Fast Fourier Transform
Compute the Discrete Fourier Transform of a sequence of (real or complex) numbers and its inverse. Power-of-two lengths use the Cooley-Tukey radix-2 FFT (O(N log N)); other lengths use the direct O(N²) definition. The tool lists each frequency bin's magnitude, phase, and power, with a magnitude bar chart, and verifies the inverse round-trips to the input. Everything runs locally in your browser.
Input sequence (one value per line)
Real numbers, or complex as a+b i / a-b i (e.g. 1+2i, 3, -0.5i).
Summary
Bins
The DFT maps a length-N sequence x[n] to frequency bins X[k] = Σ x[n]·e−2πikn/N; the inverse inverts the sign and scales by 1/N. Bin 0 is the DC (average) component; bins k and N−k are complex-conjugate pairs for real input. The magnitude |X[k]| is amplitude, the phase arg X[k] is the bin's rotation. For power-of-two N the radix-2 FFT computes the same result in O(N log N) via even/odd splitting (X[k]=E[k]+WkO[k], X[k+N/2]=E[k]−WkO[k]). Parseval: Σ|x[n]|² = (1/N)·Σ|X[k]|². Pairs with the Convolution, Waveform/Signal tools, and Calculus Toolkit. Everything runs locally — nothing leaves your browser.