Convolution theorem

Template:Short description In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the 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 variableEdit

Consider two functions <math>u(x)</math> and <math>v(x)</math> with Fourier transforms <math>U</math> and <math>V</math>:

<math>\begin{align}

U(f) &\triangleq \mathcal{F}\{u\}(f) = \int_{-\infty}^{\infty}u(x) e^{-i 2 \pi f x} \, dx, \quad f \in \mathbb{R}\\ V(f) &\triangleq \mathcal{F}\{v\}(f) = \int_{-\infty}^{\infty}v(x) e^{-i 2 \pi f x} \, dx, \quad f \in \mathbb{R} \end{align}</math>

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

<math>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.</math>

In this context the asterisk denotes convolution, instead of standard multiplication. The tensor product symbol <math>\otimes</math> is sometimes used instead.

The convolution theorem states that:<ref name=McGillem/><ref name=Weisstein/>Template:Rp

Template:Equation box 1

Applying the inverse Fourier transform <math>\mathcal{F}^{-1},</math> produces the corollary:<ref name=Weisstein/>Template:Rp

Template:Equation box 1

The theorem also generally applies to multi-dimensional functions.

Template:Collapse top Consider functions <math>u,v</math> in L^p-space <math>L^1(\mathbb{R}^n),</math> with Fourier transforms <math>U,V</math>:

<math>

\begin{align} U(f) &\triangleq \mathcal{F}\{u\}(f) = \int_{\mathbb{R}^n} u(x) e^{-i 2 \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^{-i 2 \pi f \cdot x} \, dx, \end{align} </math>

where <math>f\cdot x</math> indicates the inner product of <math>\mathbb{R}^n</math>: <math>f\cdot x = \sum_{j=1}^{n} {f}_j x_j,</math> and <math>dx = \prod_{j=1}^{n} d x_j.</math>

The convolution of <math>u</math> and <math>v</math> is defined by:

<math>r(x) \triangleq \int_{\mathbb{R}^n} u(\tau) v(x-\tau)\, d\tau.</math>

Also:

Hence by Fubini's theorem we have that <math>r\in L^1(\mathbb{R}^n)</math> so its Fourier transform <math>R</math> is defined by the integral formula:

<math>

\begin{align} R(f) \triangleq \mathcal{F}\{r\}(f) &= \int_{\mathbb{R}^n} r(x) e^{-i 2 \pi f \cdot x}\, dx\\ &= \int_{\mathbb{R}^n} \left(\int_{\mathbb{R}^n} u(\tau) v(x-\tau)\, d\tau\right)\, e^{-i 2 \pi f \cdot x}\, dx. \end{align} </math>

Note that <math>|u(\tau)v(x-\tau)e^{-i 2\pi f \cdot x}|=|u(\tau)v(x-\tau)|,</math> Hence by the argument above we may apply Fubini's theorem again (i.e. interchange the order of integration):

<math>

\begin{align} R(f) &= \int_{\mathbb{R}^n} u(\tau) \underbrace{\left(\int_{\mathbb{R}^n} v(x-\tau)\ e^{-i 2 \pi f \cdot x}\,dx\right)}_{V(f)\ e^{-i 2 \pi f \cdot \tau}}\,d\tau\\ &=\underbrace{\left(\int_{\mathbb{R}^n} u(\tau)\ e^{-i 2\pi f \cdot \tau}\,d\tau\right)}_{U(f)}\ V(f). \end{align} </math> Template:Collapse bottom

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)Edit

Consider <math>P</math>-periodic functions <math>u_{_P}</math> and <math>v_{_P},</math> which can be expressed as periodic summations:

<math>u_{_P}(x)\ \triangleq \sum_{m=-\infty}^{\infty} u(x-mP)</math> and <math>v_{_P}(x)\ \triangleq \sum_{m=-\infty}^{\infty} v(x-mP).</math>

In practice the non-zero portion of components <math>u</math> and <math>v</math> are often limited to duration <math>P,</math> but nothing in the theorem requires that.

The Fourier series coefficients are:

<math>\begin{align}

  U[k] &\triangleq \mathcal{F}\{u_{_P}\}[k] = \frac{1}{P} \int_P u_{_P}(x) e^{-i 2\pi k x/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^{-i 2\pi k x/P} \, dx, \quad k \in \mathbb{Z}
\end{align}</math>

where <math>\mathcal{F}</math> denotes the Fourier series integral.

The product: <math>u_{_P}(x)\cdot v_{_P}(x)</math> is also <math>P</math>-periodic, and its Fourier series coefficients are given by the discrete convolution of the <math>U</math> and <math>V</math> sequences:

<math>\mathcal{F}\{u_{_P}\cdot v_{_P}\}[k] = \{U*V\}[k].</math>

The convolution:

<math>\begin{align}

\{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{align}</math>

is also <math>P</math>-periodic, and is called a periodic convolution.

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<math>\begin{align}

\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{align}</math> Template:Collapse bottom

The corresponding convolution theorem is:

Template:Equation box 1

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<math>\begin{align}

\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^{-i 2\pi k x/P} \, dx\\ &= \int_P u_{_P}(\tau)\left(\frac{1}{P}\int_P v_{_P}(x-\tau)\ e^{-i 2\pi k x/P} dx\right) \, d\tau\\ &= \int_P u_{_P}(\tau)\ e^{-i 2\pi k \tau/P} \underbrace{\left(\frac{1}{P}\int_P v_{_P}(x-\tau)\ e^{-i 2\pi k (x-\tau)/P} dx\right)}_{V[k], \quad \text{due to periodicity}} \, d\tau\\ &=\underbrace{\left(\int_P\ u_{_P}(\tau)\ e^{-i 2\pi k \tau/P} d\tau\right)}_{P\cdot U[k]}\ V[k]. \end{align}</math> Template:Collapse bottom

Functions of a discrete variable (sequences)Edit

By a derivation similar to Eq.1, there is an analogous theorem for sequences, such as samples of two continuous functions, where now <math>\mathcal{F}</math> denotes the discrete-time Fourier transform (DTFT) operator. Consider two sequences <math>u[n]</math> and <math>v[n]</math> with transforms <math>U</math> and <math>V</math>:

<math>\begin{align}

U(f) &\triangleq \mathcal{F}\{u\}(f) = \sum_{n=-\infty}^{\infty} u[n]\cdot e^{-i 2\pi f n}\;, \quad f \in \mathbb{R}, \\ V(f) &\triangleq \mathcal{F}\{v\}(f) = \sum_{n=-\infty}^{\infty} v[n]\cdot e^{-i 2\pi f n}\;, \quad f \in \mathbb{R}. \end{align}</math>

The Template:Slink of <math>u</math> and <math>v</math> is defined by:

<math>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].</math>

The convolution theorem for discrete sequences is:<ref name=Proakis/><ref name=Oppenheim/>Template:Rp

Template:Equation box 1

Periodic convolutionEdit

<math>U(f)</math> and <math>V(f),</math> as defined above, are periodic, with a period of 1. Consider <math>N</math>-periodic sequences <math>u_{_N}</math> and <math>v_{_N}</math>:

<math>u_{_N}[n]\ \triangleq \sum_{m=-\infty}^{\infty} u[n-mN]</math> and <math>v_{_N}[n]\ \triangleq \sum_{m=-\infty}^{\infty} v[n-mN], \quad n \in \mathbb{Z}.</math>

These functions occur as the result of sampling <math>U</math> and <math>V</math> at intervals of <math>1/N</math> and performing an inverse discrete Fourier transform (DFT) on <math>N</math> samples (see Template:Slink). The discrete convolution:

<math>\{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]</math>

is also <math>N</math>-periodic, and is called a periodic convolution. Redefining the <math>\mathcal{F}</math> operator as the <math>N</math>-length DFT, the corresponding theorem is:<ref name="Rabiner" /><ref name="Oppenheim" />Template:Rp

Template:Equation box 1

And therefore:

Template:Equation box 1

Under the right conditions, it is possible for this <math>N</math>-length sequence to contain a distortion-free segment of a <math>u*v</math> convolution. But when the non-zero portion of the <math>u(n)</math> or <math>v(n)</math> sequence is equal or longer than <math>N,</math> some distortion is inevitable. Such is the case when the <math>V(k/N)</math> sequence is obtained by directly sampling the DTFT of the infinitely long Template:Slink impulse response.Template:Efn-ua

For <math>u</math> and <math>v</math> sequences whose non-zero duration is less than or equal to <math>N,</math> a final simplification is: Template:Equation box 1

This form is often used to efficiently implement numerical convolution by computer. (see Template:Slink and Template:Slink)

As a partial reciprocal, it has been shown <ref>Template:Cite book</ref> that any linear transform that turns convolution into a product is the DFT (up to a permutation of coefficients).

Template:Collapse top A time-domain derivation proceeds as follows:

<math>

\begin{align} \scriptstyle{\rm DFT}\displaystyle \{u_{_N} * v\}[k] &\triangleq \sum_{n=0}^{N-1} \left(\sum_{m=0}^{N-1} u_{_N}[m]\cdot v_{_N}[n-m]\right) e^{-i 2\pi kn/N}\\ &= \sum_{m=0}^{N-1} u_{_N}[m] \left(\sum_{n=0}^{N-1} v_{_N}[n-m]\cdot e^{-i 2\pi kn/N}\right)\\ &= \sum_{m=0}^{N-1} u_{_N}[m]\cdot e^{-i 2\pi km/N} \underbrace{ \left(\sum_{n=0}^{N-1} v_{_N}[n-m]\cdot e^{-i 2\pi k(n-m)/N}\right)}_{\scriptstyle{\rm DFT}\displaystyle\{v_{_N}\}[k]\quad \scriptstyle \text{due to periodicity}}\\ &= \underbrace{ \left(\sum_{m=0}^{N-1} u_{_N}[m]\cdot e^{-i 2\pi km/N}\right)}_{\scriptstyle{\rm DFT}\displaystyle\{u_{_N}\}[k]} \left(\scriptstyle{\rm DFT}\displaystyle\{v_{_N}\}[k]\right). \end{align} </math>

A frequency-domain derivation follows from Template:Slink, which indicates that the DTFTs can be written as:

<math>

\mathcal{F}\{u_{_N} * v\}(f) = \frac{1}{N} \sum_{k=-\infty}^{\infty} \left(\scriptstyle{\rm DFT}\displaystyle \{u_{_N} * v\}[k]\right)\cdot \delta\left(f-k/N\right). \quad \scriptstyle \mathsf{(Eq.5a)} </math>

<math>

\mathcal{F}\{u_{_N}\}(f) = \frac{1}{N} \sum_{k=-\infty}^{\infty} \left(\scriptstyle{\rm DFT}\displaystyle\{u_{_N}\}[k]\right)\cdot \delta\left(f-k/N\right). </math>

The product with <math>V(f)</math> is thereby reduced to a discrete-frequency function:

<math>

\begin{align} \mathcal{F}\{u_{_N} * v\}(f) &= G_{_N}(f) V(f) \\ &= \frac{1}{N} \sum_{k=-\infty}^{\infty} \left(\scriptstyle{\rm DFT}\displaystyle\{u_{_N}\}[k]\right)\cdot V(f)\cdot \delta\left(f-k/N\right)\\ &= \frac{1}{N} \sum_{k=-\infty}^{\infty} \left(\scriptstyle{\rm DFT}\displaystyle\{u_{_N}\}[k]\right)\cdot V(k/N)\cdot \delta\left(f-k/N\right)\\ &= \frac{1}{N} \sum_{k=-\infty}^{\infty} \left(\scriptstyle{\rm DFT}\displaystyle\{u_{_N}\}[k]\right)\cdot \left(\scriptstyle{\rm DFT}\displaystyle\{v_{_N}\}[k]\right) \cdot \delta\left(f-k/N\right), \quad \scriptstyle \mathsf{(Eq.5b)} \end{align} </math>

where the equivalence of <math>V(k/N)</math> and <math>\left(\scriptstyle{\rm DFT}\displaystyle\{v_{_N}\}[k]\right)</math> follows from Template:Slink. Therefore, the equivalence of (5a) and (5b) requires:

<math>\scriptstyle{\rm DFT}

\displaystyle {\{u_{_N} * v\}[k]}

= \left(\scriptstyle{\rm DFT}

\displaystyle\{u_{_N}\}[k]\right)\cdot \left(\scriptstyle{\rm DFT}\displaystyle\{v_{_N}\}[k]\right).</math>

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

<math>

\begin{align} (u_{_N} * v)[n] & = \int_{0}^{1} \left(\frac{1}{N} \sum_{k=-\infty}^{\infty} \scriptstyle{\rm DFT}\displaystyle\{u_{_N}\}[k]\cdot \scriptstyle{\rm DFT}\displaystyle\{v_{_N}\}[k]\cdot \delta\left(f-k/N\right)\right)\cdot e^{i 2 \pi f n} df \\ & = \frac{1}{N} \sum_{k=-\infty}^{\infty} \scriptstyle{\rm DFT}\displaystyle\{u_{_N}\}[k]\cdot \scriptstyle{\rm DFT}\displaystyle\{v_{_N}\}[k]\cdot \underbrace{\left(\int_{0}^{1} \delta\left(f-k/N\right)\cdot e^{i 2 \pi f n} df\right)}_{\text{0, for} \ k\ \notin\ [0,\ N)} \\ & = \frac{1}{N} \sum_{k=0}^{N-1} \bigg(\scriptstyle{\rm DFT}\displaystyle\{u_{_N}\}[k]\cdot \scriptstyle{\rm DFT}\displaystyle\{v_{_N}\}[k]\bigg)\cdot e^{i 2 \pi \frac{n}{N} k}\\ &=\ \scriptstyle{\rm DFT}^{-1} \displaystyle \bigg( \scriptstyle{\rm DFT}\displaystyle \{u_{_N}\}\cdot \scriptstyle{\rm DFT}\displaystyle \{v_{_N}\} \bigg). \end{align} </math> Template:Collapse bottom

Convolution theorem for inverse Fourier transformEdit

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

Here, "<math>\cdot</math>" represents the Hadamard product, and "<math>*</math>" represents a convolution between the two matrices.

<math>\begin{align}

&\mathcal{F}\{u*v\} = \mathcal{F}\{u\} \cdot \mathcal{F}\{v\}\\ &\mathcal{F}\{u \cdot v\}= \mathcal{F}\{u\}*\mathcal{F}\{v\} \end{align}</math>

so that

<math>\begin{align}

&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{align}</math>

Convolution theorem for tempered distributionsEdit

The convolution theorem extends to tempered distributions. Here, <math>v</math> is an arbitrary tempered distribution:

<math>\begin{align}

&\mathcal{F}\{u*v\} = \mathcal{F}\{u\} \cdot \mathcal{F}\{v\}\\ &\mathcal{F}\{\alpha \cdot v\}= \mathcal{F}\{\alpha\}*\mathcal{F}\{v\}. \end{align}</math>

But <math>u = F\{\alpha\}</math> must be "rapidly decreasing" towards <math>-\infty</math> and <math>+\infty</math> in order to guarantee the existence of both, convolution and multiplication product. Equivalently, if <math>\alpha = F^{-1}\{u\}</math> is a smooth "slowly growing" ordinary function, it guarantees the existence of both, multiplication and convolution product.<ref>Template:Cite book</ref><ref>Template:Cite book</ref><ref>Template:Cite book</ref>

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 <math>1</math> are smooth "slowly growing" ordinary functions. If, for example, <math>v\equiv\operatorname{\text{Ш}}</math> is the Dirac comb both equations yield the Poisson summation formula and if, furthermore, <math>u\equiv\delta</math> is the Dirac delta then <math>\alpha \equiv 1</math> is constantly one and these equations yield the Dirac comb identity.