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. 

How to Use Memory Properly

In a recent post, we addressed setting sample rate for serial data acquisition, but let’s look again at how time per division (time/div), memory length and sample rate all interact, and what you can do to optimize your use of oscilloscope capture memory when setting up your timebase.

Figure 1: Sample rate as a function of time/div for three different memory lengths.
Longer memory extends the range of time/div settings that support the highest sample rate.

About Data Truncation in Fast Fourier Transforms

Figure 1: The first precondition of using the Fourier transform
is a repetitive signal

Our last post discussed how time-domain signals acquired by an oscilloscope might be translated into the frequency domain using the discrete Fourier transform. We noted that using the Fourier transform only works if our signal is repetitive (Figure 1), and that it allows us to identify only harmonics of the first harmonic frequency, which is 1/acquisition window. Moreover, the discrete Fourier transform, if used on a large number of data points, is relatively slow to calculate.

How Much Transmission-Line Loss is Too Much?

Figure 1: This plot represents the differential insertion-loss
profile for a 20" FR-4 microstrip trace

One of the fundamental facts of transmission lines is losses. Any effort to debug the performance of a high-speed serial data link begins there. But it begs an equally fundamental question: How much loss in a transmission line is too much? How do we quantify losses, and what is the connection between attenuation at the Nyquist frequency and the eye diagram? Is there a rule of thumb one might apply, some sort of rough estimate of how much loss might be too much for your channel to bear at a given data rate?

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.

Oscilloscope Basics: Sampling Rate

In a recent overview post on oscilloscope banner specifications,
one of the topics covered is sampling rate. Let's do a somewhat deeper dive on that topic and look at what sampling rate means to oscilloscope users.