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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 Previous << 1 .. 210 211 212 213 214 215 < 216 > 217 218 219 220 221 222 .. 254 >> Next 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, Previous << 1 .. 210 211 212 213 214 215 < 216 > 217 218 219 220 221 222 .. 254 >> Next 