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

ISBN 0-471-09517-6

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Throughout most of our work we retain the white noise assumption so singular tests never arise. Whenever the assumption is removed, it is necessary to check the model to ensure that it does not correspond to a singular test.

306 4.3 Detection and Estimation in Nonwhite Gaussian Noise

General Binary Receivers. Our discussion up to this point has considered only the simple binary detection problem. The extension to general binary receivers is straightforward. Let

ÊÎ = y/% S&) + n(t), Tt<t< T,:HU

ÊÎ = VE0s0(t) + n(t), T, < t < Tf :H0,

where 50(0 and sx(t) are normalized over the interval (0, T) and are zero elsewhere. Proceeding in exactly the same manner as in the simple binary case, we obtain the following results. One receiver configuration is shown in Fig. 4.40a. The function gA(0 satisfies

sA(t) a Vex ^(o - Ve0 s0(t)

= ¥' gA(u) Kn(t, u) du, Tt< t < Tf. (212)

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The performance is characterized by d2:

Tf

d2 = JJsA(t) Qn(t, u) sA(u) dt du. (213)

t,

The functions Kn(t, è) and Qn(t, u) were defined in (145) and (161), respectively. As an alternative, we can use the whitening realization shown in Fig. 4.406. Here hw(t, è) satisfies (158) and

$ä,(0 é f X hw(t, è) Sb(u) du, Ò³ < t < Tf. (214)

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rTf , Threshold #1 or H0

device

(a)

sAJt)

(b)

Fig. 4.40 (a) Receiver configurations: general binary problem, colored noise; (b) alternate receiver realization.

Estimation 307

The performance is characterized by the energy in the whitened difference signal:

«/2=¥×Ä0À. (215)

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The M-àãó detection case is also a straightforward extension (see Problem 4.3.5). From our discussion of white noise we would expect that the estimation case would also follow easily. We discuss it briefly in the next section.

4.3.5 Estimation

The model for the received waveform in the parameter estimation problem is

ÊÎ = s(t, A) + n(t), T{<t< 7>. (216)

The basic operation on the received waveform consists of constructing the likelihood function, for which it is straightforward to derive an expression. If, however, we look at (98-101), and (146-153), it is clear that the answer will be:

In AJKO, A] = ( ' r(z) dz Ã ' Qn(z, v) ô, A) dv

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- \ \ ' dz s{z, A) f ' Qn(z, V) s(v, A) dv. (217)

JTt JTi

This result is analogous to (153) in the detection problem. If we define

g(z, A) =( ' Qn(z, v) s(v, A) dv, Tt < z < Th (218)

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or, equivalently,

s(v, A) = f ' Kn(v, z)g(z, A) dz, Ò³ < v < T„ (219)

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(217) reduces to

In Ax[/-(/), A] = f ’ r(z) g(z, A) dz

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~ ³ ( ´ Ô’ A) g(z> A) dz- (22°)

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The discussions in Sections 4.2.2 and 4.2.3 carry over to the colored noise case in an obvious manner. We Summarize some of the important results for the linear and nonlinear estimation problems.

308 4.3 Detection and Estimation in Nonwhite Gaussian Noise

Linear Estimation. The received waveform is

r(t) = AVEs(t) + n(t), Tt< t < Th (221)

where ä(ã) is normalized [0, T] and zero elsewhere. Substituting into (218), we see that

where g(t) is the function obtained in the simple binary detection case by solving (169).

Thus the linear estimation problem is essentially equivalent to simple binary detection. The estimator structure is shown in Fig. 4.41, and the estimator is completely specified by finding g(t). If A is a nonrandom variable, the normalized error variance is

where d2 is given by (198). If A is a value of a random variable a with a Gaussian a priori density, N(0, <ra), the minimum mean-square error is

(These results correspond to (96) and (97) in the white noise case) All discussion regarding singular tests and optimum signals carries over directly.

Nonlinear Estimation. In nonlinear estimation, in the presence of colored noise, we encounter all the difficulties that occur in the white noise case. In addition, we must find either Qn(t, u) or g(t, A). Because all of the results are obvious modifications of those in 4.2.3, we simply summarize the results:

1. A necessary, but not sufficient, condition on aml:

g(t, A) = A g(t),

(222)

(223)

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