Control Engineering

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How do Fourier transforms work?
November 19, 2007

The Fourier transform is a means to extract frequency information from a waveform. In control applications, it is useful for diagnosing resonance and oscillation problems in operating machinery. It is also the basis for digital filtering. The full-blown Fourier transform gives both frequency and phase information, but this is a simplified description that only looks at how frequency information is extracted.

Theoretically, Fourier’s theorem assumes infinitely long wavetrains. In practical situations, however, we just can’t wait that long, so we deal with the finite-length wavetrains we encounter in the real world. Exactly how we deal with the fact that the wavetrain is of finite length is called “windowing,” and there are various windowing functions that are beyond the scope of this blog entry. For now, we just assume that the wavetrain starts at a node (zero crossing) and appears at its full amplitude throughout the time period t, then ends at another node.

Start with two waveforms. The first is the waveform under test (WUT), from which we wish to extract frequency information. The second is a comparison waveform (CW) representing a single tone at the frequency f. Let’s say that the CW is chosen to have a node at the same time the WUT begins. There is no constraint on what the CW’s phase is at the ending time t.

If we line up the two waveforms on a time axis, we can compare their instantaneous values at any given time between the start (t₀) and end (t). For some intermediate time t_i we can multiply these values together to form the product 



Now, integrate this product over all time values between the start and end. Mathematicians call this operation "convolving" the two waveforms.

Again, the true convolution operation integrates from negative infinity to positive infinity, but we still can’t wait that long. Similarly, convolution involves an integral, but in the practical world of digital computers, we perform a sum using a great many (thousands) of closely spaced time values.

If the WUT happens to be a single tone that coincides with the CW frequency, the two waveforms are always in phase, and the product equals the root mean square of the product of the two waveform amplitudes—times a factor. For convenience, we normalize this result by multiplying the whole mess by another factor (the normalization factor) that makes it come out to unity.

That sounds pretty stupid on the surface—to go through all that work, then force the answer to be 1.0—but it will make more sense very soon. You see, we’ll be using this part of the calculation for comparison.

Suppose that we now relax the condition that the CW frequency be the same as the WUT frequency. If the frequencies are just a little bit different, the two waveforms start out in phase, then slowly go out of phase. Succeeding products in the sum contribute less and less compared to the comparison case. Eventually, when the phase difference reaches beyond 90°, they actually start subtracting from the sum because the sign of one waveform value has the opposite sign to the value of the other. The result is a sum that, when multiplied by the normalization factor determined above, is less than one.

Finally, to complete the transform, we plot the results for all values of CW frequency within some band of interest, such as dc (zero frequency) to 100 MHz. That graph represents the relative amplitude of the WUT as a function of frequency. By our assumption that the WUT represents a single tone, the Fourier transform shows a single spike at that frequency. If, on the other hand, the WUT is the superposition of two tones, two peaks will appear in the transform. Three tones make three peaks, and so forth.

One rather interesting Fourier transform application is finding resonances in existing mechanical or electromechanical systems. The engineer creates a waveform that represents the superposition of many equally spaced tones (called a “comb”). He or she then uses a transducer to drive this signal into the system. If the system is mechanical, the transducer might be a piezoelectric transducer. If it’s a feedback-controlled system, it might be a rapid change of set point.

In any case, the system responds to each frequency according to its own special frequency-response function. The engineer records that response as a waveform, then washes it through a Fourier transform to obtain a graph of the system’s response at all of the comb frequencies. The height of each peak signal's the frequency response value at that frequency. The graph’s resolution depends on the spacing between the comb frequencies. If the system has a resonance, the graph will show a tall peak at that frequency. If it’s unresponsive at some frequency, then there will be little response there.

It is possible to get even more sophisticated. For example, by monitoring the control inputs to a system and its responses during actual operation, then Fourier analyzing both and convolving the results, you can watch the system’s frequency response function change with time as bearings wear and components age. Then, use this information to modify control-system parameters and prevent drift over time.

Also read from Control Engineering:
Frequency Domain Analysis Explained 
and
Analyzing Control Loop Behavior in the Frequency Domain

Posted by Charlie Masi on November 19, 2007 | Comments (0)