Convolution theorem

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 ${\displaystyle \mathcal{F}}$ denotes the discrete-time Fourier transform (DTFT) operator. Consider two sequences ${\displaystyle u[n]}$ and ${\displaystyle v[n]}$ with transforms ${\displaystyle U}$ and ${\displaystyle V}$:

$${\displaystyle \begin{array}{cc}U(f)& \triangleq \mathcal{F}\{u\}(f)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}u[n]\cdot e^{-i2\pi fn},f\in \mathbb{R},\\ V(f)& \triangleq \mathcal{F}\{v\}(f)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}v[n]\cdot e^{-i2\pi fn},f\in \mathbb{R}.\end{array}}$$

The § Discrete convolution of ${\displaystyle u}$ and ${\displaystyle v}$ is defined by:

$${\displaystyle r[n]\triangleq (u\ast v)[n]=\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}u[m]\cdot v[n-m]=\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}u[n-m]\cdot v[m].}$$

The convolution theorem for discrete sequences is:

${\displaystyle R(f)=\mathcal{F}\{u\ast v\}(f)=U(f)V(f).}$ ()

Periodic convolution

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

$${\displaystyle u_{{}_N}[n]\triangleq \sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}u[n-mN]}$$ and $${\displaystyle v_{{}_N}[n]\triangleq \sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}v[n-mN],n\in \mathbb{Z}.}$$

These functions occur as the result of sampling ${\displaystyle U}$ and ${\displaystyle V}$ at intervals of ${\displaystyle 1/N}$ and performing an inverse discrete Fourier transform (DFT) on ${\displaystyle N}$ samples (see § Sampling the DTFT). The discrete convolution:

$${\displaystyle \{u_{{}_N}\ast v\}[n]\triangleq \sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{u}_{{}_N}[m]\cdot v[n-m]\equiv \sum _{m=0}^{N-1}{u}_{{}_N}[m]\cdot v_{{}_N}[n-m]}$$

is also ${\displaystyle N}$-periodic, and is called a periodic convolution. Redefining the ${\displaystyle \mathcal{F}}$ operator as the ${\displaystyle N}$-length DFT, the corresponding theorem is:

${\displaystyle \mathcal{F}\{u_{{}_N}\ast v\}[k]=\underset{U(k/N)}{\underbrace{\mathcal{F}\{u_{{}_N}\}[k]}}\cdot \underset{V(k/N)}{\underbrace{\mathcal{F}\{v_{{}_N}\}[k]}},k\in \mathbb{Z}.}$ ()

And therefore:

${\displaystyle \{u_{{}_N}\ast v\}[n]={\mathcal{F}}^{-1}\{\mathcal{F}\{u_{{}_N}\}\cdot \mathcal{F}\{v_{{}_N}\}\}.}$ ()

Under the right conditions, it is possible for this ${\displaystyle N}$-length sequence to contain a distortion-free segment of a ${\displaystyle u\ast v}$ convolution. But when the non-zero portion of the ${\displaystyle u(n)}$ or ${\displaystyle v(n)}$ sequence is equal or longer than ${\displaystyle N,}$ some distortion is inevitable. Such is the case when the ${\displaystyle V(k/N)}$ sequence is obtained by directly sampling the DTFT of the infinitely long § Discrete Hilbert transform impulse response.

For ${\displaystyle u}$ and ${\displaystyle v}$ sequences whose non-zero duration is less than or equal to ${\displaystyle N,}$ a final simplification is:

Circular convolution

${\displaystyle \{u_{{}_N}\ast v\}[n]={\mathcal{F}}^{-1}\{\mathcal{F}\{u\}\cdot \mathcal{F}\{v\}\}.}$ ()

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 that any linear transform that turns convolution into a product is the DFT (up to a permutation of coefficients).

Convolution theorem for inverse Fourier transform

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

Here, "${\displaystyle \cdot }$" represents the Hadamard product, and "${\displaystyle \ast }$" represents a convolution between the two matrices.

$${\displaystyle \begin{array}{cc}& \mathcal{F}\{u\ast v\}=\mathcal{F}\{u\}\cdot \mathcal{F}\{v\}\\ & \mathcal{F}\{u\cdot v\}=\mathcal{F}\{u\}\ast \mathcal{F}\{v\}\end{array}}$$

so that

$${\displaystyle \begin{array}{cc}& u\ast v={\mathcal{F}}^{-1}\left\{\mathcal{F}\{u\}\cdot \mathcal{F}\{v\}\right\}\\ & u\cdot v={\mathcal{F}}^{-1}\left\{\mathcal{F}\{u\}\ast \mathcal{F}\{v\}\right\}\end{array}}$$

Convolution theorem for tempered distributions

The convolution theorem extends to tempered distributions. Here, ${\displaystyle v}$ is an arbitrary tempered distribution:

$${\displaystyle \begin{array}{cc}& \mathcal{F}\{u\ast v\}=\mathcal{F}\{u\}\cdot \mathcal{F}\{v\}\\ & \mathcal{F}\{u\cdot v\}=\mathcal{F}\{u\}\ast \mathcal{F}\{v\}.\end{array}}$$

But ${\displaystyle \alpha =F\{u\}}$ must be "rapidly decreasing" towards ${\displaystyle -\mathrm{\infty }}$ and ${\displaystyle +\mathrm{\infty }}$ in order to guarantee the existence of both, convolution and multiplication product. Equivalently, if ${\displaystyle u=F^{-1}\{\alpha \}}$ 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 ${\displaystyle 1}$ are smooth "slowly growing" ordinary functions. If, for example, ${\displaystyle v\equiv \text{Ш}}$ is the Dirac comb both equations yield the Poisson summation formula and if, furthermore, ${\displaystyle u\equiv \delta }$ is the Dirac delta then ${\displaystyle \alpha \equiv 1}$ is constantly one and these equations yield the Dirac comb identity.

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