2.5. Z-transforms of the autocorrelation and intercorrelation functions – Digital Filters Design for Signal and Image Processing

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 the following connection between the transfer function Hz (z) of the system and its interspectral functions Sxy(z) and Sxx(z):