  We consider the design of multiple-input multiple-output communication
systems with a linear precoder at the transmitter, zero-forcing decision
feedback equalization (ZF-DFE) at the receiver, and a low-rate feedback channel
that enables communication from the receiver to the transmitter. The channel
state information (CSI) available at the receiver is assumed to be perfect, and
based on this information the receiver selects a suitable precoder from a
codebook and feeds back the index of this precoder to the transmitter. Our
approach to the design of the components of this limited feedback scheme is
based on the development, herein, of a unified framework for the joint design
of the precoder and the ZF-DFE under the assumption that perfect CSI is
available at both the transmitter and the receiver. The framework is general
and embraces a wide range of design criteria. This framework enables us to
characterize the statistical distribution of the optimal precoder in a standard
Rayleigh fading environment. Using this distribution, we show that codebooks
constructed from Grassmann packings minimize an upper bound on an average
distortion measure, and hence are natural candidates for the codebook in
limited feedback systems. We also show that for any given codebook the
performance of the proposed limited feedback schemes is an upper bound on the
corresponding schemes with linear zero-forcing receivers. Our simulation
studies show that the proposed limited feedback scheme can provide
significantly better performance at a lower feedback rate than existing schemes
in which the detection order is fed back to the transmitter.
