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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 .. 218 219 220 221 222 223 < 224 > 225 226 227 228 229 230 .. 254 >> Next where w(t) is white (No/2). Find the optimum unrealizable demodulator and plot the mean-square error as a function of m.
Problem 6.4.7 (continuation). Consider the model in Problem 6.4.6. Let
Find the optimum unrealizable demodulator and plot the mean-square error as a function of m.
Problem 6.4.8. Consider the example on p. 583. Assume that
1. Find an expression for the mean-square error using an unrealizable demodulator designed to be optimum for the known-phase case.
2. Approximate the integral in part 1 for the case in which Am » 1.
Problem 6.4.9 (continuation). Consider the model in Problem 6.4.8. Let
where
Sa(°J"> O)2 + k2
SM = 2JP’ Hs
0, elsewhere.
and
Sa(*) = H ^ 2TTIV,
0, elsewhere.
Repeat Problem 6.4.8.
614 6.7 Problems
Fig. P6.15
Problem 6.4.10. Consider the model in Problem 6.4.7. The demodulator is shown in Fig. P6.15. Assume m « 1 and
ŕä = 2JP’ 2nWi < \w\ < 2n(W, + W),
0,
elsewhere,
where 2tt( Wx 4- W) « o)c.
Choose hL(r) to minimize the mean-square error. Calculate the resulting error gP.
P6.6 Related Issues
In Problems 6.6.1 through 6.6.4, we show how the state-variable techniques we have developed can be used in several important applications. The first problem develops a necessary preliminary result. The second and third problems develop a solution technique for homogeneous and nonhomogeneous Fredholm equations (either vector or scalar). The fourth problem develops the optimum unrealizable filter. A complete development is given in . The model for the four problems is
x(0 = F(;)x(0 + G(0u(0 y(0 = C(0x(0 ?[u(0ut(t)] = Q8(t - r)
We use a function ?(r) to agree with the notation of . It is not related to the variance matrix %P(t).
Problem 6.6.1. Define the linear functional
(*T
5(0 = Ă Ęő(/, t) s(t) dr,
JTi
where s(0 is a bounded vector function.
We want to show that when Kx(/, r) is the covariance matrix for a state-variable random process x(t) we can represent this functional as the solution to the differential equations
dm _
and
dt
= F(0?(0 + G(0 QGr(0 r)(0
with the boundary conditions
×Ň,) = 0,
%(T,) = Po-niT,),
Po = ĘŐ(Ă„ Tt).
and
where
Related Issues 615
1. Show that we can write the above integral as
?(/) = ? Ô(/, r) Kx(r, t) s(t) dr + J(T/ Ęő(/, /) Ôň(ň, Î s(r) rfr.
Hint. See Problem 6.3.16.
2. By using Leibnitz’s rule, show that
Ůđ = m 5(0 + G(/)QGr(/) [f*T(r, t) S(r) dr.
Hint. Note that Kx(f, t) satisfies the differential equation
= F(') K*('> '> + Kx(', ')FT(0 + G(t)QGT(t), (text 273)
with ĘŐ(Ă|, Tt) = P0, and Ôă(ň, /) satisfies the adjoint equation; that is,
with Ô(Ă„ Ă,) = I.
3. Define a second functional r\(t) by
Ô) = j*f<t>4r,t)S(r)dr.
Show that it satisfies the differential equation
4. Show that the differential equations must satisfy the two independent boundary
r\(Tf) = 0, Ř) = P0*lą).
5. By combining the results in parts 2, 3, and 4, show the desired result.
Problem 6.6.2. Homogeneous Fredholm Equation. In this problem we derive a set of differential equations to determine the eigenfunctions for the homogeneous Fredholm equation. The equation is given by
conditions
Ň{ < t < 7},
or
for Ë > 0.
Define
so that
ô(/) = ł C(t) %(t).
áłá 6.7 Problems
1. Show that 5(0 satisfies the differential equations
d ĂŃ(01 =
dt UcoJ
¦ F(/) G(f)QGT(/)
--- --- CT(t) C(t) -F T(t)
A
Č.
U(oJ
with
úň = đ0ň\(Ňł),
r\(Tf) = 0.
(Use the results of Problem 6.6.1.)
2. Show that to have a nontrivial solution which satisfies the boundary conditions we need
det [4V7}, Ă(:Ë)Đî + yjT,, Tt: A)] = 0, where V(t, Tt: A) is given by
ňă.ň
d Tt : A) j VUt, r.rA)!
dt te'7r,:A)"iA)J
Tt: Ŕ)"
Tt -.X) 'Vnr.it, T, : A)
F(0 : G(0 QGT(f)
j"'”;""'"
and ^(71, 71: A) = I. The values of A which satisfy this equation are the eigenvalues.
3. Show that the eigenfunctions are given by
Ô(/, r(:A) = Tr.X)P0 + Tt: Ŕ)] 4(7i)
where ó)(7ł) satisfies the orthogonality relationship Previous << 1 .. 218 219 220 221 222 223 < 224 > 225 226 227 228 229 230 .. 254 >> Next 