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

ISBN 0-471-09517-6

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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

àä = 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 = Ã Êõ(/, 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

×Ò,) = 0,

%(T,) = Po-niT,),

Po = ÊÕ(Ã„ 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/ Êõ(/, /) Ôò(ò, Î s(r) rfr.

Hint. See Problem 6.3.16.

2. By using Leibnitz’s rule, show that

Ùð = 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 ÊÕ(Ã|, Tt) = P0, and Ôã(ò, /) satisfies the adjoint equation; that is,

with Ô(Ã„ Ã,) = 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, Ø) = P0*l¹).

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

Ò{ < t < 7},

or

for Ë > 0.

Define

so that

ô(/) = ³ C(t) %(t).

á³á 6.7 Problems

1. Show that 5(0 satisfies the differential equations

d ÃÑ(01 =

dt UcoJ

¦ F(/) G(f)QGT(/)

--- --- CT(t) C(t) -F T(t)

A

È.

U(oJ

with

úò = ð0ò\(Ò³),

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}, Ã(:Ë)Ðî + yjT,, Tt: A)] = 0, where V(t, Tt: A) is given by

òã.ò

d Tt : A) j VUt, r.rA)!

dt te'7r,:A)"iA)J

Tt: À)"

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: À)] 4(7i)

where ó)(7³) satisfies the orthogonality relationship

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