Digital Signal Processing Reference
In-Depth Information
DEFINITION 2.6
A stochastic process
X
(
t
)
is said to be cyclostationary with
period
T
if and only if
p
X(t
1
)
,
...
,
X(t
k
)
(
x
1
,
...
,
x
k
)
is periodic in
t
with period
T
,
i.e.,
p
X(t
1
)
,
...
,
X(t
K
)
(
x
1
,
...
,
x
K
)
=
p
X(t
1
+
T)
,
...
,
X(t
K
+
T)
(
x
1
,
...
,
x
K
)
It is also possible to establish two possibilities: strict-sense cyclostation-
arity, which corresponds to the above definition, and wide-sense (weak)
cyclostationarity. A stochastic process
X
is wide-sense cyclostationary
if its mean and autocorrelation function are periodic in
t
with some
period
T
, i.e.,
(
t
)
κ
1
(
X
,
t
+
T
)
=
κ
1
(
X
,
t
)
(2.102)
R
X
t
2
E
X
t
2
X
∗
t
2
R
X
t
T
(2.103)
τ
2
,
t
τ
τ
τ
τ
2
+
τ
2
+
+
−
=
+
−
=
+
T
,
t
−
for all τ
∈
(
−
T
,
T
)
.
2.4.5 Discrete-Time Random Signals
A discrete-time random process is a particular kind of random process in
which the time variable is of a discrete nature. Formally, a discrete-time
random process is defined as follows.
DEFINITION 2.7
A discrete-time stochastic process
X
is a collection, or
ensemble, of functions engendered by a rule that assigns a sequence
x
(
n
)
(
n
, ω
i
)
or simply
x
i
(
, called sample of the stochastic process, to each possible
outcome of a sample space.
n
)
Similarly to its continuous-time counterpart, it is possible to character-
ize a discrete-time random process by means of first- and second-order
moments. Thus, we define the mean of a discrete-time random process as
the mean value of the corresponding random variable produced when the
time index
n
is fixed, i.e.,
κ
1
(
X
,
n
)
=
E
{
X
(
n
)
}
∞
=
xp
X(n)
(
x
)
dx
(2.104)
−∞
where
p
X(n)
(
x
)
is the first-order pdf of the process.
