Fourier Transform Calculator
Compute X(f) = ∫ x(t)·e−j2πft dt numerically for a menu of standard signals, with the closed-form analytical result alongside. Explore the time–frequency duality interactively.
| x(t) | X(f) | Notes |
|---|
The transform pairs every signal-processing course expects you to know. The pattern: narrow in time → wide in frequency. A rectangular pulse becomes a sinc, a Gaussian remains a Gaussian, an impulse becomes a constant, and a cosine becomes a pair of delta functions.
| Property | Time domain | Frequency domain | Meaning |
|---|
These properties are the reason the Fourier transform is so powerful: operations that are hard in one domain become trivial in the other. Differentiation becomes multiplication, convolution becomes pointwise product, and the uncertainty principle is a consequence of the time–bandwidth product.
| Parameter | Value | Meaning |
|---|
The continuous integral is approximated by a Riemann sum over the time grid. Accuracy improves with more samples and a wider time window; the trade-off is computation time, but for these signals the numerical and analytical results agree to 4–5 significant figures even at moderate sample counts.
Time–frequency duality
Narrow in time → wide in frequency. A very short pulse has an almost flat spectrum; a very long pure tone has a spectrum concentrated at a single frequency. This is not a limitation of the measurement — it is a mathematical identity.
Parseval's theorem
Energy is preserved by the Fourier transform: ∫|x(t)|²dt = ∫|X(f)|²df. You can compute the total energy in either domain and get the same answer. This calculator verifies the identity as a sanity check on the numerical integration.
Gaussian is its own transform
The Gaussian pulse is unique: its Fourier transform is also a Gaussian, and its time-bandwidth product achieves the theoretical minimum. This is why Gaussian pulses are used in radar, communications and quantum optics — they concentrate energy most efficiently in both domains simultaneously.
Real signals have even magnitude
If x(t) is real-valued, |X(f)| = |X(−f)| — the magnitude spectrum is even. The phase is odd. This is why we usually only plot positive frequencies: for a real signal, the negative half is redundant, and the two-sided spectrum is symmetric.
Pick a signal type, set its parameters, and click Compute. The Fourier transform is calculated numerically from the definition and matched against the closed-form analytical result.
Choose a signal
Eight standard signals are available: rectangular pulse, triangular pulse, Gaussian, two-sided exponential, causal exponential, sinc, cosine burst, and sine burst. Each has a known closed-form Fourier transform so you can compare against the numerical result.
Set the parameters
Amplitude A and a width parameter (T, σ, or τ depending on the signal) are always present. Cosine and sine bursts additionally expose a carrier frequency f₀. All values are in consistent units — usually seconds for time, Hz for frequency.
Click Compute
The page smoothly scrolls to the result card, which flashes briefly. Both time-domain and frequency-domain plots render, along with the analytical formula and numerical verification.
Read the result card
The closed-form X(f) appears at top, followed by the peak magnitude, peak frequency, DC value, bandwidth, and energy in both domains. A Parseval check confirms the numerical integration matches.
Examine the charts
Time + magnitude shows both domains together. |X(f)| only zooms in on the spectrum. Phase & components shows the phase response and the real and imaginary parts of X(f) — useful for understanding why the magnitude alone does not determine the signal.
Advanced mode
Exposes the numerical integration parameters (time window, frequency span, sample counts), the full parameter table for the current computation, and the reference tables for transform pairs and properties.
Analytical + numerical
The closed-form transform is shown alongside the numerically computed spectrum, so you can see where they agree and understand how the integral converges.
Dual-domain plotting
Time-domain x(t) and frequency-domain |X(f)| are plotted together on shared axes, plus separate views for magnitude, phase, and the real and imaginary components.
Step-by-step method
Every computation shows the integration method (trapezoidal Riemann sum), the time grid, the frequency grid, and a verification against the closed-form at several test frequencies.
Parseval verification
Time-domain energy ∫|x|²dt is computed and compared against frequency-domain energy ∫|X|²df. A match to numerical precision confirms both the transform and the integration grid.
Reference tables
Common transform pairs (rect, tri, Gaussian, exponential, sinc, cosine, impulse, step) and the transform properties (linearity, shifting, scaling, differentiation, convolution, Parseval).
Configurable resolution
Sample counts and window sizes are adjustable in Advanced mode. Increase them to see the numerical result converge to the analytical answer, or decrease them to see how coarse sampling degrades the spectrum.
The Fourier transform is the mathematical operation that decomposes a signal into its constituent frequencies. Formally, X(f) = ∫−∞∞ x(t)·e−j2πft dt, where x(t) is the time-domain signal, X(f) is the complex-valued frequency-domain representation, and f is frequency in Hz. The result X(f) tells you both the magnitude and the phase of each frequency component. The inverse transform, x(t) = ∫ X(f)·e+j2πft df, recovers the original signal from its spectrum — the pair is a bijection, so no information is lost. The Fourier transform is the continuous-time analog of the Fourier series (which handles periodic signals only), and its discrete counterpart — the DFT — is what digital signal processors and computers actually implement. What makes the Fourier transform so powerful is not just that it exists, but that it turns difficult operations in one domain into easy ones in the other: differentiation in time becomes multiplication by j2πf in frequency; convolution in time becomes pointwise multiplication in frequency; and the energy of a signal is preserved (Parseval's theorem). This duality underlies almost everything in modern signal processing — filter design, spectral analysis, modulation, sampling theory, image compression, and the convolutional neural networks that dominate machine learning. Different signal shapes have very different spectra: a rectangular pulse has a sinc-shaped spectrum with slowly decaying sidelobes; a Gaussian pulse has a Gaussian spectrum with no sidelobes at all; a pure cosine has a two-line spectrum at ±f₀; and a Dirac impulse has a completely flat spectrum. These pairings are the standard examples that reveal how the shape of a signal in one domain determines the shape in the other — narrow in time always means wide in frequency, and vice versa.
For each frequency f, it computes the complex number X(f) that best describes how much of that frequency is present in the signal. The magnitude |X(f)| tells you the amplitude of that frequency component, and the phase ∠X(f) tells you how much it is delayed relative to the origin. Equivalently, it is the projection of the signal onto a complex sinusoid of frequency f — a correlation between the signal and e−j2πft. Wherever that projection is large, the signal resembles a sinusoid at that frequency.
Convention. The forward transform uses e−j2πft and the inverse uses e+j2πft. The sign is arbitrary — you could swap them and get a mirror-image spectrum. The negative sign in the forward transform is chosen to match the physics of a wave traveling in the positive direction with ej(ωt−kx) convention. Some authors use the opposite sign; the magnitude and physical interpretation are identical either way. What matters is consistency: once you fix the sign of the forward transform, the inverse must use the opposite sign.
Because the integral of a rect(t/T) times a complex exponential is straightforward: ∫−T/2T/2 A·e−j2πft dt = A·[e−jπfT − e+jπfT]/(−j2πf) = A·T·sin(πfT)/(πfT) = A·T·sinc(fT). The sinc function appears naturally because the integral of a complex exponential over a finite interval produces a sine wave divided by its argument. The sinc's slow 1/f sidelobe decay is the reason rectangular windows cause spectral leakage in practical spectrum analysis — the sharp edges in time create ringing in frequency.
The Fourier series decomposes a periodic signal into a discrete sum of harmonically related sinusoids — the spectrum consists of isolated lines at f₀, 2f₀, 3f₀, etc. The Fourier transform handles non-periodic (and general) signals by integrating over a continuous range of frequencies — the spectrum is a continuous function of f. The transform is the natural limit of the series as the period goes to infinity: the harmonic spacing becomes infinitesimally fine, and the discrete sum becomes an integral. For a periodic signal, the transform still exists (in the sense of distributions) as a train of Dirac impulses at the harmonic frequencies.
For a real-valued signal x(t), the spectrum satisfies X(−f) = X*(f) — the conjugate symmetry property. The magnitude is even and the phase is odd. As a result, |X(f)| for negative f contains no new information; it is a mirror of the positive half. When computing total energy or power, you either integrate over all f (both signs) or double the positive-frequency result. In practical spectrum analysis, engineers often plot only f ≥ 0 and label the axis with "one-sided" magnitude. This is a convention, not a physical distinction — negative frequencies are a mathematical consequence of using complex exponentials, not a sign of "unphysical" waves.
The continuous integral is approximated by a Riemann sum: X(f) ≈ Σ x(tn)·e−j2πftn·Δt, where the time grid tn is uniformly spaced with step Δt. For each frequency f, the sum runs over all time samples. This is a direct implementation of the trapezoidal rule, which converges as O(Δt²) for smooth signals. The signal is windowed by its own finite support for pulse-shaped signals (rect, tri, Gaussian), or by a chosen time window for slowly-decaying signals (exponentials). Truncation error is largest for the two-sided exponential, which never reaches zero; increasing the window size in Advanced mode reduces this error.
What it means
The Fourier transform is a mathematical operation that decomposes a time‑domain signal x(t) into its frequency components X(f) = ∫ x(t) e^(−j2πft) dt. It provides the amplitude and phase of each frequency in the signal. The inverse transform recovers the original signal from its spectrum. The Fourier transform is fundamental in signal processing, communications, and control systems. It is used to analyse filter responses, design equalisers, and perform spectral analysis. In practice, the Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT) are used for digital data. Example: A rectangular pulse in the time domain transforms into a sinc function (sin(x)/x) in the frequency domain, showing that a short pulse contains a wide range of frequencies. This is used in pulse shaping and modulation.
Worked example
Fourier Transform – Practical Example
Real‑World| Parameter | Value |
|---|---|
| Time signal | rect(t/T) |
| Frequency signal | T·sinc(fT) |
| Formula | X(f) = ∫x(t)·e−j2πft dt |
Common mistakes
Watch unit consistency and the assumptions behind the formula; misapplying it outside its valid conditions is the most frequent error.Applications
The Fourier transform decomposes a time‑domain signal into its frequency components, enabling analysis in the frequency domain. It is essential for signal processing, communication, and control systems. Engineers use it to design filters, to analyse system stability, and to process audio and image data. The Fourier transform is the basis for many algorithms, including FFT, and is used in spectrum analysers and digital signal processors. Understanding the Fourier transform is crucial for any work involving signals and systems.
- Signal processing – filtering, spectral analysis, and denoising
- Communication system modulation and demodulation
- Control system frequency response analysis
- Image processing and pattern recognition
- Educational foundation of frequency‑domain analysis
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