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overview
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A quick read of Harry Nyquist's classic 1924 Bell System Technical Journal article isn't enough to understand what the code that bears his name really means. At the time, Nyquist was working on the transmission of telegraph signals over bandwidth-constrained channels. When dealing with data sampling systems, a detailed understanding of modern interpretations of the Nyquist criterion is essential. This guide will explain how the Nyquist criterion applies to baseband sampling, undersampling, and oversampling applications in an easy-to-understand manner
overall architecture process
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It is a typical real-time data sampling system block diagram. Before the actual analog-to-digital conversion, the analog signal usually passes through some kind of signal conditioning circuit that amplifies, attenuates, and filters. Whereas to remove interfering signals outside the bandwidth of interest and prevent aliasing, a low-pass/band-pass filter is required
Explanation of technical terms
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ADC:
Analog-to-digital converter, or A/D converter, or ADC for short, usually refers to an electronic component that converts an analog signal into a digital signal. A common analog-to-digital converter converts an input voltage signal into an output digital signal. Since the digital signal itself has no practical significance, it only represents a relative size. Therefore, any analog-to-digital converter needs a reference analog quantity as a conversion standard, and the more common reference standard is the largest convertible signal size. The output digital quantity represents the magnitude of the input signal relative to the reference signal
technical details
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The system shown is a real-time system, that is, it continuously samples the signal input to the ADC at a rate of fs, and the ADC provides new samples to the DSP at the same rate. To maintain real-time operation, the DSP must complete all necessary calculations within the sampling interval (1/fs) and provide output samples to the DAC before the next sample from the ADC arrives. A typical example of a DSP function is a digital filter. Note that the DAC is only required if the DSP data must be converted back to analog (for example, in voiceband or audio applications). In many applications, after the initial analog-to-digital conversion, the signal still exists entirely in digital format. Similarly, in some applications, the DSP is only responsible for generating the input signal to the DAC. If a DAC is used, it must be followed by an analog anti-imaging filter to remove image frequencies. Finally, in some slower industrial process control systems, the sampling rate is much slower. Regardless of the system, the basics of sampling theory still apply
The actual analog-to-digital and digital-to-analog conversion process involves two key concepts: discrete-time sampling and limited amplitude resolution due to quantization. This tutorial discusses discrete-time sampling.
The general data sampling system shown assumes some type of AC signal at the input. It should be noted that this is not a hard requirement (for example, a modern digital voltmeter (DVM) or ADC optimized for DC measurements), but this discussion assumes that the input signal has a certain upper frequency fa. Most of today's ADCs have a built-in sample-and-hold function, which allows them to handle ac signals. Such ADCs are called "sampling ADCs". However, many early ADCs, such as the industry standard AD574 from Analog Devices, were not of this type of sampling, but simply encoders as shown. If the input signal to a SAR ADC (assuming no SHA capability) changes by more than 1LSB within a conversion time (eg 8µs), the output data will have large errors depending on the code position. With the possible exception of flash converters with matching comparators, most ADC architectures suffer from this type of error to some degree.
This means that even though the 8µs ADC supports a sampling frequency of 100kSPS (thus giving an extra 2µs to allow the external SHA to reacquire the signal after exiting hold mode), there will still be conversion errors once the input frequency exceeds 9.7Hz. In order to process AC signals, a sample-and-hold (SHA) function needs to be added, as shown in the figure. The ideal SHA is a simple switch driving a hold capacitor followed by a high input impedance buffer. The input impedance of the buffer must be high enough so that the capacitor can discharge in less than 1LSB during the hold-up time. SHA samples the signal in sample mode and holds the signal constant during hold mode. Also adjust the timing so that the encoder can perform transitions within the hold time. Therefore, a sampling ADC can handle fast signals, and the upper frequency limit depends on the SHA aperture jitter, bandwidth, distortion, etc., not the encoder. In the example given, the sample-and-hold function acquires the signal in 2µs and the encoder transitions in 8µs, resulting in a total sampling period of 10µs. In this way, the sampling frequency is equal to 100kSPS, and can handle the input frequency up to 50kHz.
It is important to understand the subtle differences between a true sample-and-hold amplifier (SHA) and a track-and-hold amplifier (T/H or THA). Strictly speaking, the output of the sample-and-hold function is not defined during sample mode, but the output of the track-and-hold function tracks the signal during sample or track mode. In practice, the function is typically configured as track-and-hold, and the terms "track-and-hold" and "sample-and-hold" are often used interchangeably. The waveforms shown in the figure are those related to the track-and-hold function.
summary
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Continuous analog signals are sampled at discrete time intervals ts=1/fs, which must be carefully chosen to ensure that the sampled data accurately describe the original analog signal. Obviously, the more samples taken (and the faster the sample rate), the more accurate the digital representation of the analog signal will be. If there are fewer samples (slower sampling rate), there will be a point at which critical information of the analog signal will be lost due to lack of samples. While working at Bell Telephone Laboratories, Harry Nyquist published two classic papers in 1924 and 1928 that laid the mathematical foundations of sampling. Shortly thereafter, RVLHartley added to Nyquist's original work. These papers formed the basis for work on PCM in the 1940s, before Claude Shannon wrote his classic paper on communication theory in 1948