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 Rxx(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 Rxy(k).
When x and y are real, it can also be demonstrated that Sxy(z) = Syx (z−1).
Inverse transforms allow us to find intercorrelation and autocorrelation functions from Sxy(z) and Sxx (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 Sxy(z) when it exists:
If permutation between the mathematical expectation and summation is possible:
Now, as the signal x is real, Rxx (−n) = Rxx (n). Since and , we thus establish ...
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