課程名稱︰消息理論 課程性質︰選修 課程教師︰林茂昭 開課學院：電資學院 開課系所︰電信所 考試日期（年月日）︰2020.6.16 考試時限（分鐘）：100 min 試題： 1. Consider an additive Gaussian noise channel. (a) What is the minimum siganl to noise tatio needed for achieving transmission rate R = 1/2? (b) In the literaturem it is reported that using a very long rate 1/2 binary low density parity density (LDPC) code, a very low bit-error-rate can be achieved at SNR only 0.04dB away from the Shannon channel. Is this Shanon limit the SNR derived in part (a)? Explain it. 2. Calculate the capacity of the folowing channel. (a) The binary erasure channel with erasure probability α. (b) The channel with input and output given by X = Y = {1,2,3}, and the probability transition matrix is: p(y|x) = 1 2 3 1 0.5 0.5 0 2 0 0.5 0.5 3 0.5 0 0.5 3. (a)(5%) Find the differential entropy h(X) = -∫f㏑f for the random variable with the exponential density, f(x) = λexp(-λx), x≧0. (b)(5%) Please show that the exponential distribution with mean 1/λ is the maximum entropy distribution among all continuous disstributions supported in [0,∞] that have a mean of 1/λ. (c)(5%) Let Yi = Xi + Zi, where Zi is i.i.d. exponential distribution with mean μ. Assume that we have a mean constraint on the signal (i.e. EXi ≦ λ). Show that the capacity of such a channel is C = ㏒(1+λ/μ). 4. Consider an additive noise fading channel, with input X and output Y = XV + Z, where Z is the additive noise, V is a random variable representing fading, and Z and V are independent of each other and of X. Please show that: I(X;Y|V) ≧ I(X;Y) 5. For the case of a continuous random variable X with mean zero and variance σ^2 and square error distortion, show that h(X) - 1/2(log(2πeD)) >= R(D) >= 1/2(log(2πeσ^2/D)) 6. Consider a random variable with tenary alphabet {A,B,C} with P(A) = P(B) = 0.4 , P(C) = 0.2. Use arithmetic coding to encode the sequence ACB, where we assume A<B<C 7. Assume that we have a sender of power P and two distant receivers. The model of the channel is Y1 = X + Z1 and Y2 = X + Z2, where Z1 and Z2 are arbitrarily correlated Gaussian random variables with variances N1 and N2 respectively and N1 < N2. (a) (7%) Describe the capacity region of this Gaussian broadcast channel and the procedure of decoding. (b) (8%) Show that the rate pairs in the capacity region can be better than the simple time-sharing. 8. Use the sliding window Lempel-Ziv algorithm to encode he sequence 0000001101010000011010111. 9. (a)Describe the capacity for the multiple access channel (X1 x X2, p(y|x1,x2), Y) for a given product distribution p1(x1)p2(x2) on X1 x X2 (b)Describe the capacity region in 9a if y = x1+x2+z, where z is a zero mean Gaussian random variable with variance N and power constraint on xj is Pj. 10. Suppose that the encoder does not have the knowledge of the typical set for Xn but have the knowledge of the entropy rate H(X). How can we implement the source coding for the sequence in Xn? --

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