Fast Fourier Transform

Audio is a complex signal.
It is a mix of different frequencies.
A Fourier Transform can decompose this signal in individual frequencies.
Likewise if you sum these individual frequencies you get the original sound.

The fast Fourier transform (FFT), is a computer algorithm that computes the discrete Fourier transform much faster than other algorithms

 

FFT graphs are used to display the frequency content of a small sample.

It displays the frequency content (Hz) and the intensity in dB.

You can analyze the performance of a DAC using FFT.

Send a 11 kHz test signal to the DAC and analyze the frequency content of the analog out.

 

TFFT (Temporal Frequency Analysis) plots the frequencies over time.

A color intensity scale represents the intensity of the frequency.

Demo

Paul Falstad wrote an applet demonstrating how very different functions like a triangle or a square can be generated from a sum of cosine terms.

 

References

1. The Fundamentals of FFT-Based Audio Measurements in SmaartLive - Paul D. Henderson
2. Fast Fourier transform - WikipediA
3. WavePad FFT Sound Analyzer - NCH
4. Mastering The Fourier Transform in One Day - Bernsee
5. Fourier applet - Paul Falstad