# Consider the following discret

Consider the following discrete memoryless channel: Y = X + Z,where Pr{Z = 0} = Pr{Z = a} = 1/2 and the alphabet for x is {0, 1}.Assume that Z is independent of X.

a) Find the capacity of the channel in terms of a.

b) Explain how capacity is affected by a in terms ofsignal-to-noise ratio (SNR).

Answer:

a) Here depending on the value of (a) various case can bemade

Case 1 —> a =0

For this case Y = X and max I(X:Y) = max I(X)=1. So the capacityhere is 1

Case 2 —> a 0

For this case Y has four possible value 0,1,a,1+a. Knowing Y, weknow the X which was sent, and hence H(XY) = 0 . Hence max I(X; Y ) = max H(X) = 1, achieved for an uniformdistribution on the input X

Case 3 —> a =1

In this case, Y has three possible output values, 0, 1, and 2.The channel is identical to the binary erasure channel with a =1/2. The capacity of this channel is 1/2 bit per transmission.

Case 4 —> a =1

In this case, Y has three possible output values, 0, 1, and 2.The channel is identical to the binary erasure channel with a =1/2. The capacity of this channel is 1/2 bit per transmission.

Case 4 —> a = -1

This is similar to the case when a = 1 and the capacity here isalso 1/2 bit per transmission.

b) Capacity and SNR relation is given by

So when the SNR iincrease by then capacity is increse by 1