Distribution of Inner Product of Complex Gaussian Random Vectors and its Applications

Let X and Y be two independent L x 1 complex Gaussian random vectors distributed as CN (m×, σ²× I_L) and CN (m_Y, σ_Y² I_L), respectively, where I_L denotes the L x L identity matrix. The joint characteristic function (c.f.) of the real and imaginary parts of the inner product Y^H X is derived in closed form, with (·)^H denoting the conjugate transpose. Based on this joint c.f., a unified analytical framework for the derivation of the average symbol error probability (ASEP) of a multibranch diversity reception system over flat correlated fading using M-ary phase-shift keying is developed. The receiver employs maximal-ratio combining with least squares channel estimation by means of pilot symbols. The optimal average pilot-to-noise power ratio is obtained in closed form, and the analytical framework is applied to Nakagami and Rice fading. For Nakagami fading, closed form ASEP expressions are obtained for the cases of high signal-to-noise (SNR) and binary phase-shift keying, while for Rice fading, high SNR approximate expressions are obtained in terms of a single integral under the constant correlation model and for independent and identically distributed channels. Both analytical and computer simulation results are presented and compared in order to verify the validity of the proposed analysis.