Monday, September 18, 2017

filters - Autocorrelation of the product of deterministic and random signal


I was wondering how to calculate the autocorrelation of a deterministic signal x(t) multiplied by a stochastic process M(t), whose autocorellation RM(τ) is known a priori. In my case, x(t) is a truncated monolateral exponentially decaying function.


I suppose that the result of such multiplication y(t)=x(t)M(t) is again a stochastic process, but when approaching the calculation of the autocorrelation of y(t) I obtain something that is not even, therefore I suppose I am making some mistakes. I know that the definition of autocorrelation for deterministic signals is different from the one of stochastic processes, but I do not know how to connect the two of them.



Answer



As correctly pointed out in the comments, in general the process Y(t)=x(t)M(t) is not wide-sense stationary (WSS), i.e. its autocorrelation function depends not only on the time difference parameter τ, but also on the absolute time t:


RY(τ,t)=E[Y(t+τ)Y(t)]=E[x(t+τ)M(t+τ)x(t)M(t)]==x(t+τ)x(t)E[M(t+τ)M(t)]=x(t+τ)x(t)RM(τ)


In your case you just need to evaluate (1) with the given function x(t). However, as expected you will end up with a function of two variables because Y(t) is not WSS.


There are a few special cases in which the resulting process is indeed WSS or can be easily made WSS. The first case is modulation by a complex exponential:


x(t)=ejωct



in which case


x(t+τ)x(t)=ejωc(t+τ)ejωct=ejωcτ


only depends on τ and not on t. Another case of interest is the case where x(t) is T-periodic. In this case the process x(t)M(t) can be made WSS by introducing a random phase epoch Θ with uniform distribution in the interval [0,T] which is independent of M(t):


Y(t)=x(t+Θ)M(t)


The autocorrelation function of Y(t) is then given by


RY(τ)=RM(τ)1TT0x(α+τ)x(α)dα


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