Oscilloscope sampling rate and memory depth

When selecting an oscilloscope, engineers first need to determine the bandwidth required for measurement. However, once the bandwidth of the oscilloscope is determined, it is the sampling rate and storage depth that interact and restrict each other that affect the actual measurement. Figure 1 is a simplified diagram of the working principle of a digital oscilloscope.

Figure 1 Principle block diagram of digital storage oscilloscope

The input voltage signal first enters the front-end amplifier of the oscilloscope. The amplifier amplifies or attenuates the signal to adjust the dynamic range of the signal. The output signal 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 simple working process of the digital storage oscilloscope.

Sampling, sampling rate
Since computers can only process discrete digital signals, the first problem faced by analog voltage signals after entering the oscilloscope is the digitization (analog/digital conversion) of continuous signals.

The sampling of DSO is to measure the voltage amplitude of the waveform with equal time intervals and convert the voltage into digital information represented by 8-bit binary code (see Figure 2). The smaller the time interval between each two samples, the closer the reconstructed waveform is to the original signal. The sampling rate is the reciprocal of 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, 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.

According to the Nyquist theorem, for an oscilloscope with a maximum sampling rate of 10GSa/s, it can measure a signal with a maximum frequency of 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 completely different from the oscilloscope bandwidth (analog bandwidth) we often mention.

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

Sampling Modes
Sampling techniques are broadly 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 and uses fixed time intervals for sampling. After triggering once, the oscilloscope continuously samples the voltage and then reconstructs the signal waveform based on the sampling points.

Equivalent-Time Sampling is to sample the 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 oscilloscope works in real-time sampling mode most of the time, and the bandwidth of the oscilloscope depends on the maximum sampling rate of the ADC and the interpolation algorithm used. Therefore, the real-time bandwidth of the oscilloscope is related to the interpolation algorithm used by the DSO.

The effective storage bandwidth (BWa) is usually used to characterize the actual bandwidth of the DSO, which is defined as: BWa = maximum sampling rate / K. For a single signal, the maximum sampling rate refers to the maximum real-time sampling rate, 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 1GSa/s is 250MHz. 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. Generally speaking, the higher the sampling rate, the better.

Figure 3 Waveform display of different interpolation methods

Storage, storage depth
In an oscilloscope, the eight-bit binary waveform information after A/D digitization is stored in the high-speed CMOS memory of the oscilloscope, which is the storage of the oscilloscope. The capacity of the memory (storage depth) is very important. When the storage depth is constant, the faster the storage speed, the shorter the storage time, and the relationship between them is inversely proportional. Therefore:
Storage depth = sampling rate × sampling time

It can be seen that increasing the storage depth of an oscilloscope can indirectly increase its sampling rate: when you want to capture a longer waveform, since the storage depth is fixed, you can only reduce the sampling rate, but this will inevitably cause a decrease in waveform quality; if you increase the storage depth, you can use a higher sampling rate to obtain an undistorted waveform.

Therefore, memory depth determines the DSO's ability to analyze both high-frequency and low-frequency phenomena simultaneously, including high-frequency noise on slow-speed signals and low-frequency modulation on high-speed signals.

Once you understand the sampling rate and memory depth, it is very easy to understand the impact of these two parameters on actual measurements.

1 Importance of long storage in power supply measurement
In common switching power supply tests, the switching frequency is generally around 200kHz or faster. Since there is often power frequency modulation in the switching signal, engineers need to capture a quarter cycle or half cycle, or even multiple cycles of the power frequency signal. The typical rise time of the switching signal is about 100ns. To ensure accurate reconstruction of the waveform, there must be more than 5 sampling points on the rising edge of the signal, that is, the sampling rate must be at least 5/100ns=50MSa/s, that is, the time interval between two sampling points must be less than 100/5=20ns. For the requirement of capturing at least one power frequency cycle, it means that a 20ms long waveform needs to be captured, so the required storage depth per channel of the oscilloscope can be calculated = 20ms/20ns=1M. Similarly, in analyzing the maximum value of the voltage stress borne by the power device during the soft start of the power supply, it is necessary to capture the entire power-on process (more than ten milliseconds), and the required oscilloscope sampling rate and storage depth are even higher.

2 The impact of storage depth on FFT results
In DSO, the spectrum of a signal can be obtained through fast Fourier transform (FFT), and then a signal can be analyzed in the frequency domain. For example, the measurement of power supply harmonics requires the use of FFT to observe the spectrum. In the measurement of high-speed serial data, FFT is often used to analyze the noise and interference that cause system failure.

For FFT operation, the memory depth will determine both the maximum range of observable signal components (Nyquist frequency) and the frequency resolution △f. If the Nyquist frequency is 500MHz and the resolution is 10kHz, to obtain a resolution of 10kHz, the acquisition time must be at least: T = 1/△f = 1/10kHz = 100ms

For a digital oscilloscope with 1M memory, the highest frequency that can be analyzed is:
△f×N/2=10kHz×1M/2=5GHz

Therefore, long memory can produce better FFT results, increasing both frequency resolution and improving signal-to-noise ratio.

It should be pointed out that FFT analysis of long waveforms requires the oscilloscope to have super data processing capabilities, which often exceeds the computing limit of ordinary oscilloscopes. LeCroy oscilloscopes can perform up to 128M point FFT.

Figure 4 Eye diagram/jitter measurement of 18M data using LeCroy oscilloscope

3 High-speed serial signal analysis requires truly long storage
When using an oscilloscope for jitter testing, the high-speed acquisition memory length is a key indicator for the oscilloscope to perform jitter testing. The memory depth not only determines the number of samples in a jitter test, but also determines the jitter frequency range that the oscilloscope can test. For example, using an oscilloscope with a 20GSa/s sampling rate and 1M memory depth to capture a 2.5Gb/s signal, a 50μs long waveform can be obtained, which means that a 20kHz low-frequency jitter cycle can be captured. If the memory depth is increased to 100M at the same sampling rate, a 200Hz low-frequency jitter cycle can be captured.

In the eye diagram measurement, as the data rate of high-speed serial bus is getting higher and higher, the oscilloscope needs to have stronger data processing capabilities to perform real-time eye diagram analysis on a large number of data samples. For example, the eye diagram analysis of PCIE-G2 requires measuring 1M UI data at a time, capturing continuous 1M UI data samples, that is, 200μs. At a sampling rate of 40GSa/s, the required storage depth reaches 8M. The processing of this amount of data can easily cause the oscilloscope to process very slowly or even freeze! Therefore, some brands of oscilloscopes can only be completed with the help of software, but the efficiency of software in making eye diagrams is very low, and it is not a good tool for positioning and debugging.

At present, the fourth-generation oscilloscope based on the X-Stream II architecture released by LeCroy has first proposed the concept of "analyzable storage depth". With high sampling and long storage, its operation and eye diagram measurement speed is 2 to 50 times faster than other oscilloscopes! It can easily cope with the debugging and analysis of current and next-generation high-speed serial buses.