7.2 The Fourier transform

The Fourier series of 7.1 handles functions that repeat with period $T$ . Most signals in real applications don’t repeat — a footstep, a single struck note, a Gaussian pulse, the pressure history of a passing aircraft. For these we need the Fourier transform, which is what the Fourier series becomes when the period goes to infinity.

This lesson develops the transform, surveys its operational properties (linearity, time-shift, scaling, differentiation), tabulates the most-useful pairs, and ends with the uncertainty principle that links the widths of $f$ and $\tilde f$ .

Limit of a Fourier series

Take a function $f(t)$ that is zero outside $[-T/2, T/2]$ — a transient of finite extent. Make a periodic extension of $f$ with period $T$ . As long as $T$ is much larger than the support of $f$ , the periodic extension looks just like $f$ inside one period.

The Fourier series of the periodic extension is

f(t) \;=\; \sum_{n=-\infty}^{\infty} c_n\, e^{i\, 2\pi n t / T}, \qquad c_n \;=\; \frac{1}{T} \int_{-T/2}^{T/2} f(t)\, e^{-i\, 2\pi n t / T}\, dt.

Define the continuous frequency $\omega_n \equiv 2\pi n / T$ . Adjacent values of $\omega_n$ differ by $\Delta \omega = 2\pi / T$ . As $T \to \infty$ , $\Delta \omega \to 0$ and the discrete set $\{\omega_n\}$ fills up the real line.

▶ From series to transform: the T → ∞ limit Derivation

Let $\tilde f(\omega) \equiv T\, c_n$ when $\omega = \omega_n$ . (Take the limit-preserving rescaling: as $T$ grows, the coefficients shrink like $1/T$ , so multiplying by $T$ keeps the quantity finite.)

The coefficient formula becomes

\tilde f(\omega_n) \;=\; \int_{-T/2}^{T/2} f(t)\, e^{-i \omega_n t}\, dt.

In the limit $T \to \infty$ this is the Fourier transform:

\boxed{\;\tilde f(\omega) \;\equiv\; \int_{-\infty}^{\infty} f(t)\, e^{-i \omega t}\, dt.\;}

Conversely, the series formula

f(t) \;=\; \sum_n c_n\, e^{i \omega_n t} \;=\; \sum_n \tilde f(\omega_n)\, e^{i \omega_n t}\, \frac{\Delta \omega}{2\pi}

(using $1/T = \Delta \omega / (2\pi)$ ) becomes, in the limit, a Riemann sum that converges to the inverse Fourier transform:

\boxed{\;f(t) \;=\; \frac{1}{2\pi} \int_{-\infty}^{\infty} \tilde f(\omega)\, e^{i \omega t}\, d\omega.\;}

The discrete sum over harmonics has become a continuous integral over frequency. The period $T$ has dropped out; $\omega$ is now a continuous variable, not a discrete set of harmonics.

The factor of $1/(2\pi)$ on the inverse transform is a convention. Different textbooks split it differently:

Physics convention (the one we use): $\tilde f = \int f\, e^{-i\omega t}\, dt$ , $f = \tfrac{1}{2\pi} \int \tilde f\, e^{i\omega t}\, d\omega$ .
Symmetric convention: $1/\sqrt{2\pi}$ on each side.
Engineering convention with linear frequency: $\tilde f(\nu) = \int f\, e^{-i 2\pi \nu t}\, dt$ , with no $2\pi$ anywhere. Here $\nu = \omega / (2\pi)$ is frequency in Hz.

All three are mathematically equivalent; they shuffle factors of $2\pi$ around. Pick one and stick with it. When reading a textbook or paper, always check which convention it uses before applying any formula.

A function and its transform

f(t):

width Δt = 1.00 carrier ω₀ = 0.00

Time width Δt ≈ 1.00 · Frequency width Δω ≈ 1.00 · product Δt·Δω ≈ 1.00 (uncertainty principle: Δt·Δω ≥ ½).

Pick a function family — Gaussian, rectangle, windowed cosine, one-sided exponential — slide the width parameter, and watch $f(t)$ and $|\tilde f(\omega)|$ negotiate. The reciprocal-width relationship is universal: making $f$ narrower in time makes $\tilde f$ broader in frequency, and vice versa. The Gaussian is the unique (up to scaling) function that’s its own Fourier transform — the “fixed point” of the transform operator, and the function that saturates the uncertainty bound.

Operational properties

The transform’s algebraic properties are what make it indispensable for solving linear differential equations. A short list (proofs are routine substitutions):

| Property | Time-domain | Frequency-domain | |---|---|---| | Linearity | $a f + b g$ | $a \tilde f + b \tilde g$ | | Time shift | $f(t - t_0)$ | $e^{-i \omega t_0}\, \tilde f(\omega)$ | | Frequency shift / modulation | $e^{i \omega_0 t}\, f(t)$ | $\tilde f(\omega - \omega_0)$ | | Time scaling | $f(a t)$ | $\frac{1}{\lvert a \rvert}\, \tilde f(\omega / a)$ | | Differentiation | $\dot f(t)$ | $i \omega\, \tilde f(\omega)$ | | Higher derivative | $f^{(n)}(t)$ | $(i \omega)^n\, \tilde f(\omega)$ | | Multiplication by $t$ | $t f(t)$ | $i\, d\tilde f / d\omega$ | | Complex conjugation | $f^*(t)$ | $\tilde f^*(-\omega)$ |

The differentiation property is the cash value of the entire transform for differential equations. A constant-coefficient linear ODE in time becomes a polynomial equation in $\omega$ . A constant-coefficient linear PDE in $(x, t)$ becomes an algebraic equation in $(k, \omega)$ . The transform converts calculus into algebra.

Useful Fourier-transform pairs

| $f(t)$ | $\tilde f(\omega)$ | |---|---| | Rectangular pulse, width $T$ centred at 0 | $T\, \mathrm{sinc}(\omega T / 2)$ | | Gaussian, $e^{-t^2 / 2 \sigma^2}$ | $\sigma \sqrt{2\pi}\, e^{-\sigma^2 \omega^2 / 2}$ | | One-sided exponential decay, $e^{-\alpha t}\, \theta(t)$ | $1 / (\alpha + i \omega)$ | | Two-sided exponential, $e^{-\alpha \lvert t \rvert}$ | $2\alpha / (\alpha^2 + \omega^2)$ — a Lorentzian | | Pure sinusoid, $\cos(\omega_0 t)$ | $\pi\, [\delta(\omega - \omega_0) + \delta(\omega + \omega_0)]$ | | Pure sinusoid, $\sin(\omega_0 t)$ | $-i\pi\, [\delta(\omega - \omega_0) - \delta(\omega + \omega_0)]$ | | Delta function, $\delta(t)$ | $1$ — flat spectrum | | Constant, $1$ | $2\pi\, \delta(\omega)$ — pure DC | | Sign function, $\mathrm{sgn}(t)$ | $2 / (i \omega)$ | | Heaviside step, $\theta(t)$ | $\pi \delta(\omega) + 1/(i\omega)$ |

A few observations worth pulling out:

The Gaussian is the only one of its own Fourier transform (the symmetric convention with $1/\sqrt{2\pi}$ makes this exact).
A narrower time signal has a wider spectrum, as the scaling property predicts. A delta function (infinitely narrow in time) has an infinitely wide (flat) spectrum. A constant (infinitely wide) has a delta-function (infinitely narrow) spectrum.
Pure sinusoids transform to delta functions in frequency — the cleanest possible spectrum.

▶ Fourier transform of a decaying exponential (damped oscillation) Worked Example

Let $f(t) = e^{-\alpha t}\,\theta(t)$ with $\alpha = 3\,\text{s}^{-1}$ (a sound that decays with time constant $1/3$ s). Compute $\tilde f(\omega)$ :

\tilde f(\omega) = \int_0^\infty e^{-\alpha t}\, e^{-i\omega t}\, dt = \int_0^\infty e^{-(\alpha + i\omega)t}\, dt = \frac{1}{\alpha + i\omega}.

The magnitude spectrum is $|\tilde f(\omega)| = 1/\sqrt{\alpha^2 + \omega^2} = 1/\sqrt{9 + \omega^2}$ . At $\omega = 0$ : $|\tilde f| = 1/3$ . At $\omega = 3$ : $|\tilde f| = 1/(3\sqrt{2}) \approx 0.236$ , half-power point. The bandwidth (half-power) is $\Delta\omega = \alpha = 3\,\text{rad/s}$ , confirming that faster decay $\Rightarrow$ broader spectrum.

▶ Fourier transform of a Gaussian pulse Worked Example

Let $f(t) = e^{-t^2/(2\sigma^2)}$ with $\sigma = 2\,\text{ms}$ (a brief acoustic click). Using the Gaussian FT pair:

\tilde f(\omega) = \sigma\sqrt{2\pi}\, e^{-\sigma^2 \omega^2/2} = 2\sqrt{2\pi}\times 10^{-3}\, e^{-2\times10^{-6}\,\omega^2}.

The spectral width (where the amplitude falls to $1/e$ ) is $\Delta\omega = 1/\sigma = 500\,\text{rad/s}$ , i.e. $\Delta f \approx 80\,\text{Hz}$ . Doubling $\sigma$ to 4 ms halves the bandwidth to $\sim 40\,\text{Hz}$ — the uncertainty relation at work.

The uncertainty principle

A function and its Fourier transform cannot both be sharply localised. With suitable definitions of widths $\Delta t$ and $\Delta \omega$ (typically the standard deviations of $|f|^2$ and $|\tilde f|^2$ ),

\boxed{\;\Delta t \cdot \Delta \omega \;\geq\; \frac{1}{2}.\;}

Equality holds only for Gaussians; every other function strictly exceeds it. The proof is a Cauchy–Schwarz argument on the inner product of $t f(t)$ with $\dot f(t)$ .

This is the same inequality as Heisenberg’s $\Delta x \cdot \Delta p \geq \hbar / 2$ in quantum mechanics, via $p = \hbar k$ and momentum-eigenstates-are-plane-waves. The uncertainty principle is not a fact about quantum physics; it is a fact about the Fourier transform applied to a function-space inner product. Quantum mechanics inherits it because its states live in such a space.

The practical consequence for acoustics: a short pulse (small $\Delta t$ ) has a wide spectrum (large $\Delta \omega$ ). A sharply defined pitch (small $\Delta \omega$ ) requires listening over a long window (large $\Delta t$ ). This is the central design tension of spectrogram analysis (Sound 8.2) and is what limits how precisely a click and a pitch can simultaneously be discerned.

What we use this for

The transform is the workhorse of frequency-domain physics:

Solving linear PDEs by transform (Foundations 6.7 — Helmholtz) — the spatial Fourier transform converts $\partial_x$ into $ik$ , reducing the PDE to algebra.
The wave equation in space is what gives plane-wave solutions $e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)}$ with $\omega = c |\mathbf{k}|$ (Sound 5.1 — Plane harmonic waves).
Sound spectra and timbre — Sound 8.1.
Diffraction patterns are the Fourier transform of the aperture function — Sound 7.5.
Cochlear frequency analysis — the ear is, to leading order, a continuous-frequency spectrum analyser — Hearing Ch 4.

The next lesson, 7.3 — Convolution and Parseval, develops the two algebraic identities that turn the Fourier transform from a calculation tool into the foundation of linear-systems theory.