Theorem in mathematics
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
and
with Fourier transforms
and
:
![{\displaystyle {\begin{aligned}U(f)&\triangleq {\mathcal {F}}\{u\}(f)=\int _{-\infty }^{\infty }u(x)e^{-i2\pi fx}\,dx,\quad f\in \mathbb {R} \\V(f)&\triangleq {\mathcal {F}}\{v\}(f)=\int _{-\infty }^{\infty }v(x)e^{-i2\pi fx}\,dx,\quad f\in \mathbb {R} \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/703940743483974572c895a7d66ec110b7defe64)
where
denotes the Fourier transform operator. The transform may be normalized in other ways, in which case constant scaling factors (typically
or
) will appear in the convolution theorem below. The convolution of
and
is defined by:
![{\displaystyle r(x)=\{u*v\}(x)\triangleq \int _{-\infty }^{\infty }u(\tau )v(x-\tau )\,d\tau =\int _{-\infty }^{\infty }u(x-\tau )v(\tau )\,d\tau .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8314d97f6d495649c11749a1926cde4f3ede3ec5)
In this context the asterisk denotes convolution, instead of standard multiplication. The tensor product symbol
is sometimes used instead.
The convolution theorem states that:[1][2]: eq.8
| | (Eq.1a) |
Applying the inverse Fourier transform
produces the corollary:[2]: eqs.7, 10
Convolution theorem | | (Eq.1b) |
The theorem also generally applies to multi-dimensional functions.
Multi-dimensional derivation of Eq.1 |
Consider functions in Lp-space with Fourier transforms : ![{\displaystyle {\begin{aligned}U(f)&\triangleq {\mathcal {F}}\{u\}(f)=\int _{\mathbb {R} ^{n}}u(x)e^{-i2\pi f\cdot x}\,dx,\quad f\in \mathbb {R} ^{n}\\V(f)&\triangleq {\mathcal {F}}\{v\}(f)=\int _{\mathbb {R} ^{n}}v(x)e^{-i2\pi f\cdot x}\,dx,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d907799d3682481d3c4b15170ebd87e6b6d6e72) where indicates the inner product of : and The convolution of and is defined by: ![{\displaystyle r(x)\triangleq \int _{\mathbb {R} ^{n}}u(\tau )v(x-\tau )\,d\tau .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d2104bdd12346d8556f666b969e1e6a2edcb9b7) Also: ![{\displaystyle \iint |u(\tau )v(x-\tau )|\,dx\,d\tau =\int \left(|u(\tau )|\int |v(x-\tau )|\,dx\right)\,d\tau =\int |u(\tau )|\,\|v\|_{1}\,d\tau =\|u\|_{1}\|v\|_{1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e00d7e0eef523b5e1c44bdad869baf506adaf2c) Hence by Fubini's theorem we have that so its Fourier transform is defined by the integral formula: ![{\displaystyle {\begin{aligned}R(f)\triangleq {\mathcal {F}}\{r\}(f)&=\int _{\mathbb {R} ^{n}}r(x)e^{-i2\pi f\cdot x}\,dx\\&=\int _{\mathbb {R} ^{n}}\left(\int _{\mathbb {R} ^{n}}u(\tau )v(x-\tau )\,d\tau \right)\,e^{-i2\pi f\cdot x}\,dx.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/003615289e661249b5ef2e87901e5e6f49e5d3b8) Note that Hence by the argument above we may apply Fubini's theorem again (i.e. interchange the order of integration): ![{\displaystyle {\begin{aligned}R(f)&=\int _{\mathbb {R} ^{n}}u(\tau )\underbrace {\left(\int _{\mathbb {R} ^{n}}v(x-\tau )\ e^{-i2\pi f\cdot x}\,dx\right)} _{V(f)\ e^{-i2\pi f\cdot \tau }}\,d\tau \\&=\underbrace {\left(\int _{\mathbb {R} ^{n}}u(\tau )\ e^{-i2\pi f\cdot \tau }\,d\tau \right)} _{U(f)}\ V(f).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35e9c8b29e77f6461fcaa36ec3db49a846b09641) |
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
-periodic functions
and
which can be expressed as periodic summations:
and ![{\displaystyle v_{_{P}}(x)\ \triangleq \sum _{m=-\infty }^{\infty }v(x-mP).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddc35a9959b087ea6ea897e6ef15b831efe7966f)
In practice the non-zero portion of components
and
are often limited to duration
but nothing in the theorem requires that.
The Fourier series coefficients are:
![{\displaystyle {\begin{aligned}U[k]&\triangleq {\mathcal {F}}\{u_{_{P}}\}[k]={\frac {1}{P}}\int _{P}u_{_{P}}(x)e^{-i2\pi kx/P}\,dx,\quad k\in \mathbb {Z} ;\quad \quad \scriptstyle {\text{integration over any interval of length }}P\\V[k]&\triangleq {\mathcal {F}}\{v_{_{P}}\}[k]={\frac {1}{P}}\int _{P}v_{_{P}}(x)e^{-i2\pi kx/P}\,dx,\quad k\in \mathbb {Z} \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51b732f1fe5d5074f423f1b96c4eea07631284ed)
where
denotes the Fourier series integral.
- The pointwise product:
is also
-periodic, and its Fourier series coefficients are given by the discrete convolution of the
and
sequences:
![{\displaystyle {\mathcal {F}}\{u_{_{P}}\cdot v_{_{P}}\}[k]=\{U*V\}[k].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab30eb678559f102051cedba4be0fc07f346a4c0)
![{\displaystyle {\begin{aligned}\{u_{_{P}}*v\}(x)\ &\triangleq \int _{-\infty }^{\infty }u_{_{P}}(x-\tau )\cdot v(\tau )\ d\tau \\&\equiv \int _{P}u_{_{P}}(x-\tau )\cdot v_{_{P}}(\tau )\ d\tau ;\quad \quad \scriptstyle {\text{integration over any interval of length }}P\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/128f2029c16f9a5db5689ae5a6c29adb6f2d90f1)
is also
-periodic, and is called a periodic convolution.
Derivation of periodic convolution |
![{\displaystyle {\begin{aligned}\int _{-\infty }^{\infty }u_{_{P}}(x-\tau )\cdot v(\tau )\,d\tau &=\sum _{k=-\infty }^{\infty }\left[\int _{x_{o}+kP}^{x_{o}+(k+1)P}u_{_{P}}(x-\tau )\cdot v(\tau )\ d\tau \right]\quad x_{0}{\text{ is an arbitrary parameter}}\\&=\sum _{k=-\infty }^{\infty }\left[\int _{x_{o}}^{x_{o}+P}\underbrace {u_{_{P}}(x-\tau -kP)} _{u_{_{P}}(x-\tau ),{\text{ by periodicity}}}\cdot v(\tau +kP)\ d\tau \right]\quad {\text{substituting }}\tau \rightarrow \tau +kP\\&=\int _{x_{o}}^{x_{o}+P}u_{_{P}}(x-\tau )\cdot \underbrace {\left[\sum _{k=-\infty }^{\infty }v(\tau +kP)\right]} _{\triangleq \ v_{_{P}}(\tau )}\ d\tau \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1575bd74f86366816ffc18d0fabd05dd5cd1269) |
The corresponding convolution theorem is:
| | (Eq.2) |
Derivation of Eq.2 |
![{\displaystyle {\begin{aligned}{\mathcal {F}}\{u_{_{P}}*v\}[k]&\triangleq {\frac {1}{P}}\int _{P}\left(\int _{P}u_{_{P}}(\tau )\cdot v_{_{P}}(x-\tau )\ d\tau \right)e^{-i2\pi kx/P}\,dx\\&=\int _{P}u_{_{P}}(\tau )\left({\frac {1}{P}}\int _{P}v_{_{P}}(x-\tau )\ e^{-i2\pi kx/P}dx\right)\,d\tau \\&=\int _{P}u_{_{P}}(\tau )\ e^{-i2\pi k\tau /P}\underbrace {\left({\frac {1}{P}}\int _{P}v_{_{P}}(x-\tau )\ e^{-i2\pi k(x-\tau )/P}dx\right)} _{V[k],\quad {\text{due to periodicity}}}\,d\tau \\&=\underbrace {\left(\int _{P}\ u_{_{P}}(\tau )\ e^{-i2\pi k\tau /P}d\tau \right)} _{P\cdot U[k]}\ V[k].\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/252a143162a38ecc1ee2c1780ed231edf64ed597) |
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
denotes the discrete-time Fourier transform (DTFT) operator. Consider two sequences
and
with transforms
and
:
![{\displaystyle {\begin{aligned}U(f)&\triangleq {\mathcal {F}}\{u\}(f)=\sum _{n=-\infty }^{\infty }u[n]\cdot e^{-i2\pi fn}\;,\quad f\in \mathbb {R} ,\\V(f)&\triangleq {\mathcal {F}}\{v\}(f)=\sum _{n=-\infty }^{\infty }v[n]\cdot e^{-i2\pi fn}\;,\quad f\in \mathbb {R} .\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/244dd8c93040378c9c8849ab6483c6582445d1b7)
The § Discrete convolution of
and
is defined by:
![{\displaystyle r[n]\triangleq (u*v)[n]=\sum _{m=-\infty }^{\infty }u[m]\cdot v[n-m]=\sum _{m=-\infty }^{\infty }u[n-m]\cdot v[m].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22aac1f7102502f6b39153be6aacd9bce5fc7cdd)
The convolution theorem for discrete sequences is:[3][4]: p.60 (2.169)
| | (Eq.3) |
Periodic convolution
and
as defined above, are periodic, with a period of 1. Consider
-periodic sequences
and
:
and ![{\displaystyle v_{_{N}}[n]\ \triangleq \sum _{m=-\infty }^{\infty }v[n-mN],\quad n\in \mathbb {Z} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62b3cb372f16be04589aa1f446455da9d8bcd8f0)
These functions occur as the result of sampling
and
at intervals of
and performing an inverse discrete Fourier transform (DFT) on
samples (see § Sampling the DTFT). The discrete convolution:
![{\displaystyle \{u_{_{N}}*v\}[n]\ \triangleq \sum _{m=-\infty }^{\infty }u_{_{N}}[m]\cdot v[n-m]\equiv \sum _{m=0}^{N-1}u_{_{N}}[m]\cdot v_{_{N}}[n-m]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/907f89efecfca8b8c1519230c0efe93f721d0ad3)
is also
-periodic, and is called a periodic convolution. Redefining the
operator as the
-length DFT, the corresponding theorem is:[5][4]: p. 548
| | (Eq.4a) |
And therefore:
| | (Eq.4b) |
Under the right conditions, it is possible for this
-length sequence to contain a distortion-free segment of a
convolution. But when the non-zero portion of the
or
sequence is equal or longer than
some distortion is inevitable. Such is the case when the
sequence is obtained by directly sampling the DTFT of the infinitely long § Discrete Hilbert transform impulse response.[A]
For
and
sequences whose non-zero duration is less than or equal to
a final simplification is:
Circular convolution | | (Eq.4c) |
This form is often used to efficiently implement numerical convolution by computer. (see § Fast convolution algorithms and § Example)
As a partial reciprocal, it has been shown [6] that any linear transform that turns convolution into pointwise product is the DFT (up to a permutation of coefficients).
There is also a convolution theorem for the inverse Fourier transform:
Here, "
" represents the Hadamard product, and "
" represents a convolution between the two matrices.
![{\displaystyle {\begin{aligned}&{\mathcal {F}}\{u*v\}={\mathcal {F}}\{u\}\cdot {\mathcal {F}}\{v\}\\&{\mathcal {F}}\{u\cdot v\}={\mathcal {F}}\{u\}*{\mathcal {F}}\{v\}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37ddecd7286fc974d3451bcde4d7ff808ec5a289)
so that
![{\displaystyle {\begin{aligned}&u*v={\mathcal {F}}^{-1}\left\{{\mathcal {F}}\{u\}\cdot {\mathcal {F}}\{v\}\right\}\\&u\cdot v={\mathcal {F}}^{-1}\left\{{\mathcal {F}}\{u\}*{\mathcal {F}}\{v\}\right\}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b190c5722a0514e83731d0e42be4df3eed7b499)
Convolution theorem for tempered distributions
The convolution theorem extends to tempered distributions. Here,
is an arbitrary tempered distribution:
![{\displaystyle {\begin{aligned}&{\mathcal {F}}\{u*v\}={\mathcal {F}}\{u\}\cdot {\mathcal {F}}\{v\}\\&{\mathcal {F}}\{\alpha \cdot v\}={\mathcal {F}}\{\alpha \}*{\mathcal {F}}\{v\}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4daa86d0f58079ef77237bd1a2ca428456deca8)
But
must be "rapidly decreasing" towards
and
in order to guarantee the existence of both, convolution and multiplication product. Equivalently, if
is a smooth "slowly growing" ordinary function, it guarantees the existence of both, multiplication and convolution product.[7][8][9]
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
are smooth "slowly growing" ordinary functions. If, for example,
is the Dirac comb both equations yield the Poisson summation formula and if, furthermore,
is the Dirac delta then
is constantly one and these equations yield the Dirac comb identity.
See also
Notes
- ^ An example is the MATLAB function, hilbert(u,N).
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.
- ^ a b Weisstein, Eric W. "Convolution Theorem". From MathWorld--A Wolfram Web Resource. Retrieved 8 February 2021.
- ^ 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
- ^ a b 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.
- ^ 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.
- ^ Amiot, Emmanuel (2016). Music through Fourier Space. Computational Music Science. Zürich: Springer. p. 8. doi:10.1007/978-3-319-45581-5. ISBN 978-3-319-45581-5. S2CID 6224021.
- ^ 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.
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: