[試題] 99下 林茂昭 消息理論 期末考

[OhLoveU (Love U)]

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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