The Diversity-Multiplexing Tradeoff of Lognormal Channels as a Function of the dB Spread

In this paper, we investigate the dependency of the diversity-multiplexing tradeoff (DMT) of lognormal channels on their dB spread for both single-input-single-output (SISO) and multiple-input-multiple-output (MIMO) channels. We tackle both the finite and the asymptotically high signal-to-noise ratio (SNR) regimes. The finite SNR DMT for a SISO lognormal channel is shown to be strongly dependent on the dB spread, [10/ln(10)]σx, where σ
_x
is the shaping parameter of the lognormal distribution. For different values of σ
_x
, the outage probability versus SNR curves on a log-log scale have different slopes and significant flooring for high values of σ
_x
. In order to capture the DMT dependency on the dB spread, the SNR gap between the outage curves of lognormal channels with different σ
_x
is derived. Moreover, the relative slopes of the outage probability curves are derived. Given that in the limiting-case of high SNR, the DMT for MIMO channels are dictated by the tail behaviour of their probability density function [1] and based on the similarities between the tail behavior of lognormal and normal random variables for very small σ
_x
≪
1 dB, and lognormal and Gamma random variables for small to medium σ
_x
≤ 1.5 dB, we analyze the dependency of the high-SNR DMT for MIMO lognormal channels on σ
_x
. We show that for σ
_x
≪ 1, the lognormal fading channel behaves like a Gaussian one where the power control techniques are sufficient, whereas for medium to high values of σ
_x
, the lognormal channels act like regular multipath fading channels, and thus, diversity techniques should be utilized to improve communication reliability over such channels