Including the Factor of (1/T) in Going from the Laplace to Z-Transform and in Recovering the Autocorrelation from the PSD
Consider the Laplace to z-transform pairs below.
1. / / /2. / / /
Commentary on the Above Pairs
Example 1.
Consider the impulse response with transfer function . For a sampling interval, T, where . In this case, we have
where .
Hence, the above transforms 1. and 2. are not entirely correct, as the z-transforms are missing a factor of T.
Now, suppose that we have a random process X(t) with autocorrelation function . We can express this as:
where is the Kronecker delta function. It is not the Dirac delta function. The psd is
.
.
Letting , we then have
Now, since , The sampled process has psd:
where
is the z-transform of ; which is what the table that gave 1. and 2. above would give. Even though continuous-time white noise does not exist, the above psd for the sampled process can be viewed a discrete-time white noise (which does exist) passing through the filter . This fictitious white noise, call it U(kT), has variance .
Conclusion 1: It is essential that the factor T be included when going from Laplace to z-transforms.
Using Matlab’s fft and ifft Codes in Relation to Autocorrelations & PSDs
For a continuous-time process X(t), the Fourier transform pair relation between and is:
The sampled process has the frequency range where the sampling frequency is . Hence, the sampled process transform pair is:
Now, suppose that we wish to evaluate at N uniformly spaced frequencies over the interval . Then where . In this case, we have
.
Note that , and so , and . It follows that . Define the normalized frequency . Then if we assume that for , we have
.
The Matlab fft.m and ifft.m commands refer to the following pairs:
.
Hence, once again we see that the factor of T is missing!
Example 1 continued.
Now, suppose that we have computed the psd via . Then to recover we cannot simply use the ifft.m command applied to . Instead, we must first multiply by 1/T.
CONCLUSION: In going from a continuous-time to a discrete-time random process modeled as white noise passing through a filter, the sampling interval T must be accounted for in relation to (i) computing the white noise variance associated with the normalized filter obtained from a Laplace to z-transform table that does not include that factor, and (ii) in using the ifft.m command to recover the the autocorrelation function from a given expression for the psd.