The autocorrelation matrix is used in various digital signal processing algorithms. It consists of elements of the discrete autocorrelation function,
R
x
x
(
j
)
arranged in the following manner:
R
x
=
E
[
x
x
H
]
=
[
R
x
x
(
0
)
R
x
x
∗
(
1
)
R
x
x
∗
(
2
)
⋯
R
x
x
∗
(
N
−
1
)
R
x
x
(
1
)
R
x
x
(
0
)
R
x
x
∗
(
1
)
⋯
R
x
x
∗
(
N
−
2
)
R
x
x
(
2
)
R
x
x
(
1
)
R
x
x
(
0
)
⋯
R
x
x
∗
(
N
−
3
)
⋮
⋮
⋮
⋱
⋮
R
x
x
(
N
−
1
)
R
x
x
(
N
−
2
)
R
x
x
(
N
−
3
)
⋯
R
x
x
(
0
)
]
This is a Hermitian matrix and a Toeplitz matrix. If
x
is wide-sense stationary then its autocorrelation matrix will be positive definite.
The autocovariance matrix is related to the autocorrelation matrix as follows:
C
x
=
E
[
(
x
−
m
x
)
(
x
−
m
x
)
H
]
=
R
x
−
m
x
m
x
H
Where
m
x
is a vector giving the mean of signal
x
at each index of time.
