Wireless Transceiver Architecture

Appendix 1 Correlation

Autocorrelation and cross-correlation functions are often required for analytical derivations throughout this book. For instance, autocorrelation functions are used to derive power spectral densities, while cross-correlation functions are required for checking the non-correlation between two signals. In the latter case the aim may be to check that a given unwanted signal can effectively be considered as an additive noise term for the derivation of an SNR budget, for instance.

Analytical derivations involving RF bandpass signals may be carried out using their lowpass complex envelope representations. It is thus of interest to examine the conditions that such RF bandpass signals need to fulfill for equivalence to hold between their non-correlation and the non-correlation of their corresponding complex envelopes; we do this in the next section. We can then detail some general properties of the cross-correlation and autocorrelation functions that are useful for our analytical derivations.

A1.1 Bandpass Signals Correlations

Let us first focus on the correspondence between the non-correlation of two bandpass signals and the non-correlation of their complex envelopes. We begin with the case of two deterministic bandpass signals. The reasons for this will become apparent later on when making the link with the stochastic case.

Suppose that we want to check the non-correlation between two deterministic bandpass signals, x(t) and y(t), and derive an equivalent ...