Bandwidth, sampling rate, and memory depth: three key indicators of oscilloscopes explained in detail

Bandwidth, sampling rate and memory depth are the three key indicators of digital oscilloscopes. Compared with engineers' familiarity and attention to oscilloscope bandwidth, sampling rate and memory depth are often overlooked in oscilloscope selection, evaluation and testing. The purpose of this article is to help engineers better understand the important characteristics of these two indicators and their impact on actual testing by briefly introducing the relevant theories of sampling rate and memory depth combined with common applications. At the same time, it helps us master the trade-off method of selecting oscilloscopes and establish the correct concept of using oscilloscopes.

Before we begin to understand the concepts related to sampling and storage, let us first review the working principle of the digital storage oscilloscope.

The input voltage signal is sent to the front-end amplifier after the coupling circuit, and the front-end amplifier amplifies the signal to improve the sensitivity and dynamic range of the oscilloscope. The signal output by the amplifier is sampled by the sampling/holding circuit and digitized by the A/D converter. After A/D conversion, the signal is converted into digital form and stored in the memory. The microprocessor processes the digitized signal waveform in the memory accordingly and displays it on the display. This is the working process of the digital storage oscilloscope.

Sampling, sampling rate

We know that computers can only process discrete digital signals. The first problem faced after the analog voltage signal enters the oscilloscope is the digitization (analog/digital conversion) of the continuous signal. The process from continuous signal to discrete signal is generally called sampling. Continuous signals must be sampled and quantized before they can be processed by computers. Therefore, sampling is the basis for digital oscilloscopes to perform waveform operations and analysis. The sampling of digital storage oscilloscopes is to measure the voltage amplitude of waveforms with equal time intervals and convert the voltage into digital information represented by eight-bit binary codes. The smaller the time interval between the sampled voltages, the closer the reconstructed waveform is to the original signal. The sampling rate is the sampling time interval. For example, if the sampling rate of the oscilloscope is 10G times per second (10GSa/s), it means that a sample is taken every 100ps.

Figure 2 Oscilloscope sampling

According to the Nyquist sampling theorem, when sampling a band-limited signal with a maximum frequency of f, the sampling frequency SF must be greater than twice f to ensure that the original signal can be completely reconstructed from the sampled values. Here, f is called the Nyquist frequency, and 2f is the Nyquist sampling rate. For a sine wave, at least two samples are required per cycle to ensure that the digitized pulse sequence can accurately restore the original waveform. If the sampling rate is lower than the Nyquist sampling rate, aliasing will occur.

Figure 3 Sampling rate SF < 2 f, aliasing distortion

The waveforms shown in Figure 4 and Figure 5 look very similar, but the frequency measurement results are very different. Which one is correct? A careful observation will reveal that the trigger position and trigger level in Figure 4 do not correspond, and the sampling rate is only 250MS/s, while the sampling rate in Figure 5 is 20GS/s. It can be determined that the waveform shown in Figure 4 deceives us. This is an example of aliasing caused by too low a sampling rate.

Therefore, in actual measurements, for higher frequency signals, engineers should always keep an eye on the oscilloscope's sampling rate to prevent the risk of aliasing. We recommend that engineers fix the oscilloscope's sampling rate before starting the measurement to avoid undersampling. LeCroy oscilloscopes provide this option in the Time Base menu for easy setting.

From the Nyquist theorem, we know that for an oscilloscope with a maximum sampling rate of 10GS/s, the highest frequency that can be measured is 5GHz, which is half of the sampling rate. This is the digital bandwidth of the oscilloscope, and this bandwidth is the upper frequency limit of the DSO. The actual bandwidth cannot reach this value. The digital bandwidth is derived from theory and is the theoretical value of the DSO bandwidth. It is a completely different concept from the oscilloscope bandwidth (analog bandwidth) that we often mention.

So in an actual digital storage oscilloscope, for a specific bandwidth, what sampling rate should be selected? It is usually related to the sampling mode used by the oscilloscope.   

Sampling Mode

When the signal enters the DSO, all input signals need to be sampled before A/D conversion. Sampling techniques are generally divided into two categories: real-time mode and equivalent-time mode.

The real-time sampling mode is used to capture non-repetitive or single-shot signals, using fixed time intervals for sampling. After a trigger, the oscilloscope continuously samples the voltage and then reconstructs the signal waveform based on the sampling points.

Equivalent-time sampling is to sample a periodic waveform in different periods, and then splice the sampling points to reconstruct the waveform. In order to obtain enough sampling points, multiple triggers are required. Equivalent-time sampling includes sequential sampling and random repeated sampling. Two prerequisites must be met to use the equivalent-time sampling mode: 1. The waveform must be repeated; 2. It must be able to trigger stably. 

The bandwidth of the oscilloscope in real-time sampling mode depends on the maximum sampling rate of the A/D converter and the interpolation algorithm used. That is, the real-time bandwidth of the oscilloscope is related to the A/D and interpolation algorithm used by the DSO.

Here we mention the concept of real-time bandwidth, which is also called effective storage bandwidth. It is the bandwidth of a digital storage oscilloscope when it adopts real-time sampling. So many bandwidth concepts may drive you crazy. Let me summarize it here: the bandwidth of a DSO is divided into analog bandwidth and storage bandwidth. Usually, the bandwidth we often talk about refers to the analog bandwidth of the oscilloscope, that is, the nominal bandwidth on the oscilloscope panel. The storage bandwidth is the theoretical digital bandwidth calculated according to the Nyquist theorem, which is only a theoretical value.

Usually we use effective storage bandwidth (BWa) to characterize the actual bandwidth of DSO, which is defined as: BWa = maximum sampling rate / k. The maximum sampling rate refers to the maximum real-time sampling rate for a single signal, that is, the maximum rate of the A/D converter; for a repetitive signal, it refers to the maximum equivalent sampling rate. K is called the bandwidth factor, which depends on the interpolation algorithm used by the DSO. The interpolation algorithms used by the DSO generally include linear interpolation and sinusoidal (sinx/x) interpolation. K is about 10 when using linear interpolation and about 2.5 when using sinusoidal interpolation. K=2.5 is only suitable for reproducing sine waves. For pulse waves, k=4 is generally taken. At this time, the effective storage bandwidth of a DSO with a sampling rate of 1GS/s is 250MHz.

Figure 6 Waveform display of different interpolation methods

The theoretical relationship between interpolation and maximum sampling rate is not the focus of this article. We only need to understand the following conclusions: When using sinusoidal interpolation, in order to accurately reproduce the signal, the oscilloscope's sampling rate must be at least 2.5 times the highest frequency component of the signal. When using linear interpolation, the oscilloscope's sampling rate should be at least 10 times the highest frequency component of the signal. This also explains why the maximum sampling rate of an oscilloscope is usually four times or more of its rated analog bandwidth when used for real-time sampling.

After discussing the sampling rate, there is another concept closely related to the A/D of the DSO, which is the vertical resolution of the oscilloscope. The vertical resolution determines the minimum voltage increment that the DSO can distinguish, and is usually expressed by the number of bits of the A/D n. As mentioned earlier, the A/D converters of the current DSO are all 8-bit encoded, so the minimum quantization unit of the oscilloscope is 1/256, (2 to the 8th power), which is 0.391%. It is very important to understand this. For the voltage amplitude measurement, if the current vertical scale of your oscilloscope is set to 1v/div, it means that your measured value has an error within 8V*0.391%=31.25mV, which is normal! ! ! Because the voltage less than 31.25mV can no longer be distinguished by the oscilloscope at this level, if only 4 bits are used, the measured error is even more amazing! Therefore, it is recommended that when measuring the waveform, adjust the waveform as much as possible to make it fill the entire screen and make full use of the 8-bit resolution. We often hear engineers complain that the oscilloscope cannot accurately measure his voltage or the measurement results are inconsistent. In fact, in most cases, the engineers have not yet understood the impact of the oscilloscope's vertical resolution on the measurement results. By the way, regarding the measurement accuracy of the oscilloscope, one thing must be clarified - the oscilloscope itself is not a measuring instrument! ! ! It is the "engineer's eyes" to help you understand the characteristics of your circuit more deeply. Advertisement: Engineers who often do power supply measurements or ripple measurements, or want to understand the quantization error of the oscilloscope in depth, you can refer to an article on my colleague Frankie's blog "Oscilloscope is not a vertical measurement tool" https://blog.sina.com.cn/s/blog_521262a301009ryp.html