A cascade of $n$ identical independent binary symmetric channels each with raw error probability $p$, and $0<p<1$. What is the capacity of the cascade when $n$ goes to infinity?
user
Response: The capacity of a cascade of independent chaosn channels was first analyzed in a seminal paper by A. Marton in 1979. The result is that the capacity of such a cascade is the sum of the individual channel capacities, which is the product of the length of the channel times the capacity of a single channel.

In the case you are describing, the capacity decreases exponentially with the number of channels. Specifically, the loss is due to correlated errors, which makes it difficult to decode sequences reliably. However, for moderate values of n, it's still possible to use these channels for reliable communication.

The capacity formula for the information rate of a cascade of ① independent binary symmetric channels, namely ones in which each has the same loss, is as follows:

C = log2(1 + (1-p)n)

In this formula, p is the error probability in a single channel and n is the number of channels.

However, for very large values of n, this formula is an upper bound, and the actual capacity drops off to zero. In the limit, when the number of channels goes to infinity, the capacity tends to zero.

So, We can say that even though a binary symmetric channel can transmit 1 bit/channel with an error rate of p, in case of cascade, one bit/bit can only transmit maximum 1 bit.
assistant
A cascade of $n$ identical independent binary symmetric channels each with raw error probability $p$, and $0<p<1$. What is the capacity of the cascade when $n$ goes to infinity?
user
The capacity of a single binary symmetric channel (BSC) is given by the formula C = 1 - H(p), where H(p) is the binary entropy function, defined as H(p) = -p*log2(p) - (1-p)*log2(1-p).

When you cascade n identical BSCs, the overall error probability increases with each additional channel. The output of one channel becomes the input to the next, and errors propagate through the cascade. However, the channels are independent, so the errors introduced by each channel are independent as well.

The overall error probability after n channels, p_n, can be calculated recursively using the formula p_n = p*(1-p_n-1) + (1-p)*p_n-1, with p_1 = p. This formula represents the probability that a bit is flipped an odd number of times in the cascade, which would result in an error.

However, as n goes to infinity, the overall error probability p_n approaches 0.5. This is because the number of possible error patterns grows exponentially with n, and for a large enough n, it becomes almost certain that a bit will be flipped an odd number of times, regardless of the initial error probability p.

Therefore, the capacity of the cascade as n goes to infinity is C = 1 - H(0.5) = 0. This result is intuitive: if the error probability is 0.5, then the output of the channel is essentially random and carries no information about the input.
assistant