AND Channel Capacity

Problem

Compute the capacity in bits of the channel Y = X AND Z, where X is the binary input and Z is an independent Bernoulli noise bit with P(Z = 1) = q. When Z is 0 the output is forced to 0, so the channel is an erasure-like model whose capacity is 1.0 for q <= 0.5 and H2(1 - q) otherwise.

Hint

Given X the output is deterministic, so maximize the entropy of P(Y = 1) = p(1 - q) over the input bias p.

Test cases

case 1

in: [0]

expected: 1

case 2

in: [0.25]

expected: 1

case 3

in: [0.75]

expected: 0.8112781244591328

case 4

in: [1]

expected: 0

case 5

in: [0.5]

expected: 1

Starter codePython

import math
def and_channel_capacity(q):
    # Your code here
    pass

5 test cases · run your code against them in the editor