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

ISBN 0-471-09517-6

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Attenuator Fig. 6.14 Filtering with prediction.

Errors in Optimum Systems 493

the s-plane. In this expanded form the realizable part consists of the first two terms. Thus

The use of (99b) reduces the required manipulation.

In this section we have developed an algorithm for solving the Wiener-Hopf equation and presented a simple example to demonstrate the technique. Next we investigate the resulting mean-square error.

6.2.2 Errors in Optimum Systems

In order to evaluate the performance of the optimum linear filter we calculate the minimum mean-square error. The minimum mean-square error for the general case was given in (24) of Property 4. Because the processes are stationary and the filter is time-invariant, the mean-square error will not be a function of time. Thus (24) reduces to

Because h0(r) = 0 for r < 0, we can equally well write (100) as

Substituting the inverse transform of (102) into (101), we obtain,

The part of the integral inside the brackets is just K$z(t). Thus, since Kdz(t) is real,

(99ft)

(100)

(101)

Now

where

f, = uo> - j; î^ç/Ã. «*'“4

(105)

494 6.2 Wiener Filters

The result in (106) is a convenient expression for the mean-square error. Observe that we must factor the input spectrum and perform an inverse transform in order to evaluate it. (The same shortcuts discussed above are applicable.)

We can use (106) to study the effect of a on the mean-square error. Denote the desired signal when a = 0 as d0(t) and the desired signal for arbitrary a as da(t) é. dQ(t + «). Then

E[d0(t) z(t - r)] = KdQZ{r) À ô(ò)9 (107a)

and

E[da(t) z(t - r)] = E[do(t + a) z(t - ò)] = ô(ò + a). (107*)

We can now rewrite (106) in terms of ô(ò). Letting

Kd2(t) = ô{³ + a) (108a)

in (106), we have

= Kd(0) - Ã ô\³ + a) dt = tfd(0) - Ã ô\è) du. (1086)

JO Ja

Note that ô(è) is not a function of a. We observe that because the integrand is a positive quantity the error is monotone increasing with increasing a. Thus the smallest error is achieved when a = -oo (infinite delay) and increases monotonely to unity as +oo. This result says that for any desired signal the minimum mean-square error will decrease if we allow delay in the processing. The mean-square error for infinite delay provides a lower bound on the mean-square error for any finite delay and is frequently called the irreducible error. A more interesting quantity in some cases is the normalized error. We define the normalized error as *Pn<x - (109a)

or

(^-,-òÃ*1(è)“è- (im)

We may now apply our results to the preceding example.

Example 3 (continued). For our example

(pna = <

----^=ã-2(Ãëå + 2ê'/1 + ë' + Ã o<0,

JVo(l + VI + Ë) \J« Jo '

8P_______________1_

No (1 + Vl + Ë)2

1 — 7²7- —-------------------- / _ j* ª 2kt dt, a > 0.

(110)

Errors in Optimum Systems 495

Evaluating the integrals, we have

I +2 kJ 1 + Ë a

VTTA + (1 + Vi + A)2 V1 + Ë

+ .. . ,7r—“^0. (Hi)

and

ipn°= TTVTTa (112)

,* =---------,2 -ã,--it + -¦^~ ~ e~.2kaL a > î. (èç)

(1 + Vl + A) (1 + Vl + A)

The two limiting cases for (111) and (113) are a = — ñþ and a — oo, respectively.-

fm-" = -7==- (115)

V1 + Ë

fn,- = 1, (Ï4)

A plot of ?Pn“ versus (A:a) is shown in Fig. 6.15. Physically, the quantity ka is related

to the reciprocal of the message bandwidth. If we define

TC = (116)

the units on the horizontal axis are a/rc, which corresponds to the delay measured in correlation times. We see that the error for a delay of one time constant is approximately the infinite delay error. Note that the error is not a symmetric function of a.

Before summarizing our discussion of realizable filters, we discuss the related problem of unrealizable filters.

Fig. 6.15 Effect of time-shift on filtering error.

496 6.2 Wiener Filters

6.2.3 Unrealizable Filters

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