# FAST-FOURIER-TRANSFORM ( FTT ) DESCRIPTION

By Wayne Pauly, Wayne Pauly, Inc.

Frequency is a key parameter in machinery diagnosis. Unbalance, mis-alignment, looseness, roller-bearing flaws, and many more common problems are easily recognized from a characteristic pattern in a frequency spectrum. Initially, tunable filters were used to manually sweep over the frequency range and readings were recorded; a very tedious and time consuming task!

Modern spectral conversion is accomplished using the Fast-Fourier Transform, or FFT for short. It has been known for a long time that any waveform can be transformed into its frequency spectrum, or Fourier series, by a long mathematical process called the Fourier Transform, but it was too computationally intensive and computers were too slow for it to be of much use. A mathematical shortcut called the Fast-Fourier-Transform or FFT was found that greatly reduced the computations needed and personal computers gained the speed necessary to perform the FFT quickly even on laptop computers.

The FFT transforms a waveform into a series of sines and cosines ( or amplitudes and* *phase-angles) at each frequency present in the original signal. Typically the waveform* *is* "*sampled"^{ }many times over its period, and the number of samples will affect the frequency resolution. It is a bi-directional transform and a waveform can be formed from the Inverse-Fourier Transform ( IFT ) of the spectrum if both the amplitude and phase angle is known at each frequency. A FFT takes approximately (n/2) log2(n) complex multiplications to complete, or about 5000 for a 1024 point FFT Vs over (1 million for the original Fourier Transform; a 1024 point FFT will take about 0.1 second or less on a modern personal computer.

WINDOWING. When taking samples of a periodic signal the starting and ending points of the samples will usually not line up and there will appear to be a step in the sample at that point Once a FFT is performed on the samples, there will be extra frequency components caused by the step and discrete frequency lines will smear together; this effect is referred to as spectral "leakage". It may or* *may not be significant depending on the type of waveforms sampled, but it can be greatly reduced by applying a "window" to the sample set. Having no window can be thought of as a rectangu1ar or flat-top window where all values are multiplied by one; if the window tapers to zero at both ends it will alleviate the effects of leakage, but will have other side-effects.

There are many common windows that are usually named after their originator, and usually are based on a cosine function with a bell shape; common ones are the Hamming, Hanning, and Blackman windows; they are just digital filters by another name.

ALIASING. Abasing is the effect where false spectral lines appear in a spectrum from frequencies above the measured bandwidth; it can be a major problem in data acquisition unless strong measures arc taken to prevent it. Frequencies above 1/2 the sampling frequency "fold-down into those below the 1/2 Fs point and are indistinguishable from real spectral lines. The critical 1/2 Fs frequency point is known as the Nyquist frequency.