Convolution Theorem

Equivalent forms

(f*g)(x) = \int_{-\infty}^{\infty} f(\tau)\,g(x-\tau)\, d\tau

\mathcal{F}\{f\cdot g\} = \frac{1}{2\pi}\,(\hat{f} * \hat{g})

It is why filters, blurs, Green's functions, and probability sums are all 'the same operation' — and why we compute them with an FFT instead of an O(N²) integral.

Symbol	Name	SI unit	Dimension	Range
$(f*g)$	Convolutionoutput Value of the convolution at offset x	1	1	-2 – 2
$a$	Width of f Half-width of the first box pulse	m	L	0.2 – 3
$b$	Width of g Half-width of the second box pulse	m	L	0.2 – 3
$x$	Offset Relative shift between the two pulses	m	L	-6 – 6

Unit systems

Symbol	SI	CGS	Imperial
$a$	m	cm	ft
$b$	m	cm	ft
$x$	m	cm	ft
$(f*g)$	1	1	1

Where it holds

f, g integrable enough for the convolution integral and transforms to exist

(L^{1} or

tempered distributions).

Dimensional analysis

F{f*g} \to [f]\cdot [L]\cdot [g]\cdot [L]

/ ...

f̂

\cdot

ĝ

\to ([f][L])\cdot ([g][L])

Each transform multiplies by a length; the convolution integral over

d\tau

supplies the matching length, so both sides carry

[f][g][L]^{2}

.

\checkmark

Discovery

Joseph Fourier / Vito Volterra (formalized) · 1822

The duality is implicit in Fourier's own work; it became a workhorse once the Fourier transform was rigorized. The Fast Fourier Transform (Cooley–Tukey, 1965) turned the theorem into the fastest way to convolve large signals, powering modern DSP.

Try this

Blurring a photo means smearing every pixel into its neighbors — a painful sum. The convolution theorem says: just multiply two spectra and the blur is done.

Slide two pulses past each other to build their convolution, then watch the same answer fall out of a single multiplication in frequency space.

Research status: stable

Real-world applications

Image blurring/sharpening and convolutional neural networks
Linear time-invariant system response (impulse response $\times$ input)
Fast polynomial and big-integer multiplication via FFT
Probability: the PDF of a sum of independent variables

Common misconceptions

Convolution is multiplication — it's a shift-and-integrate; it only becomes multiplication after a Fourier transform.
Convolution is not commutative — it is: $f * g = g * f$ .
The output is as narrow as the inputs — widths add, so the result is broader.

Experimental verification

Optical systems convolve a scene with the point-spread function; measured blur matches the PSF's transform (the optical transfer function). Audio reverb and seismic deconvolution rely on the theorem quantitatively.

Derivation

Transform the convolution:

\int [\int f(\tau )g(x-\tau )d\tau ]e^{-ikx}dx

.

Swap the order, substitute

u = x - \tau so e^{-ikx} = e^{-ik\tau }e^{-iku}

.

The integrals separate into

[\int f(\tau )e^{-ik\tau }d\tau ][\int g(u)e^{-iku}du] = f

̂(k)ĝ(k).

Limiting cases

g = \delta (\delta )

⟶

f * \delta = f

Convolving with a spike leaves the signal unchanged —

\delta is

the identity.

Two boxes⟶ TriangleEqual boxes convolve to a triangle; repeated convolution tends to a Gaussian (central limit theorem).

In frequency space⟶ Just a productf̂

\cdot

ĝ replaces the shift-and-integrate operation entirely.

What if…

What if I convolve a function with itself many times?

The result approaches a Gaussian — the convolution form of the central limit theorem.

What if I want correlation instead?

Cross-correlation is convolution with a flipped kernel; in frequency space it's f̂ $\cdot$ ĝ* (complex conjugate).

1

Box convolved with box

Given ·

f:: box width 2a
g:: box width 2b

Find · f * g

Steps

Overlap area grows linearly as the boxes slide in
Stays flat while one box sits inside the other
Shrinks linearly as they slide out — a triangle for $a = b$

Result · A trapezoid (triangle if

a = b)

, base 2(a+b), peak

= 2\cdot \min

(a,b)

2

Check via frequencies

Given ·

f̂:: $2a\cdot$ sinc(ka)
ĝ:: $2b\cdot$ sinc(kb)

Find · transform of f * g

Steps

Each box transforms to a sinc
Multiply the two sincs
Inverse-transforming returns the triangle — no integral needed

Result · f̂

\cdot

ĝ

= 4ab\cdot

sinc(ka)sinc(kb)

Fourier Transform→Fourier Series→Dirac Delta Function→Gaussian (Normal) Distribution→