of the estimator of the signal's amplitude must
satisfy
$$\forall k\colon \sum_{l=0}^{L-1} {h}_{\mathrm{LIN}}(l)(r(l)r(k))=(r(k))$$ Assuming that the signal's amplitude has zero mean and is
statistically independent of the zero-mean noise, the expectedvalues in this equation are given by
$$(r(l)r(k))={}_{}^{2}s(l)s(k)+{K}_{n}(k, l)$$
$$(r(k))={}_{}^{2}s(k)$$ where
${K}_{n}(k, l)$ is the covariance function of the noise. The
equation that must be solved for the unit-sample response
${h}_{\mathrm{LIN}}()$ of the optimal linear MMSE estimator of signal
amplitude becomes
$\forall k\colon \sum_{l=0}^{L-1} {h}_{\mathrm{LIN}}(l){K}_{n}(k, l)={}_{}^{2}s(k)(1-\sum_{l=0}^{L(1)} {h}_{\mathrm{LIN}}(l)s(l))$

This equation is easily solved once phrased in matrix
notation. Letting
${K}_{n}$ denote the covariance matrix of the noise,
$s$ the signal vector, and
${h}_{\mathrm{LIN}}$ the vector of coefficients, this equation becomes
$${K}_{n}{h}_{\mathrm{LIN}}={}_{}^{2}(1-s^T{h}_{\mathrm{LIN}})s$$ The matched filter for colored-noise problems consisted of the
dot product between the vector of observations and
${K}_{n}^{(-1)}s$ (see the
detector
result ). Assume that the solution to the linear
estimation problem is proportional to the detectiontheoretical one:
${h}_{\mathrm{LIN}}=c{K}_{n}^{(-1)}s$ , where
$c$ is a scalar constant. This
proposed solution satisfies the equation; the MMSE estimate ofsignal amplitude corresponds to applying a matched filter to
the observations with
${h}_{\mathrm{LIN}}=\frac{{}_{}^{2}}{1+{}_{}^{2}s^T{K}_{n}^{(-1)}s}{K}_{n}^{(-1)}s$

The mean-squared estimation error of signal amplitude is given by
$$(^{2})={}_{}^{2}-(\sum_{l=0}^{L-1} {h}_{\mathrm{LIN}}(l)r(l))$$ Substituting the vector expression for
${h}_{\mathrm{LIN}}$ yields the result that the mean-squared estimation error
equals the proportionality constant
$c$ defined earlier.
$$(^{2})=\frac{{}_{}^{2}}{1+{}_{}^{2}s^T{K}_{n}^{(-1)}s}$$
Thus, the linear filter that produces the optimal estimate of
signal amplitude is equivalent to the matched filter used todetect the signal's presence. We have found this situation to
occur when estimates of unknown parameters are needed to solvethe detection problem (see
Detection
in the Presence of Uncertainties ). If we had not
assumed the noise to be Gaussian, however, thisdetection-theoretic result would be different, but the
estimator would be unchanged. To repeat, this invarianceoccurs because the linear MMSE estimator requires
*no* assumptions on the noise's amplitude
characteristics.

Let the noise be white so that its covariance matrix is
proportional to the identity matrix (
${K}_{n}={}_{n}^{2}I$ ). The weighting factor in the minimum
mean-squared error linear estimator is proportional to thesignal waveform.
$${h}_{\mathrm{LIN}}(l)=\frac{{}_{}^{2}}{{}_{n}^{2}+{}_{}^{2}}s(l)$$
$$({}_{\mathrm{LIN}})=\frac{{}_{}^{2}}{{}_{n}^{2}+{}_{}^{2}}\sum_{l=0}^{L-1} s(l)r(l)$$ This proportionality constant depends only on the relative
variances of the noise and the parameter.
*If* the noise variance can be considered
to be much smaller than the

We find the mean-squared estimation error to be
$$(^{2})=\frac{{}_{}^{2}}{1+\frac{{}_{}^{2}}{{}_{n}^{2}}}$$ This error is significantly reduced from its nominal value
${}_{}^{2}$ only when the
variance of the noise is small compared with the

In other words, the problem is difficult
in this case.

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Source:
OpenStax, Estimation theory. OpenStax CNX. May 14, 2006 Download for free at http://cnx.org/content/col10352/1.2

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