Spectral Characteristics of Random Processes - Random Signals and Processes Primer with MATLAB - page 503

(a) What does a constant part in a PSD plot indicate?
(b) What does a constant part in an autocorrelation plot indicate?
(c) What does a delta function in a PSD plot indicate?
(d) What does a delta function in an autocorrelation plot indicate?

Q.7.6. Why must a process with a DC component have a delta function in its
PSD?
Q.7.7. How we can detect from the PSD whether or not the process has a
periodical component?
Q.7.8. Can a low-pass process have a zero average power at frequency f ¼ 0?
Q.7.9. Why is the PSD of a white process usually presented as N 0 /2 instead of N 0 ?
Q.7.10. Can a white process also be a non-Gaussian?
Q.7.11. Can a Gaussian process be a nonwhite process?

7.8 Answers

A.7.1. Let us suppose the contrary that there is a WS process X ( t ) which has the
autocorrelation function given in ( 7.201 ). As such, we find the PSD of the
process as a Fourier transform of its autocorrelation function:

S XX ðoÞ¼FR XX ðtÞ f g F sin( o 0 tÞ f g: (7.202)

From Appendix E , we find:

S XX ðoÞ¼ j pdðo o 0 Þdðo þ o 0 Þ ½ :

(7.203)

Note that the power spectral characteristic is negative and imaginary.
However, the PSD can be neither negative nor imaginary. Therefore, our
assumption was wrong and as a consequence, there is no WS process which
has an autocorrelation function given in ( 7.201 ).
A.7.2. As opposed to the Fourier transform of deterministic signals, a PSD, being
a real function in frequency, does not tell us anything about the phase of the
Fourier transform of a random process. As shown in Sect. 7.1 , a PSD is an
expected value of a power and no information about the phase is involved.
A.7.3. The PSD is a Fourier transform of an autocorrelation function which is a
characteristic of a whole process. Therefore, a PSD is also a characteristic
of a whole process.
A.7.4. In contrast to a deterministic signal, a random process decomposes to
sinusoids of different frequencies that have random amplitudes and ran-
dom phases. For more details, see [KAY06, p. 569].
A.7.5. (a) A constant part in a PSD plot over a band of frequencies indicates that
equal power is distributed in this range of frequencies.