and, for an integer t > 0
Calculation of a (theoretical) autocorrelation function for a normal random discrete white noise filtered by a stationary filter with impulse response given by function h(j), j >0 can be done by using convolution.
Let x(k) denote the discrete white noise. Its autocorrelation function is a Kronecker delta. The output of the filter is given by the follwing convolution-type formula:
and, for an integer t > 0
We now compute the autocorrelation function of signal y
The value of the autocorrelation function at t = 0 is
equal to the sum of squares of the impulse
response. For t > 0 we can interpret this formula as a discrete-time
convolution of the impulse response h(j) with its mirror-image reflection
h*(i) = h(-i). For t < 0 we simply use the symmetry property
of the autocorrelation function.
Try the following Exercise:
Let the system be an IIR system with numerator = 1 and denominator with
one pole at z = a.
Then the impulse response for j > 0 is
Show that the autocorrelation function of signal
y is
(first show it for t > 0, then use symmetry). In this example,
the shape of the autocorrelation function for t > 0 is exactly the
same as the shape of the inpulse response h(j), however the
amplitude is multiplied by the sum of squares of h(j). For
a = 0.95 this autocorrelation function
is shown on Figure 2 on previous page. (note # 1)
Practical computation of the autocorrelation function for an IIR or
FIR filter can thus be done by first computing filter's impulse response,
then flipping it left-to-right (the time inversion), and then
computing the convolution of the original impulse response with the
flipped one.
As another exercise, you can try to compute autocorrelation function of a signal at the output of an oscillatory IIR filter whose poles are 0.9888+ 0.0494i and 0.9888- 0.0494i.
Discrete-time impulse response of such a filter (assuming numerator = 1) is shown on Figure 4 , and the result of the convolution computation is shown as blue curve on Figure 5 .
The red curve on that Figure represents an average of 500 sample time-averaged autocoreelation functions obtained for 500 different realizations of the input white noise.
