The usual Sampling Theorem applies to random processes, with the
spectrum of interest beign the power spectrum. If stationary
process
Xt
X
t
is bandlimited -
𝒮
X
ω=0
𝒮
X
ω
0
,
|ω|>W
ω
W
, as long as the sampling interval
TT satisfies the classic constraint
T<πW
T
W
the sequence
XlT
X
l
T
represents the original process. A sampled process is
itself a random process defined over discrete time. Hence, all
of the random process notions introduced in the previous section apply to the random sequence
X
∼
l≡XlT
X
∼
l
X
l
T
. The correlation functions of these two processes are
related as
R
X
∼
k=E
X
∼
l
X
∼
l+k=
R
X
kT
R
X
∼
k
X
∼
l
X
∼
l
k
R
X
k
T
We note especially that for distinct samples of a random process
to be uncorrelated, the correlation function
R
X
kT
R
X
k
T
must equal zero for all non-zero
kk. This requirement places severe
restrictions on the correlation function (hence the power
spectrum) of the original process. One correlation function
satisfying this property is derived from the random process
which has a bandlimited, constant-valued power spectrum over
precisely the frequency region needed to satisfy the sampling
criterion. No other power spectrum satisfying the
sampling criterion has this property. Hence,
sampling does not normally yield uncorrelated amplitudes,
meaning that discrete-time white noise is a rarity.
White noise has a correlation function given by
R
X
∼
k=σ2δk
R
X
∼
k
σ
2
δ
k
, where
δ·
δ
·
is the unit sample. The power spectrum of white noise
is a constant:
𝒮
X
∼
ω=σ2
𝒮
X
∼
ω
σ
2
.
