Optimal Noncoherent MIMO-OFDM Constellations at Low SNR

Reliable communication over the discrete input, continuous output, noncoherent multiple-input multiple-output (MIMO) orthogonal frequency division multiplexed (OFDM) channel is considered when the SNR per degree of freedom is low. When the peak power (per antenna per time slot) to the average power ratio is fixed, the mutual information behaves as O(SNR²) at low SNR. In this work, mutual information is optimized up to second order jointly over signal constellation (space-frequency) matrices and their respective probabilities of cardinality M les N+1, where N is the number of tones. Even though the problem considered is a finite dimensional non-convex optimization, it admits an elegant closed form solution. For M = N + 1, in the frequency domain, the optimal constellation consists of on-off keying with unequal probabilities in general, with on signaling comprising of sending a signal, equi-probably, across all transmit antennas at maximum allowable peak power, on a carrier selected by the message. This constellation is named "All-on, all-Off Carrier Shift Keying" (AO-CSK). In the time domain, AO-CSK is identical to a special (IDFT-) version of Space Time Orthogonal Rank-one Modulation (STORM), which was shown to be optimal for the frequency non-selective channel at low SNR in [1]. Remarkably therefore, the main result of this paper taken together with that of [1] implies that the IDFT version of STORM is optimal in mutual information up to second order at low SNR for both frequency non-selective and frequency-selective (FS) channels when OFDM is used to convert the FS channel into non-interacting, parallel flat fading channels.