Strictly, the Fourier transform is not a statistical technique. It is basically a solution of a large number of simultaneous equations. However, its application is strongly linked to both regression analysis and autocorrelation that we covered in previous chapters. As we will show, it offers the engineer another valuable diagnostic tool.

Most texts covering the Fourier transform do so in a highly mathematical way, making it difficult for the control engineer to identify where its application might be beneficial. Here we restrict its use to identifying cyclic disturbances to process measurements. Often such disturbances are not immediately obvious, presenting themselves as random noise. This is common problem with averaging level control. Applying this technique has identified many controllers that appear to be working well, using the available surge capacity, but the variation in level is in fact a very slow oscillation (disguised by process disturbances) caused by excessive integral action. The technique can also help diagnose control valve problems, such as stiction and hysteresis.

Since control engineers deal largely with process data collected at a fixed time interval we will focus on the *discrete Fourier transform (DFT)*. Strictly it is the *real DFT* where the input is restricted to real data. There is also an *imaginary DFT* but this has no application here.

Fourier showed that any signal can be decomposed into a number of sinusoidal signals. Each of these ...

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