Convolution and Correlation Mathematical Notation and Definitions

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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Date 4/28/2026

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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:

`\(\mathbb{R} = (-\infty, +\infty)\)`	The set of real numbers.
`\(\mathbb{Z} = \{0, \pm 1, \pm 2, \dots\}\)`	The set of integer numbers.
`\(\mathbb{Z}^{N} = \mathbb{Z} \times \dots \times \mathbb{Z}\)`	The set of `\(N\)`-dimensional series of integer numbers.
`\(p = (p_{1}, \dots, p_{N}) \in \mathbb{Z}^{N}\)`	N-dimensional series of integers.
`\(u : \mathbb{Z}^{N} \to \mathbb{R}\)`	Function `\(u\)` with arguments from `\(\mathbb{Z}^{N}\)` and values from `\(\mathbb{R}\)`.
`\(u(p) = u(p_1, \dots, p_N)\)`	The value of the function `\(u\)` for the argument `\((p_1, \dots, p_N)\)`.
`\(w = u \ast v\)`	Function `\(w\)` is the convolution of the functions `\(u\)` and `\(v\)`.
`\(w = u \star v\)`	Function `\(w\)` is the correlation of the functions `\(u\)` and `\(v\)`.

Given series \(p, q \in \mathbb{Z}^N\):

series \(r = p + q\) is defined as \(r_{n} = p_{n} + q_{n}\) for every \(n = 1, \dots, N\)
series \(r = p - q\) is defined as \(r_{n} = p_{n} - q_{n}\) for every \(n = 1, \dots, N\)
series \(r = sup\{p, q\}\) is defines as \(r_{n} = max\{p_{n}, q_{n}\}\) for every \(n = 1, \dots, N\)
series \(r = inf\{p, q\}\) is defined as \(r_{n} = min\{p_{n}, q_{n}\}\) for every \(n = 1, \dots, N\)
inequality \(p \le q\) means that \(p^n \le q^n\) for every \(n = 1, \dots, N\).

A function \(u(p)\) is called a finite function if there exist series \(P^{min}, P^{max} \in \mathbb{Z}^{N}\) such that:

\(u(p) \ne 0\) implies \(P^{min} \le p \le P^{max}\).

Operations of convolution and correlation are only defined for finite functions.

Consider functions \(u\), \(v\) and series \(P^{min}, P^{max}, Q^{min}, Q^{max} \in \mathbb{Z}^{N}\) such that:

\(u(p) \ne 0\) implies \(P^{min} \le p \le P^{max}\).

\(v(q) \ne 0\) implies \(Q^{min} \le q \le Q^{max}.\)

Definitions of linear correlation and linear convolution for functions \(u\) and \(v\) are given below.

Linear Convolution

If function \(w = u \ast v\) is the convolution of \(u\) and \(v\), then:

\(w(r) \ne 0\) implies \(R^{min} \le r \le R^{max}\), where \(R^{min} = P^{min} + Q^{min}\) and \(R^{max} = P^{max} + Q^{max}.\)

If \(R^{min} \le r \le R^{max}\), then:

\[w(r) = \sum_{t = T^{min}}^{T^{max}} u(t) v(r - t)\]

where \(T^{min} = sup\{P^{min}, r - Q^{max}\}\) and \(T^{max} = inf\{P^{max}, r - Q^{min}\}\).

Linear Correlation

If function \(w = u \star v\) is the correlation of \(u\) and \(v\), then:

\(w(r) \ne 0\) implies \(R^{min} \le r \le R^{max}\), where \(R^{min} = Q^{min} - P^{max}\) and \(R^{max} = Q^{max} - P^{min}\).

If \(R^{min} \le r \le R^{max}\), then:

\[w(r) = \sum_{t = T^{min}}^{T^{max}} u(t) v(r + t)\]

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 (oneMKL) convolution and correlation functions is described in Data Allocation.

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

Convolution and Correlation Mathematical Notation and Definitions

Linear Convolution

Linear Correlation