Capacity-achieving transmitter and receiver pairs for dispersive MISO channels

We focus on data transmission over continuous-time, dispersive M-input/single-output (MISO) channels where the transmitter knows all M channels. The biorthogonal Karhunen-Loeve (KL) expansion in the continuous-time domain is proposed as a unifying analytical framework for studying dispersive MISO channels. Using this KL expansion, we rigorously derive an expression for the capacity of the continuous-time dispersive MISO channel and show that a dispersive MISO channel is equivalent, in terms of channel capacity, to a dispersive single-input/multiple-output (SIMO) channel. Our derivations based on the KL expansion avoid the well-known problems associated with the "frequency splitting" approach. We show that the capacity-achieving transmitter for a dispersive MISO channel can be broken into two parts: (1) an optimal encoder for a dispersive single-input/single-output (SISO) channel and (2) an M-dimensional (M-D) matched filter (or a broadband beamformer). The M-D matched filter turns the original dispersive MISO channel into an equivalent SISO channel with exactly the same capacity as the capacity of the original MISO channel, and the one-dimensional encoder is just the well-known optimal encoder of Gallager for this equivalent SISO channel created by the M-D matched filter. As the number of transmit antennas increases, we show that the channel seen by the receiver approaches a flat-fading SISO channel (even when the individual forward channels are dispersive); hence, water filling in the transmitter can be dropped without much degradation. Second, this shows that with a large number of transmit antennas, the achievable data rate using a pre-RAKE at the transmitter and a very simple symbol-by-symbol detector at the receiver is quite close to the actual capacity of the dispersive MISO channel. With a large number of transmit antennas, this implies that the receiver for a capacity-achieving dispersive MISO system can be considerably less complex than the receiver for a dispersive SISO channel.