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京都大学 情報学研究科 通信情報システム専攻 2022年8月実施 専門基礎A [A-5]

Author​

SUN

Description​

Answer all the following questions.

(1) Consider a general communication system model, which consists of a source, destination, channel encoder, channel decoder, source encoder, source decoder, and communication channel. Draw this model as a block diagram.

(2) A stationary memoryless information source generates information symbols and with probabilities and respectively. Answer the following questions. and may be used.

  • (a) Describe the definitions of “memoryless” and “stationary”.
  • (b) Find a binary Huffman code of .
  • (c) Find the expected codeword length per symbol of the code in Question (b).
  • (d) Find the entropy of .

(3) Let be a memoryless binary symmetric channel (BSC) with crossover probability . Answer the following questions.

  • (a) Show the channel matrix of .
  • (b) Show that the channel capacity of is given by . In addition, graph it as a function of .
  • (c) Consider communications with Hamming code through . Evaluate the probability of decoding failure assuming that any correctable errors are corrected.
  • (d) Find the channel capacity of a cascade of two BSCs with crossover probabilities and .

Kai​

(1)​

Source source encoder channel encoder Communication channel channel decoder source decoder destination

(2)​

(a)​

"Memoryless" means the current output is not related to previous ones.
"stationary" means the output behaviour keeps the same over time.

(b)​

construct the Huffman code:

(c)​

(d)​

(3)​

(a)​

(b)​



will be achieved when the output probability is uniform.
. So that

The graph:

(c)​

The (7,4) Hamming code can correct 1 bit error.

(d)​

This is still a BSC :

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