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Random Processes and Spectral Analysis Chap. 6
Thus, the PDF for y is
1
f y (y) =
(6–123)
(2p) N/2 |K| |Det C x | 1/2 e-(1/2)(y-m y) T C -1 y (y-m y )
which is an N-dimensional Gaussian PDF. This completes the proof of the theorem.
If a linear system acts like an integrator or an LPF, the output random variables (of the
output random process) tend to be proportional to the sum of the input random variables.
Consequently, by applying the central limit theorem (see Appendix B), the output of the integrator
or LPF will tend toward a Gaussian random process when the input random variables are
independent with non-Gaussian PDFs.
Example 6–10 WHITE GAUSSIAN-NOISE PROCESS
Assume that a Gaussian random process n(t) is given that has a PSD of
1
n (f) = e 2 N 0, | f | … B
(6–124)
0, f otherwise
where B is a positive constant. This describes a bandlimited white Gaussian process as long as B is
finite, but becomes a completely white (all frequencies are present) Gaussian process as B → q.
The autocorrelation function for the bandlimited white process is
sin 2pBt
R n (t) = BN 0 a (6–125)
2pBt
b
The total average power is P = R n (0) = BN 0 . The mean value of n(t) is zero, since there is no d
function in the PSD at f = 0. Furthermore, the autocorrelation function is zero for t = k(2B)
when k is a nonzero integer. Therefore, the random variables n 1 = n(t 1 ) and n 2 = n(t 2 ) are
uncorrelated when t 2 - t 1 = t = k(2B), k Z 0. For other values of t, the random variables are
correlated. Since n(t) is assumed to be Gaussian, n 1 and n 2 are jointly Gaussian random variables.
Consequently, by property 3, the random variables are independent when t 2 - t 1 = k(2B).
They are dependent for other values of t 2 and t 1 . As B → q, R n (t) : 1 2 N od(t), and the random
variables n 1 and n 2 become independent for all values of t 1 and t 2 , provided that t 1 Z t 2 .
Furthermore, as B : q , the average power becomes infinite. Consequently, a white-noise
process is not physically realizable. However, it is a very useful mathematical idealization for
system analysis, just as a deterministic impulse is useful for obtaining the impulse response of a
linear system, although the impulse itself is not physically realizable.
6–7 BANDPASS PROCESSES †
Bandpass Representations
In Sec. 4–1, it was demonstrated that any bandpass waveform could be represented by
v(t) = Re{g(t)e jvct }
(6–126a)
† In some other texts these are called narrowband noise processes, which is a misnomer because they may be
wideband or narrowband.