On the Capacity-Achieving Input Covariance for Multicarrier Communications over Doubly Selective Channels

We consider pulse-shaping multicarrier (MC) communications over a (possibly rapidly varying) doubly selective channel with uncorrelated scattering. Assuming transmission free of intersymbol interference (while intercarrier interference is not constrained), we show that the statistical properties of the MC system are invariant under cyclic shifts of the subcarriers. This invariance is then used to derive structural properties of the capacity-achieving input covariance function. We show that capacity can be achieved by transmit symbols that are uncorrelated over time and cyclically stationary with respect to the frequency (subcarrier) index. We also show that capacity- achieving precoding can be equivalently realized by a suitable adaptation of the transmit pulse. For classical OFDM and transmission over a WSSUS channel, precoding can be completely avoided if pulse-shaping MC transmission is used instead.