Consider a binary symmetric communication channel, whose input source is the alphabet X = {0, 1} with probabiliíties (0.5, 0.5); whose output alphabet is Y = {0, 1}; and whose channel matrix is: 1-e e a) Calculate the conditional entropy of X given that the receiver observes Y = 0. [Note that this will be a function of the parameters p and lepsilon.] H(XIY = 0)=? b)Calculate the conditional entropy of X given that the receiver observes Y = 1. H(XIY= 1)=? (c) Calculate the conditional entropy of X given Y. (This will be the weighted average of (a) and (b).) (d) Calculate the entropy of X. H(X)=? (e) Calculate the mutual information of X and Y. I (X; Y )=? (f) What is the channel capacity of the BSC? Why? (Show how you start out with (e) above and arrive at the channel capacity expression.) What are the units of channel capacity?
Consider a binary symmetric communication channel, whose input source is the alphabet X = {0, 1} with probabiliíties (0.5, 0.5); whose output alphabet is Y = {0, 1}; and whose channel matrix is: 1-e e a) Calculate the conditional entropy of X given that the receiver observes Y = 0. [Note that this will be a function of the parameters p and lepsilon.] H(XIY = 0)=? b)Calculate the conditional entropy of X given that the receiver observes Y = 1. H(XIY= 1)=? (c) Calculate the conditional entropy of X given Y. (This will be the weighted average of (a) and (b).) (d) Calculate the entropy of X. H(X)=? (e) Calculate the mutual information of X and Y. I (X; Y )=? (f) What is the channel capacity of the BSC? Why? (Show how you start out with (e) above and arrive at the channel capacity expression.) What are the units of channel capacity?
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![Consider a binary symmetric communication channel, whose input source is the alphabet X = {0,
1} with probabilities (0.5, 0.5}; whose output alphabet is Y = {0, 1}; and whose channel matrix
is:
$1-e
e
a) Calculate the conditional entropy of X given that the receiver observes Y = 0. [Note that this will be a
function of the parameters p and lepsilon.] H(XIY = 0)=?
b)Calculate the conditional entropy of X given that the receiver observes Y = 1. H(XIY = 1)=?
(c) Calculate the conditional entropy of X given Y. (This will be the weighted average of (a) and (b).)
(d) Calculate the entropy of X. H(X)=?
(e) Calculate the mutual information of X and Y. I(X; Y )=?
(f) What is the channel capacity of the BSC? Why? (Show how you start out with (e) above and arrive at
the channel capacity expression.) What are the units of channel capacity?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fed575c54-742b-4aa8-b03d-b13e9e97ae66%2F2503923c-a85d-44bf-aa80-f07777186ab1%2Fvfb9gdb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider a binary symmetric communication channel, whose input source is the alphabet X = {0,
1} with probabilities (0.5, 0.5}; whose output alphabet is Y = {0, 1}; and whose channel matrix
is:
$1-e
e
a) Calculate the conditional entropy of X given that the receiver observes Y = 0. [Note that this will be a
function of the parameters p and lepsilon.] H(XIY = 0)=?
b)Calculate the conditional entropy of X given that the receiver observes Y = 1. H(XIY = 1)=?
(c) Calculate the conditional entropy of X given Y. (This will be the weighted average of (a) and (b).)
(d) Calculate the entropy of X. H(X)=?
(e) Calculate the mutual information of X and Y. I(X; Y )=?
(f) What is the channel capacity of the BSC? Why? (Show how you start out with (e) above and arrive at
the channel capacity expression.) What are the units of channel capacity?
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