FFT & Signal Domain

The Signal domain provides tools for Digital Signal Processing (DSP), including the Fast Fourier Transform.

Signals are everything: audio from a mic, vibrations in a bridge, or stock market prices.

• Time Domain: "How loud is it right now?"
• Frequency Domain: "What notes are being played?"

The Fast Fourier Transform (FFT) is a legendary algorithm that converts Time into Frequency. It lets you unbake a cake to find the ingredients. If you want to filter noise from audio, analyze heartbeats, or compress images, you need this domain.

Overview

use aivi.signal (fft, ifft)

// A simple signal (e.g., audio samples)
timeDomain = [1.0, 0.5, 0.25, 0.125]

// Convert to frequencies to analyze pitch
freqDomain = fft(timeDomain)

Features

Signal = { samples: List Float, rate: Float }
Spectrum = { bins: List Complex, rate: Float }

Domain Definition

domain Signal over Signal = {
  (+) : Signal -> Signal -> Signal
  (+) a b = { samples: zipWith (+) a.samples b.samples, rate: a.rate }
  
  (*) : Signal -> Float -> Signal
  (*) s k = { samples: map (\x -> x * k) s.samples, rate: s.rate }
}

Helper Functions

Function	Explanation
fft signal Signal -> Spectrum	Transforms a signal into a frequency-domain spectrum.
ifft spectrum Spectrum -> Signal	Reconstructs a time-domain signal from its spectrum.
windowHann signal Signal -> Signal	Applies a Hann window to reduce spectral leakage.
normalize signal Signal -> Signal	Scales samples so the max absolute value is 1.0.

Usage Examples

use aivi.signal
use aivi.number.complex

audio = { samples: [0.0, 0.5, 1.0, 0.5], rate: 44100.0 }
spectrum = fft audio
recon = ifft spectrum