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

ISBN 0-471-09517-6

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1. Show that an efficient estimate of A will exist if

s(t, A) = LA s(t).

2. Find an explicit solution for am&p and an expression for the resulting mean-square error.

Problem 4.5.6. Let

ê

rx(t) = 2 au Suit) + wt(0, / = 1, 2,..., M:HU

i=³

rt(0 = Wi(0, ³ = 1, 2,..M:H0.

The noise in each channel is a sample function from a zero-mean white Gaussian random process

E[Mt ) w»] = ^ I 8(/ — u).

The au are jointly Gaussian and zero-mean. The stj(t) are orthogonal. Find an expression for the optimum Bayes receiver.

Problem 4.5.7. Consider the binary detection problem in which the received signal is an M-dimensional vector:

r(0 = s(0 + nc(r) + w(/), -oo < t < oo:#i,

= nc(0 + w(0, —00 < t < oo:HQ.

The total signal energy is ME:

jT sT(t) s(/) dt = ME.

The signals are zero outside the interval (0, T).

1. Draw a block diagram of the optimum receiver.

2. Verify that

d2 = J_"e ST(jw) Sn ~ *(») S(>) g •

Problem 4.5.8. Maximal-Ratio Combiners. Let

r(0 = s(r) + w(0, 0 < t < T.

The received signal r(/) is passed into a time-invariant matrix filter with M inputs and one output y(t):

y(t) = JJ h(/ - ã) ã(ò) dr.

The subscript s denotes the output due to the signal. The subscript n denotes the output due to the noise. Define

(S\ ä ys\T)

Wout “ E[yn2{T)]

1. Assume that the covariance matrix of w(/) satisfies (439). Find the matrix filter h(r) that maximizes (S/N)out. Compare your answer with (440).

2. Repeat part 1 for a noise vector with an arbitrary covariance matrix Êc(t, u).

Multiple Channels 411

Random Phase Channels

Problem 4,5,9 [14]. Let

M

x = 2 «ë

i = 1

where each at is an independent random variable with the probability density

pai(A) = ^ exp î < A < oo,

= 0, elsewhere.

Show that

= 0, elsewhere,

where

M

P = a2 J “i2-

1 = 1

Problem 4,5,10, Generalized Q-Function.

The generalization of the ^-function to M channels is

Gm(«, P) = Je *(“) exp(-* 2 a}lM-Aax)dx.

1. Verify the relation

( 2 ³ 02\ M-1 fO\k

-----2 ) ? W /fc(ai8)‘

2. Find Gm(«, 0).

3. Find QM(0, j8).

Problem 4.5.11. On-Off Signaling: N Incoherent Channels. Consider an on-off communication system that transmits over N fixed-amplitude random-phase channels. When Hi is true, a bandpass signal is transmitted over each channel. When H0 is true, no signal is transmitted. The received waveforms under the two hypotheses are

r,(t) = V2E, / (t) cos (Wit + <?,(>) + 0,) + w(t), 0 < t < T :HU

r*(r) = w(r), 0 < t < T :#o,

³ = 1,2, ...,N.

The carrier frequencies are separated enough so that the signals are in disjoint frequency bands. The/0) and </>t(t) are known low-frequency functions. The amplitudes VEi are known. The 0t are statistically independent phase angles with a uniform distribution. The additive noise w(/) is a sample function from a white Gaussian random process (N0/2) which is independent of the 0(.

1. Show that the likelihood ratio test is where LCi and L$i are defined as in (361) and (362).

412 4.8 Problems

2. Draw a block diagram of the optimum receiver based on In A.

3. Using (371), find a good approximation to the optimum receiver for the case in which the argument of /0(-) is small.

4. Repeat for the case in which the argument is large.

5. If the Ex are unknown nonrandom variables, does a UMP test exist?

Problem 4.5.12 (continuation). In this problem we analyze the performance of the suboptimum receiver developed in part 3 of the preceding problem. The test statistic is

/= + V)i r.

i = i H°

1. Find E[LCt\Hol, E[LSi\H0], Var [Lc,|tf0], Var [Ls,|tf0], E[LCl\Hu â], E[LH\HU 6], Var lLCi\Hu â], Var [LH\HU â].

2. Use the result in Problem 2.6.4 to show that

MllHl(jv) = (1 - jvN0)-« exp (f

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