Definition (Information capacity with input cost)

The information capacity (or information capacity-cost function) of a discrete-time continuous memoryless channel $(\mathcal{X},\mathcal{Y},f_{Y|X})$ with input average cost constraint $(t(\cdot),P)$ is given by $$\tag{in bits}C(P):=\sup_{F_{X}:E[t(X)]\le P}I(X;Y)$$where the supremum is taken over all input distributions $F_{X}$ and $I(X;Y)$ is the channel’s input-output mutual information.