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

ISBN 0-471-09517-6

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whereas for small x

/î(ëã) ~ 1 + j, x « I, (370a)

and

In TJy\ ~

Observe that because In IQ(x) is monotone we can remove it by modifying

In r0(x) ~ JC« 1. (3706)

340 4.4 Signals with Unwanted Parameters

V2/(0sin[wc* + <M)l

Fig. 4.53 Optimum receiver: random phase angle

the threshold. Thus two tests equivalent to (368) are /, ^ ëã,Ë.\> «>

I³' + Ø) +L‘ ³ r

and

/2ë/?Ä2,. 2 2VEr . * ,

(“NT) ( c +^) + 2Ë*-^-³ñ|Ó.

(371a)

(3716)

x

Fig. 4.54 Plot of Iq(x).

Matched Filter-Envelope Detector 341 Linear component

Fig. 4.55 Alternate realization of optimum receiver.

Redrawing the receiver structure as shown in Fig. 4.55, we see that the optimum receiver consists of a linear component and a square-law component.

Looking at (371a), we see that the region in the Lc, Ls plane corresponding to_the decision H0 is the interior of a circle centered at (-N0Aml2VEr,0) with radius yYK We denote this region as Q,0. The probability density of Lc and Ls under H0 is a circularly symmetric Gaussian density centered at the origin. Therefore, if ó is fixed and Am is allowed to increase, Q0 wiU move to the left and the probability of being in it on Hq will decrease. Thus, to maintain a constant PF we increase ó as Am is increased. Several decision regions are shown in Fig. 4.56. In the limit, as Am -> oo, the decision boundary approaches a straight line and we have the familiar known signal problem of Section 4.2. The probability density on #x depends on 0. A typical case is shown in the figure. We evaluate PF and PD for some interesting special cases on p. 344 and in the problems. Before doing this it will be worthwhile to develop an alternate receiver realization for the case in which Am = 0. In many cases this alternate realization will be more convenient to implement.

Matched Filter-Envelope Detector Realization. When Am = 0, we must find VLc2 + Ls2. We can do so by using a bandpass filter followed by an envelope detector, as shown in Fig. 4.57. Because h(t) is the impulse response of a bandpass filter, it is convenient to write it as

h(t) = hL{t) cos [coct + ôü(0],

(372)

342 4A Signals with Unwanted Parameters

where hL(t) and ôü(³) are low-pass functions. The output at time T is

y(T) = h(T - r) r(r) dr. (373)

Using (372), we can write this equation as

y{T) = JT r(r) hL(T - r) COS [ojc(T -r) + ôü(Ò - ò)] dr

cT Jo ã(ò) hL(T - t) cos [øñò - UT - t)] dr

+ sin <ocT J r(r) hL(T — t) sin [øñã — ôü(Ò — r)] dr. (374)

= COS Ct)n

r(t) ^ h{i) y(t) . Envelope ³

detector

Observe at time

*4 t=r .

filter

Fig. 4.57 Matched filter-envelope detector for uniform phase case.

Random Phase 343

This can be written as

ó{Ò) k yc{T) cos WCT + ó IT) sin 0JcT

= Vyc\T) + y,\T) cos [«„Ã - tan -1 • (375)

Observing that

yc(T) = Re JJ r(r) hL(T - r) exp [+jwcr - J4>l{T - r)] dr (376a)

and

ÃÒ

ó IT) = Im ? r(r) hL(T - r) exp [+j<oer - j^L(T - r)] dr, (3766)

we see that the output of the envelope detector is

VõÓñ\Ò) + ys2(T) = Ã r(r) hL(T - t) exp [~jA(T - t) + jo}cr] dr . Jo

(377)

From (361) and (362) we see that the desired test statistic is

VL,2 + L2 =

(378)

JT r(r)V2 /(ã)å+Øã)å+³è>ñ' dr.

We see the two expressions will be identical if

hL(T- r) = V2/(r) (379)

and

*l(T - T) = -Ô³ã). (380)

This bandpass matched filter provides a simpler realization for the uniform phase case.

The receiver in the uniform phase case is frequently called an incoherent receiver, but the terminology tends to be misleading. We see that the matched filter utilizes all the internal phase structure of the signal. The only thing missing is an absolute phase reference. The receiver for the known signal case is called a coherent receiver because it requires an oscillator at the receiver that is coherent with the transmitter oscillator. The general case developed in this section may be termed the partially coherent case.

To complete our discussion we consider the performance for some simple cases. There is no conceptual difficulty in evaluating the error

probabilities but the resulting integrals often cannot be evaluated analyti-

cally. Because various modifications of this particular problem are frequently encountered in both radar and communications, a great deal of effort has been expended in finding convenient closed-form expressions

344 4.4 Signals with Unwanted Parameters

and in numerical evaluations. We have chosen two typical examples to illustrate the techniques employed. First we consider the radar problem defined at the beginning of this section (354-355).

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