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Detection, Estimation modulation theory part 1 - Vantress H.

Vantress H. Detection, Estimation modulation theory part 1 - Wiley & sons , 2001. - 710 p.
ISBN 0-471-09517-6
Download (direct link): sonsdetectionestimati2001.pdf
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Problem 6.2.11.
1. The received signal is a(u), < < 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 = + kr)
where kt kj\i for = 1,..., nf j = 1,..., n.
2. Now assume that the received signal is a(w), Tt < < 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 = 1,2, and = 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), 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 -
- 1
Fig. P6.2
592 6.7 Problems
Problem 6,2,13. Let
r(u) = a(u) 4- n(u), oo < < t,
where a(u) and n(u) are uncorrelated random processes with spectra
CO2
5a(w) = &() =
CO4 + 1 1
CO2 + 2'
The desired signal is a(t). Find the optimum (MMSE) linear filter and the resulting error for the limiting case in which -> 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), < < 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 (4 = ).
u(t)
1
s+k
x\(t)
1
s + k
*2(0

-(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 /û IJ^T/O'co)12(^/ct/27r) = 1, find the /() 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, ó , 2Cn/No I
"2f J - * 2 L1 + 1 + ()4
1. Do this by letting = 2cn/N0. Differentiate with respect to 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 < < t,
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