The Fading Number of IID MIMO Gaussian Fading Channels with a Scalar Line-of-Sight Component

The capacity of regular noncoherent fading channels grows like log log SNR + χ at high signal-to-noise ratios (SNR). Here, χ, denoted fading number, is a constant independent of the SNR, but dependent on the distribution of the fading process. Recently, an expression of the fading number has been derived for the situation of general memoryless multiple-input multiple-output (MIMO) fading channels. In this paper, this expression is evaluated in the special situation of an independent and identically distributed MIMO Gaussian fading channel with a scalar line-of-sight component d. It is shown that, for large |d|, the fading number grows like min{nR, nT} log |d| where nR and nT denote the number of antennas at the receiver and transmitter, respectively. As a side-product along the way, closed-form expressions are derived for the expectation of the logarithm and for the expectation of the n-th power of the reciprocal value of a noncentral chi-square random variable. It is shown that these expectations can be expressed by a family of continuous functions gm(·) and that these families have nice properties (monotonicity, concavity, etc.). Moreover, some tight upper and lower bounds are derived that are helpful in situations where the closed-form expression of gm(·) is too complex for further analysis.