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Unit 4 · Lesson 4-1
The Fourier Transform
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The Fourier Transform

From series to transform, the rectangular pulse and its sinc, exponentials and Gaussians, magnitude and phase, duality and line spectra

Munich — 1814
Joseph von Fraunhofer is an optician, and his problem is glass: to make better telescope lenses he needs to know exactly how strongly each kind of glass bends each colour. By 1814 he has built what is in effect the modern spectroscope — a prism and a small telescope that spread light into its colours and let him measure where each one falls. Pointing it at sunlight, he finds the spectrum crossed by dark lines that never move. William Hyde Wollaston had noticed a few of them in 1802; Fraunhofer maps 574 and measures their positions. Today millions of such lines are known, and astronomers use them to read the chemistry of the Sun and of stars they will never visit (Joseph von Fraunhofer; Fraunhofer lines). A prism splits light into its frequencies and shows how much of each is there. This lesson builds the same instrument for any signal at all: the Fourier transform.
A Fourier series describes a periodic signal by its harmonics, spaced ω0=2π/T\omega_0 = 2\pi/T apart. A clap, a radar pulse or a lightning strike never repeats — but we can pretend it does, with a period so long that the copies never meet, and then let the period grow without bound. The harmonics crowd together, their spacing shrinks to nothing, and the comb of discrete frequencies becomes a continuous function of frequency: the spectrum.
Derivation
From the series to the transform
Let x~(t)\tilde x(t) repeat a pulse x(t)x(t) every TT seconds. Its Fourier coefficients, with the integral taken over the period centred on the pulse, are
ak=1T∫−T/2T/2x(t) e−jkω0t dt.a_k = \frac{1}{T}\int_{-T/2}^{T/2} x(t)\,e^{-jk\omega_0 t}\,dt .
Define X(jω)=∫−∞∞x(t) e−jωt dtX(j\omega) = \int_{-\infty}^{\infty} x(t)\,e^{-j\omega t}\,dt. Because the pulse fits inside one period, ak=1TX(jkω0)a_k = \frac{1}{T}X(jk\omega_0): the coefficients are samples of one fixed function, spaced ω0\omega_0 apart. The synthesis equation becomes
x~(t)=∑k1TX(jkω0) ejkω0t=12π∑kX(jkω0) ejkω0t ω0,\tilde x(t) = \sum_k \frac{1}{T}X(jk\omega_0)\,e^{jk\omega_0 t} = \frac{1}{2\pi}\sum_k X(jk\omega_0)\,e^{jk\omega_0 t}\,\omega_0 ,
using 1/T=ω0/2π1/T = \omega_0/2\pi. As T→∞T \to \infty, x~→x\tilde x \to x, the spacing ω0→dω\omega_0 \to d\omega, and the sum becomes an integral:
x(t)=12π∫−∞∞X(jω) ejωt dω.x(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} X(j\omega)\,e^{j\omega t}\,d\omega .
X(jω)=∫−∞∞x(t) e−jωt dt,x(t)=12π∫−∞∞X(jω) ejωt dωX(j\omega) = \int_{-\infty}^{\infty} x(t)\,e^{-j\omega t}\,dt, \qquad x(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} X(j\omega)\,e^{j\omega t}\,d\omega

The Fourier transform and its inverse. X(jω) says how much of each frequency the signal contains; the inverse rebuilds the signal from them.

📐A rectangular pulse and its sinc
Let x(t)=1x(t) = 1 for ∣t∣<τ/2|t| < \tau/2 and 00 otherwise — a pulse of width τ\tau. Then X(jω)=∫−τ/2τ/2e−jωt dt=ejωτ/2−e−jωτ/2jω=2sin⁡(ωτ/2)ω=τ sinc⁡ ⁣(ωτ2π),X(j\omega) = \int_{-\tau/2}^{\tau/2} e^{-j\omega t}\,dt = \frac{e^{j\omega\tau/2} - e^{-j\omega\tau/2}}{j\omega} = \frac{2\sin(\omega\tau/2)}{\omega} = \tau\,\operatorname{sinc}\!\left(\frac{\omega\tau}{2\pi}\right), with sinc⁡(u)=sin⁡(πu)/(πu)\operatorname{sinc}(u) = \sin(\pi u)/(\pi u). The spectrum's value at ω=0\omega = 0 is the pulse's area τ\tau; its first zeros are at ω=±2π/τ\omega = \pm 2\pi/\tau. Halve the pulse and the main lobe doubles in width. A 1 µs radar pulse occupies roughly 2 MHz of spectrum between its first zeros; a 1 ns pulse, 2 GHz.
📐Exponentials and Gaussians
  • The decaying exponential e−atu(t)e^{-at}u(t), a>0a > 0: X(jω)=1a+jωX(j\omega) = \dfrac{1}{a + j\omega}, with ∣X∣=1/a2+ω2|X| = 1/\sqrt{a^2 + \omega^2}. Its magnitude falls to 1/21/\sqrt2 of its peak at ω=a\omega = a — the RC circuit's cut-off, since e−t/RCu(t)/RCe^{-t/RC}u(t)/RC is the RC circuit's impulse response.
  • The Gaussian e−t2/2σ2e^{-t^2/2\sigma^2}: X(jω)=σ2π e−σ2ω2/2X(j\omega) = \sigma\sqrt{2\pi}\,e^{-\sigma^2\omega^2/2}. A Gaussian transforms into a Gaussian, with the width inverted: narrow in time, wide in frequency.
💡Magnitude, phase and real signals
X(jω)X(j\omega) is complex. Its magnitude ∣X(jω)∣|X(j\omega)| says how much of each frequency is present; its phase ∠X(jω)\angle X(j\omega) says how each component is shifted in time. For a real signal, X(−jω)=X∗(jω)X(-j\omega) = X^*(j\omega): the magnitude is even and the phase odd, so the negative frequencies carry no new information — they are the conjugate partners that make the sum real, exactly as a−k=ak∗a_{-k} = a_k^* did for series.
💡Duality
The transform and its inverse have almost the same form, so every pair works both ways: a rectangle in time gives a sinc in frequency, and a sinc in time gives a rectangle in frequency. That second reading matters in Lesson 4-4: an ideal low-pass filter, a rectangle in frequency, has a sinc for its impulse response — infinitely long and starting before t=0t = 0.
Which signals have transforms? Any absolutely integrable signal, and any signal of finite energy. Periodic signals have neither, but we admit them through impulses in frequency:
δ(t)↔1,1↔2π δ(ω),cos⁡ω0t↔π[δ(ω−ω0)+δ(ω+ω0)].\delta(t) \leftrightarrow 1, \qquad 1 \leftrightarrow 2\pi\,\delta(\omega), \qquad \cos\omega_0 t \leftrightarrow \pi\left[\delta(\omega - \omega_0) + \delta(\omega + \omega_0)\right].
An impulse contains every frequency equally; a constant contains only zero frequency; a cosine is two lines at ±ω0\pm\omega_0. Periodic signals have line spectra, single events have continuous ones — like the sharp emission lines of a lamp against the smooth glow of a hot body.
as bodies excite sounds of various tones and consequently vibrations, in the air of various bignesses, so when rays of light by impinging on the stiff refracting superficies excite vibrations in the ether, these rays excite vibrations of various bignesses
– Isaac Newton, Hypothesis explaining the Properties of Light (1675), quoted on Wikiquote
🕰The lines that read the stars
Fraunhofer was measuring glass, not the Sun. His dark lines, fixed in position, gave opticians exact reference colours for testing lenses, and later became the most powerful tool of astronomy: each line is a fingerprint of an element absorbing light at its own frequencies, and the spectrum of a star tells you what it is made of (Joseph von Fraunhofer).
⚠Conventions differ
Engineers often use ordinary frequency ff in hertz: X(f)=∫x(t) e−j2πftdtX(f) = \int x(t)\,e^{-j2\pi f t}dt and x(t)=∫X(f) ej2πftdfx(t) = \int X(f)\,e^{j2\pi f t}df, with no 1/2π1/2\pi anywhere. Physicists sometimes split the 1/2π1/2\pi symmetrically, and SymPy's fourier_transform uses the hertz convention. Check which one a table or a library uses before comparing numbers — factors of 2π2\pi are the commonest bug in spectral work.
💡Where you will use this
Lesson 4-2 gives the properties that make the transform useful; Lessons 4-3 and 4-4 read and design frequency responses. Sampling a spectrum is the DFT of Lesson 6-2, and sampling a signal replicates its spectrum (Lesson 5-1). Radar range resolution, radio bandwidth and spectroscopy all rest on the width relations you met here.
📚Further reading
  • Oppenheim and Willsky, Signals and Systems, sections 4.1–4.2 — the continuous-time Fourier transform, from the series.
  • Ronald N. Bracewell, The Fourier Transform and Its Applications (3rd ed., McGraw-Hill, 2000) — the classic engineer's tour of transform pairs and pictures.
  • Fraunhofer lines on Wikipedia.
↓But there is a problem
We can now take apart a single pulse into its frequencies. The rectangle and the Gaussian already hinted at a rule: narrow in time means wide in frequency. Is that a property of these two examples, or a law? And what happens to a spectrum when a signal is delayed, compressed, multiplied by a carrier, or passed through a system? In 1946 Dennis Gabor wrote down the rule as an inequality — a limit no signal can beat.