Comparison of waveform reconstruction methods

When using electronic measuring instruments, waveform viewing is a commonly used function, so how do you generally achieve waveform acquisition and reconstruction? Two typical instruments in acquisition methods are oscilloscopes and power analyzers. Briefly introduce the common waveform acquisition methods of transient and steady-state measuring instruments.

According to the (Nyquist) sampling theorem, the sampling frequency of the reconstructed waveform that can be completed should be at least 2 times the signal frequency, and when the oscilloscope sampling rate exceeds 2 times the measured signal frequency, the oscilloscope "scanning" acquisition is far enough Sample points to construct accurate images. This is the common acquisition method of digital oscilloscopes-real-time sampling. Real-time sampling is a way to use an oscilloscope to capture fast signals, single-shot signals, and transient signals.

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When the sampling process does not meet the (Nyquist) sampling theorem, you can consider using another acquisition method-equivalent sampling. The basic principle of equivalent sampling is to convert high-frequency and fast signals into low-frequency, slow-repetitive signals for collection. In order to achieve the purpose of low-speed sampling and restoration of high-frequency signals, the measured signal must be periodically changed. If each sampling point is arranged in a different signal period and taken from a different position in the waveform, instead of in the same period, The sampling frequency can be greatly reduced. Through mathematical methods, the sampling points in multiple cycles are restored to one cycle, and the measured signal is reconstructed.

Such equivalent sampling can use a sampling frequency lower than twice the frequency of the original signal to sample without distortion and restore the original signal, which is suitable for sampling and analysis of high-frequency periodic signals. For example, when measuring high-frequency signals, when the sampling rate is not enough, you cannot collect enough samples in the scan. Equivalent time sampling can be used to accurately collect signals whose frequency exceeds the acquisition rate/2.5. Equivalent time sampling constructs an image of a repetitive signal by capturing a small amount of information from each repetition. The waveform is slowly constructed, like a string of lights, which lights up one after another. The oscilloscope can accurately capture signals whose frequency components are much higher than the sampling rate of the oscilloscope.

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Equivalent sampling can be divided into sequential equivalent sampling and random equivalent sampling.

Sequential equivalent sampling is to capture a sample value at an interval of K cycles, and obtain a sample value after a slight delay Δt after each k cycles. Assuming that k=1, the equivalent sampling and reconstruction process of sampling N points per cycle. Piece the collected data into one cycle to realize the reconstruction of the original input signal waveform. The reconstructed sampling frequency becomes the reciprocal of the small delay Δt. By controlling the size of this Δt, the frequency of equivalent sampling can be controlled. The actual sampling frequency can be adjusted by controlling the size of K. The larger the K, the smaller the actual sampling frequency; and the smaller the △t, the higher the equivalent sampling frequency. This achieves the purpose of low-speed sampling of high-frequency signals.

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Figure 1 Sequential equivalent sampling

Random equivalent sampling uses an internal clock, which is not synchronized with the input signal and the clock triggered by the signal. The sample value is continuously obtained, and is independent of the trigger position. By recording the time difference between the sampling data and the trigger position, the position of the sampling point in the signal is determined to reconstruct the waveform. This creates the problem of accurately measuring the position relative to the sampling trigger point. Although the sampling is continuous in time, it is random relative to the flip-flop, which gives rise to the term "random" equivalent time sampling.

Different from oscilloscopes, instruments that analyze steady-state signals, such as power analyzers, can use the concept of equivalent sampling to perform sampling operations. It is required that the measured signal must be a stable periodic signal, otherwise the measurement result will have a relatively large error. Therefore, the power analyzer is a steady-state measuring instrument with weak transient analysis capability.

When the instrument uses a fixed sampling rate for sampling, the sampling point may appear at a fixed position on the measured signal. Therefore, consider introducing random sampling, that is, dynamically modifying the sampling rate, so that the method can guarantee the measured signal and the sampling rate. There is no integer multiple relationship between them, which can prevent the sampling point from appearing on the fixed position of the signal under test and leading to inaccurate measurement results.

When the sampling rate of the instrument is lower than the frequency of the input signal, the high frequency components contained in the signal will be lost. At this time, according to the sampling theorem of, there will be a phenomenon that the high-frequency components in the instrument signal are mistakenly processed into low-frequency data. This phenomenon is called aliasing. Random sampling is to solve the problem of aliasing.

Realization principle: By increasing the sampling rate, and then sampling randomly, it will eventually become the same as the sampling rate of the instrument. This is equivalent to moving sampling.

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Figure 2 Frequency aliasing phenomenon

When the relationship between the frequency of the measured signal and the sampling rate is an integer multiple, for the entire update period, the sampling rate is fixed at a fixed position of the measured signal, and the amount of information obtained by sampling the measured signal is very high. For a limited part, it is impossible to obtain all the information of the measured signal, and as a result, measurement inaccuracy and jitter will occur. When there is no integer multiple relationship between the sampling rate and the measured signal, the sampling point will appear on all the waveforms of the measured signal with equal probability according to the time of each cycle, and then all the effective information of the measured signal can be obtained. Calculate the accurate measurement result of the measured signal.

For example, when the input signal frequency is higher than 100kHz, a sampling point of 5 microseconds (200kHz) is not enough to describe an input signal cycle. However, the envelope formed by multiple cycle sampling points is consistent with the input signal amplitude, and the frequency is reduced. The measured effective value is consistent with the actual effective value, so signals higher than fs/2 can be measured. This process of aliasing can be simply understood as "frequency conversion". However, it is necessary to avoid the situation where the signal frequency is an integer multiple of fs/2, fin= n * (fin/2), n=1,2,3... At 200kHz, these frequencies are 100kHz, 200kHz, 300kHz..., around these frequencies, random sampling is required.

In addition, the power analyzer cannot fully reflect the details because the display pixels are much smaller than the number of sampling points in one update cycle. When the power analyzer displays the waveform, it provides two extraction methods: equal interval extraction and peak extraction;

Equal interval sampling is the sampling point 2M, the number of display points is 2k, and the actual display is 1k point sampling, that is, one point is displayed, the 1001th point, the 2001 point, and so on.

Peak extraction means that the sampling points are 2M, the number of display points is 2k, and the actual display is 2k points extracted, and the sum value is between 0 and 2000 points, and then the sum value is between 2001 and 4000, and so on, and the waveform is reconstructed . As shown below:

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Figure 3 Two extraction methods of power analyzer

In summary, it can be seen that oscilloscopes that measure signal transients are more inclined to depict more "dense" points and display stronger transient capabilities; in contrast to oscilloscopes, power analyzers for steady-state measurement do not The dead zone mainly focuses on the effective value measurement, and the waveform extraction focuses on depicting the characteristics and amplitude of the signal. This method is more suitable for steady-state applications. In addition, with the advancement of technology at any time, ZLG Zhiyuan Electronics’ PA8000 power analyzer has a basic power accuracy of 0.01%, a bandwidth of 5M, and a sampling rate of 2MS/s. There has been a qualitative leap in waveform acquisition and capabilities. Get closer to the oscilloscope to show more powerful capabilities.