Distribution of the Product of a Complex Gaussian Matrix and Vector and Its Sum with a Complex Gaussian Vector

In this paper, we derive the distribution of the product of a complex Gaussian matrix and a complex Gaussian vector. Further, we calculate the distribution of the sum of this product and a complex Gaussian vector, which generalizes the recent results where a complex Gaussian scalar is considered instead of a complex Gaussian matrix. The exact probability density functions (pdf) are derived for both the product and the sum. The pdf and cumulative distribution function (cdf) of the square norm of the sum are also provided. Then, we apply the derived results to analyze the performance of the energy detector for a multiple-input-multiple-output (MIMO) communication system. Our analytical results are verified via numerical examples.

1. INTRODUCTION

The product of Gaussians occurs frequently in radar and communication systems, such as the keyhole or pinhole channel model [1] [2], time reversal detection [3], and reflection coefficients in over-the-horizon radars [4]. The distributions of the products involving real Gaussian random variables (RVs) under certain conditions are summarized in [5]. Recently, the distribution of the product of two correlated real Gaussian RVs with zero means has been studied in [6] and that with arbitrary means was derived in [7].