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

ISBN 0-471-09517-6

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ffi2 + a22 CTi2 + a22’ C or2

Then

Ï = ååÓ/a, v'A) - exp (-/„(VS), (F.3.2)

or

Ð* = ÃÒ75 [1 “ Q(Vb, ^a)] + Q(Va, Vb), (F.3.3)

or

p* = Ml - Q(Vb, Va) + Q(Va, Vb)] - A^-Z-iexp (~5^) hi.^ab). (F.3.4)

Problem 4.4.1, Q-function Properties. Marcum’s 0-function appears frequently in the calculation of error probabilities:

Q(a, P) = ? * exp [-i(*2 + a2)] I0(ax) dx.

Verify the following properties:

1. G(«, 0) = 1,

2. (2(0, jS) = *-*2'2.

3. G(a, /3) = «-'«’+/•’>« j. /„(a/3), a < ft

= 1 - ª-<“à+»2>'2 ² (2)Ï/„ÍÇ), /3 < a.

n = l W

396 4.8 Problems

4. à», Ä + Oft «) - 1 + (e-<«2+«2»'2)/o(ot^).

5. C(«, w = ³ - ^ (i)V*-"2'2), »»a»1.

6. G(«, 0 ~ ^ (JL) /»» « » 1.

Problem 4.4,2. Let jc be a Gaussian random variable 7V(m*, a*).

1. Prove that

Mx*(jv) Ä ß[åõð (+ë>*2)] =

ä , A ._2M - exp L/pmx2/(l - 2jvax2)]

(1 - 2jvax2)i/2 Hint.

Mx2(Jv) = [M,2(;i;) Afya(/®)]*,

where >> is an independent Gaussian random variable with identical statistics.

2. Let z be a complex number. Modify the derivation in part 1 to show that

3. Let

2M

Ó2 = 2 w, i = l

where the Xt are statistically independent Gaussian variables, N(miy at).

Find My2(jv) and ?jexp (+ zy2)]. What condition must be imposed on Re (z) in order for the latter expectation to exist.

4. Consider the special case in which A4 = 1 and a*2 = a2. Verify that the probability density of y2 is

= 0, elsewhere,

where 5 = 2?=Mi mia. (See Erdelyi [75], p. 197, eq. 18.)

Problem 4.4.3. Let Q(x) be a quadratic form of correlated Gaussian random variables,

Q(x) A xTAx.

1. Show that the characteristic function of Q is

MQ(jv) Ä = exp {-imxTA-4I -(^-2jvAA)-']mx}

2. Consider the special case in which Ë-1 = A and mx = 0. What is the resulting density ?

3. Extend the result in part 1 to find Äå20), where z is a complex number. What restrictions must be put on Re (z) ?

Problem 4.4.4. [76] Let xu x2, *3, x4 be statistically independent Gaussian random variables with identical variances. Prove

Pr (.X!2 + X22 > Õç2 + X42) = ±[1 - <2(0, a) + G(a, fl)L

where

Signals with Unwanted Parameters 397

Random Phase Channels

Problem. 4.4.5. On-Off Signaling: Partially Coherent Channel. Consider the hypothesis testing problem stated in (357) and (358) with the probability density given by (364). From (371) we see that an equivalent test statistic is

(0 + Lc)2 + L? \ y,

Ho

where

ft Ë ^0

P ~ 2 VEr

1. Express PF as a Q-function.

2. Express PD as an integral of a g-function.

Problem 4.4.6. M-orthogonal Signals: Partially Coherent Channel. Assume that each of the M hypotheses are equally likely. The received signals at the output of a random phase channel are

r(t) = V2E, f(t) cos Ê ³ + Ø + 0] + èit), 0 < / < Ã:ß„ i = 1,2,..M,

where pe(0) satisfies (364) and w(t) is white with spectral height N0/2. Find the LRT and draw a block diagram of the minimum probability of error receiver.

Problem 4.4.7 (continuation). Error Probability; Uniform Phase. [18] Consider the special case of the above model in which the signals are orthogonal and â has a uniform density.

1. Show that

Pr(<|0) = 1 - b([i - exp (-Sli-Z?)]"'1}.,

where x and ó are statistically independent Gaussian random variables with unit variance.

E[x |0] = V2E,IN0 cos â,

?Û0] = V2¨Æî sin â.

The expectation is over jc and ó, given â.

2. Show that

pr w = Y 1)ê+1^ð³-(ÅÆ)ò + ³)]ó

Problem 4.4.8. In the binary communication problem on pp. 345-348 we assumed that the signals on the two hypotheses were not phase-modulated. The general binary problem in white noise is

r(t) = V2Er Mt) COS [<oct + Ô^²) + â] + w(t), 0 <. t < T :#b

r(t) = \ 2Er f0(t) cos [<act + <t>0(t) + â] + w(t), 0 < t < T :H0,

where Er is the energy received in the signal component. The noise is white with spectral height N0/2, and pe(&) satisfies (364). Verify that the optimum receiver structure is as shown in Fig. P4.7 for / = 0, 1, and that the minimum probability of error test is

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