A binary symmetric channel (BSC) with crossover probability p is shown in the figure below.
Two BSCs with the same crossover probability are cascaded such that the output of channel-1 is fed as input to channel-2. The overall capacity of the equivalent channel in terms of the entropy function H(p) is
2H(p(1−p))
H(2p(1−p))
1−2H(p(1−p))
1−H(2p(1−p))
To determine the overall capacity of the equivalent channel in terms of the entropy function H(p), we need to find the capacity of each individual channel and then combine them.
Let's start with channel-1. The capacity of a binary symmetric channel with crossover probability p can be calculated using the formula:
C1 = 1 - H(p)
Here, H(p) is the binary entropy function, which is given by:
H(p) = -p * log2(p) - (1 - p) * log2(1 - p)
Now, let's move on to channel-2, which is fed with the output of channel-1. Since channel-2 has the same crossover probability as channel-1, we can use the same formula to calculate its capacity:
C2 = 1 - H(p)
To find the overall capacity of the cascaded channels, we need to consider the worst-case scenario, which is when both channels introduce errors. In this case, the crossover probabilities of both channels are multiplied:
p' = p * p = p^2
Since the capacity of the equivalent channel is determined by the worst-case scenario, we can calculate its capacity using the formula:
Ceq = 1 - H(p')
Substituting p' = p^2, we get:
Ceq = 1 - H(p^2)
Therefore, the correct answer is 1 - H(p^2).