The basic principle and derivation of signal cyclostationarity (Matlab implementation)
-
Introduction
The cyclostationarity of a signal is an important property for the statistical analysis of signals in the time domain. It means that the statistical properties of the signal do not change over time, that is, the statistical properties of the signal are periodic. This article will introduce the basic principle and derivation of the cyclostationarity of the signal, and use Matlab to realize it. -
Definition of cyclostationarity
Let x(t) be a random process, t represents time, and it is a time index in discrete cases. If for any positive integer m and any constant n, the relationship is as follows:
E[x(t)] = E[x(t+m)], for all n
where E[ ] represents the expectation operator. This means that the expectation of the signal after time translation by m units is equal to the expectation of the original signal. If the above relation holds true for all t and m, the signal is said to satisfy cyclostationarity.
- Derivation of cyclostationarity
To better understand cyclostationarity, we will start with the derivation from the autocorrelation function of the signal. Let x(t) be a cyclostationary signal, then its autocorrelation function R_x(m) can be expressed as:
R_x(m) = E[x(t)x(t+m)]
According to the definition of cyclostationarity, we have:
E[x(t)] = E[x(t+m)]
Substituting the above equation into the expression for the autocorrelation function, we get:
R_x(m) = E[x(t)x(t+m)] = E[x(t+m)x(t+2m)]
Continuing to deduce, we can get:
R_x(m) = E[x(t)x(t+m)]
= E[x(t+m)x(t+2m)]
= E[x(t+2m)x(t+3m)]
…
Through the above derivation, we can