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 開課系所︰電信所 考試日期（年月日）︰2015.6.23 考試時限（分鐘）：100分鐘 試題 : Final Exam of Information Theory June 23, 2015 1. Consider the discrete-time memoryless channel, for which the output Y is the sum of the input X and tje noise Z, where X and Y are real numbers and Z is a Gaussian random variable with zero mean and variance N. Suppose the power constraint of X is E[x^2] ≦ P. (a)(5%) What is the channel capacity of this channel? (b)(5%) What is the least signal-to-noise ratio needed to achieve coding rate of 1/2 bits per transmission? (c)(5%) Suppose that X = {±1, ±3} and p(x) = 1/4 for x ∈ {±1,±3}. What is the capacity for large signal-to-noise ratios. 2. (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). 3. (a)(5%) Consider a binary symmetric channel with transition probability ε. Please find its channel capacity. (b)(5%) A channel with input alphabet X = {a1, a2, a3, a4, a5, a6} and output alphabet Y = {b1, b2, b3} and the transition probabilities, P(b1|a1) = 1, P(b1|a2) = 1, P(b2|a3) = 1, P(b2|a4) = 1, P(b2|a5) = 1, P(b3|a6) = 1. 4. (10%) Please find the rate distortion function of the N(0,σ^2) Gaussian source X with squared error distortion. 5. (a)(5%) Use the sliding window Lempel-Ziv algorithm to encode the sequence 0000001101010000011010111. (window size = 4 bits) (b)(5%) Use the tree-structured Lempel-Ziv algorithm to encode the sequence 0000001101010000011010111. 6. (10%) Describe th watering filling principle for the parallel Gaussian channels. 7. Consider two random variables X and Y with X = Y = {0,1} and joint probability p(x,y) given p(y|x) = Y| 0 1 X | ──────── 0 | 0.72 0.18 | 1 | 0.02 0.08 10000 10000 (a)(5%) Please given an example of a joint typical sequence in X Y (b)(5%) Suppose that source A and source B are separately encode and jointly decoded. What is the achievable rate region for source A and B? (c)(5%) Please give a heuristic explanation for the method to achieve the rate region given in (b). 8. (8%) Consider a Gaussian multiple-access channel with two users. Two senders , X1 and X2 communicate to a single receiver, Y. The received signal at time i is Yi=X1i+X2i+Zi where {Zi} is a sequence of i.i.d. zero mean Gaussian random variables with variance N. Assume that there is a power constraint Pj on sender j; that is, we have n 1/n(Σ(x_ji)^2 (w_j)) <= Pj, wj belongs to {1,2,...,2^nR}, j=1,2. i=1 Please describe the achievable regions of R1 and R2. 9. (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. Describe the capacity region of this Gaussian broadcast channel and the procedure of decoding. --

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