Fourier was obsessed with heat after his return to France from the warmer shores of Egypt where he partook in Napoleons' Campaign. This probably explains his fervent fascination in the phenonomem of heat transfer. In his principal composition, "The Analytic Theory of Heat", Joseph Fourier established the partial differential equations governing heat diffusion and solved it by using an infinite series of trigonometric functions, the Fourier series. A major development which revolutionised the computational implementation of the Fourier transform was the introduction of the fast Fouriertransform (FFT) by Cooley and Tukey in 1965, which enabled the implementation of the first real time spectral analysers. The FFT improved the computational efficiency
of the Fourier transform of a signal represented by n discrete data points, from an order of n x n to n x log(n) arithmetic operations. Despite the functionality of the Fourier transform, especially in regard to obtaining the spectral analysis of a signal, there are several shortcomings of this technique. The first of these is the inability of the Fourier transform to accurately represent functions that have non-periodic components that are localised in time or space, such as transient impulses. This is due to the Fourier transform being based on the assumption that the signal to be transformed is periodic in nature and of infinite length. Another deficiency is its inability to provide any information about the time dependence of a signal, as results are averaged over the entire duration of the signal. This is a problem when analysing signals of a non-stationary nature, where it is often beneficial to be able to acquire a correlation between the time and frequency domains of a signal. This is often the case when monitoring machine vibrations. Some typical examples include;
Engine start up or shutdown (Variable speed and resonant transients)
Vibrations from cranes or excavators (Variable load and speed, transient vibrations, slow rotational speeds)
Helicopter gear transmission systems (Variable transmission path, variable speed)
Internal combustion engines (Variable transmission path, variable speed)
To demonstrate the deficiencies of the Fourier transform, two examples are shown below. Figures 1 and 2 show how two signals, which bear no resemblance in the time domain can produce almost identical power spectrums. The blue line is a stationary signal created by excitation of a linear system by white noise, whereas the red line represents the same system being driven by an initial impulse. As it can be clearly seen it would be impossible to reconstruct or even determine the original time domain signals from their power spectrums.

TIME DOMAIN SIGNALS

POWER SPECTRUM