Convolution theorem

In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two signals is the pointwise product of their Fourier transforms. In other words, 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. Let ${\displaystyle f}$ and ${\displaystyle g}$ be two functions with convolution ${\displaystyle f*g}$. (Note that the asterisk denotes convolution in this context, not standard multiplication. The tensor product symbol ${\displaystyle \otimes }$ is sometimes used instead.)

If ${\displaystyle {\mathcal {F}}}$ denotes the Fourier transform operator, then ${\displaystyle {\mathcal {F}}\{f\}}$ and ${\displaystyle {\mathcal {F}}\{g\}}$ are the Fourier transforms of ${\displaystyle f}$ and ${\displaystyle g}$, respectively. Then

${\displaystyle {\mathcal {F}}\{f*g\}={\mathcal {F}}\{f\}\cdot {\mathcal {F}}\{g\}}$[1]

where ${\displaystyle \cdot }$ denotes point-wise multiplication. It also works the other way around:

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

By applying the inverse Fourier transform ${\displaystyle {\mathcal {F}}^{-1}}$, we can write:

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

and:

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

The relationships above are only valid for the form of the Fourier transform shown in the Proof section below. 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 relationships above.

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.

This formulation is especially useful for implementing a numerical convolution on a computer: The standard convolution algorithm has quadratic computational complexity. With the help of the convolution theorem and the fast Fourier transform, the complexity of the convolution can be reduced from ${\displaystyle O\left(n^{2}\right)}$ to ${\displaystyle O\left(n\log n\right)}$, using big O notation. This can be exploited to construct fast multiplication algorithms, as in Multiplication algorithm § Fourier transform methods.

Proof

The proof here is shown for a particular normalization of the Fourier transform. As mentioned above, if the transform is normalized differently, then constant scaling factors will appear in the derivation.

Let ${\displaystyle f,g}$ belong to the L^p-space ${\displaystyle L^{1}(\mathbb {R} ^{n})}$. Let ${\displaystyle F}$ be the Fourier transform of ${\displaystyle f}$ and ${\displaystyle G}$ be the Fourier transform of ${\displaystyle g}$:

${\displaystyle {\begin{aligned}F(\nu )&={\mathcal {F}}\{f\}(\nu )=\int _{\mathbb {R} ^{n}}f(x)e^{-2\pi ix\cdot \nu }\,dx,\\G(\nu )&={\mathcal {F}}\{g\}(\nu )=\int _{\mathbb {R} ^{n}}g(x)e^{-2\pi ix\cdot \nu }\,dx,\end{aligned}}}$

where the dot between ${\displaystyle x}$ and ${\displaystyle \nu }$ indicates the inner product of ${\displaystyle \mathbb {R} ^{n}}$. Let ${\displaystyle h}$ be the convolution of ${\displaystyle f}$ and ${\displaystyle g}$

${\displaystyle h(z)=\int _{\mathbb {R} ^{n}}f(x)g(z-x)\,dx.}$

Also

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

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

${\displaystyle {\begin{aligned}H(\nu )={\mathcal {F}}\{h\}&=\int _{\mathbb {R} ^{n}}h(z)e^{-2\pi iz\cdot \nu }\,dz\\&=\int _{\mathbb {R} ^{n}}\int _{\mathbb {R} ^{n}}f(x)g(z-x)\,dx\,e^{-2\pi iz\cdot \nu }\,dz.\end{aligned}}}$

Note that ${\displaystyle |f(x)g(z-x)e^{-2\pi iz\cdot \nu }|=|f(x)g(z-x)|}$ and hence by the argument above we may apply Fubini's theorem again (i.e. interchange the order of integration):

${\displaystyle H(\nu )=\int _{\mathbb {R} ^{n}}f(x)\left(\int _{\mathbb {R} ^{n}}g(z-x)e^{-2\pi iz\cdot \nu }\,dz\right)\,dx.}$

Substituting ${\displaystyle y=z-x}$ yields ${\displaystyle dy=dz}$. Therefore

${\displaystyle H(\nu )=\int _{\mathbb {R} ^{n}}f(x)\left(\int _{\mathbb {R} ^{n}}g(y)e^{-2\pi i(y+x)\cdot \nu }\,dy\right)\,dx}$

${\displaystyle =\int _{\mathbb {R} ^{n}}f(x)e^{-2\pi ix\cdot \nu }\left(\int _{\mathbb {R} ^{n}}g(y)e^{-2\pi iy\cdot \nu }\,dy\right)\,dx}$

${\displaystyle =\int _{\mathbb {R} ^{n}}f(x)e^{-2\pi ix\cdot \nu }\,dx\int _{\mathbb {R} ^{n}}g(y)e^{-2\pi iy\cdot \nu }\,dy.}$

These two integrals are the definitions of ${\displaystyle F(\nu )}$ and ${\displaystyle G(\nu )}$, so:

${\displaystyle H(\nu )=F(\nu )\cdot G(\nu ),}$

QED.

Convolution theorem for inverse Fourier transform

A similar argument, as the above proof, can be applied to the convolution theorem for the inverse Fourier transform;

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

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

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

and:

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

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. [2] [3] [4].

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.

Functions of discrete variable sequences

The analogous convolution theorem for discrete sequences ${\displaystyle x}$ and ${\displaystyle y}$ is:

${\displaystyle \scriptstyle {\rm {DTFT}}\displaystyle \{x*y\}=\ \scriptstyle {\rm {DTFT}}\displaystyle \{x\}\cdot \ \scriptstyle {\rm {DTFT}}\displaystyle \{y\},}$[5][lower-alpha 1]

where DTFT represents the discrete-time Fourier transform.

There is also a theorem for circular and periodic convolutions:

${\displaystyle x_{_{N}}*y\ \triangleq \sum _{m=-\infty }^{\infty }x_{_{N}}[m]\cdot y[n-m]\equiv \sum _{m=0}^{N-1}x_{_{N}}[m]\cdot y_{_{N}}[n-m],}$

where ${\displaystyle x_{_{N}}}$ and ${\displaystyle y_{_{N}}}$ are periodic summations of sequences ${\displaystyle x}$ and ${\displaystyle y}$:

${\displaystyle x_{_{N}}[n]\ \triangleq \sum _{m=-\infty }^{\infty }x[n-mN]}$   and   ${\displaystyle y_{_{N}}[n]\ \triangleq \sum _{m=-\infty }^{\infty }y[n-mN].}$

The theorem is:

${\displaystyle \scriptstyle {\rm {DFT}}\displaystyle \{x_{_{N}}*y\}=\ \scriptstyle {\rm {DFT}}\displaystyle \{x_{_{N}}\}\cdot \ \scriptstyle {\rm {DFT}}\displaystyle \{y_{_{N}}\},}$[6][lower-alpha 2]

where DFT represents an N-length Discrete Fourier transform.

And therefore:

${\displaystyle x_{_{N}}*y=\ \scriptstyle {\rm {DFT}}^{-1}\displaystyle {\big [}\ \scriptstyle {\rm {DFT}}\displaystyle \{x_{_{N}}\}\cdot \ \scriptstyle {\rm {DFT}}\displaystyle \{y_{_{N}}\}{\big ]}.}$

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

Circular convolution

${\displaystyle x_{_{N}}*y\ =\ \scriptstyle {\rm {DFT}}^{-1}\displaystyle \left[\scriptstyle {\rm {DFT}}\displaystyle \{x\}\cdot \ \scriptstyle {\rm {DFT}}\displaystyle \{y\}\right]}$

Under certain conditions, a sub-sequence of ${\displaystyle x_{_{N}}*y}$ is equivalent to linear (aperiodic) convolution of ${\displaystyle x}$ and ${\displaystyle y}$, which is usually the desired result. (see Example)  And when the transforms are efficiently implemented with the Fast Fourier transform algorithm, this calculation is much more efficient than linear convolution.

Convolution theorem for Fourier series coefficients

Two convolution theorems exist for the Fourier series coefficients of a periodic function:

• The first convolution theorem states that if ${\displaystyle f}$ and ${\displaystyle g}$ are in ${\displaystyle L^{1}([-\pi ,\pi ])}$, the Fourier series coefficients of the 2π-periodic convolution of ${\displaystyle f}$ and ${\displaystyle g}$ are given by:

${\displaystyle [{\widehat {f*_{2\pi }g}}](n)=2\pi \cdot {\widehat {f}}(n)\cdot {\widehat {g}}(n),}$[upper-alpha 1]

where:

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

• The second convolution theorem states that the Fourier series coefficients of the product of ${\displaystyle f}$ and ${\displaystyle g}$ are given by the discrete convolution of the ${\displaystyle {\hat {f}}}$ and ${\displaystyle {\hat {g}}}$ sequences:

${\displaystyle \left[{\widehat {f\cdot g}}\right](n)=\left[{\widehat {f}}*{\widehat {g}}\right](n).}$

See also

• Moment-generating function of a random variable

Notes

1. The scale factor is always equal to the period, 2π in this case.

Page citations

1. Oppenheim and Schafer, p 60 (2.169).
2. 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. Horváth, John (1966). Topological Vector Spaces and Distributions. Reading, MA: Addison-Wesley Publishing Company.
3. Barros-Neto, José (1973). An Introduction to the Theory of Distributions. New York, NY: Dekker.
4. Petersen, Bent E. (1983). Introduction to the Fourier Transform and Pseudo-Differential Operators. Boston, MA: Pitman Publishing.
5. 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
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.

7. 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
• Weisstein, Eric W. .html "Convolution Theorem" Check |url= value (help). MathWorld.
• 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