Fourier / Signal Math Lab
Build signals from sine waves (or square/saw/triangle), run the FFT, and see the waveform and frequency spectrum live: harmonic analysis, ideal filter preview, sampling-rate & Nyquist math and audio frequency (note ↔ Hz) math. For audio, electronics and data.
How to Use
- In <strong>Spectrum analyzer</strong>, list the sine components — each line is <code>frequency amplitude [phase°]</code> (e.g. <code>8 1</code>, <code>20 0.5</code>). The tool sums them, samples at your sample rate, runs the FFT, and draws the waveform and its frequency spectrum.
- Hit a <strong>waveform preset</strong> (sine / square / sawtooth / triangle) to fill the builder with that wave's Fourier series — a live demonstration of how harmonics add up to a shape.
- It marks the peaks, finds the fundamental and harmonics with a THD figure, and warns about <strong>aliasing</strong> when a component sits above the Nyquist frequency (fₛ / 2).
- <strong>Filter preview</strong> applies an ideal low-pass / high-pass / band-pass / notch filter and overlays the original (grey) and filtered (pink) waveform and spectrum.
- <strong>Sampling & Nyquist</strong> is a calculator: enter sample rate and FFT size for the duration, bin width Δf and Nyquist limit, plus note ↔ frequency, period and wavelength math.
Waveform (time domain)
Spectrum (frequency domain)
Signal & Fourier math
About the Fourier / Signal Math Lab
The Fourier / Signal Math Lab is a free tool for everyday maths and number work. It runs right in your web browser, so there is nothing to download. Build signals from sine waves (or square/saw/triangle), run the FFT, and see the waveform and frequency spectrum live: harmonic analysis, ideal filter preview, sampling-rate & Nyquist math and audio frequency (note ↔ Hz) math. For audio, electronics and data.
How it works
Type your numbers into the boxes. The answer shows up right away — you do not have to press a button. If you change a number, the answer changes too. So you can try different numbers and watch what happens, or check an answer you worked out yourself. Just make sure each box has the right kind of number in it.
Want the deeper story? The Knowledge Base explains the ideas behind the tools in more detail.
Frequently Asked Questions
What does the FFT show me?
The Fast Fourier Transform converts your signal from the <em>time domain</em> (amplitude vs time — the wiggly waveform) to the <em>frequency domain</em> (amplitude vs frequency — the spectrum). A pure 8 Hz sine of amplitude 1 shows a single spike at 8 Hz of height 1. Adding components adds spikes. It's how you see which frequencies a signal is made of — the basis of audio analysis, vibration testing and spectral data work.
What is the Nyquist frequency and aliasing?
To capture a signal correctly you must sample at more than twice its highest frequency; that limit, fₛ / 2, is the <strong>Nyquist frequency</strong>. A frequency above it can't be represented and instead masquerades as a lower one — <strong>aliasing</strong>. A 200 Hz tone sampled at 256 Hz folds down to 56 Hz, for example. The lab flags this and tells you the apparent frequency.
How do the waveform presets work?
Square, sawtooth and triangle waves are sums of sine harmonics with specific amplitudes (a square wave is the odd harmonics with amplitudes 4/π·1/n). Picking a preset fills the builder with those components, so you can watch the harmonics combine into the shape and see them stand up in the spectrum — Fourier-series decomposition you can play with.
What kind of filter is the preview?
An <strong>ideal (brick-wall) filter</strong> in the frequency domain: it FFTs the signal, zeroes the bins outside the pass-band, and inverse-FFTs back. Low-pass keeps frequencies below the cutoff, high-pass above it, band-pass between two cutoffs, notch removes a band. It's the cleanest way to <em>see</em> what a filter does; real-time analog/IIR filters have gentler roll-off.
What is THD?
Total Harmonic Distortion — the energy in the harmonics (2nd, 3rd, …) relative to the fundamental, as a percentage. A pure sine has 0% THD; a square wave has a lot. It's a standard measure of how 'clean' a tone or amplifier output is.
Does it use my microphone or a server?
Neither — this is a math lab that builds and analyses <em>synthetic</em> signals entirely in your browser (for live microphone spectra, see the Audio Spectrum Analyzer). Nothing is uploaded and it works offline.
How do I use the Fourier / Signal Math Lab?
Just type your numbers. The answer shows up right away — there is no button to press. Change anything and it updates by itself.
Is it free? Does it work without internet?
Yes to both. It is free with no sign-up, and once the page has loaded it keeps working even with no internet.
Where does my data go?
Nowhere — every calculation runs on your own device. Nothing you enter is uploaded, logged, or stored.
Common Use Cases
Audio & music
See harmonics, THD, and note ↔ frequency; understand timbre and tuning.
Electronics & DSP
Sampling rate, Nyquist, aliasing and filter behaviour before you build.
Data & vibration analysis
Decompose a signal into its frequency components with the FFT.
Teaching Fourier
Watch square/saw/triangle waves build from sine harmonics.
Filter design intuition
Preview low/high/band/notch filters in the frequency domain.
Signal experiments
Try aliasing, leakage and resolution by changing fₛ and N.
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