Convolution theorem

In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution is the pointwise product of 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 $\ f$ and $\ g$ be two functions with convolution $\ f*g$. (Note that the asterisk denotes convolution in this context, and not multiplication. The tensor product symbol $\otimes$ is sometimes used instead.) Let $\ \mathcal{F}$ denote the Fourier transform operator, so $\ \mathcal{F}\{f\}$ and $\ \mathcal{F}\{g\}$ are the Fourier transforms of $\ f$ and $\ g$, respectively. Then

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

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

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

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

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

Note that the relationships above are only valid for the form of the Fourier transform shown in the Proof section below. The transform may be normalised in other ways, in which case constant scaling factors (typically $\ 2\pi$ or $\ \sqrt{2\pi}$) 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 to O(n log n). This can be exploited to construct fast multiplication algorithms.

Proof

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

Let f, g belong to L¹(Rⁿ). Let F be the Fourier transform of f and G be the Fourier transform of g:

$F(\nu) = \mathcal{F}\{f\} = \int_{\mathbb{R}^n} f(x) e^{-2 \pi i x\cdot\nu} \, \mathrm{d}x$

$G(\nu) = \mathcal{F}\{g\} = \int_{\mathbb{R}^n}g(x) e^{-2 \pi i x\cdot\nu} \, \mathrm{d}x,$

where the dot between x and ν indicates the inner product of Rⁿ. Let h be the convolution of f and g

$h(z) = \int\limits_{\mathbb{R}^n} f(x) g(z-x)\, \mathrm{d} x.$

Now notice that

$\int\!\!\int |f(z)g(x-z)|\,dx\,dz=\int |f(z)| \int |g(z-x)|\,dx\,dz = \int |f(z)|\,\|g\|_1\,dz=\|f\|_1 \|g\|_1.$

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

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

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

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

Substitute y = z − x; then dy = dz, so:

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

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

$=\int_{\mathbb{R}^n} f(x)e^{-2\pi i x\cdot \nu}\,dx \int_{\mathbb{R}^n} g(y) e^{-2 \pi i y\cdot\nu}\,dy.$

These two integrals are the definitions of F(ν) and G(ν), so:

$H(\nu) = F(\nu) \cdot G(\nu),$

QED.

The proof is trivial in linear algebra where convolution is represented by an infinite-dimensional Toeplitz matrix, h, which are known to have the Fourier eigenbasis, F. This means that h can be represented by a diagonal one, H = (F h F^-1),

$\overrightarrow y = h \overrightarrow x = (F^{-1} F) h (F^{-1} F) \overrightarrow x = F^{-1} (F h F^{-1}) (F \overrightarrow x) = F^{-1} (H \overrightarrow X)$

or

$F(h \overrightarrow x) = H \overrightarrow X.$

References

• Katznelson, Yitzhak (1976), An introduction to Harmonic Analysis, Dover, ISBN 0-486-63331-4 
• Weisstein, Eric. http://mathworld.wolfram.com/ConvolutionTheorem.html Convolution Theorem, MathWorld, November 19, 2010. Retrieved on November 19, 2010.
• Crutchfield, Steve (October 9, 2010), "The Joy of Convolution", Johns Hopkins University, http://www.jhu.edu/signals/convolve/index.html, retrieved November 19, 2010 

Additional resources

• For 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