Spectral Characteristics of Random Processes - Random Signals and Processes Primer with MATLAB - page 457

Fig. 7.6 PSDs for low-pass, band-pass, and band-limited processes

The obtained result shows that the autocorrelation function of a white noise is a
delta function (Fig. 7.8b ) in its origin. This means that there is no correlation
between values of a process, for any t , except t ¼ 0. Consequently, this process
is a maximally unpredictable process.
The area under a spectral density is infinite, indicating that a mean power of a
white noise is infinite. As a process with infinite power cannot really exist, we can
conclude that there is no white process in the real world.
Instead, consider a low-pass process, which has a flat PSD in the frequency band
[ B , B ], as shown in Fig. 7.9 . The corresponding autocorrelation function is:

ð ð

B B

N 0
2

sin 2 pBt
2 pBt :

S XX ð f Þ e j 2 pf t d f ¼ e j 2 pf t d f ¼N 0 B

R XX ðtÞ¼ (7.52)

B B