channel_capacity.m
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channel_capacity.m
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			% Exercise 4.57: Capacity of a communication channel
			% Boyd & Vandenberghe "Convex Optimization"
			% Joëlle 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_ilog_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.'])