% Exercise 4.57: Capacity of a communication channel % Boyd & Vandenberghe "Convex Optimization" % JoĆ«lle Skaf - 04/24/08 % % We consider a discrete memoryless communication channel, with input % X(t) \in {1,...,n}, and output Y(t) \in {1,...,m}, for t = 1,2,... % The relation between the input and output is given statistically: % p_ij = Prob(Y(t)=i|X(t)=j), i=1,...,m, j=1,...,n % The matrix P is called the channel transition matrix. % The channel capacity C is given by % C = sup{ I(X;Y) | x >= 0, sum(x) = 1}, % I(X;Y) is the mutual information between X and Y, and it can be shown % that: I(X;Y) = c'*x - sum_{i=1}^m y_i*log_2(y_i) % where c_j = sum_{i=1}^m p_ij*log_2(p_ij), j=1,...,m % Input data rand('state', 0); n = 15; m = 10; P = rand(m,n); P = P./repmat(sum(P),m,1); c = sum(P.*log2(P))'; % Channel capacity cvx_begin variable x(n) y = P*x; maximize (c'*x + sum(entr(y))/log(2)) x >= 0; sum(x) == 1; cvx_end C = cvx_optval; % Results display(['The channel capacity is: ' num2str(C) ' bits.'])