MIMO Precoding with Limited Rate Feedback: Simple Quantizers Work Well

Transmitter precoding is a crucial technique for harnessing the potential of multiple-input multiple-output (MIMO) fading channels. In many practical wireless systems, a limited amount of feedback from the receiver is available at the transmitter, which can be used to direct the choice of the precoder from a codebook to match the channel state. Assuming noiseless, limited-rate feedback, this work studies the design of simple, efficient quantization and feedback schemes which achieve near-optimal ergodic channel capacity. In the case the precoder takes the form of a beamforming vector for modulating a single symbol stream, it is found that simple scalar quantization of the elements of the vector is nearly optimal over a wide range of feedback rates; it typically costs a fraction of a dB higher SNR to achieve the same capacity as that of far more sophisticated vector quantization schemes. In the case a precoding matrix consisting of multiple beams is used to modulate multiple symbol streams, separate encoding of the beams using scalar quantization also performs well. Roughly speaking, the rate loss due to separate encoding of the beams increases linearly with the number of beams but appears to be constant over a wide range of SNRs. The loss can be reduced substantially by more sophisticated encoding of each beam, e.g., two-state trellis coded quantization. The complexity of such quantization schemes is linear in the number of antennas and the number of feedback bits.