Six Principles of FFT Analysis Using Real-time Oscilloscopes

Figure 1. A 100 MHz sine wave in the time domain
and its spectrum in the frequency domain showing
the one peak at 100 MHz. Click on any image to enlarge.

By Prof. Eric Bogatin,
Teledyne LeCroy Fellow

The following piece was published in Signal Integrity Journal and is excerpted here by permission of Signal Integrity Journal.

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We live in the time domain. This is where we measure all digital performance. But sometimes, we can get to an answer faster by taking a detour through the frequency domain. With these six principles, we can understand how an oscilloscope transforms time domain measurements into a frequency domain view. All six principles are applied “under the hood” by oscilloscopes with a built-in FFT function. (Our note: Also by software packages designed for spectral analysis, such as the SPECTRUM-1 and SPECTRUM-PRO-2R options.)

1. The spectrum is a combination of sine wave components

In the frequency domain, the only waveforms we are allowed to consider are sine waves. There are other special waveforms combinations of which can describe any time-domain waveform, such as Legendre polynomials, Hermite polynomials or even wavelets. The reason we single out sine waves for a frequency domain description, is that sine waves are solutions to second order, linear, differential equations—the equations found so often in electrical circuits involving resistor, capacitor and inductor elements. This means signals that arise or have interacted with RLC circuits are described more simply when using combinations of sine waves than any other function because sine waves naturally occur. 

Back to Basics: What is an FFT?

Figure 1: An FFT of a 300-kHz square wave.

In an earlier post, we discussed the basics of setting up a fast-Fourier transform (FFT) on an oscilloscope, and why you'd want to use an FFT to get a frequency-domain view of a time-domain signal in the first place. It might be a good idea to take a step back and dig into just what an FFT is (Figure 1).

Oscilloscope Basics: Setting Up FFTs

Figure 1: Capture time determines the
frequency resolution, Δf.

For most of their history, oscilloscopes have been thought of chiefly as a time-domain instrument. That is, an oscilloscope facilitates the observation of changes in a signal's amplitude over time. However, many modern digital and mixed-signal oscilloscopes provide spectral analysis capabilities based on fast Fourier transforms (FFTs) that convert a time-domain waveform into the frequency domain. There are lots of good reasons for taking advantage of this capability. Perhaps the most important is to gain insight into characteristics of the signal that simply are not apparent from a time-domain perspective.