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

ISBN 0-471-09517-6

**Download**(direct link)

**:**

**124**> 125 126 127 128 129 130 .. 254 >> Next

Unwanted parameters appear in both detection and estimation problems. Because of the inherent similarities, it is adequate to confine our present

334 4.4 Signals with Unwanted Parameters

discussion to the detection problem. In particular, we shall discuss general binary detection. In this case the received signals under the two hypotheses are

r(t) = s^t, 6) + n(t), Ò³ < t < Òã:Í³,

r(t) = s0(t, ª) + n(t), Tt< t < Tf:H0. K )

The vector 0 denotes an unwanted vector parameter. The functions .Óî(/, 0) and Si(t, 0) are conditionally deterministic (i.e., if the value of 0 were known, the values of s0(t, 0) and s^t, 0) would be known for all t in the observation interval). We see that this problem is just the waveform counterpart to the classical composite hypothesis testing problem discussed in Section 2.5. As we pointed out in that section, three types of situations can develop:

1. 0 is a random variable with a known a priori density;

2. 0 is a random variable with an unknown a priori density;

3. 0 is a nonrandom variable.

We shall confine our discussion here to the first situation. At the end of the section we comment briefly on the other two. The reason for this choice is that the two physical problems encountered most frequently in practice can be modeled by the first case. We discuss them in detail in Sections 4.4.1 and 4.4.2, respectively.

The technique for solving problems in the first category is straightforward. We choose a finite set of observables and denote them by the ËÃ-dimensional vector r. We construct the likelihood ratio and then let Ê->oo.

Ë[ã(0] Ä lim (350)

K-> 00 /?Ã|ß0(Ê|^0>)

The only new feature is finding /^ÿÄÊ]#!) and /?r|Ho(R|#o) *n presence of 0. If 0 were known, we should then have a familiar problem. Thus an obvious approach is to write

^³ÿÄÂÄ) = f (351)

J*e

and

/>ã|ßî(Ê|ß0) = f Ðïâ,íîòÍ0)ðÂ²Íî(Ü\Í0)ñ³Â. (352)

Substituting (351) and (352) into (350) gives the likelihood ratio. The tractability of the procedure depends on the form of the functions to be integrated. In the next two sections we consider two physical problems in which the procedure leads to easily interpretable results.

Random Phase Angles 335

4.4.1 Random Phase Angles

In this section we look at several physical problems in which the uncertainty in the received signal is due to a random phase angle. The first problem of interest is a radar problem. The transmitted signal is a bandpass waveform which may be both amplitude- and phase-modulated. We can write the transmitted waveform as

VzEtfit) cos [cv + ô(³)1 0 <t<T, (353)

0, elsewhere.

Two typical waveforms are shown in Fig. 4.50. The function fit) corresponds to the envelope and is normalized so that the transmitted energy is Et. The function ô³³) corresponds to a phase modulation. Both functions are low frequency in comparison to coc.

For the present we assume that we simply want to decide whether a target is present at a particular range. If a target is present, the signal will be reflected. In the simplest case of a fixed point target, the received waveform will be an attenuated version of the transmitted waveform with a random phase angle added to the carrier. In addition, there is an additive white noise component wit) at the receiver whether the target is present or

(a)

(b)

Fig. 4.50 Typical envelope and phase functions.

336 4.4 Signals with Unwanted Parameters

not. If we define #! as the hypothesis that the target is present and H0 as the hypothesis that the target is absent, the following detection problem results:

Hi\r{t) = V2E.f(t — r) cos (<oc(t — r) + ô{¿ - r) + 6)

+ w(t), t < t < ã + T,

= w(t)9 Ò³ < t < ò9 t + T < t < Tf,

A sr(t — ã, 0) + w(t), Ò³ < t < Tf. (354a)

H0:r(t) = w(t); Ti< t < Tf. (354b)

Because the noise is white, we need only observe over the interval t < t < ã + T. Under the assumption that we are interested only in a

particular t, the model is the same if we let ò = 0. Thus we need only

consider the problem

Hi.r(t) = sr{t9 â) + í’(/), 0 < t < T, (355a)

H0\r(t) = w(t), 0 < t < T. (3556)

**124**> 125 126 127 128 129 130 .. 254 >> Next