Training-Codes for the Noncoherent Multi-Antenna Block-Rayleigh-Fading Channel

We consider signal design for the noncoherent block-Rayleigh-fading channel, in which neither the transmitter nor the receiver know the fading coefficients. We show that several recently proposed signal designs for this channel can be interpreted as training-based, i.e., part of the coherence interval is used to transmit known symbols to learn the channel and in the remainder a “coherent” information symbol is transmitted. At the receiver, the estimated channel is then used as if correct, which allows to employ efficient implementations (like the sphere-decoder) for the coherent decision rule. We establish conditions for which such a strategy is equivalent to the optimal noncoherent decision rule. Motivated by the good performance of the training schemes even when the coherence time is very small (indeed minimal), we propose the paradigm of training-codes to build signalconstellations for the noncoherent block-Rayleigh-fading channel. Training-codes leverage the advances made in coherent codes and thus can often address the signal-design and efficient-decoding problems simultaneously.