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 $f(x)$ and $g(x)$ with Fourier transforms $F$ and $G$ :

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

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 $\otimes$ is sometimes used instead.

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

H(\nu )\triangleq {\mathcal {F}}\{h\}(\nu )=F(\nu )G(\nu ).\quad \nu \in \mathbb {R}

And by applying the inverse Fourier transform ${\mathcal {F}}^{-1}$ , we have the corollary:^[b]

Convolution theorem

$h(x)=\{f*g\}(x)={\mathcal {F}}^{-1}\{F\cdot G\},\quad \scriptstyle {\text{where }}\displaystyle \cdot \scriptstyle {\text{ denotes point-wise multiplication}}$

The theorem also generally applies to multi-dimensional functions. A general proof can be viewed here:

Proof of convolution theorem

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

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

The convolution of $f$ and $g$ is defined by:

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

Also:

\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 $h\in L^1(\mathbb{R}^n)$ so its Fourier transform $H$ is defined by the integral formula:

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

{\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 $P$ -periodic functions $f_{_{P}}$ and $g_{_{P}}$ :

f_{_{P}}(x)\ \triangleq \sum _{m=-\infty }^{\infty }f(x-mP)

and

g_{_{P}}(x)\ \triangleq \sum _{m=-\infty }^{\infty }g(x-mP),

where the non-zero portion of components $f$ and $g$ are not necessarily limited to duration $P$ . The Fourier series coefficients are:

{\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 ${\mathcal {F}}$ denotes the Fourier series integral. The convolution:

{\begin{aligned}\{f_{_{P}}*g\}(x)\ &\triangleq \int _{-\infty }^{\infty }f_{_{P}}(\tau )\cdot g(x-\tau )\ d\tau \\&\equiv \int _{P}f_{_{P}}(\tau )\cdot g_{_{P}}(x-\tau )\ d\tau \end{aligned}}

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

{\mathcal {F}}\{f_{_{P}}*g\}[k]=\ P\cdot F[k]\ G[k].

Proof of periodic convolution theorem

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

{\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 $f$ and $g$ is defined by:

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:

H(\nu )={\mathcal {F}}\{f*g\}(\nu )=\ F(\nu )G(\nu ).

^[2]^[c]

Periodic convolution

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

f_{_{N}}[n]\ \triangleq \sum _{m=-\infty }^{\infty }f[n-mN]

and

g_{_{N}}[n]\ \triangleq \sum _{m=-\infty }^{\infty }g[n-mN],\quad n\in \mathbb {Z} ,

where the non-zero portion of components $f$ and $g$ are not necessarily limited to duration $N$ . The discrete convolution:

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

{\mathcal {F}}\{f_{_{N}}*g\}[k]=\ {\mathcal {F}}\{f_{_{N}}\}[k]\cdot {\mathcal {F}}\{g_{_{N}}\}[k],\quad k\in \mathbb {Z} .

And therefore:

\{f_{_{N}}*g\}[n]=\ {\mathcal {F}}^{-1}\{{\mathcal {F}}\{f_{_{N}}\}\cdot {\mathcal {F}}\{g_{_{N}}\}\}.

Proof of periodic convolution theorem

A time-domain proof proceeds as follows:

{\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 proof follows from § Periodic data, which indicates that $F_{_{N}}(\nu )$ can be written as:

{\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 $G(\nu)$ is thereby reduced to a discrete-frequency function:

{\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)\end{aligned}}

={\frac {1}{N}}\sum _{k=-\infty }^{\infty }\underbrace {\left(\scriptstyle {\rm {DFT}}\displaystyle \{f_{_{N}}\}[k]\right)\cdot \left(\scriptstyle {\rm {DFT}}\displaystyle \{g_{_{N}}\}[k]\right)} _{\scriptstyle {\rm {DFT}}\displaystyle \{f_{_{N}}*g\}[k]}\cdot \delta \left(\nu -k/N\right),

Q.E.D.

where the equivalence of $G(k/N)$ and $\left(\scriptstyle {\rm {DFT}}\displaystyle \{g_{_{N}}\}[k]\right)$ follows from § Sampling the DTFT, and the underbrace follows by application of § Periodic data to ${\mathcal {F}}\{f_{_{N}}*g\}(\nu ).$

We can also compute the inverse DTFT:

{\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}\end{aligned}}

=\ \scriptstyle {\rm {DFT}}^{-1}\displaystyle {\bigg (}\scriptstyle {\rm {DFT}}\displaystyle \{f_{_{N}}\}\cdot \scriptstyle {\rm {DFT}}\displaystyle \{g_{_{N}}\}{\bigg )}.

Q.E.D.

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

Circular convolution

$\{f_{_{N}}*g\}[n]=\ {\mathcal {F}}^{-1}\{{\mathcal {F}}\{f\}\cdot {\mathcal {F}}\{g\}\}.$

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

Convolution theorem for inverse Fourier transform

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

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

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

so that

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

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, $g$ is an arbitrary tempered distribution (e.g. the Dirac comb)

{\mathcal {F}}\{f*g\}={\mathcal {F}}\{f\}\cdot {\mathcal {F}}\{g\}

{\mathcal {F}}\{\alpha \cdot g\}={\mathcal {F}}\{\alpha \}*{\mathcal {F}}\{g\}

but $f=F\{\alpha \}$ must be "rapidly decreasing" towards $-\infty$ and $+\infty$ in order to guarantee the existence of both, convolution and multiplication product. Equivalently, if $\alpha =F^{-1}\{f\}$ 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 $1$ are smooth "slowly growing" ordinary functions. If, for example, $g\equiv \operatorname {III}$ is the Dirac comb both equations yield the Poisson Summation Formula and if, furthermore, $f\equiv \delta$ is the Dirac delta then $\alpha \equiv 1$ is constantly one and these equations yield the Dirac comb identity.

Notes

Page citations

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

References

^ 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.
^ 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
^ 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.
^ Horváth, John (1966). Topological Vector Spaces and Distributions. Reading, MA: Addison-Wesley Publishing Company.
^ Barros-Neto, José (1973). An Introduction to the Theory of Distributions. New York, NY: Dekker.
^ Petersen, Bent E. (1983). Introduction to the Fourier Transform and Pseudo-Differential Operators. Boston, MA: Pitman Publishing.

Weisstein, Eric W. "Convolution Theorem". From MathWorld--A Wolfram Web Resource. Retrieved 8 February 2021.
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

Additional resources

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

Johns Hopkins University's Java-aided simulation: http://www.jhu.edu/signals/convolve/index.html

[2] Weisstein, eq (8).

[3] Weisstein, eqs (7) and (10).

[5] Oppenheim and Schafer, p 60 (2.169).

[7] Oppenheim and Schafer, p 548.

[McGillem-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.

[Proakis-4] 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

[Rabiner-6] 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.

[8] Horváth, John (1966). Topological Vector Spaces and Distributions. Reading, MA: Addison-Wesley Publishing Company.

[9] Barros-Neto, José (1973). An Introduction to the Theory of Distributions. New York, NY: Dekker.

[10] Petersen, Bent E. (1983). Introduction to the Fourier Transform and Pseudo-Differential Operators. Boston, MA: Pitman Publishing.

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