On the distribution of the squared norm of non-circular complex Gaussian random variables with applications

We present a novel relationship between the distribution of circular and non-circular complex Gaussian random variables. Specifically, we show that the distribution of the squared norm of a non-circular complex Gaussian random variable, usually referred to as squared Hoyt distribution, can be constructed from a conditional exponential distribution. Leveraging this approach, we provide novel results on the Shannon capacity of adaptive transmission techniques in Hoyt fading channels, showing that the asymptotic capacity loss per bandwidth unit in the high-SNR regime is up to 1.83 bps/Hz, compared to the AWGN case.