Properties of the Fourier Transform

The property table is the daily working language of imaging, communication, and instrumentation. Magnetic-resonance imaging acquires its data in the frequency domain: the scanner samples $k$-space and the image is the inverse transform, so the translation theorem is the tool radiologists use to register images, and the scaling theorem sets the field-of-view/resolution trade-off. Every 4G, 5G, and Wi-Fi link runs on orthogonal frequency-division multiplexing — the modulation theorem applied thousands of times per symbol, with the FFT doing the transform in silicon. Gravitational-wave observatories dig signals out of noise by matched filtering — convolution against a template — and the convolution theorem turns that expensive integral into a cheap multiplication. The shift theorem's deepest consequence remains a live open problem: in X-ray crystallography and coherent diffractive imaging you measure only the magnitude $|\hat f(k)|$, while all the structure sits in the phase you cannot record. Recovering that lost phase — the phase-retrieval problem — is "position lives in the phase" turned into a decades-long computational challenge, and why every protein structure solved by crystallography is a triumph over the missing phase.

Think of the kernel $e^{-ikx}$ as a meter reporting how much of frequency $k$ a signal contains. Slide the whole signal by $a$ and you have not changed which frequencies are present, only their relative timing — so the magnitude spectrum $|\hat f|$ is untouched and only a phase $e^{-ika}$ appears. All information about where a feature sits is banished into the phase. The first figure makes this literal: a pulse and its shifted copy have identical magnitude spectra, and the entire record of the displacement is a straight-line phase ramp of slope $-a$.

Multiplying the signal by an oscillation $e^{ik_0x}$ does the dual thing: it relabels every frequency by $+k_0$, so the whole spectrum slides to $\hat f(k-k_0)$ — the mirror of translation, and the reason AM radio, heterodyne receivers, and frequency combs all reduce to one theorem in one domain or the other. The deepest property is scaling: squeeze a pulse in time and it must contain higher frequencies over a wider band, since you cannot localize a wave sharply in both domains at once. The reciprocal law $\widehat{f(ax)}=|a|^{-1}\hat f(k/a)$ is the algebraic engine of the uncertainty relation, and the Gaussian is the one shape that trades off optimally, saturating the bound at every width.

Finally, differentiation turns into multiplication by $ik$ for a reason at the heart of linear physics: $\partial_x e^{ikx}=ik\,e^{ikx}$, so the exponentials are eigenfunctions of the derivative. Because every translation-invariant operator is built from derivatives (equivalently, is a convolution), the transform diagonalizes all of them at once — which is why the Fourier basis is the natural coordinate system for constant-coefficient differential equations, filters, and the normal modes of a translationally symmetric system.

Everything follows from the defining pair, stated in the symmetric convention where $1/\sqrt{2\pi}$ appears both ways and the transform is unitary on $L^2$.

where: $f(x)$ — function of the spatial or time variable $x$; $\hat f(k)$ — its transform, a function of angular frequency $k$; $k$ — angular frequency (radians per unit length or time); $1/\sqrt{2\pi}$ — symmetric normalization. Read $\hat f(k)$ as the overlap of $f$ with the probe $e^{ikx}$, and $f$ as a superposition of those probes. Sliding $f$ by a real distance $a$ multiplies the transform by a pure phase; the magnitude is untouched.

where: $a$ — real shift in the spatial domain; $e^{-ika}$ — linear phase factor of unit modulus. Reading: a translation is invisible to $|\hat f|$, appearing only as a phase slope. Its dual: multiplying $f$ by a carrier $e^{ik_0x}$ translates the whole spectrum.

where: $k_0$ — carrier angular frequency; $\hat f(k-k_0)$ — the spectrum shifted by $k_0$. Reading: modulation in $x$ is the exact mirror of translation in $k$. Stretching the argument by $a$ stretches the spectrum reciprocally and rescales its height.

where: $a$ — real nonzero dilation of the argument; $1/|a|$ — amplitude factor. Reading: narrowing in $x$ ($a>1$) widens in $k$ — the uncertainty trade-off. Differentiation becomes multiplication by $ik$; the $n$-th derivative a degree-$n$ polynomial.

where: $f^{(n)}$ — the $n$-th derivative, assumed to decay so boundary terms vanish; $(ik)^n$ — the polynomial multiplier replacing the operator. Reading: calculus in $x$ becomes arithmetic in $k$. The dual rule trades multiplication by $x$ for differentiation in $k$.

where: $-ix$ — multiplication operator in $x$; $\mathrm{d}/\mathrm{d}k$ — differentiation in $k$. Reading: the pair $(x,\partial_x)$ maps to $(i\partial_k,ik)$ — the Fourier fingerprint of the canonical commutator. Convolution and product are dual, with a convention-dependent $\sqrt{2\pi}$.

where: $(f*g)(x)=\int f(y)g(x-y)\,\mathrm{d}y$ — the convolution; $\sqrt{2\pi}$ — artifact of the symmetric convention. Reading: smearing in one domain is a pointwise product in the other. Reality and evenness of $f$ constrain the transform.

where: $\overline{\;\cdot\;}$ — complex conjugation. Reading: real $f$ has a Hermitian spectrum; real-and-even $f$ has a real-and-even spectrum, i.e. a cosine transform. The master duality: transforming twice reflects the argument.

where: $\mathcal F$ — the transform operator; $\mathcal P$ — the parity operator $f(x)\mapsto f(-x)$. Reading: the transform is a fourth root of the identity, so its eigenvalues are $\{1,i,-1,-i\}$ and its eigenbasis is the Hermite functions.

How the table specializes in the key regimes:

1 · A shifted, modulated Gaussian (confidence builder)

From the self-dual pair $f(x)=e^{-x^2/2}\leftrightarrow\hat f(k)=e^{-k^2/2}$, find the transform of $g(x)=e^{i5x}f(x-3)$ and locate where its magnitude peaks.

Step 1. Handle the shift first with the translation theorem $(2)$: the transform of $f(x-3)$ is $e^{-3ik}\hat f(k)=e^{-3ik}e^{-k^2/2}$ — magnitude untouched, only a phase ramp of slope $-3$.

Step 2. Now the modulation theorem $(3)$: multiplying by the carrier $e^{i5x}$ shifts the whole spectrum by $k_0=5$, replacing $k$ by $k-5$ everywhere. Thus $\hat g(k)=e^{-3i(k-5)}\,e^{-(k-5)^2/2}$. Result: $\hat g(k)=e^{-3i(k-5)}e^{-(k-5)^2/2}$.

Step 3. The magnitude $|\hat g(k)|=e^{-(k-5)^2/2}$ is a Gaussian peaking at $k=5$ with value $1$: modulation slid the spectral peak from $0$ to $5$, while the shift contributed only the phase $e^{-3i(k-5)}$, which vanishes at the peak. The spectrum peaks at $k=5$ with $|\hat g(5)|=1$; position lives in the phase, carrier frequency in the peak location.

2 · Rectangular pulse, the sinc, and scaling made numerical

A rectangular pulse $f(x)=1$ for $|x|\le T$ and $0$ otherwise has transform $\hat f(k)=\sqrt{2/\pi}\,\sin(kT)/k$. For $T=1$, find the zero-frequency value and the first spectral zero, then predict what happens when the pulse is halved to $T=\tfrac12$.

Step 1. Zero-frequency value: as $k\to0$, $\sin(kT)/k\to T$, so $\hat f(0)=\sqrt{2/\pi}\,T$ — the pulse area divided by $\sqrt{2\pi}$. For $T=1$ this is $\sqrt{2/\pi}=\mathbf{0.798}$. $\hat f(0)=0.798$.

Step 2. First zero: $\sin(kT)=0$ first at $kT=\pi$, so $k=\pi/T$, giving a main lobe of half-width $\pi/T$. For $T=1$ the spectrum first crosses zero at $k=\pi=\mathbf{3.14}$. First zero at $k=\pi\approx 3.14$.

Step 3. Halve the pulse, $T\to\tfrac12$: this is the scaling law $(4)$ with $a=2$. The first zero moves to $k=\pi/T=2\pi\approx\mathbf{6.28}$, twice as far out, and the amplitude $\hat f(0)$ halves to $0.399$. A pulse squeezed by two in space has a spectrum stretched by two and lowered by two — the reciprocal scaling of the uncertainty trade-off, made arithmetic. The Gaussian version of this trade-off is exact rather than approximate: for $f_\sigma=e^{-x^2/2\sigma^2}$ one finds $\Delta x=\sigma/\sqrt2$ and $\Delta k=1/(\sigma\sqrt2)$, so $\Delta x\,\Delta k=\tfrac12$ at every width (as the second figure confirms for $\sigma=0.5,1.0,1.5$). Halving the pulse doubles the spectral width to $k=2\pi$ and halves the peak; the Gaussian ties the uncertainty bound at every width.

The property table is a portrait of one man's idea outrunning its own rigor. Fourier asserted, to an unconvinced Paris Academy, that any function could be built from sines and cosines — a claim so bold his examiners refused it publication for years. The transform and its algebra are the mature form of that assertion, and it took another century of Dirichlet, Riemann, Plancherel, and finally Schwartz to put every property on a rigorous foundation.

Texts. Stone & Goldbart, Mathematics for Physics, §2.2 · Arfken, Weber & Harris, Mathematical Methods for Physicists, 7th ed., Ch. 20 · Hassani, Mathematical Physics, 2nd ed., Ch. 8 · Byron & Fuller, Mathematics of Classical and Quantum Physics, Ch. 5 · Riley, Hobson & Bence, Mathematical Methods for Physics and Engineering, 3rd ed., §13.1 · Folland, Fourier Analysis and Its Applications, Ch. 7 (rigorous, on Schwartz space).

Provenance. Conventions and property formulas follow Stone & Goldbart §2.2 (symmetric $1/\sqrt{2\pi}$). A verifier should confirm: the self-dual pair $e^{-x^2/2}\leftrightarrow e^{-k^2/2}$ and the symmetric-convention pair $1/(k^2+\lambda^2)\leftrightarrow\sqrt{\pi/2}\,\lambda^{-1}e^{-\lambda|x|}$ (not $\tfrac{1}{2\lambda}e^{-\lambda|x|}$, which is the physicist-convention form); the uncertainty constant $\tfrac12$ with Gaussian saturation $\Delta x\,\Delta k=0.500$ for $\sigma=0.5,1.0,1.5$; the rectangular-pulse values $\hat f(0)=\sqrt{2/\pi}=0.798$ and first zero at $k=\pi$; and $\mathcal F^4=\mathrm{id}$. All $2\pi$ factors are convention-dependent and stated explicitly.

Prerequisites. The Fourier Transform · Fourier Series and Periodic Functions · The Dirac Delta Function. Leads to. The Convolution Theorem · Parseval's and Plancherel's Theorems · Solving Differential Equations by Transform Methods · Tempered Distributions and Fourier Transforms.

Ask the tutor: Ask me to run any signal through the property table step by step — shift it, scale it, modulate it, or feed it to a differential equation — and I will show you the transform emerge one rule at a time.