Fourier Transforms (DFT/FFT and IFT)

Fourier transforms, named for 19th century French mathematician and physicist Jean-Baptiste Joseph Fourier, are based on the idea that complex signals (such as speech or music) can be constructed from, or broken down into sinewaves of varying amplitude and phase relationships. Fourier transforms are used extensively in audio analysis to find the spectral content of time domain signals. Inverse Fourier transforms (IFTs) reconstruct time-domain signals from spectral data. 

There are several different types of Fourier transforms, but the type that we concern ourselves with in Smaart is the discrete Fourier transform (DFT), which works on time domain signals of finite length. The term fast Fourier transform (FFT) refers to methods for calculating a DFT more efficiently, most commonly requiring the chunk of signal being analyzed to be a power of two (2n ) samples in length, e.g., 4096 (4K), 8192 (8K), 16384 (16K)… (212, 213, 214…). All FFTs are DFTs, but not all DFTs are fast. 

Most DFTs in Smaart are power-of-two FFTs (also called radix 2 FFTs or just FFTs). We use arbitrary length DFTs for some things, notably for impulse response analysis, but since FFTs generally execute much faster, they are very much preferred for real-time operations in particular, or any application where the accompanying restrictions on the precise length of the time record are not a problem.

Fourier analysis. The discrete Fourier transform (DFT or FFT) analyzes a complex time-domain signal to find the magnitude and phase of the component sinewaves that make up the complex waveform. The magnitude of each component sinewave can be plotted on a frequency-domain graph to form a picture of the spectral content of the complex signal – the phase data is really only of interest if we have a reference signal to compare it to, or want to synthesize a replica of the original time-domain signal using an inverse Fourier transform (IFT).