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). Versions of the convolution theorem are true for 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:

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

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:

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

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

Convolution theorem

$h(x)=f*g={\mathcal {F}}^{-1}\{F\cdot G\},$

where $\cdot$ denotes point-wise multiplication.

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

Proof of Convolution Theorem

Let functions $f,g$ belong to the Lp-space $L^{1}(\mathbb {R} ^{n})$. Let $F$ be the Fourier transform of $f$ and $G$ be the Fourier transform of $g$: ${\begin{aligned}F(\nu )&={\mathcal {F}}\{f\}(\nu )=\int _{\mathbb {R} ^{n}}f(x)e^{-i2\pi \nu \cdot x}\,dx,\quad \nu \in \mathbb {R} ^{n}\\G(\nu )&={\mathcal {F}}\{g\}(\nu )=\int _{\mathbb {R} ^{n}}g(x)e^{-i2\pi \nu \cdot x}\,dx,\end{aligned}}$ where the dot between $\nu$ and $x$ indicates the inner product of $\mathbb {R} ^{n}$. Let $h$ be the convolution of $f$ and $g$: $h(x)=\int _{\mathbb {R} ^{n}}f(\tau )g(x-\tau )\,d\tau .$ 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: ${\begin{aligned}H(\nu )={\mathcal {F}}\{h\}&=\int _{\mathbb {R} ^{n}}h(x)e^{-i2\pi \nu \cdot x}\,dx\\&=\int _{\mathbb {R} ^{n}}\left(\int _{\mathbb {R} ^{n}}f(\tau )g(x-\tau )\,d\tau \right)\,e^{-i2\pi \nu \cdot x}\,dx.\end{aligned}}$ 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): $H(\nu )=\int _{\mathbb {R} ^{n}}f(\tau )\left(\int _{\mathbb {R} ^{n}}g(x-\tau )e^{-i2\pi \nu \cdot x}\,dx\right)\,d\tau .$ Substituting $y=x-\tau$ and $dy=dx$ yields: ${\begin{aligned}H(\nu )&=\int _{\mathbb {R} ^{n}}f(\tau )\left(\int _{\mathbb {R} ^{n}}g(y)e^{-i2\pi \nu \cdot (y+\tau )}\,dy\right)\,d\tau \\&=\int _{\mathbb {R} ^{n}}f(\tau )e^{-i2\pi \nu \cdot \tau }\left(\int _{\mathbb {R} ^{n}}g(y)e^{-i2\pi \nu \cdot y}\,dy\right)\,d\tau \\&=\left(\int _{\mathbb {R} ^{n}}f(\tau )e^{-i2\pi \nu \cdot \tau }\,d\tau \right)\left(\int _{\mathbb {R} ^{n}}g(y)e^{-i2\pi \nu \cdot y}\,dy\right)\\&=\underbrace {\left(\int _{\mathbb {R} ^{n}}f(x)e^{-i2\pi \nu \cdot x}\,dx\right)} _{F(\nu )}\underbrace {\left(\int _{\mathbb {R} ^{n}}g(x)e^{-i2\pi \nu \cdot x}\,dx\right)} _{G(\nu )}.\end{aligned}}$

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.

Functions of a discrete variable (sequences)

Consider two sequences $f[n]$ and $g[n]$ with discrete-time Fourier transforms (DTFT) $F$ and $G$:

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

where now ${\mathcal {F}}$ denotes the DTFT operator. 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:

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

There is also a theorem for circular and N-periodic convolutions:

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

where $f_{_{N}}$ and $g_{_{N}}$ are periodic summations of sequences $f$ and $g$:

$f_{_{N}}[n]\ \triangleq \sum _{m=-\infty }^{\infty }f[n-mN]$   and   $g_{_{N}}[n]\ \triangleq \sum _{m=-\infty }^{\infty }g[n-mN].$

The theorem is:

$\scriptstyle {\rm {DFT}}\displaystyle \{f_{_{N}}*g\}=\ \scriptstyle {\rm {DFT}}\displaystyle \{f_{_{N}}\}\cdot \ \scriptstyle {\rm {DFT}}\displaystyle \{g_{_{N}}\},$

where DFT represents an N-length Discrete Fourier transform.

And therefore:

$f_{_{N}}*g=\ \scriptstyle {\rm {DFT}}^{-1}\displaystyle \left[\ \scriptstyle {\rm {DFT}}\displaystyle \{f_{_{N}}\}\cdot \ \scriptstyle {\rm {DFT}}\displaystyle \{g_{_{N}}\}\right].$

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: ${\mathcal {F}}\{f_{N}\}(\nu )={\frac {1}{N}}\sum _{k=-\infty }^{\infty }\left(\scriptstyle {\rm {DFT}}\displaystyle \{f_{N}\}[k]\right)\cdot \delta \left(\nu -k/N\right).$ (Eq.1) The product with $G(\nu )$ is thereby reduced to a discrete-frequency function: ${\begin{aligned}{\mathcal {F}}\{f_{N}*g\}(\nu )&=F_{N}(\nu )G(\nu )\\&={\frac {1}{N}}\sum _{k=-\infty }^{\infty }\left(\scriptstyle {\rm {DFT}}\displaystyle \{f_{N}\}[k]\right)\cdot G(\nu )\cdot \delta \left(\nu -k/N\right)\\&={\frac {1}{N}}\sum _{k=-\infty }^{\infty }\left(\scriptstyle {\rm {DFT}}\displaystyle \{f_{N}\}[k]\right)\cdot G(k/N)\cdot \delta \left(\nu -k/N\right)\\&={\frac {1}{N}}\sum _{k=-\infty }^{\infty }\underbrace {\left(\scriptstyle {\rm {DFT}}\displaystyle \{f_{N}\}[k]\right)\cdot \left(\scriptstyle {\rm {DFT}}\displaystyle \{g_{N}\}[k]\right)} _{\scriptstyle {\rm {DFT}}\displaystyle \{f_{N}*g\}[k]}\cdot \delta \left(\nu -k/N\right),\end{aligned}}$ 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 comparison with Eq.1. We can also compute the inverse DTFT: ${\begin{aligned}(f_{N}*g)[n]&=\int _{0}^{1}\left({\frac {1}{N}}\sum _{k=-\infty }^{\infty }\scriptstyle {\rm {DFT}}\displaystyle \{f_{N}\}[k]\cdot \scriptstyle {\rm {DFT}}\displaystyle \{g_{N}\}[k]\cdot \delta \left(\nu -k/N\right)\right)\cdot e^{i2\pi \nu n}d\nu \\&={\frac {1}{N}}\sum _{k=-\infty }^{\infty }\scriptstyle {\rm {DFT}}\displaystyle \{f_{N}\}[k]\cdot \scriptstyle {\rm {DFT}}\displaystyle \{g_{N}\}[k]\cdot \underbrace {\left(\int _{0}^{1}\delta \left(\nu -k/N\right)\cdot e^{i2\pi \nu n}d\nu \right)} _{{\text{0, except for}}\ {0\ \leq \ k\ <\ N}}\\&={\frac {1}{N}}\sum _{k=0}^{N-1}{\bigg (}\scriptstyle {\rm {DFT}}\displaystyle \{f_{N}\}[k]\cdot \scriptstyle {\rm {DFT}}\displaystyle \{g_{N}\}[k]{\bigg )}\cdot e^{i2\pi {\frac {n}{N}}k}\\&=\ \scriptstyle {\rm {DFT}}^{-1}\displaystyle \left[\scriptstyle {\rm {DFT}}\displaystyle \{f_{N}\}\cdot \scriptstyle {\rm {DFT}}\displaystyle \{g_{N}\}\right].\end{aligned}}$

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

Circular convolution

$f_{_{N}}*g\ =\ \scriptstyle {\rm {DFT}}^{-1}\displaystyle \left[\scriptstyle {\rm {DFT}}\displaystyle \{f\}\cdot \ \scriptstyle {\rm {DFT}}\displaystyle \{g\}\right]$

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. .

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.

Convolution theorem for Fourier series coefficients

Two convolution theorems exist for the Fourier series coefficients of a periodic function. Consider two functions $f(x)$ and $g(x)$ in $L^{1}([-\pi ,\pi ])$, with Fourier series coefficients $F$ and $G$:

${\begin{aligned}F[n]&\triangleq {\mathcal {F}}\{f\}[n]={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x)e^{-inx}\,dx,\quad n\in \mathbb {Z} \\G[n]&\triangleq {\mathcal {F}}\{g\}[n]={\frac {1}{2\pi }}\int _{-\pi }^{\pi }g(x)e^{-inx}\,dx,\quad n\in \mathbb {Z} \end{aligned}}$

where ${\mathcal {F}}$ denotes the Fourier series integral. The 2π-periodic convolution of $f$ and $g$ is given by:

${\begin{aligned}\left\{f*_{2\pi }g\right\}(x)\ &\triangleq \int _{-\pi }^{\pi }f(u)\cdot g[{\text{pv}}(x-u)]\,du,&&{\big (}{\text{and }}\underbrace {{\text{pv}}(x)\ \triangleq \arg \left(e^{ix}\right)} _{\text{principal value}}{\big )}\\&=\int _{-\pi }^{\pi }f(u)\cdot g(x-u)\,du,&&\scriptstyle {\text{when }}g(x){\text{ is 2}}\pi {\text{-periodic.}}\\&=\int _{2\pi }f(u)\cdot g(x-u)\,du.&&\scriptstyle {\text{when both functions are 2}}\pi {\text{-periodic, and the integral is over any 2}}\pi {\text{ interval.}}\end{aligned}}$

The first convolution theorem states that the Fourier series coefficients of the periodic convolution are:

${\mathcal {F}}\{f*_{2\pi }g\}[n]=2\pi \ F[n]G[n].$

The second convolution theorem states that the Fourier series coefficients of the product of $f$ and $g$ are given by the discrete convolution of the $F$ and $G$ sequences:

${\mathcal {F}}\{f\cdot g\}[n]=\{F*G\}[n].$

See also

• Moment-generating function of a random variable

References

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

Further reading

• 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 November 19, 2010

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