Calculator
Formula used
This calculator approximates the continuous Fourier transform over a finite time window:
- X(f) = ∫ x(t) e^{-i2πft} dt (frequency in Hz)
- X(ω) = ∫ x(t) e^{-iωt} dt (angular frequency in rad/s)
Because the integral is computed numerically, the time domain is sampled on Nt points from t_start to t_end. A window (Hann/Hamming/Blackman) can reduce truncation artifacts. The integral is then approximated using trapezoidal or Simpson weights:
- Trapezoidal: endpoints get half weight, scaled by dt.
- Simpson: alternating weights 4 and 2, scaled by dt/3 (requires odd Nt).
How to use this calculator
- Pick a preset or type your own x(t) using t.
- Set a time window wide enough to capture the signal’s energy.
- Choose Nt for time resolution; larger gives smoother spectra.
- Select frequency units and set the frequency range and Nf.
- Use a window when truncation causes ripples (spectral leakage).
- Press Compute transform. Results appear above the form.
Example data table
Sample values for x(t)=exp(-t^2) (Gaussian) across a short window. Your run may differ based on settings.
| t | x(t) |
|---|---|
| -2 | 0.0183156 |
| -1.5 | 0.105399 |
| -1 | 0.367879 |
| -0.5 | 0.778801 |
| 0 | 1 |
| 0.5 | 0.778801 |
| 1 | 0.367879 |
| 1.5 | 0.105399 |
| 2 | 0.0183156 |
Time window and practical resolution
The transform is computed on a finite window of length T = t_end − t_start. A practical resolution limit is about Δf ≈ 1/T (or Δω ≈ 2π/T). With t ∈ [−8, 8], T = 16, so Δf ≈ 0.0625 and Δω ≈ 0.3927. Doubling the window halves the minimum resolvable spacing.
Sampling density and usable bandwidth
Time sampling sets the highest meaningful frequency. The step is dt = T/(Nt−1), giving a Nyquist limit f_N ≈ 1/(2dt) (or ω_N ≈ π/dt). For Nt = 1001 over T = 16, dt ≈ 0.016 and f_N ≈ 31.25. Keep f_max well below f_N to reduce aliasing and numerical noise.
Frequency grid selection
The frequency grid is user‑defined: df = (f_max − f_min)/(Nf−1). For f ∈ [−5, 5] with Nf = 801, df = 0.0125. Smaller df smooths plots, but it cannot beat the window‑limited Δf. Use an odd Nf to place a point exactly at 0. Choose symmetric ranges to inspect positive and negative frequencies consistently. A practical workflow is to start with df near 1/T, then refine around peaks.
Window choice and leakage control
Truncating x(t) multiplies it by a window, which spreads energy in frequency. A rectangular window has the narrowest main lobe but strong sidelobes. Hann typically drops the first sidelobe to about −31 dB, Hamming to about −43 dB, and Blackman to about −58 dB, at the cost of wider peaks. Use stronger windows when small tones sit near large components.
DC removal and gain compensation
Removing DC subtracts the mean of the sampled, windowed signal, which prevents a large zero‑frequency spike from masking nearby structure. Gain compensation divides by the mean window value so amplitudes remain comparable across window choices. Typical mean values are about 1.00 (none), 0.50 (Hann), 0.54 (Hamming), and 0.42 (Blackman).
Reading magnitude and phase together
Magnitude indicates how strongly each frequency contributes; phase describes alignment and delays. For real‑valued x(t), |X| is even in frequency and the imaginary part is odd under ideal conditions. A Gaussian input often yields a real, positive spectrum, so phase stays near 0°. When phase wraps at ±180°, interpret jumps carefully. Export CSV to compare runs, verify symmetry, and document settings with your results more reliably.
FAQs
Does this calculator produce symbolic transforms?
No. It numerically approximates the integral on a finite time window. For closed‑form transforms, use a computer algebra system and then compare the numeric spectrum here.
Why do I see ripples or extra peaks?
Finite windows cause spectral leakage and ringing. Increase the time span, apply Hann/Hamming/Blackman windows, and ensure f_max stays below the sampling Nyquist limit.
What do Nt and Nf change in practice?
Nt controls time sampling and the integration accuracy; larger Nt reduces discretization error. Nf controls how finely the frequency axis is evaluated and plotted, not the underlying resolution.
When should I use f (Hz) versus ω (rad/s)?
Use f for engineering spectra and ω for many math and physics conventions. They are linked by ω = 2πf, so the shapes match after rescaling the horizontal axis.
Why does Simpson integration require an odd Nt?
Simpson’s rule fits parabolas over pairs of intervals, which needs an even number of sub‑intervals. That means Nt must be odd so (Nt−1) is even.
How can I improve accuracy for slow decay signals?
Expand the time window, raise Nt, and consider removing DC. If the signal is not near zero at the boundaries, use a window to reduce edge discontinuities before integrating.