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

ISBN 0-471-09517-6

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Ó

or

/• - J2EJNq cos â 1 / r2\

Pr(e|0) = L VT,exp\2)dz- (394)

Using (364),

PrW-J*.''-T^sSilpr(‘m‘"' <M5>

This can be integrated numerically. The results for two particular values of d2 are shown in Figs. 4.62 and 4.63.f The results for other p were also evaluated in [44] and are given in these figures. We see that for Am greater than about 2 the negatively correlated signals become more efficient than orthogonal signals. For Am > 10 the difference is significant. The physical significance of Am will become clearer when we study phase estimating systems in Chapter II.2.

In this section we have studied a particular case of an unwanted parameter, a random phase angle. By using a family of densities we were able to demonstrate how to progress smoothly from the known signal case to the uniform phase case. The receiver consisted of a weighted sum of a linear operation and a quadratic operation. We observe that the specific receiver structure is due to the precise form of the density chosen. In many cases the probability density for the phase angle would not correspond to any of these densities. Intuitively we expect that the receiver developed here should be “almost” optimum for any single-peaked density with the same variance as the member of the family for which it was designed.

We now turn to a case of equal (or perhaps greater) importance in which both the amplitude and phase of the received signal vary.

t The values of d2 were chosen to give a Pr (e) = 10“3 and 10“5, respectively, at A. = oo.

Random Phase 349

Ë/ï ^

Fig. 4.62 Pr (e), partially coherent binary system (10 "3 asymptote) [44].

4.4.2 Random Amplitude and Phase

As we discussed in Section 4.1, there are cases in which both the amplitude and phase of the received signal vary. In the communication context this situation is encountered in ionospheric links operating above the maximum usable frequency (e.g., [51]) and in some tropospheric links (e.g., [52]). In the radar context it is encountered when the target’s aspect or effective radar cross section changes from pulse to pulse (e.g., Swerling [53]).

Experimental results for a number of physical problems indicate that when the input is a sine wave, V2 sin o)ct9 the output (in the absence of additive noise) is

K0 = fcb0)sin êt + ejt)l (396)

An exaggerated sketch is shown in Fig. 4.64a. The envelope and phase vary continuously. The envelope vch(t) has the Rayleigh probability density shown in Fig. 4.646 and that the phase angle 0ch(O has a uniform density.

350 4.4 Signals with Unwanted Parameters

A m

Fig. 4.63 Pr (c), partially coherent binary system (10" 5 asymptote) [44].

There are several ways to model this channel. The simplest technique is to replace the actual channel functions by piecewise constant functions (Fig. 4.65). This would be valid when the channel does not vary significantly

Fig. 4.64 Narrow-band process at output of channel, and the probability of its

envelope.

(a)

(b)

t

Fig. 4.65 Piecewise constant approximation: (a) actual envelope; (b) piecewise

constant model.

351

352 4.4 Signals with Unwanted Parameters

in a T second interval. Given this “slow-fading” model, two choices are available. We can process each signaling interval independently or exploit the channel continuity by measuring the channel and using the measurements in the receiver. We now explore the first alternative.

For the simple binary detection problem in additive white Gaussian noise we may write the received signal under the two hypotheses as|

= vV2 f{t) cos [a>ct + ô³ã) + â] + wit), 0 < t < T,

H0:rit) = wit), 0 < t <T, Vó }

where v is a Rayleigh random variable and â is a uniform random variable.

We can write the signal component equally well in terms of its quadrature components:

V2 v fit) cos [a)ct + <?(/) -f 6] = axVl f{t) cos [a)ct 4- <?(/)]

+ a2V2 f(t) sin [a)ct + ô(³)],

0 <t<T, (398)

where ax and a2 are independent zero-mean Gaussian random variables with variance a2 (where E[v2] = 2a2; see pp. 158-161 of Davenport and Root [2]). We also observe that the two terms are orthogonal. Thus the signal out of a Rayleigh fading channel can be viewed as the sum of two orthogonal signals, each multiplied by an independent Gaussian random variable. This seems to be an easier way to look at the problem. As a matter of fact, it is just as easy to solve the more general problem in which the received waveform on Hx is,

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