Maximizing data rate over M-input/1-output channels

We focus on transmit diversity schemes for a communication system with M transmit antennas and one receive antenna (i.e. an M-input/single-output (MISO) system) where the transmitter knows all these M channels, and we derive a transmission scheme that maximizes the data rate, in an information-theoretic sense, that can be reliably transmitted over this MISO channel. An important example of the type of system we consider is a cellular wireless system on the downlink with M transmit antennas at the base station and one receive antenna at the mobile station. Our main result is the identification of the structure of the capacity-achieving transmitter. The capacity-achieving transmitter can be broken into two parts: (I) an optimal encoder for a single-input/single-output (SISO) channel, and (II) an M-dimensional matched filter. We show that the M-dimensional matched filter turns the original MISO channel into a SISO channel with exactly the same capacity as the capacity of the original MISO channel, and the 1-dimensional encoder is just the well known optimal encoder of Gallager (1966) for the SISO channel created by the M-dimensional matched filter. We also derive a closed-form expression for the channel capacity of MISO channels which is valid under a rather general set of assumptions: (1) each channel can be dispersive or flat, and (2) the additive noise at the receiver can be a Gaussian process with arbitrary power spectrum. In a typical urban environment, to achieve 1% block error rate at 4 Mbits/second, the MISO system with our transmission scheme requires 10 dB less SNR than today´s SISO system.