Contents

# 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. ZN = Z× ... ×Z The set of N-dimensional series of integer numbers. p = (p1, ..., pN) ∈ZN N-dimensional series of integers. u:ZN→R Function u with arguments from ZN and values from R. u(p) = u(p1, ..., pN) The value of the function u for the argument (p1, ..., pN). 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
` Pmin≤p≤ Pmax.`
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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