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 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 ...