Generalized Degrees Freedom of Noncoherent MIMO Channels with Asymmetric Link Strengths

We study the generalized degrees of freedom (gDoF) of block-fading noncoherent multiple input multiple output (MIMO) channels with asymmetric distributions of link strengths and a coherence time of T symbol durations. We derive the optimal signaling structure for communication for the asymmetric MIMO channel, which is distinct from that for the MIMO channel 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 the SIMO channel, we prove that for T=1, the gDoF is zero for MIMO channels with arbitrary link strengths., extending the result for MIMO with i.i.d. links 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 2X2 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 gDoF, in contrast to the results for the 2X2 MIMO channel with i.i.d. links. We show that having weaker crosslinks, gives gDoF gain compared to the case with i.i.d. links. For the noncoherent MIMO channel 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 2X2 MIMO channel, the traditional training is not gDoF-optimal. We extend this observation to a larger MX M MIMO channel by demonstrating a strategy that can achieve larger gDoF than a traditional training-based method.