Fourier transform

The formula

Forward transform, ordinary-frequency (Hz) convention:

X(f) = ∫ x(t)·e^(−j·2π·f·t) dt, integrated over all time — slide a complex sinusoid of frequency f along the signal and add up how well they line up. X(f) is a complex number per frequency: magnitude says how much, angle says when.

Inverse transform:

x(t) = ∫ X(f)·e^(+j·2π·f·t) df, over all frequencies — reassemble the signal by summing every frequency back in with its stored magnitude and phase. Nothing lost in either direction.

The angular-frequency convention, X(ω) = ∫ x(t)·e^(−jωt) dt with x(t) = (1/2π)·∫ X(ω)·e^(jωt) dω, carries the same content with the 2π parked in the inverse. Textbooks split roughly evenly between the two ^[1].

The two theorems that do the daily work:

y(t) = h(t) * x(t) ⇔ Y(f) = H(f)·X(f) — convolution in time is multiplication in frequency. Every filter, channel, and transducer response calculation runs on this line.

∫ |x(t)|² dt = ∫ |X(f)|² df — Rayleigh's energy theorem (Plancherel's, in the general case): the energy in the waveform equals the energy under the spectrum. Same joules, two bookkeeping systems ^[5].

Where you meet it

EMC precompliance, watching a clock line on the analyzer. A trapezoidal edge has a spectrum that rolls off as 1/f out to the corner set by the rise time, then 1/f² beyond it. That is why the fix for a radiated-emissions failure at 300 MHz is usually a slower edge or a series resistor, not a different clock frequency — the fundamental was never the problem.
Radar and comms pulse design. A rectangular RF pulse of width τ has a sinc-shaped spectrum with nulls at multiples of 1/τ: a 1 µs pulse puts its first nulls at ±1 MHz, with 90.3% of the pulse energy inside them (checked numerically). Shorten the pulse for range resolution and the spectrum widens in exact proportion — the trade the spectrum-management reviewer will make you defend.
Shock and vibration test. A pyroshock or drop transient doesn't repeat, so Fourier series doesn't apply; the transform (in practice an FFT of the captured record) gives the energy-versus-frequency picture that feeds the shock response spectrum and tells you which structural modes the event can excite.
Anywhere you characterize a box by frequency response. H(f) — the transform of the impulse response — is what the network analyzer measures, what the Bode plot draws, and what you multiply an input spectrum by to predict an output.

How it works

The series-to-transform step is a limit. A Fourier series describes a repeating signal with harmonics spaced 1/T apart; let the period grow without bound and the line spacing goes to zero, the discrete comb fills in, and the coefficient list becomes a continuous function X(f). The units shift with it: series coefficients are volts per line, the transform is volts per hertz — a density. That units change is real, and it is the root of most spectrum-scaling confusion on the bench.

The pairs worth carrying in your head, each verified numerically:

rect pulse, width τ, height 1   ⇔   τ·sinc(f·τ)          (sinc(x) = sin(πx)/(πx))
e^(−t/τ)·u(t)                   ⇔   τ / (1 + j·2π·f·τ)   (−3 dB at f = 1/(2πτ))
e^(−π·t²)                       ⇔   e^(−π·f²)            (the Gaussian is its own transform)
δ(t)                            ⇔   1                     (an impulse contains every frequency equally)
cos(2π·f₀·t)                    ⇔   ½·δ(f−f₀) + ½·δ(f+f₀)

The sinc pair is the one engineers live with. First null at 1/τ, first sidelobe at −13.26 dB (checked) — which is why an unwindowed rectangular gate anywhere in a system, hardware or software, leaves a −13 dB skirt on everything it touches.

Scaling is the law with no exceptions: x(a·t) ⇔ (1/|a|)·X(f/a). Compress a signal in time by ten and its spectrum stretches by ten. The formal version is the uncertainty principle, σ_t·σ_f ≥ 1/(4π) with the RMS widths, and only the Gaussian hits the bound exactly (verified to machine precision). Every "narrow pulse, narrow spectrum" request that comes through a design review is asking the mathematics for something it does not sell.

Duality is the two-for-one: because forward and inverse transforms differ only by a sign, every pair reads both ways. Rect in time gives sinc in frequency, so a brick-wall (rect) filter in frequency has a sinc impulse response — ringing before and after the edge. Non-causal preshoot on an ideal-filtered step isn't a defect in your scope's DSP; it's the transform of the passband shape.

Strictly, the integral converges when the signal is absolutely integrable — it must die out. A sine wave that runs forever doesn't qualify; its transform only exists as delta functions, which is the mathematical statement behind the fact that a pure tone shows as a line and a transient shows as a smear. For steady-state or growing signals with initial conditions, the Laplace transform is the right tool; Fourier is Laplace evaluated on the jω axis when that axis is inside the region of convergence.

The mistakes people actually make:

Dropping or doubling a 2π. The Hz and rad/s conventions differ by factors of 2π in the formulas and the tables. Mixing pairs from two textbooks is the classic way to be wrong by 6.28 in a bandwidth calculation. Pick a convention per analysis and write it at the top of the page.
Reading an FFT as if it were the transform. The FFT sees a finite record and implicitly repeats it. If the signal isn't periodic in the record, energy leaks across bins — windowing trades that leakage against resolution, and the window's own gain has to be corrected before any amplitude on the screen means anything.
Confusing spectrum with spectral density. A tone is volts (or dBm); noise is volts per √Hz (or dBm/Hz). An analyzer displays both in the same trace, scaled by its resolution bandwidth. Comparing a tone level against a noise level without normalizing by RBW is a factor-of-√RBW error that survives into far too many test reports.
Throwing away phase. Magnitude spectra are what get plotted; phase is what preserves waveform shape. Two signals with identical magnitude spectra can be a crisp pulse and a smeared chirp. If time-domain fidelity matters — matched filters, pulse compression, distortion analysis — the phase is half the data.

History

Joseph Fourier presented his memoir on heat propagation to the Paris Institute in December 1807 and collected objections from Lagrange and Laplace over the trigonometric series at its core ^[1]^[2]. The Institute made heat the subject of its 1811 prize; Fourier's winning resubmission is where the integral form — the transform proper, for problems on infinite domains where no period exists — first appears ^[3]^[4]. The full treatment reached print in his 1822 Théorie analytique de la chaleur ^[1]^[2].

He wasn't alone. Cauchy published a version of the integral formula in 1815, and Poisson used the inversion formula in 1816 in his memoir on water waves — three mathematicians arriving at the same tool independently within a few years, each chasing a different physics problem ^[3]^[4]. Fourier got the name; Cauchy and Poisson supplied most of the early convergence proofs ^[4].

The rigorous foundation for the energy theorem waited a century: Michel Plancherel proved in 1910 that the transform conserves energy for any finite-energy signal, series or no series ^[5]^[6]. And the reason working engineers touch this mathematics daily is computational: Cooley and Tukey published the fast Fourier transform in 1965, cutting the cost of a discrete transform from N² to N·log N operations — a few hundred times fewer at 4,096 points ^[7]^[8]. Gauss had worked out essentially the same algorithm around 1805 for asteroid-orbit interpolation and never published it; it surfaced posthumously, in Latin, and stayed unnoticed until after 1965 ^[7]^[8]. The most-run algorithm in signal processing spent 160 years in a drawer.

Related tools

/tools/rc-filter — the one-pole H(f) above is exactly the RC low-pass response
/tools/frequency-period — the 1/τ reciprocity between pulse width and spectral nulls
/tools/q-factor-bandwidth — resonator bandwidth read straight off the frequency-domain response
/tools/thermal-noise-floor — noise density in dBm/Hz, the spectral-density side of the scaling trap
/tools/snr-enob — quantization noise spread across the Nyquist band is a spectral-density argument
/tools/db-voltage-ratio — converting spectrum magnitudes to the dB values on the analyzer
/tools/vibration-natural-freq — the structural modes a transient's spectrum can excite
/tools/lc-resonance — where a filter's H(f) peaks, and why its impulse response rings

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →