Fourier Transform Calculator

The Fourier transform answers a question that ordinary calculus does not ask: not "what is this signal's value?" but "what frequencies is this signal made of?" It rewrites a function of time (or space) as a function of frequency, revealing the pure tones hidden inside a vibration, an audio waveform, an image, or any other varying quantity. The integral involved is genuinely heavy to evaluate by hand, which is why this calculator computes Fourier transforms directly — for checking transform pairs, coursework in signals and systems, and building intuition for spectra.

Quick Answer

The Fourier transform of f(t) is F(ω) = ∫₋∞^∞ f(t) · e^(−iωt) dt, converting a time-domain function into its frequency-domain spectrum. The inverse transform reconstructs f(t) from F(ω), with the 1/(2π) factor placed by convention.

A Prism for Functions

The analogy worth keeping is a prism: white light enters, a spectrum of colours leaves. The transform does the same to a signal, decomposing it into sinusoids of every frequency, with F(ω) recording how much of frequency ω is present and with what phase. Because e^(−iωt) = cos(ωt) − i·sin(ωt), the single complex integral simultaneously tests the signal against both a cosine and a sine of each frequency — the real part of F tracks the cosine content, the imaginary part the sine content, and the magnitude |F(ω)| gives the strength regardless of phase.

Conventions are the quiet complication of this subject. Some texts define the transform with angular frequency ω and put 1/(2π) in the inverse; others split it symmetrically as 1/√(2π) in both directions; engineers working in cycles per second (Hz) instead use F(ν) = ∫ f(t) e^(−2πiνt) dt, with no stray factors anywhere. All three describe the same decomposition, but numerical values differ between them, so a transform is only fully specified when its convention is. When comparing your result with a table or a classmate's answer, check the convention before concluding anyone is wrong.

How to Use the Fourier Transform Calculator

1. Enter the time-domain function f(t) exactly as written, noting any piecewise definition or multiplying factor such as a unit step.
2. Confirm which variable the function uses and which frequency variable you expect in the answer.
3. Run the calculation and read the spectrum F(ω).
4. Check a landmark value by hand — F(0), which equals the total area under f(t), is the easiest.
5. If your course uses the Hz convention, expect the argument scaling to differ by 2π from the ω form; compare shapes before comparing numbers.
6. For real-valued signals, confirm the symmetry you should see: the real part of F even in ω, the imaginary part odd.

A Worked Example: The Transform of e^(−a|t|)

Take f(t) = e^(−a|t|) with a > 0 — a two-sided exponential decay, the textbook's favourite example. Splitting the integral at zero and evaluating the two halves gives the classical result F(ω) = 2a / (a² + ω²). Check the landmark: F(0) = 2a/a² = 2/a, and the area under f(t) is 2∫₀^∞ e^(−at) dt = 2/a ✓. With a = 1 the spectrum is 2/(1 + ω²), so F(0) = 2 and F(1) = 1. Read the shape: a smooth, even, bell-like curve centred at zero — a signal concentrated near t = 0 has a spectrum spread broadly across frequencies, and the faster the decay (larger a), the wider the spectrum. That reciprocal spreading, narrow in one domain meaning wide in the other, is the deep principle behind the uncertainty limits in signal processing and quantum mechanics alike.

Transform Errors That Convention Differences Hide

The most common source of "wrong" answers is not wrong at all: mismatched conventions between your computation and your reference table, differing by 2π factors or by ω versus ν in the argument. The second is forgetting that the transform of a real signal must obey conjugate symmetry, F(−ω) = conjugate of F(ω) — an answer violating it signals an algebra slip. A third is ignoring convergence: the defining integral must converge (at least in the generalised sense), and functions that grow without bound have no ordinary transform. Sign errors in the exponent flip the spectrum left-to-right, which matters the moment phase matters. The sibling transform for differential equations is the Laplace Transform Calculator, and for the integration machinery underneath both, the Integral Calculator covers the underlying techniques.

A Second Landmark Pair: The Rectangular Pulse

The example to set beside the exponential is the rectangular pulse — f(t) = 1 for −1 ≤ t ≤ 1, and 0 elsewhere — because its transform can be checked by pure geometry. Integrating directly, F(ω) = ∫₋₁¹ e^(−iωt) dt = 2 sin(ω) / ω. Three facts about this spectrum repay attention. At ω = 0 it equals 2, the pulse's width times its height — the area rule again. It crosses zero whenever sin(ω) = 0 at ω = ±π, ±2π, …, so a pulse of width 2 has its first spectral nulls at ±π: double the pulse's width and the nulls move in to ±π/2, the reciprocal-spreading principle made concrete. And it decays only slowly, like 1/ω, with endless ripples — the frequency-domain price of the pulse's sharp edges, the same effect that produces ringing (the Gibbs phenomenon) when such a spectrum is cut off. Comparing the two examples side by side teaches the subject's central intuition: smooth, spread-out signals have compact, smooth spectra; abrupt, localised signals spray energy across all frequencies.

A practical note on scope: this calculator works with the continuous transform of a mathematical function. Real digital systems — audio software, spectrum analysers, image processors — almost always compute its discrete cousin, the discrete Fourier transform, usually through the fast Fourier transform algorithm, because their input is a finite list of samples rather than a formula. The ideas transfer directly (frequencies, magnitudes, phases, spectral leakage), but the outputs are not interchangeable: a DFT bin is not a value of F(ω), and comparing the two without accounting for sampling rate and windowing produces confusion that looks like error. Know which transform your problem is actually about before comparing numbers.

Thinking in Frequencies

Once you start reading spectra — why a filter removes hiss, why an image compresses, why a bridge resonates — the transform stops being an exam topic and becomes a way of seeing. Compute your transforms here, sanity-check them at ω = 0, and browse the math calculators collection for the analysis tools around it.