Convolution Theorem
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Convolution Theorem

In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain). Other versions of the convolution theorem are applicable to various Fourier-related transforms.

## Functions of a continuous-variable

Consider two functions ${\displaystyle f(x)}$ and ${\displaystyle g(x)}$ with Fourier transforms ${\displaystyle F}$ and ${\displaystyle G}$:

{\displaystyle {\begin{aligned}F(\nu )&\triangleq {\mathcal {F}}\{f\}(\nu )=\int _{-\infty }^{\infty }f(x)e^{-i2\pi \nu x}\,dx,\quad \nu \in \mathbb {R} \\G(\nu )&\triangleq {\mathcal {F}}\{g\}(\nu )=\int _{-\infty }^{\infty }g(x)e^{-i2\pi \nu x}\,dx,\quad \nu \in \mathbb {R} \end{aligned}}}

where ${\displaystyle {\mathcal {F}}}$ denotes the Fourier transform operator. The transform may be normalized in other ways, in which case constant scaling factors (typically ${\displaystyle 2\pi }$ or ${\displaystyle {\sqrt {2\pi }}}$) will appear in the convolution theorem below. The convolution of ${\displaystyle f}$ and ${\displaystyle g}$ is defined by:

${\displaystyle h(x)=\{f*g\}(x)\triangleq \int _{-\infty }^{\infty }f(\tau )g(x-\tau )\,d\tau =\int _{-\infty }^{\infty }f(x-\tau )g(\tau )\,d\tau .}$

In this context the asterisk denotes convolution, instead of standard multiplication. The tensor product symbol ${\displaystyle \otimes }$ is sometimes used instead.

The convolution theorem states that:[1][a]

Applying the inverse Fourier transform ${\displaystyle {\mathcal {F}}^{-1}}$, produces the corollary:[b]

The theorem also generally applies to multi-dimensional functions.

Multi-dimensional derivation of Eq.1

Consider functions ${\displaystyle f,g}$ in Lp-space ${\displaystyle L^{1}(\mathbb {R} ^{n})}$, with Fourier transforms ${\displaystyle F,G}$:

{\displaystyle {\begin{aligned}F(\nu )&\triangleq {\mathcal {F}}\{f\}(\nu )=\int _{\mathbb {R} ^{n}}f(x)e^{-i2\pi \nu \cdot x}\,dx,\quad \nu \in \mathbb {R} ^{n}\\G(\nu )&\triangleq {\mathcal {F}}\{g\}(\nu )=\int _{\mathbb {R} ^{n}}g(x)e^{-i2\pi \nu \cdot x}\,dx,\end{aligned}}}

where ${\displaystyle \nu \cdot x}$ indicates the inner product of ${\displaystyle \mathbb {R} ^{n}}$:   ${\displaystyle \nu \cdot x=\sum _{j=1}^{n}{\nu }_{j}x_{j},}$   and   ${\displaystyle dx=\prod _{j=1}^{n}dx_{j}.}$

The convolution of ${\displaystyle f}$ and ${\displaystyle g}$ is defined by:

${\displaystyle h(x)\triangleq \int _{\mathbb {R} ^{n}}f(\tau )g(x-\tau )\,d\tau .}$

Also:

${\displaystyle \iint |f(\tau )g(x-\tau )|\,dx\,d\tau =\int \left(|f(\tau )|\int |g(x-\tau )|\,dx\right)\,d\tau =\int |f(\tau )|\,\|g\|_{1}\,d\tau =\|f\|_{1}\|g\|_{1}.}$

Hence by Fubini's theorem we have that ${\displaystyle h\in L^{1}(\mathbb {R} ^{n})}$ so its Fourier transform ${\displaystyle H}$ is defined by the integral formula:

{\displaystyle {\begin{aligned}H(\nu )\triangleq {\mathcal {F}}\{h\}(\nu )&=\int _{\mathbb {R} ^{n}}h(x)e^{-i2\pi \nu \cdot x}\,dx\\&=\int _{\mathbb {R} ^{n}}\left(\int _{\mathbb {R} ^{n}}f(\tau )g(x-\tau )\,d\tau \right)\,e^{-i2\pi \nu \cdot x}\,dx.\end{aligned}}}

Note that ${\displaystyle |f(\tau )g(x-\tau )e^{-i2\pi \nu \cdot x}|=|f(\tau )g(x-\tau )|}$ and hence by the argument above we may apply Fubini's theorem again (i.e. interchange the order of integration):

{\displaystyle {\begin{aligned}H(\nu )&=\int _{\mathbb {R} ^{n}}f(\tau )\underbrace {\left(\int _{\mathbb {R} ^{n}}g(x-\tau )\ e^{-i2\pi \nu \cdot x}\,dx\right)} _{G(\nu )\ e^{-i2\pi \nu \cdot \tau }}\,d\tau \\&=\underbrace {\left(\int _{\mathbb {R} ^{n}}f(\tau )\ e^{-i2\pi \nu \cdot \tau }\,d\tau \right)} _{F(\nu )}\ G(\nu ).\end{aligned}}}

This theorem also holds for the Laplace transform, the two-sided Laplace transform and, when suitably modified, for the Mellin transform and Hartley transform (see Mellin inversion theorem). It can be extended to the Fourier transform of abstract harmonic analysis defined over locally compact abelian groups.

### Periodic convolution (Fourier series coefficients)

Consider ${\displaystyle P}$-periodic functions ${\displaystyle f_{_{P}}}$ and ${\displaystyle g_{_{P}},}$ which can be expressed as periodic summations:

${\displaystyle f_{_{P}}(x)\ \triangleq \sum _{m=-\infty }^{\infty }f(x-mP)}$   and   ${\displaystyle g_{_{P}}(x)\ \triangleq \sum _{m=-\infty }^{\infty }g(x-mP).}$

In practice the non-zero portion of components ${\displaystyle f}$ and ${\displaystyle g}$ are often limited to duration ${\displaystyle P,}$ but nothing in the theorem requires that. The Fourier series coefficients are:

{\displaystyle {\begin{aligned}F[k]&\triangleq {\mathcal {F}}\{f_{_{P}}\}[k]={\frac {1}{P}}\int _{P}f_{_{P}}(x)e^{-i2\pi kx/P}\,dx,\quad k\in \mathbb {Z} ;\quad \quad \scriptstyle {\text{integration over any interval of length }}P\\G[k]&\triangleq {\mathcal {F}}\{g_{_{P}}\}[k]={\frac {1}{P}}\int _{P}g_{_{P}}(x)e^{-i2\pi kx/P}\,dx,\quad k\in \mathbb {Z} \end{aligned}}}

where ${\displaystyle {\mathcal {F}}}$ denotes the Fourier series integral.

• The pointwise product: ${\displaystyle f_{_{P}}(x)\cdot g_{_{P}}(x)}$

is also ${\displaystyle P}$-periodic, and its Fourier series coefficients are given by the discrete convolution of the ${\displaystyle F}$ and ${\displaystyle G}$ sequences:

${\displaystyle {\mathcal {F}}\{f_{_{P}}\cdot g_{_{P}}\}[k]=\{F*G\}[k].}$
• The convolution:
{\displaystyle {\begin{aligned}\{f_{_{P}}*g\}(x)\ &\triangleq \int _{-\infty }^{\infty }f_{_{P}}(x-\tau )\cdot g(\tau )\ d\tau \\&\equiv \int _{P}f_{_{P}}(x-\tau )\cdot g_{_{P}}(\tau )\ d\tau ;\quad \quad \scriptstyle {\text{integration over any interval of length }}P\end{aligned}}}[A]

is also ${\displaystyle P}$-periodic, and is called a periodic convolution. The corresponding convolution theorem is:

Derivation of Eq.2
{\displaystyle {\begin{aligned}{\mathcal {F}}\{f_{_{P}}*g\}[k]&\triangleq {\frac {1}{P}}\int _{P}\left(\int _{P}f_{_{P}}(\tau )\cdot g_{_{P}}(x-\tau )\ d\tau \right)e^{-i2\pi kx/P}\,dx\\&=\int _{P}f_{_{P}}(\tau )\left({\frac {1}{P}}\int _{P}g_{_{P}}(x-\tau )\ e^{-i2\pi kx/P}dx\right)\,d\tau \\&=\int _{P}f_{_{P}}(\tau )\ e^{-i2\pi k\tau /P}\underbrace {\left({\frac {1}{P}}\int _{P}g_{_{P}}(x-\tau )\ e^{-i2\pi k(x-\tau )/P}dx\right)} _{G[k],\quad {\text{due to periodicity}}}\,d\tau \\&=\underbrace {\left(\int _{P}\ f_{_{P}}(\tau )\ e^{-i2\pi k\tau /P}d\tau \right)} _{P\cdot F[k]}\ G[k].\end{aligned}}}

## Functions of a discrete variable (sequences)

By a derivation similar to Eq.1, there is an analogous theorem for sequences, such as samples of two continuous functions, where now ${\displaystyle {\mathcal {F}}}$ denotes the discrete-time Fourier transform (DTFT) operator. Consider two sequences ${\displaystyle f[n]}$ and ${\displaystyle g[n]}$ with transforms ${\displaystyle F}$ and ${\displaystyle G}$:

{\displaystyle {\begin{aligned}F(\nu )&\triangleq {\mathcal {F}}\{f\}(\nu )=\sum _{n=-\infty }^{\infty }f[n]\cdot e^{-i2\pi \nu n}\;,\quad \nu \in \mathbb {R} \\G(\nu )&\triangleq {\mathcal {F}}\{g\}(\nu )=\sum _{n=-\infty }^{\infty }g[n]\cdot e^{-i2\pi \nu n}\;.\quad \nu \in \mathbb {R} \end{aligned}}}

The § Discrete convolution of ${\displaystyle f}$ and ${\displaystyle g}$ is defined by:

${\displaystyle h[n]\triangleq (f*g)[n]=\sum _{m=-\infty }^{\infty }f[m]\cdot g[n-m]=\sum _{m=-\infty }^{\infty }f[n-m]\cdot g[m].}$

The convolution theorem for discrete sequences is:[2][c]

### Periodic convolution

Consider ${\displaystyle N}$-periodic sequences ${\displaystyle f_{_{N}}}$ and ${\displaystyle g_{_{N}}}$:

${\displaystyle f_{_{N}}[n]\ \triangleq \sum _{m=-\infty }^{\infty }f[n-mN]}$   and   ${\displaystyle g_{_{N}}[n]\ \triangleq \sum _{m=-\infty }^{\infty }g[n-mN],\quad n\in \mathbb {Z} .}$

In practice the non-zero portion of components ${\displaystyle f}$ and ${\displaystyle g}$ are often limited to duration ${\displaystyle N,}$ but nothing in the theorem requires that. The discrete convolution:

${\displaystyle \{f_{_{N}}*g\}[n]\ \triangleq \sum _{m=-\infty }^{\infty }f_{_{N}}[m]\cdot g[n-m]\equiv \sum _{m=0}^{N-1}f_{_{N}}[m]\cdot g_{_{N}}[n-m]}$

is also ${\displaystyle N}$-periodic, and is called a periodic convolution. In this case, the ${\displaystyle {\mathcal {F}}}$ operator can be redefined as the much simpler ${\displaystyle N}$-length Discrete Fourier transform (DFT). And the corresponding theorem is:[3][d]

And therefore:

For ${\displaystyle f}$ and ${\displaystyle g}$ sequences whose non-zero duration is less than or equal to N, a final simplification is:

This form is especially useful for implementing a numerical convolution on a computer. (see § Fast convolution algorithms) Under certain conditions, a sub-sequence of ${\displaystyle f_{_{N}}*g}$ is equivalent to linear (aperiodic) convolution of ${\displaystyle f}$ and ${\displaystyle g,}$ which is usually the desired result. (see § Example)

Derivations of Eq.4

A time-domain derivation proceeds as follows:

{\displaystyle {\begin{aligned}\scriptstyle {\rm {DFT}}\displaystyle \{f_{_{N}}*g\}[k]&\triangleq \sum _{n=0}^{N-1}\left(\sum _{m=0}^{N-1}f_{_{N}}[m]\cdot g_{_{N}}[n-m]\right)e^{-i2\pi kn/N}\\&=\sum _{m=0}^{N-1}f_{_{N}}[m]\left(\sum _{n=0}^{N-1}g_{_{N}}[n-m]\cdot e^{-i2\pi kn/N}\right)\\&=\sum _{m=0}^{N-1}f_{_{N}}[m]\cdot e^{-i2\pi km/N}\underbrace {\left(\sum _{n=0}^{N-1}g_{_{N}}[n-m]\cdot e^{-i2\pi k(n-m)/N}\right)} _{\scriptstyle {\rm {DFT}}\displaystyle \{g_{_{N}}\}[k]\quad \scriptstyle {\text{due to periodicity}}}\\&=\underbrace {\left(\sum _{m=0}^{N-1}f_{_{N}}[m]\cdot e^{-i2\pi km/N}\right)} _{\scriptstyle {\rm {DFT}}\displaystyle \{f_{_{N}}\}[k]}\left(\scriptstyle {\rm {DFT}}\displaystyle \{g_{_{N}}\}[k]\right).\end{aligned}}}

A frequency-domain derivation follows from § Periodic data, which indicates that the DTFTs can be written as:

${\displaystyle {\mathcal {F}}\{f_{_{N}}*g\}(\nu )={\frac {1}{N}}\sum _{k=-\infty }^{\infty }\left(\scriptstyle {\rm {DFT}}\displaystyle \{f_{_{N}}*g\}[k]\right)\cdot \delta \left(\nu -k/N\right).\quad \scriptstyle {\mathsf {(Eq.5a)}}}$
${\displaystyle {\mathcal {F}}\{f_{_{N}}\}(\nu )={\frac {1}{N}}\sum _{k=-\infty }^{\infty }\left(\scriptstyle {\rm {DFT}}\displaystyle \{f_{_{N}}\}[k]\right)\cdot \delta \left(\nu -k/N\right).}$

The product with ${\displaystyle G(\nu )}$ is thereby reduced to a discrete-frequency function:

{\displaystyle {\begin{aligned}{\mathcal {F}}\{f_{_{N}}*g\}(\nu )&=F_{_{N}}(\nu )G(\nu )\\&={\frac {1}{N}}\sum _{k=-\infty }^{\infty }\left(\scriptstyle {\rm {DFT}}\displaystyle \{f_{_{N}}\}[k]\right)\cdot G(\nu )\cdot \delta \left(\nu -k/N\right)\\&={\frac {1}{N}}\sum _{k=-\infty }^{\infty }\left(\scriptstyle {\rm {DFT}}\displaystyle \{f_{_{N}}\}[k]\right)\cdot G(k/N)\cdot \delta \left(\nu -k/N\right)\\&={\frac {1}{N}}\sum _{k=-\infty }^{\infty }\left(\scriptstyle {\rm {DFT}}\displaystyle \{f_{_{N}}\}[k]\right)\cdot \left(\scriptstyle {\rm {DFT}}\displaystyle \{g_{_{N}}\}[k]\right)\cdot \delta \left(\nu -k/N\right),\quad \scriptstyle {\mathsf {(Eq.5b)}}\end{aligned}}}

where the equivalence of ${\displaystyle G(k/N)}$ and ${\displaystyle \left(\scriptstyle {\rm {DFT}}\displaystyle \{g_{_{N}}\}[k]\right)}$ follows from § Sampling the DTFT. Therefore, the equivalence of (5a) and (5b) requires:

${\displaystyle \scriptstyle {\rm {DFT}}\displaystyle {\{f_{_{N}}*g\}[k]}=\left(\scriptstyle {\rm {DFT}}\displaystyle \{f_{_{N}}\}[k]\right)\cdot \left(\scriptstyle {\rm {DFT}}\displaystyle \{g_{_{N}}\}[k]\right).}$

We can also compute the inverse DTFT of (5b):

{\displaystyle {\begin{aligned}(f_{_{N}}*g)[n]&=\int _{0}^{1}\left({\frac {1}{N}}\sum _{k=-\infty }^{\infty }\scriptstyle {\rm {DFT}}\displaystyle \{f_{_{N}}\}[k]\cdot \scriptstyle {\rm {DFT}}\displaystyle \{g_{_{N}}\}[k]\cdot \delta \left(\nu -k/N\right)\right)\cdot e^{i2\pi \nu n}d\nu \\&={\frac {1}{N}}\sum _{k=-\infty }^{\infty }\scriptstyle {\rm {DFT}}\displaystyle \{f_{_{N}}\}[k]\cdot \scriptstyle {\rm {DFT}}\displaystyle \{g_{_{N}}\}[k]\cdot \underbrace {\left(\int _{0}^{1}\delta \left(\nu -k/N\right)\cdot e^{i2\pi \nu n}d\nu \right)} _{{\text{0, for}}\ k\ \notin \ [0,\ N)}\\&={\frac {1}{N}}\sum _{k=0}^{N-1}{\bigg (}\scriptstyle {\rm {DFT}}\displaystyle \{f_{_{N}}\}[k]\cdot \scriptstyle {\rm {DFT}}\displaystyle \{g_{_{N}}\}[k]{\bigg )}\cdot e^{i2\pi {\frac {n}{N}}k}\\&=\ \scriptstyle {\rm {DFT}}^{-1}\displaystyle {\bigg (}\scriptstyle {\rm {DFT}}\displaystyle \{f_{_{N}}\}\cdot \scriptstyle {\rm {DFT}}\displaystyle \{g_{_{N}}\}{\bigg )}.\end{aligned}}}

## Convolution theorem for inverse Fourier transform

There is also a convolution theorem for the inverse Fourier transform:

${\displaystyle {\mathcal {F}}^{-1}\{f*g\}={\mathcal {F}}^{-1}\{f\}\cdot {\mathcal {F}}^{-1}\{g\}}$
${\displaystyle {\mathcal {F}}^{-1}\{f\cdot g\}={\mathcal {F}}^{-1}\{f\}*{\mathcal {F}}^{-1}\{g\}}$

so that

${\displaystyle f*g={\mathcal {F}}\left\{{\mathcal {F}}^{-1}\{f\}\cdot {\mathcal {F}}^{-1}\{g\}\right\}}$
${\displaystyle f\cdot g={\mathcal {F}}\left\{{\mathcal {F}}^{-1}\{f\}*{\mathcal {F}}^{-1}\{g\}\right\}}$

## Convolution theorem for tempered distributions

The convolution theorem extends to tempered distributions. Here, ${\displaystyle g}$ is an arbitrary tempered distribution (e.g. the Dirac comb)

${\displaystyle {\mathcal {F}}\{f*g\}={\mathcal {F}}\{f\}\cdot {\mathcal {F}}\{g\}}$
${\displaystyle {\mathcal {F}}\{\alpha \cdot g\}={\mathcal {F}}\{\alpha \}*{\mathcal {F}}\{g\}}$

but ${\displaystyle f=F\{\alpha \}}$ must be "rapidly decreasing" towards ${\displaystyle -\infty }$ and ${\displaystyle +\infty }$ in order to guarantee the existence of both, convolution and multiplication product. Equivalently, if ${\displaystyle \alpha =F^{-1}\{f\}}$ is a smooth "slowly growing" ordinary function, it guarantees the existence of both, multiplication and convolution product. .[4][5][6]

In particular, every compactly supported tempered distribution, such as the Dirac Delta, is "rapidly decreasing". Equivalently, bandlimited functions, such as the function that is constantly ${\displaystyle 1}$ are smooth "slowly growing" ordinary functions. If, for example, ${\displaystyle g\equiv \operatorname {III} }$ is the Dirac comb both equations yield the Poisson Summation Formula and if, furthermore, ${\displaystyle f\equiv \delta }$ is the Dirac delta then ${\displaystyle \alpha \equiv 1}$ is constantly one and these equations yield the Dirac comb identity.

## Notes

1. ^ Proof:
{\displaystyle {\begin{aligned}\int _{-\infty }^{\infty }f_{_{P}}(x-\tau )\cdot g(\tau )\,d\tau &=\sum _{k=-\infty }^{\infty }\left[\int _{x_{o}+kP}^{x_{o}+(k+1)P}f_{_{P}}(x-\tau )\cdot g(\tau )\ d\tau \right]\quad x_{0}{\text{ is an arbitrary parameter}}\\&=\sum _{k=-\infty }^{\infty }\left[\int _{x_{o}}^{x_{o}+P}\underbrace {f_{_{P}}(x-\tau -kP)} _{f_{_{P}}(x-\tau ),{\text{ by periodicity}}}\cdot g(\tau +kP)\ d\tau \right]\quad {\text{substituting }}\tau \rightarrow \tau +kP\\&=\int _{x_{o}}^{x_{o}+P}f_{_{P}}(x-\tau )\cdot \underbrace {\left[\sum _{k=-\infty }^{\infty }g(\tau +kP)\right]} _{\triangleq \ g_{_{P}}(\tau )}\ d\tau \end{aligned}}}

## Page citations

1. ^ Weisstein, eq (8).
2. ^ Weisstein, eqs (7) and (10).
3. ^ Oppenheim and Schafer, p 60 (2.169).
4. ^ Oppenheim and Schafer, p 548.

## References

1. ^ McGillem, Clare D.; Cooper, George R. (1984). Continuous and Discrete Signal and System Analysis (2 ed.). Holt, Rinehart and Winston. p. 118 (3-102). ISBN 0-03-061703-0.
2. ^ Proakis, John G.; Manolakis, Dimitri G. (1996), Digital Signal Processing: Principles, Algorithms and Applications (3 ed.), New Jersey: Prentice-Hall International, p. 297, Bibcode:1996dspp.book.....P, ISBN 9780133942897, sAcfAQAAIAAJ
3. ^ Rabiner, Lawrence R.; Gold, Bernard (1975). Theory and application of digital signal processing. Englewood Cliffs, NJ: Prentice-Hall, Inc. p. 59 (2.163). ISBN 978-0139141010.
4. ^ Horváth, John (1966). Topological Vector Spaces and Distributions. Reading, MA: Addison-Wesley Publishing Company.
5. ^ Barros-Neto, José (1973). An Introduction to the Theory of Distributions. New York, NY: Dekker.
6. ^ Petersen, Bent E. (1983). Introduction to the Fourier Transform and Pseudo-Differential Operators. Boston, MA: Pitman Publishing.
1. Weisstein, Eric W. "Convolution Theorem". From MathWorld--A Wolfram Web Resource. Retrieved 2021.
2. Oppenheim, Alan V.; Schafer, Ronald W.; Buck, John R. (1999). Discrete-time signal processing (2nd ed.). Upper Saddle River, N.J.: Prentice Hall. ISBN 0-13-754920-2.  Also available at https://d1.amobbs.com/bbs_upload782111/files_24/ourdev_523225.pdf

• Katznelson, Yitzhak (1976), An introduction to Harmonic Analysis, Dover, ISBN 0-486-63331-4
• Li, Bing; Babu, G. Jogesh (2019), "Convolution Theorem and Asymptotic Efficiency", A Graduate Course on Statistical Inference, New York: Springer, pp. 295-327, ISBN 978-1-4939-9759-6
• Crutchfield, Steve (October 9, 2010), "The Joy of Convolution", Johns Hopkins University, retrieved 2010

For a visual representation of the use of the convolution theorem in signal processing, see: