## 2.5. Z-transforms of the autocorrelation and intercorrelation functions

The spectral density in *z* of the sequence {*x*(*k*)} is represented as the z-transform of the autocorrelation function *R*_{xx}(*k*) of {*x*(*k*)}, a variable we saw in the previous chapter:

We can also introduce the concept of a discrete interspectrum of sequences {*x*(*k*)} and {*y*(*k*)} as the z-transform of the intercorrelation function *R _{xy}*(

*k*).

When *x* and *y* are real, it can also be demonstrated that *S*_{xy}(*z*) = *S*_{yx} (*z*^{−1}).

Inverse transforms allow us to find intercorrelation and autocorrelation functions from *S*_{xy}(*z*) and *S _{xx} (z)*:

Specific case:

Now let us look at a system with a real input {*x*(*k*)}, an output {*y*(*k*)}, and an impulse response *h*(*k*).

We then calculate *S _{xy}*(

*z*) when it exists:

If permutation between the mathematical expectation and summation is possible:

Now, as the signal *x* is real, *R*_{xx} (−*n*) = *R _{xx}* (

*n*). Since and , we thus establish the following connection between the transfer function

*H*(

_{z}*z*) of the system and its interspectral functions

*S*(

_{xy}*z*) and

*S*(

_{xx}*z*):