FFTs and oscilloscopes: A practical guide
The FFT (Fast Fourier Transform) first appeared when microprocessors entered commercial design in the 1970s. Today almost every oscilloscope from high-priced laboratory models to the lowest-priced hobby models offer FFT analysis. The FFT is a powerful tool, but using it effectively requires some study. I'll show you how to set up and use the FFT effectively. We’ll skip the technical description of the FFT, because its already implemented in the instruments. Instead I'll focus on the practical aspect of using this great tool.
The FFT is an algorithm that reduces the calculation time of the DFT (Discrete Fourier Transform), an analysis tool that lets you view acquired time domain (amplitude vs. time) data in the frequency domain (amplitude and phase vs. frequency). In essence, the FFT adds spectrum analysis to a digital oscilloscope.
If you look at upper trace in Figure 1, you'll see an amplitude-modulated carrier that uses a trapezoidal pulse as the modulation function. If you look at the time-domain view in Fig. 1 and I ask you to tell me the bandwidth of the signal, you'd have a hard time. But take the FFT of this signal and you get another point of view. The signal has a linearly swept frequency and the bandwidth, marked by the cursors, is 4.7 MHz. That’s how the FFT adds to the capability of the oscilloscope, it provides another point of view for the same data.
Figure 1. The time domain view in the top grid shows a pulse modulated RF carrier while the frequency domain view in the lower grid shows a uniform distribution of the carrier frequency between 997 MHz and 1002 MHz.
FFT frequency span and resolution bandwidth
In your earliest circuits course, you learned the frequency (frequency domain) of a periodic signal is the reciprocal of the period (time domain). That same relationship appears throughout the FFT setup.
The best place to start with setting up the FFT is choosing the RBW (resolution bandwidth) because it's related to a single setting. The RBW (Δf) is the incremental step in displaying the FFT frequency axis. In the time domain, the sampling period determines the time between samples. In the frequency domain, RBW is the frequency difference between adjacent "cells" in the spectral view. The RBW is the reciprocal of the time domain record length, also called the capture time as shown in Figure 2. You can control this with the oscilloscope's horizontal scale or time/division setting. The acquisition duration in Fig. 1 is 20 µs. Thus, the spectrum view's RBW is the reciprocal of that number, or 50 kHz.
Figure 2. The resolution bandwidth of the frequency spectrum is the reciprocal of the time domain record length, or capture time.
The next step in the FFT setup is to determine the span of the frequency-domain view—the difference between the highest and lowest frequency in the FFT. Note that FFTs generally start at 0 Hz and go out to the span. This is very different from the RF spectrum analyzer, which I'll explain shortly.
The span in the FFT is one half of the oscilloscope's effective sample rate (Figure 3). The smallest time increment in the time domain—the sample period—determines the largest element in the frequency domain. Similarly, the smallest increment in the frequency domain is a function of the largest duration in the time record. This is that reciprocal relationship between the time and frequency domain.
Figure 3. The span of the frequency Spectrum is half of the oscilloscope's effective sample rate.
To achieve finer resolution in the frequency domain, you must increase the amount of data captured by increasing the time/division setting. That's the opposite of what you would do to increase the time resolution in the oscilloscope's time-domain view.
From a practical point of view, the time domain record length is set by the oscilloscope's time/division control. Once you have selected that to achieve the desired resolution bandwidth, the only way to control the sample rate to achieve the required span is to change the oscilloscope's acquisition memory length. Now that may seem complicated, and it can be.
In the recent past, most high-end oscilloscope manufacturers have changed the FFT user interface to more closely resemble a standard RF spectrum analyzer, setting center frequency and span with resolution bandwidth as a parameter. While this type of interface makes the FFT easier to use, it does hide the underlying functionality of the FFT resulting in having to accept the combination of time/division, sample rate, and memory length set by the oscilloscope. With the rules discussed in this section, you can manually set up the FFT and achieve a great deal more freedom in your settings.