8.12.1. MINFFT-01 — Real FFT

The minfft module wraps the minfft library — a small, fast FFT/DCT kernel. It works on power-of-two lengths. This tutorial covers the real FFT: turning a real signal into its frequency spectrum and back.

8.12.1.1. Forward transform

A real signal of length N transforms to N/2+1 complex bins (the spectrum of a real signal is symmetric, so the upper half is redundant and not stored). A tone that completes exactly k cycles over the N samples lands on bin k with raw magnitude N/2 times its amplitude:

let signal <- [for (i in range(64));
    sin(TWO_PI * 4.0f * float(i) / 64.0f) +
    0.5f * sin(TWO_PI * 12.0f * float(i) / 64.0f)]

var spectrum : array<float2>
fft_real_forward(signal, spectrum)   // 64 samples -> 33 complex bins

var mag : array<float>
fft_calculate_magnitudes(spectrum, mag)   // peaks: 32 at bin 4, 16 at bin 12

8.12.1.2. Magnitude helpers

Each complex bin (re, im) carries amplitude sqrt(re*re + im*im). Three helpers fill a float array from the complex spectrum:

fft_calculate_magnitudes — raw amplitude.
fft_calculate_normalized_magnitudes — raw amplitude scaled by 1/(N/2+1).
fft_calculate_log_magnitudes — log scale, handy for spectrum displays.

8.12.1.3. Inverse transform and normalization

minfft transforms are unnormalized: fft_real_inverse(fft_real_forward(x)) returns N*x, not x. Divide by N to recover the original signal:

var back : array<float>
fft_real_inverse(spectrum, back)
// back[i] / float(N) == signal[i]

This unnormalized convention runs through the whole library — the DCT pair in the next tutorial scales by 2N per axis.