42 Fourier Transforms
With a Fourier Series that includes enough terms, you can model complex signals or multidimensional shapes as combinations of many sine and cosine signals. (video 8:24)
This is a mathematically difficult process that you don’t need to master in order to take advantage of the concepts. Signals contain high frequency elements related to the sudden changes and lower frequency elements related to changes that take place over time. You can also go in the opposite direction to decompose a time series signal into many Fourier components to identify dominant frequencies.
Finite Fourier Transforms of Time Series
If you have a finite number of data points in a time series, then you can generate a curve that runs through each of the points with a finite number of terms in a Fourier Series. The ability to draw the curve is not especially useful, as there are lots of easier ways to draw a curve through some data, however, a really valuable byproduct comes in determining the size of the contribution from different frequency components.
In a simple system we might find the natural frequency of oscillation by counting zero crossings, or peaks in the oscillating signal in response to a step function input. In a more realistic situation we may not be able to observe a step function response and there may be multiple frequencies present. For machine condition monitoring we might use dedicated hardware with the FFT capability built in. Changes in frequency characteristics give an early warning diagnostic and let us take a machine offline for maintenance before it fails – crucial for high reliability systems. (video 1:48)
https://www.youtube.com/watch?v=qz0MLVh7Gok
Fast Fourier Transform (FFT) Functions
If your time series data is uniformly spaced in time and the number of points is a power of 2, e.g. 64, 128, 256, 512, etc. points, you can take advantage of some symmetry to perform the transform calculations quickly (Cooley-Tukey 1965 if you really want details). Most engineers find these algorithms very useful without need to know their inner workings. The full FFT is done with complex (real and imaginary) numbers with a full floating point calculation that maintains phase information (separate sine and cosine series) while integer implementations that ignore the phase can run much faster. This video shows an example based on this code that shows an integer calculation completing about 20 times faster on a SAMD M0 processor. (video 7:01)
The signal at left shows three distinct frequencies, followed by the FFT trace in the middle shows three matching peaks. The red line is a full floating point complex FFT, while the green is the much faster 16 bit integer real version.
Choosing Sampling Parameters
The sampling frequency for the data defines the maximum detectable frequency. Since each cycle of a sine wave would require at least one point up and one point down to be visible, the maximum detectable frequency is half the sampling frequency, refer