Solution_8[1] - Chapter 9 Gaussian Channel 1. A channel...

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Unformatted text preview: Chapter 9 Gaussian Channel 1. A channel with two independent looks at Y. Let Y1 and Y2 be conditionally independent and conditionally identically distributed given X. (a) Show I(X;Y1,Y3)= 2I(X;Y1) — I(Yngg)_ (b) Conclude that the capacity of the channel X fl can) is less than twice the capacity of the channel Solution: Channel with two independent looks at Y. (a) I(X;Y1,Y2) = Hahn) — mnnlx) (9-1) H(Y1)+H(Y2)—I(Y1;Y2)—H(K!X)WH(YL2EX) {9-2) (since 1’1 and Y2 are conditionally independent given X )(9.3) I(X;Y1)+ Incl/'2) — 1(Y1;Y2) (9-4) ) 2I(X; Y1) — I (Y1; Yg) (since Y; and Y2 are conditionally idfiné tically distributed) (in) The capacity of the single look channel X 6 Y1 is Cl = m(a;§:I(X;Y1). (9.6) p :1: 217 if g .1, i 21 l __ W ‘.__.,,___._._...-— 218 Gaussian channel The capacity of the channel X —> (Y1,Y2) is 02 = fi§I(X;Y1,Y2) (9.7) = figZI£X3Y1)-*I(Y1;lfi) (9.8) g fifimxm) (9.9) = 201. (9.10) Hence, two independent looks cannot be more than twice as good as one iook. 2. The two—look Gaussian channel. X (YbYZ) Consider the ordinary Gaussian channel with two correlated looks at X, i.e., Y 2 (Y1: 1 Where Y1 X+ Z1 (9.11) = X—l— Z2 {9.12) II with a power constraint P on X, and (Z1, Z2) ~ N2“), K), where h N Np K; [ Np N (9.13) Find the capacity C for (a) p =.»1 (bl p = 0 (c) p =-1 Solution: T he two look Gaussian channel. It is clear that the input distribution that maximizes the capacity is X N N(0,P). Evaluating the mutual information for this distribution, 02, = maxI{X;Y1,Y2) (9.14) = M37133} _ h(Z11Z2lX) [9-16) = flu/1,172) — M21, Z2) (9-17) Now since / r T {21, 22) ~ N tow Ifip if i (9.18) Gaussian channel 219 we have 1 1 . M21, 22) : 5 log(27re)2]KZl = 5 log(27re)2N2(1 — p2). (9.19) Since Y1 =X+Z1, and Y2=X+Zg, we have P + N P + pN ' (Yl’YZ)NN(O’[P+pN P+N (9‘20) and man) = gamma: = gamma e EH2}?er — p». (921) Hence the capacity is 02 : hiyiayfl — h(31:22} (9-22) 1 BF " = 510g (1 + - _ (9-23) (a) p = 1. In this case, C = %log(l + §), which is the capacity of a single look channel. This is not Surprising, since in this case Y1 = Y2. (b) p :' 0. In this case1 l 2P 0 = 5 log (1 + , which corresponds to using twice the power in a single iook. The capacity is the same as the capacity of the channel X “>- (Y1 + Y2). (c) p = —1. In this case, 0 = 00, which is not surprising since if we add Y1 and Yg, we can recover X exactly. Note that the capacity of the above channel in all cases is the same as the capacity of the channel X —+ Yl + Y2. Output power constraint. Consider an additive white Gaussian noise channel with an expected output power constraint P. Thus Y = X + Z , Z N N (0,02), Z is independent of X , and BY2 3 P. Find the channel capacity. 90 Solution: Output power constraint O = max I(X; Y) (9.25) f(X):E{erZ)ZSP = max (MY) e h(Z)) (9.27) f{X):E(X+z)2 g (9.28) Given a constraint on the output power of Y, the maximum differential entropy is achieved by a normal distribution, and we can achieve this by have X N Nfi), P —— N J , and in this case, 1 l P V C = log 2rreP M 5 log 27TeN : — log w. (9.29) 1 E 2 N Gaussian channel 221 5. Fading channel. Consider an additive noise fading channel YzXV+Z where Z is additive noise, V is a random variable representing fading1 and Z and V are independent of each other and of X . Argue that knowledge of the fading factor V improves capacity by showing 1(X;YEV) 2.10533”)- Solution: Fading Channel Expanding I (X ; Y, V) in two ways, weget I(X;Y,V) = I(X;V)+I(X;Y[V) (9.42) = I(X;Y) + I(X;V!Y) (9.43) i.e. I(X;V)+I(X;Y[V) -— I(X;Y)+I(X;V|Y) I(X; YEV) = I(X; Y) + _I{X; VlY) (9.44) I(X;YEV) 2 I(X;Y) 9.45) where {9.44) follows from the independence of X and V, and (9.45) follows from I (X ;VIY) 2 0. ' 6. Parallel channels and waterfilling. Consider a pair of parallel Gaussian channels, Hm Y X1 a was» (was “D a ’ 0 fi ’ ’ and there is a power constraint E(Xf-+ g 2P. Assume that 0% > 0%. At what power does the channel stop behaving like a single channel with noise variance 0% , and begin behaving like a pair of channels? where 222 Gaussian channel WW Solution: Parallel channels and waterfill'mg. By the result of Section 10.4, it follows that we will put all the signal power into the channel with less noise until the total power of noise + signal in that channel equals the noise power in the other channel. After that, we will split any additional power evenly between the two channels. Thus the combined channel begins to behave like a pair of parallel channels when the 2 signal power is equal to the difference of the two noise powers, i.e., when 2P : 0'1 — 0'3. 7. Multipath Gaussian channel. Consider a Gaussian noise channel of power contraint P, where the signal takes two different paths and the received noisy signals are added together at the antenna. 1 Y1 X >©w Y2 ‘ ' 2 r . (a) Find the capacity of this channel if Zl and Z2 are jointly normal with covariance matrix Kg 2 l 02 p02 p02 02 ‘ (b) What is the capacity for p = 0, p = 1, p = —1? Solution: Mnltipath Gaussian channel. The channel reduces to the following channel: The power constraint on the input 2X SD is Z1 + Z2. Then 2X Var(Zl —l— Zg) Z1+Zg is 4P. Z1 and Zg are zero mean, and therefore = Ellzl + Z2l2l = E[le + z; + 22122] 202 + 2,002. Gaussian channel Thus the noise distribution is N(O, 202(1 + p}. 223 (a) Plugging the noise and power values into-the formula for the one—diinensional (P? N ) channel capacity, C’ = %log(1 + g) , we get 1 4P 0 — ilog (1+ ——~202(1+p)) 2 - 02(1+p) (b) 1. When p20, C=%log(1+2§§ . ii. Whenp=1, C=%log(l+§;). iii. Whenp=—l, 0:00. 8. Parallel Gaussian channels Consider the following parallel Gaussian channel 21 X1 Yr Z2~N(OJN2) X2 Y2 1 2P = ~log (1+—-). where Z; N N(O,N1) and Z2 ~ N(O,N2) are independent Gaussian random variables and Y”; = Xi + Zr. We wish to allocate power to the two parallel channels. Let fil and 52 be fixed. Consider a total cost constraint [31131 + 62132 3 fl, Where B- is the power allocated to the ith channel and fit; is the cost per unit power in that channel. Thus P1 2 U and P2 2 U can be chosen subject to the cost constraint fl . (a) For What value of [3 does the channel stop acting like a single channel and start acting like a‘pair of channels? 2,N1=3,N :2 and sale. the capacity and find Pth that achieve capacity for $1 1: 1’32 Solution: Parallel channels When we have cost constraints on the power, we need to optimize the total capacity of the two parallel channels 1 P1 1 P2 ) o = — 1 _ —1 1 ~— .4 2log(-i—Nl)+20g(+N2 (9 8) subject to the constraint that 151131 + 52P2 S 5 (9.49) Using the methods of Section 9.4, we set 1 P- J(P1, P2) 2 Z 5 log (1 + + M: an) (9.50) 2 and differentiating with respect to P5 , we have 1 1 — A ~ = . 1 21Di+m+ it 0, (95) or V a = (— — Nat (9.52) fit or an : (1/ ; par-Nip. (9.53) (a) It follows that we will put all the signal power into the channel with less weighted noise (fit-Ni) until the total weighted power of noise + signal in that channel equals the weighted noise power in the other channel. After that, we will split any additional power between the two channels according to their weights. Thus the combined channel begins to behave like a pair of parallel channels when the signal power is equal to the difference of the two weighted noise powers, i.e., when 515 = 521V — 2 — )61N1- (b) In this case, 611V; < fizNg, so We would put power into channel 1 until ,6 = 1. After that we would put power according to their weights, i.e. we would divide remaining power of 9 in the ratio 2 late 1. Thus we would set P1 = 6 + 1 and P2 2 3, and so that 1/ = 10 in the equation above. The capacity in this case is 1 . 0,: E log(1 + 7/3) + élogfl + 3/2) = 1.53 bits. (9.54) 15. Discrete input continuous output channel. Let Pr{X = 1} = p, Pr{X = 0} z 1 — p, and let Y = X + Z , where Z is uniform over the interval [0,a], a > 1, and Z is independent of X . (a) Calculate ‘ I(X;Y):H(X)—H(X[Y). (b) New calculate I (X; Y) the other way by I(X; Y) = hm — h(Y|X). (c) Calculate the capacity of this channel by maximizing over 1) Solution: Discrete input Continuous Output channel (a) Since 91H 0 < flYlX I 0) Z { D othefwis: (9'92) and ‘ _ [l—p)% Ogy<l f(Y|X = 1) = i 1 5 y S a (9.93) 10% a<y<l+a Therefore, (1—plé OSy<1 Hy): g 1gyga (9.94) pi a<y< 1+0, (bl Hlxl = H{X!Y = y) is nonzero only for 1 S y S a, and by Bayes rule, conditioned on Y, the probabilty that X = 1 is M = wow: = 1) W = P (9-95) P(X: lley) = and hence H(X|Y) = P(l g Y S a)H(p) = Therefore I(X;Y) : Hoe — Home : imp). (C) flYlX = 0) ~ UH], a) , and hence h(Y|X = O) = logo, and Similarly for X :vl, so that h(Y[X) : loge. The difFegential entropy h(Y) can be calculated from (9.94) as 1 1 1—}? a 1 1+ap p Y z — — — — — _ a _1 ~ . h( ) foo malog a dy flalogady fa a ogadaeea) 1 = Elwplogpc{1-p)10s(1—P))+10ga (9.97) 1 = EHlleoea (9.98) and again I(X; Y) 2 My) — h(Y[X) = imp). (d) The mutual information is maximized for p = 0.5, and the corresponding capacity - of the channel is i . ...
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Solution_8[1] - Chapter 9 Gaussian Channel 1. A channel...

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