Books
in black and white
Main menu
Share a book About us Home
Books
Biology Business Chemistry Computers Culture Economics Fiction Games Guide History Management Mathematical Medicine Mental Fitnes Physics Psychology Scince Sport Technics
Ads

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
= 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 [54]. 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 [54]. It is not related to the variance matrix %P(t).
Problem 6.6.1. Define the linear functional
(*T
5(0 = (/, 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
,) = 0,
%(T,) = Po-niT,),
Po = (Ä Tt).
and
where
Related Issues 615
1. Show that we can write the above integral as
?(/) = ? (/, r) Kx(r, t) s(t) dr + J(T/ (/, /) (, s(r) rfr.
Hint. See Problem 6.3.16.
2. By using Leibnitzs rule, show that
= 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 (|, Tt) = P0, and (, /) satisfies the adjoint equation; that is,
with (Ä ,) = I.
3. Define a second functional r\(t) by
) = 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, ) = P0*l).
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
{ < t < 7},
or
for > 0.
Define
so that
(/) = C(t) %(t).
6.7 Problems
1. Show that 5(0 satisfies the differential equations
d (01 =
dt UcoJ
F(/) G(f)QGT(/)
--- --- CT(t) C(t) -F T(t)
A
.
U(oJ
with
= 0\(ҳ),
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}, (:) + yjT,, Tt: A)] = 0, where V(t, Tt: A) is given by
.
d Tt : A) j VUt, r.rA)!
dt te'7r,:A)"iA)J
Tt: )"
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
(/, r(:A) = Tr.X)P0 + Tt: )] 4(7i)
where )(7) satisfies the orthogonality relationship
Previous << 1 .. 218 219 220 221 222 223 < 224 > 225 226 227 228 229 230 .. 254 >> Next