# Detection, Estimation modulation theory part 1 - Vantress H.

ISBN 0-471-09517-6

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<n = (1 + ñ’a2d2)~\

(224)

0 = Ã' dt Ã du [r(t) - s(t, A)] Qn(t, è) ÙãÃ^ • (225)

Jr, JTi Î A A = dml

r(t)

amI or amap (depending on choice of gain)

g(t)

Fig. 4.41 Linear estimation, colored noise.

Solution Techniques for Integral Equations 309

2. A necessary, but not sufficient, condition on amap (assuming that a has a Gaussian a priori density):

3. A lower bound on the variance of any unbiased estimate of the nonrandom variable A:

4. A lower bound on the mean-square error in the estimate of a zero-mean Gaussian random variable a :

5. A lower bound on the variance of any unbiased estimate of a nonrandom variable for the special case of an infinite observation interval and a stationary noise process:

As we discussed in 4.2.3, results of this type are always valid, but we must always look at the over-all likelihood function to investigate their usefulness. In other words, we must not ignore the threshold problem.

The only remaining issue in the matter of colored noise is a closed form solution for Qn(t, u) or g(t). We consider this problem in the next section.

4.3.6 Solution Techniques for Integral Equations

[map

-à2 Ã dt [r(0 - s(t, À)] Ã du Qn(t, è) ä-^ÜË . (226)

JTi JTt OA ë = 4òàð

ã 77 1-³

Var (a - A) > ff Qn(t, è) dt du > (227a)

Ò³

or, equivalently

(228)

Var (a - A) > |J

Ã Ã 00

dS*(ja>, A) c _lf , 8S(jw, A) duA 1 —dA----------5" (ø) dA............ úË

Ë -1

] - (229)

where

As we have seen above, to specify the receiver structure completely we must solve the integral equation for g(t) or Qn(t, u).

310 43 Detection and Estimation in Nonwhite Gaussian Noise

In this section we consider three cases of interest:

1. Infinite observation interval; stationary noise process.

2. Finite observation interval; separable kernel.

3. Finite observation interval; stationary noise process.

Infinite Observation Interval; Stationary Noise. In this particular case T{ = —oo, Tf = oo, and the covariance function of the noise is a function only of the difference in the arguments. Then (161) becomes /•00

S(z - v) = Qn(x — z) Kn(v —x)dx, —ñî < v, z < oo, (230)

J — ñî

where we assume that we can find a Qn(x, z) of this form. By denoting the Fourier transform of KJj) by Sn(w) and the Fourier transform of Qn(r) by S0(fc>) and transforming both sides of (230) with respect to r = z — v, we obtain

S0(u>) =

1

S„(w)

(231)

We see that SQ(w) is just the inverse of the noise spectrum. Further, in the stationary case (152) can be written as

/•00

Qn(z - v) = hju - z) hju - v) du.

J — 00

(232)

By denoting the Fourier transform of hw(j) by Hw(joj), we find that (232) implies

_k_ = S0H = \Hw(Jw)\2. (233)

Finally, for the detection and linear estimation cases (154) is useful. Transforming, we have

GooOV) = VE Sq((0) S(Jw) =

S(Jw)Ve^

Sn(“>)

(234)

where the subscript oo indicates that we are dealing with an infinite interval. To illustrate the various results, we consider some particular examples.

Example 1. We assume that the colored noise component has a rational spectrum. A typical case is

2kan2

Solution Techniques for Integral Equations 311

Then

(237)

where A = 4on2lkN0. Writing

5<3(a>) =

(jw + k)( —jio + k)

(238)

(N0/2)(ju> + kVl + A)(-jco + kV\ + A)

we want to choose an Hw(jco) so that (233) will be satisfied. To obtain a realizable whitening filter we assign the term (jcn + k(l + Ë)1/*) to and its conjugate to

tf*t/<o). The term (jco 4- k) in the numerator can be assigned to Hw(j<*>) or H*(jaj). Thus there are two equally good choicest for the whitening filter:

Thus the optimum receiver (detector) can be realized in the whitening forms shown in Fig. 4.42. A sketch of the waveforms for the case in which s(t) is a rectangular pulse is also shown. Three observations follow:

1. The whitening filter has an infinite memory. Thus it uses the entire past of r(t) to generate the input to the correlator.

2. The signal input to the multiplier will start at / = 0, but even after time t = T the input will continue.

3. The actual integration limits are (0, oo), because one multiplier input is zero before / = 0.

It is easy to verify that these observations are true whenever the noise consists of white noise plus an independent colored noise with a rational spectrum. It is also true, but less easy to verify directly, when the colored noise has a nonrational spectrum. Thus we conclude that under the above conditions an increase in observation interval will always improve the performance. It is worthwhile to observe that if we use HWl(ju>) as the whitening filter the output of the filter in the bottom path will be ßñÄÎ» the minimum mean-square error realizable point estimate of nc(t). We shall verify that this result is always true when we study realizable estimators in Chapter 6.

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