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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
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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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