(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)
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Author:Robert L. Boylestad
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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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