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Convolution and Correlation Mathematical Notation and Definitions

The following notation is necessary to explain the underlying mathematical definitions used in the text:
R
= (-
, +
)
The set of real numbers.
Z
= {0,
±
1,
±
2, ...}
The set of integer numbers.
Z
N
=
Z
×
...
×
Z
The set of N-dimensional series of integer numbers.
p
= (
p
1
, ...,
p
N
)
Z
N
N-dimensional series of integers.
u
:
Z
N
R
Function
u
with arguments from
Z
N
and values from
R
.
u
(
p
) =
u
(
p
1
, ...,
p
N
)
The value of the function
u
for the argument (
p
1
, ...,
p
N
).
w
=
u
*
v
Function
w
is the convolution of the functions
u
,
v
.
w
=
u
v
Function
w
is the correlation of the functions
u
,
v
.
Given series
p
,
q
Z
N
:
  • series
    r
    =
    p
    +
    q
    is defined as
    r
    n
    =
    p
    n
    +
    q
    n
    for every
    n
    =1,...,N
  • series
    r
    =
    p
    -
    q
    is defined as
    r
    n
    =
    p
    n
    -
    q
    n
    for every
    n
    =1,...,N
  • series
    r
    = sup{
    p
    ,
    q
    }
    is defines as
    r
    n
    = max{
    p
    n
    ,
    q
    n
    }
    for every
    n
    =1,...,N
  • series
    r
    = inf{
    p
    ,
    q
    }
    is defined as
    r
    n
    = min{
    p
    n
    ,
    q
    n
    }
    for every
    n
    =1,...,N
  • inequality
    p
    q
    means that
    p
    n
    q
    n
    for every
    n
    =1,...,N
    .
A function
u
(
p
)
is called a finite function if there exist series P
min
, P
max
Z
N
such that:
u
(
p
)
0
implies
P
min
p
P
max
.
Operations of convolution and correlation are only defined for finite functions.
Consider functions
u
,
v
and series P
min
, P
max
Q
min
, Q
max
Z
N
such that:
u
(
p
)
0
implies
P
min
p
P
max
.
v
(
q
)
0
implies
Q
min
q
Q
max
.
Definitions of linear correlation and linear convolution for functions
u
and
v
are given below.

Linear Convolution

If function
w
=
u
*
v
is the convolution of
u
and
v
, then:
w
(
r
)
0
implies
R
min
r
R
max
,
where
R
min
= P
min
+ Q
min
and
R
max
= P
max
+ Q
max
.
If
R
min
r
R
max
, then:
w
(
r
) =
u
(
t
v
(
r
t
)
is the sum for all
t
Z
N
such that
T
min
t
T
max
,
where
T
min
= sup{P
min
,
r
Q
max
}
and
T
max
= inf{P
max
,
r
Q
min
}.

Linear Correlation

If function
w
=
u
v
is the correlation of
u
and
v
, then:
w
(
r
)
0
implies
R
min
r
R
max
,
where
R
min
= Q
min
- P
max
and
R
max
= Q
max
- P
min
.
If
R
min
r
R
max
, then:
w
(
r
) =
u
(
t
v
(
r
+
t
)
is the sum for all
t
Z
N
such that
T
min
t
T
max
,
where
T
min
= sup{P
min
, Q
min
r
}
and
T
max
= inf{P
max
, Q
max
r
}
.
Representation of the functions
u
,
v
,
w
as the input/output data for the
Intel® oneAPI Math Kernel Library
convolution and correlation functions is described in theData Allocation.

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