Digital oscilloscopes: When things go wrong
DSOs (digital oscilloscopes) offer a great many advantages over their analog equivalents but as they say, "There’s no such thing as a free lunch." Digital scopes sample, digitize, and store waveforms and let you for measure, analyze, and archive signals. But, that sampling process brings a few issues along as "baggage."
Aliasing (this page), synchronous sampling (page 2), and interpolator (page 3) errors can cause you to misinterpret the measurement results unless you understand these issues. As you might expect, most DSO manufacturers don’t spend a lot of time talking about negative issues so learning about them is a discovery experience. Let's examine these problems and discuss how to detect and, hopefully, work around them.
The sampling theorem, which rules over all digital instruments and systems, requires that a signal be sampled at a rate that is greater than twice the maximum frequency contained in the signal. If the signal is properly sampled, then an oscilloscope can reconstruct it from the samples with no loss of information. Under sampling, or sampling at less than twice the highest frequency component, results in a recovered signal with lower component frequencies than the original signal, this unwanted signal is called an alias. Half of the sample rate is called the Nyquist Frequency, which marks the highest frequency that can be digitized at that sample rate.
Figure 1 provides an example of aliasing. The waveform in the upper left grid is a 400 MHz sine wave sampled at 1 (GSamples/s. There are 2½ samples per cycle as seen in the horizontally expanded zoom trace shown in the second grid from the top on the left side. Note that this is the raw sampled data with no interpolation. In the trace third from the top on the left side sin(x)/x interpolation has been applied. This is what most DSO's will display as this is their default display interpolator.
Figure 1. When a 400 MHz signal is undersampled, it loses signal fidelity and aliasing will occur.
The bottom trace on the left side is the FFT (Fast Fourier Transform) of the input signal showing the spectral or frequency domain view of the signal. It shows a spectral peak at 400 MHz as expected for this signal.
The waveform in the top right grid is the same 400 MHz sine sampled at 500 Msamples/s. The sample rate is below twice the signal frequency and the signal is aliased. The second grid from the top on the right side is the zoom view of the aliased trace. Note that the signal frequency is lower. In this case it is 100 MHz. The next lower trace is the aliased signal with interpolation applied. The FFT of the aliased trace has a frequency peak at 100 MHz. Note that the FFT trace is truncated at 250 MHz, the Nyquist frequency for the 500 MS/s sample rate.
Because Fig. 1 is an unanimated graphic, the aliased waveform appears to have a stable trigger, but but it doesn't. The trigger level is set for zero volts, and a positive slope and the non-aliased waveform shows the correct trigger level. The aliased waveform only has every other sample point of the non-aliased waveform and will hop between samples adjacent to the trigger point. This results in a trace with horizontal "jitter."
Probably the best way to investigate aliasing is to view it in the frequency domain. Sampling is similar to an analog mixing process. It essentially multiplies the sampled waveform by the sample clock, which is usually a very narrow pulse. The sampling clock is rich in harmonics. The sampling/mixing process produces frequency components that include the original baseband signal being sampled, the sample clock and all its harmonics, and lower and upper sideband images of the sampled signal about each sample clock harmonic as shown in the upper view in Figure 2.
Figure 2. The sampling process viewed in the frequency domain shows both correct and aliased sampling.
The baseband signal component approximates the frequency response of a typical DSO. The bandwidth is generally specified at the "knee" of the response with a rapidly attenuated "roll off" response above the bandwidth limit. Because there can be spectral components above the oscilloscope's bandwidth, most manufacturers sample at 2.5 times or greater than the bandwidth to prevent aliased components from this region.
Lowering the sample rate moves the sampling frequency component of the spectrum and all its harmonics to the left in the frequency domain display. Aliasing occurs when the lower sideband component about the sample frequency intersects the baseband signal as shown in the lower diagram. Once spectral components overlap, it's no longer possible to filter the resulting waveform to recover the original baseband signal.
Oscilloscope designers generally try to limit aliasing in several ways. First, they chose a maximum sampling frequency that is much greater than the minimum required over sampling. Rates of 3 to 20 times the Nyquist frequency are not uncommon. Next, they lengthen the acquisition memory. This keeps the sampling rate high even when long acquisitions are used. When choosing a DSO, you should know the maximum duration acquisitions you need to make and then choose an instrument with enough memory to support the required sampling rate for the bandwidth your signal requires.
Figure 3 illustrates how acquisition memory length affects the sample rate of an oscilloscope. This chart plots sample rate as a function of the oscilloscope's time/division setting with acquisition memory length as a parameter.
Figure 3. Chart of sample rate vs. time/division setting for a 1 GHz bandwidth scope with a maximum sample rate of 20 Gsamples/s. Note that once the sample rate falls to 2 Gsamples/s or lower, the oscilloscope will alias signals at 1 GHz.
The oscilloscope in this example has a maximum sample rate of 20 Gsamples/s and a bandwidth of 1 GHz. As long as the sample rate is above 2 Gsamples/s, the acquired data is valid. If sampling drops to exactly 2 Gsamples/s or less, the data may be aliased. The sampling rate stays at the maximum 20 Gsamples/s as the time/division setting is increased until all the acquisition memory is engaged. Beyond that point, the sample rate drops. So for an acquisition memory length of 10 ksamples, the sample rate falls to 2 Gsamples/s at 50 ns/division. With a memory length of 100 ksamples, the oscilloscope can reach 5 µs/division before the sampling rate falls to 2 Gsamples/s. As the acquisition memory increases the sample rate remains above the critical 2 Gsamples/s over more time/division settings. So the longer the acquisition memory, the less chance of aliasing.
When it comes to operating a digital oscilloscope, you should start with the fastest sweep speed available—the lowest time per division setting to detect and avoid aliasing. Doing so will result in the highest sampling rate. As you increase the time/division setting, keep your eyes on the waveform. If aliasing occurs, the waveform's frequency will drop suddenly; it's quite dramatic when it happens. If you do run into aliasing, see if you can increase the depth of the acquisition memory to increase the sampling rate.