mathematics//signal processing//Fourier analysis

Fourier analysis is the mathematical method that writes a signal as a sum of sinusoids of different frequencies, each with its own amplitude and phase, and engineers use it to find which causes are present in a measurement: in a rotating machine, faults that are tangled together in time separate cleanly in frequency. A vibration record from a pump looks like noise; its spectrum shows a peak at the shaft speed (imbalance), one at twice that speed (misalignment or looseness) and, with the right processing, a line at the rate at which balls strike a damaged race (vibration analysis).

Fourier analysis is the mathematical method that writes a signal as a sum of sinusoids of different frequencies, each with its own amplitude and phase, and engineers use it to find which causes are present in a measurement: in a rotating machine, faults that are tangled together in time separate cleanly in frequency. A vibration record from a pump looks like noise; its spectrum shows a peak at the shaft speed (imbalance), one at twice that speed (misalignment or looseness) and, with the right processing, a line at the rate at which balls strike a damaged race (vibration analysis).

The spectrum says how much of each frequency the signal holds, and it is computed on recorded samples by the discrete Fourier transform. For a signal whose character changes over time (a machine running up, a fault that appears), short spectra are computed window by window and stacked into a time-frequency image, the spectrogram. Both are reading tools; a digital filter is the shaping tool that acts on the same frequency picture.

Sinusoids are the natural language of linear systems.

A sine that enters a linear time-invariant system comes out a sine of the same frequency, only scaled and shifted, exactly as an eigenvector of a matrix AAA comes out only stretched (modes, eigenvalue). So a system can be described by what it does to each frequency, its frequency response, and a signal's spectrum times that response is the output's spectrum.

That property is why the frequency domain dominates control and filtering. Gain and phase per frequency are the Bode plot; a delay is a phase that grows with frequency; the Laplace transform extends the same idea to growing and decaying signals and gives the transfer function.

The idea reaches beyond time signals. The eigenvectors of a graph's Laplacian play the role of sinusoids on a network, from smooth patterns to ones that alternate between neighbours (graph Laplacian), and reading any system by its natural modes is a pattern of its own (spectral reading).

Its boundary is stationarity. The transform assumes the signal keeps its character within the window; a short transient spreads over all frequencies and is clearer in time, and a machine changing speed smears its lines unless the axis is normalised by shaft speed (orders).

It is often more than the problem needs. When only the overall level of vibration matters, its RMS value is an excellent tool, and it is what severity standards measure; the spectrum pays off when the question is what is failing (flyswatter rule).