Fourier Transform: From Time Domain to Frequency Domain

The Fourier Transform is one of the most powerful tools in mathematics and engineering. It decomposes any signal into its constituent frequencies, revealing the hidden frequency components that make up complex waveforms. This calculator implements the Discrete Fourier Transform (DFT) algorithm to analyze sampled signals and display their frequency spectra.

1. How the DFT Works

The Discrete Fourier Transform converts a sequence of N time-domain samples into N frequency-domain coefficients. Each coefficient represents the amplitude and phase of a sinusoidal component at a specific frequency. The formula is: X[k] = Σ(n=0 to N-1) x[n] · e^(-j·2π·k·n/N). The magnitude |X[k]| tells you how much of frequency k is present in the signal, and the angle arg(X[k]) gives the phase.

2. Key Concepts

Sample Rate: The number of samples taken per second (Hz). Higher sample rates capture higher frequencies. Nyquist Frequency: Half the sample rate — the maximum frequency that can be accurately represented. Frequency Resolution: The spacing between frequency bins, equal to sample_rate / num_samples. More samples or lower sample rates give finer resolution.

3. Signal Types

This calculator supports five signal types: (1) Sine — a pure tone at the specified frequency; (2) Square — a periodic square wave rich in odd harmonics; (3) Triangle — a periodic triangle wave with odd harmonics that fall off as 1/n²; (4) Sawtooth — a periodic sawtooth wave with all harmonics falling off as 1/n; (5) Noise — random samples with uniform distribution. You can combine two frequencies with different amplitudes to study multi-tone signals.