More about FFTs

Created on 2013-01-31 22:59:00

The Fast Fourier Transform, or FFT, is a mathematical technique that deconstructs a complex waveform into its component sine waves. Presented with a segment of data which represents the amplitude vs. time relationships of a complex waveform, an FFT will derive two sets of results: the amplitude spectrum (amplitude vs. frequency) and the phase spectrum (phase vs. frequency) of the signal. This information is sufficient to describe the relationships of the sine waves which make up the waveform, and with further computation the FFT results can be interpreted for several different types of display.

Much of the signal analysis in APx500 is performed using FFT techniques. These results are typically displayed in the frequency domain, as level vs. frequency or phase vs. frequency spectrum graphs. Some results (such as Impulse Response and Scope) are displayed in the time domain, as level vs. time.

FFT length

The length (in samples) of the audio data being transformed is called the FFT length, or record length. Since this length consists of samples, its duration in time varies with the sample rate. See Table of actual FFT Lengths.

FFT bandwidth

The bandwidth of the FFT is from DC to 1/2 the sample rate.

FFT bins and bin width

The bandwidth of the FFT is divided into bins, the number of which is 1/2 the FFT length. The bin width (also called line spacing) defines the frequency resolution of the FFT. The FFT provides amplitude and phase values for each bin. The bin width is stated in hertz. The bin width can be calculated by dividing the sample rate by the FFT length; or by dividing the bandwidth by the number of bins (which is equal to 1/2 the FFT length).

Example

As an example, at a sample rate of 48 kHz and an FFT length of 16,384 (the 16K setting), the bandwidth would be DC to 24 kHz, the number of bins would be 8,192, and the bin width would be 24 kHz / 8,192, or 2.93 Hz.

With care, you can select sample rates and FFT lengths that will result in a specific bin width; for example, an FFT length of 65,536 samples (APx menu setting of 64K) and a sample rate of 65,536 Hz (we’re assuming a digital input here, with the signal at that sample rate) provides a bin width of exactly 1 Hz.