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(a) What is the overall capacity for a cascade of 3 independent binary symmetric channels? Express your answer as a decimal with 5 digits of precision. (b) What is the overall capacity for a cascade of 4 independent binary symmetric channels? Express your answer as a decimal with 5 digits of precision.

(a) What is the overall capacity for a cascade of 3 independent binary symmetric channels? Express your answer as a decimal with 5 digits of precision. (b) What is the overall capacity for a cascade of 4 independent binary symmetric channels? Express your answer as a decimal with 5 digits of precision.

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
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The following binary symmetric channel has a capacity of C = 1 + 0.1 log₂ 0.1 + 0.9 log₂ 0.9 0.53100 bits per channel use. input bit input bit 1 1 0 0.1 0.1 1 However, if we must go through a cascade of such channels, we reduce the overall capacity to carry information. For example, if we have two such channels in series, with independent probabilities of bits flipping (or not flipping), we can view this as one overall binary symmetric channel: 0.1 0.1 0.9 0.9 0.9 two cascaded binary symmetric channels 0.9 0.9 0.1 0.1 2.0.9 0.1 0.18 2.0.9 0.1 0.18 0.9² +0.1² = 0.82 0 0.9² +0.1² 0.82 = 1 0.9 one overall binary symmetric channel 0 1 0 1 output bit output bit The overall binary symmetric channel has a capacity of only 1+0.18 log₂ 0.18+0.82 log2 0.82≈ 0.31992 bits per overall channel use. (a) What is the overall capacity for a cascade of 3 independent binary symmetric channels? Express your answer as a decimal with 5 digits of precision. (b) What is the overall capacity for a cascade of 4 independent binary symmetric channels? Express your answer as a decimal with 5 digits of precision.
Transcribed Image Text:The following binary symmetric channel has a capacity of C = 1 + 0.1 log₂ 0.1 + 0.9 log₂ 0.9 0.53100 bits per channel use. input bit input bit 1 1 0 0.1 0.1 1 However, if we must go through a cascade of such channels, we reduce the overall capacity to carry information. For example, if we have two such channels in series, with independent probabilities of bits flipping (or not flipping), we can view this as one overall binary symmetric channel: 0.1 0.1 0.9 0.9 0.9 two cascaded binary symmetric channels 0.9 0.9 0.1 0.1 2.0.9 0.1 0.18 2.0.9 0.1 0.18 0.9² +0.1² = 0.82 0 0.9² +0.1² 0.82 = 1 0.9 one overall binary symmetric channel 0 1 0 1 output bit output bit The overall binary symmetric channel has a capacity of only 1+0.18 log₂ 0.18+0.82 log2 0.82≈ 0.31992 bits per overall channel use. (a) What is the overall capacity for a cascade of 3 independent binary symmetric channels? Express your answer as a decimal with 5 digits of precision. (b) What is the overall capacity for a cascade of 4 independent binary symmetric channels? Express your answer as a decimal with 5 digits of precision.
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