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

ISBN 0-471-09517-6

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Problem 6.2.11.

1. The received signal is a(u), — ñþ < è < t. The desired signal is

d(t) = a(t + a), a > 0.

Find H0(joi) to minimize the mean-square error

mm - dim,

where

d(t) = J h0(t — u) a(u) du.

The spectrum of a(t) is

5a(“0 = Ï + kr)

where kt ô kj\i ô ó for ³ = 1,..., nf j = 1,..., n.

2. Now assume that the received signal is a(w), Tt < è < t, where T{ is a finite number. Find h0(t, t) to minimize the mean-square error.

d(t) = f h0(t, u) a(u) du.

JTi

3. Do your answers to parts 1 and 2 enable you to make any general statements about pure prediction problems in which the message spectrum has no zeros ?

Problem 6.2.12. The message is generated as shown in Fig. P6.2, where u(t) is a white noise process (unity spectral height) and ah ³ = 1,2, and Ëü ³ = 1, 2, are known positive constants. The additive white noise w(t)(N0l2) is uncorrelated with u(t).

1. Find an expression for the linear filters whose outputs are the MMSE realizable estimates of xt(t), ³ — 1,2.

2. Prove that

a(t) = J (0-

t = l

3. Assume that

d(t) = 2

i = 1

Prove that

d(t) = 2 di Ø-

³- 1

Fig. P6.2

592 6.7 Problems

Problem 6,2,13. Let

r(u) = a(u) 4- n(u), — oo < è < t,

where a(u) and n(u) are uncorrelated random processes with spectra

CO2

5a(w) = &(«) =

CO4 + 1 1

CO2 + ˆ2'

The desired signal is a(t). Find the optimum (MMSE) linear filter and the resulting error for the limiting case in which º -> 0. Sketch the magnitude and phase of H0(jw).

Problem 6,2,14, The received waveform r(u) is

r(u) = a(u) + w(u), —ñî < è < t, where a(u) and w(w) are uncorrelated random processes with spectra

2koa2

Sa(«) = 5n(«) =

Oi2 + k2

No

Let

d(t) A j a(u) du, a > 0.

1. Find the optimum (MMSE) linear filter for estimating d(t),

2. Find

Problem 6,2,15 (continuation). Consider the same model as Problem 6.2.14. Repeat that problem for the following desired signals:

1. d(t) = - f a(u)dut a > 0.

a Jt-a

2. d(t) = 3—!— f a(u) du, a > 0, jS > 0, jS > a.

P a Jt + a

What happens as (p — a) -* 0 ?

+1

3. d(t) = 2 a^ ~ a > 0.

n= -1

Problem 6,2,16, Consider the model in Fig. P6.3. The function u(t) is a sample function from a white process (unity spectral height). Find the MMSE realizable linear estimates, Jci(0 and x2(t). Compute the mean-square errors and the cross correlation between the errors (Ã4 = —ñî).

u(t)

1

s+k

x\(t)

1

s + k

*2(0

ãªæ©

-«(f)

r(t)

Fig. P6.3

Stationary Processes, Infinite Past, (Wiener Filters) 593

VFa(t)

kf(r)

Fig. P6.4

Problem 6.2.17. Consider the communication problem in Fig. P6.4. The message a(t) is a sample function from a stationary, zero-mean Gaussian process with unity variance. The channel kf(r) is a linear, time-invariant, not necessarily realizable system. The additive noise n(t) is a sample function from a zero-mean white Gaussian process (AV2).

1. We process r(t) with the optimum unrealizable linear filter to find a(t). Assuming /Ã» IJ^T/O'co)12(^/ct»/2“7r) = 1, find the ê/(ò) that minimizes the minimum mean-square error.

2. Sketch for

2k

SaO) =

+ k2

Closed Form Error Expressions Problem 6.2.18. We want to integrate

, No P du, Ã³ , 2Cn/No I

"2f J - * 2¯ã L1 + 1 + (×ê)4

1. Do this by letting ó = 2cn/N0. Differentiate with respect to ó and then integrate with respect to u). Integrate the result from 0 to y.

2. Discuss the conditions under which this technique is valid.

Problem 6.2.19. Evaluate

dw cn

I

¦-F.

2n 1 + (a>/k)2n + (2/N0)cn

Comment. In the next seven problems we develop closed-form error expressions for some interesting cases. In most of these problems the solutions are difficult. In all problems

r(u) = a(u) + n(u), —oo < è < t,

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