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

ISBN 0-471-09517-6

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Example 1 (Uniform Phase). Because this model corresponds to a radar problem, the uniform phase assumption is most realistic. To construct the ROC we must compute PF and PD. (Recall that PF and PD are the probabilities that we will exceed the threshold ó when noise only and signal plus noise are present, respectively.)

Looking at Fig. 4.55, we see that the test statistic is

/ = L2 + Ls2, (381)

where Lc and Ls are Gaussian random variables. The decision region is shown in Fig. 4.56. We can easily verify that

Ho: E(LC) = ?(!,) = 0; Var (Lc) = Var (L„) = y°.

H^.EiLc|â) = VTr cos 0; E(L,|0) = V^sin 0; Var(Lc) = Var (?s) = ^°- (382)

Then

Pf & Pr [/ > y| tf0] = //(2w y) 'exp (--cl±-,a) dU dL,. (383)

no

Changing to polar coordinates and evaluating, we have

Ë-«ô(-³). (384)

Similarly, the probability of detection for a particular â is

ãì - (3S5)

«0

Letting Lc = R cos p,Ls = R sin ft and performing the integration with respect to ft

we obtain _

L+b),«Ì. (,ù

As we expected, PD does not depend on 0. We can normalize this expression by letting z = V2/N0 R. This gives

Pd = Ã Z exp (-?l±-^f) Io{zd) dZt (387)

•"a Wo ' 2 /

where d2 A 2Er/No.

This integral cannot be evaluated analytically. It was first tabulated by Marcum [46, 48] in terms of a function commonly called Marcum’s Q function:

G(«, /3) — J" z exp (- /2 2 a2)^az) dz- (388)

This function has been studied extensively and tabulated for various values of a, /} (e.g., [48], [49], and [50]). Thus

(389>

Random Phase 345

Fig. 4.58 Receiver operating characteristic, random phase with uniform density.

This can be written in terms of PF. Using (384), we have

Pd = Q(d, Võ — 2 In PF). (390)

The ROC is shown in Fig. 4.58. The results can also be plotted in the form of PD versus d with PF as a parameter. This is done in Fig. 4.59. Comparing Figs. 4.14 and 4.59, we see that a negligible increase of d is required to maintain the same PD for a fixed PF when we go from the known signal model to the uniform phase model for the parameter ranges shown in Fig. 4.59.

The second example of interest is a binary communication system in which some phase information is available.

Example 2. Partially Coherent Binary Communication. The criterion is minimum probability of error and the hypotheses are equally likely. We assume that the signals under the two hypotheses are

H,\r(t) = V2Er MO cos (øñ1 + â) + w(t), 0 <t<T,

H0:r(t) = ë/Ù f0(t) cos (wct + ff) + w(t), 0 < t < T,

346 4A Signals with Unwanted Parameters

where f0(t) and/i(0 are normalized and

fo MO AO) dt = p; -I < p <\. (392)

The noise spectral height is N0I2 and pe(0) is given by (364). The likelihood ratio test is obtained by an obvious modification of the simple binary problem and the receiver structure is shown in Fig. 4.60.

We now look at Pr (c) as a function of p, d2, and Am. Intuitively, we expect that as Am —> oo we would approach the known signal problem, and p = — 1 (the equal and opposite signals of (39)) would give the best result. On the other hand, as Am 0,

Fig. 4.59 Probability of detection vs d, uniform phase.

Random Phase 347

Linear component

Hi

Ho

2\f0(t)) sinfwcO Fig. 4.60 Receiver: binary communication system.

the phase becomes uniform. Now, any correlation (+ or —) would move the signal points closer together. Thus, we expect that p = 0 would give the best performance. As we go from the first extreme to the second, the best value of p should move from -1 to 0.

We shall do only the details for the easy case in which p — — 1; p = 0 is done in Problem 4.4.9. The error calculation for arbitrary p is done in [44].

When p = -1, we observe that the output of the square-law section is identical on both hypotheses. Thus the receiver is linear. The effect of the phase error is to rotate the signal points in the decision space as shown in Fig. 4.61.

Using the results of Section 4.2.1 (p. 257),

(x - VEr cos â)2] No J

dx

(393)

348 4.4 Signals with Unwanted Parameters

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