Digital Signal Processing Reference
In-Depth Information
•
i.i.d. elements
: a collection of
N
random variables is
independent
and identically distributed
(i.i.d.) if the variables are independent and
each random variable has the same distribution:
l
g
r
,
y
i
d
.
,
©
,
L
s
f
X
i
(
x
i
)=
f
(
x
i
)
,
i
=
0,1,...,
N
−
1
Random vectors represent the generalization of finite-length, discrete-time
signals to the space of random signals.
Expectation and Second Order Statistics.
For random vectors, the
definitions given, in the case of random variables, extend immediately to
the multidimensional case. The mean of a
N
-element random vector
X
is
simply the
N
-element vector:
]=
E
X
N−
1
]
T
E
[
X
[
X
0
]
E
[
X
1
]
... E
[
=
m
X
0
m
X
1
...
m
X
N−
1
T
=
m
X
The correlation of two
N
-element random vectors is the
N
×
N
matrix:
XY
T
]
where the expectation operator is applied individually to all the elements of
the matrix
XY
T
. The covariance is again:
K
XY
=
E
(
X
−
m
X
)(
Y
−
m
Y
)
R
XY
=
E
[
T
and it coincides with the correlation for zero-mean random vectors. Note
that the general element
R
XY
indicates the correlation between the
k
-
th element of
X
and the
l
-th element of
Y
.Inparticular,
R
XX
(
k
,
l
)
indicates
the correlation between elements of the random vector
X
; if the elements
are uncorrelated, then the correlation matrix is diagonal.
(
k
,
l
)
Example: Jointly Gaussian Random Vector.
An important type of
vector random variable is the Gaussian random vector of dimension
N
.To
define its pdf, we need a length-
N
vector
m
and a positive definite matrix
Λ
of size
N
×
N
. Then, the
N
-dimensional Gaussian pdf is given by
1
1
2
(
Λ
−
1
e
−
x
−
m
)
T
(
x
−
m
)
,
N
f
(
x
)=
x
∈
(8.3)
N
(
2
π
)
|
Λ
|
where
|
Λ
|
is the determinant of
Λ
. Clearly,
m
is the vector of the means of
the single elements of the Gaussian vector while
is the autocorrelation
matrix. A diagonal matrix implies the decorrelation of the random vector's
elements; in this case, since all the elements are Gaussian variables, this also
means that the elements are independent. Note how, for
N
Λ
=
1and
Λ
=
σ
2
,
this reduces to the usual Gaussian distribution of (8.1).
