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

ISBN 0-471-09517-6

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To find the worst case we choose sˆ(t) to make Ad as negative as possible.

2. Let the actual noise be

«a(0 = n(t) + nˆ(t) (322a)

whose covariance function is

Kna(t, u) = Kn(t, u) + Knˆ(t, è), (322b)

We assume that ne{t) has finite energy in the interval

E Ã' èÄÎ dt < Ä„. (323a)

This implies that

Tf

J J KJ{t, è) dt du < Äï. (323b)

Tt

To find the worst case we choose Knˆ(t, u) to make Ad as negative as possible.

Various other perturbations and constraints are also possible. We now consider a simple version of the first problem. The second problem is developed in detail in [42].

We assume that the noise process is stationary with a spectrum Sn(co) and that the observation interval is infinite. The optimum receiver, using

a whitening realization (see Fig. 4.38a), is shown in Fig. 4.48a. The

328 4.3 Detection and Estimation in Nonwhite Gaussian Noise

(a)

Ã2

E(l\H0)

7

¦VEs(t)

- Decision line

Var [l\H0]=d2

--------H

ÊÅ(1\Íã)

(b)

Ã2

^Decision line

r

\

, ³ V..

7 ³

(c)

Fig. 4.48 Sensitivity analysis: (a) filter with nominal input; (b) nominal decision space;

(c) actual design space.

Sensitivity 329

corresponding decision space is shown in Fig. 4.48É. The nominal performance is

/* oo

?(/|ÿ,) - åäÿ,)

- [Var(/|ß0)]É - ( ’

or

Ã ËÎÎ -I y2

d = [J CO S*(0 dt\ ‘ (325)

We let the actual signal be

sa(t) = VE s(t) + VEˆ se(t), —oo < t < ñî, (326)

where s(t) and se(t) have unit energy. The output of the whitening filter will be

r*a(r) A s*(t) + 5»?(0 + è*(0> -°° < t < oo, (327)

and the decision space will be as shown in Fig. 4.48c. The only quantity that changes is ?(/à|ß³). The variance is still the same because the noise covariance is unchanged. Thus

Ad = -d J" **«(#) **(0 dt. (328)

To examine the sensitivity we want to make Ad as negative as possible. If we can make Ad = —d, then the actual operating characteristic will be

the PD = PF line which is equivalent to a random test. If Ad < —d, the

actual test will be worse than a random test (see Fig. 2.9a). It is important to note that the constraint is on the energy in sˆ(t), not s*f(t). Using Parseval’s theorem, we can write (328) as

dw

Arf = ³ J°°^ S*<(>) SJO'o>) (329)

This equation can be written in terms of the original quantities by observing that

SUM = VFt Hw(jco) SeO) (330)

and

Thus

A d =

S0(jw) = Vehju*) SO)- (331)

Vee]

d

j*_ ò³ s*0)

VEE-<**>

330 43 Detection and Estimation in Nonwhite Gaussian Noise

The constraint in (321) can be written as

|seO)|2^ = 1. (333)

To perform a worst-case analysis we minimize Û subject to the constraint in (333) by using Lagrange multipliers. Let

F=Ad+ A [J^ |Se(»|2 ? - l] • (334)

Minimizing with respect to Sˆ(joj), we obtain

9 (ioA = ^EEz S(ja³)

Se0U“>) 2Xd (335)

(the subscript î denotes optimum). To evaluate A we substitute into the constraint equation (333) and obtain

â-ãæà*,, òà

4WJ-. 2ò '• ,,m>

If the integral exists, then

VIE,

2A =

d

Substituting into (335) and then (332), we have

"--ØÃ<->

(Observe that we could also obtain (338) by using the Schwarz inequality in (332).) Using the frequency domain equivalent of (325), we have

rrr ³ÿì_2^Ã/2]

^ _ _ (¸<\ V U— 2ni [ m<n

d~ U) rr ^

L L J - oo 5„(<o) 2òã J J

In the white noise case the term in the brace reduces to one and we obtain the same result as in (82). When the noise is not white, several observations are important:

1. If there is a white noise component, both integrals exist and the term in the braces is greater than or equal to one. (Use the Schwarz inequality on the denominator.) Thus in the colored noise case a small signal perturbation may cause a large change in performance.

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