Exact Expressions for the Condition Number Distribution of Complex Wishart Matrices

The standard condition number (SCN) is a fundamental metric in the context of multiple-input multiple-output communication systems, linear detection and classical linear algebra. Hence, in this paper we propose a novel generic framework for the SCN distribution of three different classes of Wishart matrices which leads to new results and insights. In particular, our analysis covers both central and non-central Wishart distributions of arbitrary dimension and therefore is applicable to uncorrelated/semi-correlated Rayleigh fading and Ricean fading scenarios. For the special case of dual semi-correlated central Wishart matrices, we derive exact and asymptotic polynomial expressions for the SCN distributions. All analytical results are validated via Monte-Carlo simulations with the attained accuracy being excellent in all cases. The impact of the model parameters on channel conditioning is also investigated in detail.