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® MKL
convolution and correlation functions is described in theData Allocation.

#### Product and Performance Information

1

Intel's compilers may or may not optimize to the same degree for non-Intel microprocessors for optimizations that are not unique to Intel microprocessors. These optimizations include SSE2, SSE3, and SSSE3 instruction sets and other optimizations. Intel does not guarantee the availability, functionality, or effectiveness of any optimization on microprocessors not manufactured by Intel. Microprocessor-dependent optimizations in this product are intended for use with Intel microprocessors. Certain optimizations not specific to Intel microarchitecture are reserved for Intel microprocessors. Please refer to the applicable product User and Reference Guides for more information regarding the specific instruction sets covered by this notice.

Notice revision #20110804