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︰電機系 電信所 考試日期（年月日）︰2011/07/21 考試時限（分鐘）：1 hr 50 min 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題： Final Exam of Information Theory June 21, 2011 1. Consider two correlated Gaussian random variables X and Y with correlation ρ, i.e., (X,Y)～N(0,K), where ＿ ＿ ∣ 2 2∣ ∣σ ρσ∣ K =∣ 2 2∣ ∣ρσ σ∣ ∣ ∣ ¯ ¯ (a) (5%) Please Find the mutual information between X and Y. (b) (5%) Explain the result for the case of ρ= 1. 2. (10%) 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) 3. (10%) Consider a discrete time memoryless channel with additive noise of variance N and input constraint EX^2 ≦ P. Please show that １ １ Ｐ — log[2 πe ( P + N )] - H(Z) ≧ C ≧ — log(1 + —) ２ ２ Ｎ 4. Let the output of a continuous time channel be given by the sum of the input and white Gaussian noise of spectral density No Let the input be — ２. power constrained to a power S and let the bandwidth be constrained to (-W,W). (a) (5%) What is the capacity of this channel? (b) (5%) What is the capacity of this channel if two-dimensional transmission is used? 5. (10%) Consider a source X uniformly distributed a on the set {1, 2, ..., m}. Find the rate distortion function for this source with Hamming distortion; i.e., ╱ ︿ ︿ ▕ 0, if ｘ＝ｘ d(ｘ,ｘ) = ＜ ︿ ▕ 1, if ｘ≠ｘ ╲ nR 6. (10%) Please show that there exists a sequence of (2 , n) universal source codes such that (n) for every source Q such that H(Q) < R. Ｐe → 0 7. (a) (7%) Use the sliding window Lempel-Ziv algorithm to encode he sequence 0000001101010000011010111. (b) (8%) Use the tree-structured Lempel-Ziv algorithm to encode the sequence 0000001101010000011010111. 8. (10%) Describe the Slepian-Wolf Theorem for the disributed source coding for two correlated sources X and Y. 9. 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. --

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