Fourier Transform

Equivalent forms

f(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty} \hat{f}(k)\, e^{i k x}\, dk

\hat{f}(\nu) = \int_{-\infty}^{\infty} f(t)\, e^{-2\pi i \nu t}\, dt

It diagonalizes differentiation: ∂/∂x ↔ ik. Differential equations become algebra, and the uncertainty principle falls out as a theorem about Δx·Δk.

Symbol	Name	SI unit	Dimension	Range
$\hat{f}$	Spectrumoutput Amplitude of frequency k in the signal	1	1	-2 – 2
$k_0$	Carrier frequency Center frequency of the signal	1/m	L^-1	0 – 20
$\sigma$	Pulse width Spatial width of the wave packet	m	L	0.1 – 4
$k$	Frequency Frequency at which the spectrum is evaluated	1/m	L^-1	-20 – 20

Unit systems

Symbol	SI	CGS	Imperial
$k_0$	1/m	1/cm	1/ft
$\sigma$	m	cm	ft
$k$	1/m	1/cm	1/ft
$\hat{f}$	1	1	1

Where it holds

f absolutely integrable

(L^{1})

or square-integrable

(L^{2})

; extends to tempered distributions

for \delta

, constants, and sinusoids.

Dimensional analysis

f̂

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

\int f e^{-ikx}dx \to [f]\cdot [L]

The integral over dx multiplies f's units by length; the exponent kx is dimensionless since

[k] = 1

/length.

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Discovery

Joseph Fourier · 1822

Extending his heat-equation series to non-periodic functions, Fourier let the period grow without bound, turning the discrete harmonic sum into a continuous integral. The transform now anchors signal processing, quantum mechanics, and crystallography.

Try this

Hand a chord to the Fourier transform and it hands back the sheet music — every signal, split into the exact frequencies hiding inside it.

Slide a signal's frequency and width and watch its spectrum shift and breathe — narrow in time means broad in frequency.

Research status: stable

Real-world applications

Signal and image processing, filtering, compression
Quantum mechanics: position ↔ momentum representations
Crystallography and diffraction (structure from |f̂| $^{2})$
Solving linear PDEs via algebraic transforms

Common misconceptions

The transform of a real signal is real — it is generally complex (magnitude and phase both matter).
A short pulse has a sharp frequency — the opposite: shorter in time means wider in frequency.
Different $2\pi /\sqrt{2\pi }$ conventions change the physics — they only rescale; the content is the same.

Experimental verification

X-ray crystallography literally measures |f̂

(k)|^{2} of

electron density; MRI reconstructs images from sampled k-space; every spectrum analyzer and FFT in a phone validates it daily.

Derivation

Start from the Fourier series of a function on

[-L

, L] with coefficients

c_n = (1/2L)\int f e^{-in\pi x/L}dx

.

Let L\to \infty so

the spacing

\Delta k = \pi /L \to 0

; the

sum \Sigma c_n e^{ikx}

becomes

(1/2\pi )\int f

̂(k)e^{ikx}dk with f̂

(k) = \int f(x)e^{-ikx}dx

.

Limiting cases

f(x) = \delta (x)

⟶ f̂

(k) = 1

A spike in space contains all frequencies equally.

f(x) = 1

(constant)⟶ f̂

(k) = 2\pi \delta (k)

A pure DC signal is all zero-frequency.

Narrow Gaussian (small

\sigma )

⟶ Broad spectrum (large

\Delta k)

Time–frequency uncertainty:

\Delta x\cdot \Delta k \ge 1/2

.

What if…

What if I transform twice?

You get the original function reflected: $F[F[f]](x) = 2\pi f(-x)$ — the transform is almost its own inverse.

What if the signal is sampled, not continuous?

You use the discrete Fourier transform, computed in O(N log N) by the FFT.

1

Transform of a Gaussian

Given ·

f:: $e^{-x^{2}/2\sigma ^{2}}$

Find · f̂(k)

Steps

Complete the square in the exponent $of \int e^{-x^{2}/2\sigma ^{2}}e^{-ikx}dx$
The Gaussian integral gives another Gaussian
Width in k is $1/\sigma$ — narrow packet → broad spectrum

Result · f̂

(k) = \sigma \sqrt{2\pi }\cdot e^{-\sigma ^{2}k^{2}/2}

2

Derivative rule

Given ·

g:: f'(x)

Find · ĝ(k)

Steps

Integrate $\int f'(x)e^{-ikx}dx$ by parts
Boundary terms vanish for decaying f
Each derivative pulls down a factor ik

Result · ĝ

(k) = ik\cdot f

̂(k)

Fourier Series→inverse fourier transformConvolution Theorem→Dirac Delta Function→