Generalized Degrees Freedom of Noncoherent MIMO Channels With Asymmetric Link Strengths

Abstract

We study the generalized degrees of freedom (gDoF) of the block-fading noncoherent multiple input multiple output (MIMO) channel with asymmetric distributions of link strengths and a coherence time of T symbol durations. We derive the optimal signaling structure for communication for asymmetric MIMO, which is distinct from that for the MIMO with independent and identically distributed (i.i.d.)links. We extend the existing results for the single input multiple output (SIMO) channel with i.i.d. links to the asymmetric case, proving that selecting the statistically best antenna is gDoF-optimal. Using the gDoF result for SIMO, we prove that for T = 1, the gDoF is zero for MIMO channels with arbitrary link strengths. We show that selecting the statistically best antenna is gDoF-optimal for the multiple input single output (MISO) channel. We also derive the gDoF for the 2 × 2 MIMO channel with different exponents in the direct and cross links. In this setting, we show that it is always necessary to use both the antennas to achieve the optimal gDoF, in contrast to the results for the 2 × 2 MIMO with i.i.d. links. We show that having weaker crosslinks, gives gDoF gain compared to the case with i.i.d. links. For noncoherent MIMO with i.i.d. links, the traditional method of training each transmit antenna independently is degrees of freedom (DoF) optimal, whereas we observe that for the asymmetric 2 × 2 MIMO, the traditional training is not gDoF-optimal. We extend this observation to a larger M × M MIMO by demonstrating a strategy that can achieve larger gDoF than a traditional training-based method.

Generalized Degrees of Freedom of Noncoherent Diamond Networks

“…The above works considered a DoF framework for the noncoherent model, in the sense that for high-SNR, the link strengths are not significantly different, i.e., the links scale with the same SNR-exponent. The gDoF framework for noncoherent MIMO was considered in [12], [13] and it was shown that several insights from the DoF framework may not carry on to the gDoF framework. It was shown that a standard training-based scheme is not gDoF optimal and that all antennas may have to be used for achieving the gDoF, even when the coherence time is low, in contrast to the results for the MIMO with i.i.d.…”

“…The DoF was shown to be M * (1 − M * /T ) in that case. The case of the noncoherent MIMO channel with asymmetric statistics on the link strengths was recently studied in [15], [16]. In this work, the authors showed that the gDoF for single-input-multiple-output (SIMO) and multiple-input-single-output (MISO) channels can be achieved by using only the strongest link.…”

“…The upper bounding techniques in this paper are influenced by the methods developed in [15], [16] for the noncoherent MIMO: there the authors discretized the upper bound (without losing the gDoF) and used linear programming techniques to reduce the upper bound further. We analyze the upper bound from Theorem 1 and obtain a loosened upper bound in Theorem 4.…”

“…Subsequently, in Theorem 6, we reduce the optimization problem for choosing the two mass points, to a bilinear optimization problem, and we solve it explicitly. The bilinear optimization does not arise in the noncoherent MIMO case [15], [16]. In [15], [16], there was only a piecewise linear optimization.…”

“…The bilinear optimization does not arise in the noncoherent MIMO case [15], [16]. In [15], [16], there was only a piecewise linear optimization.…”