Wireless Transceiver Architecture

Appendix 3 Moments of Normal Random Vectors

Real or complex Gaussian processes are defined so that the vector composed of the random variables corresponding to the time samples of those processes are respectively real or complex normal random vectors. Such processes are of particular importance as they can model both real bandpass signals and lowpass complex envelopes often encountered in wireless transceivers. This applies for instance to most of the analog noises as discussed in Section 1.2, and to most of the wideband modulating waveforms discussed in Section 1.3.3. From the system design point of view, such processes make it possible to carry out analytical derivations with the Gaussian distribution. This is for instance the case when investigating the distortion experienced by a Gaussian signal that goes through a nonlinearity (see Chapter 5).

But to do so, we need to deal with higher order moments of normal random vectors. It is therefore the purpose of this appendix to recall the results on that topic. We start by considering real normal random vectors in order to use the corresponding results to derive moments of complex normal random vectors that are used to represent the time samples of complex envelopes of Gaussian bandpass processes. We also give general results in order to further highlight the simplifications linked to the stationarity of the processes as classically encountered in wireless transceivers (see the discussion in Appendix 2).

A3.1 Real Normal Random ...