Channel Capacity with a Constraint on the Smoothness of the Transmitted Signal

The authors determine the asymptotic behavior of the capacity of a channel with white Gaussian noise under constraints on the mean power and the smoothness of the periodic transmitted signal $S(t)$. It is shown that the capacity depends very strongly on the degree of smoothness of $S(t)$. In particular, for a periodic function $S(t)$ whose derivative of order $\gamma$ is bounded in the mean square by some constant, the capacity $C$ is asymptotically proportional to $T^{1/(2\gamma+1)}$ if the transmission time $T$ is large.