Nyquist -Shannon采样定理：超过Nyquist率

In thefirst articleof this series, we explored this concept by thinking in the time domain, and in the第二篇文章，我们从频域的角度进行了处理。

Now, we need to consider this theorem’s role in guiding the decisions of electrical engineers whose goal is to design functional circuits and systems.

shannon’s sampling theorem states the following:

If a system uniformly samples an analog signal at a rate that exceeds the signal’s highest frequency by at least a factor of two, the original analog signal can be perfectly recovered from the discrete values produced by sampling.

Theory informs practice but does not specify it. In other words, Shannon’s theorem doesn’t tell us how to design a sampled system; rather, it helps us tounderstandsampled systems and provides a framework that orients and supports the work of the engineer. Thus, it’s important to know where theory and practice diverge, and in the case of sampling theory, perhaps the most important divergence is that of the required sampling rate.

采样和别名

在上一篇文章中，我们看到在采样频率时发生混溶发生（f_s）小于最大信号频率的两倍（f_MAX），使得物质重叠。

I think that most of us naturally interpret the term “aliasing” as inherently negative, i.e., as a potential problem that must be avoided. However, aliasing in the broader sense is an integral part of converting a signal from a continuous waveform into a sequence of discrete values.

我一直在使用“ sexpectra”一词来参考通过采样创建的光谱副本，但更正式的名称就是aliases。

We create aliases—i.e., the original signal frequencies “disguised” as different frequencies—every time we perform analog-to-digital conversion, regardless of the sampling rate. When sampled data are converted from digital back to analog, these aliases become part of the analog signal, and consequently, D/A conversion results in an analog signal that is not identical to the sampled signal. Thus, if we wish to perfectly reconstruct the original analog signal, we must eliminate the effect of the aliases.

如我们所知，为了防止别名引起的信号损坏，我们需要按照或高于奈奎斯特率进行样品。如果我们不遵守这一基本要求，那么我们就不会有机会反对这些别名，甚至在我们浏览采样数据的时候，这些别名已经与原始频谱永久混合在一起。我们无能为力将真实频率与冒名顶替者区分开。

but signal reconstruction only begins with adequate sampling frequency. A second fundamental requirement is low-pass filtering.

reconstruction via Filtering

如果我们以高于奈奎斯特的速度进行样品，我们仍然有别名，但是现在真实频谱和别名光谱之间存在差距：

This allows us to recover the original signal by converting the digitized waveform into an analog waveform and then applying a low-pass filter:

The low-pass filter that is intended to eliminate (or more realistically, mitigate) the effect of aliases in the reconstructed analog signal is called areconstruction filter。This is ananalog应用的过滤器D/A转换后。I’m using italics here to emphasize two things.

First, we can’t remove aliases by means of digital filtering. Aliases are inherent to the nature of sampled, quantized data and therefore can’t be eliminated in the digital realm (though oversampling and interpolation can make the analog filtering requirements less severe).

second, a reconstruction filter is designed to remove aliases, but it is not an anti-aliasing filter! The term “anti-aliasing filter” refers to a low-pass filter that is appliedbefore A/D conversion。

Ideal Filter vs. Real Filter

如果您暂时考虑上图，您可能会开始理解为什么香农定理不是设计混合信号电子系统的“操作方法”指南。如果我们将采样率降低到理论限制，则傅立叶变换看起来像：

在理想化的数学领域中，我们仍然可以将真实频谱与别名分开。但是，物理组件不能创建“砖墙”类型的频率响应类型，该频率响应将直接切成薄片，从而完美过滤掉不需要的频率内容：

Furthermore, we typically prefer to avoid the cost, complexity, and board space required for filters that come anywhere near the brick-wall response. Instead, we use过采样。

by sampling a signal at a rate that is much higher than the Nyquist rate, we ensure that there will be a large frequency gap between the authentic spectrum and the nearest alias. This large gap makes it much easier for us to build an effective reconstruction filter because the magnitude response can roll off slowly and still produce significant attenuation at the alias frequencies. With generous oversampling, even a first-orderRC低通滤波器can provide adequate alias suppression.

There’s no fixed rule for how much oversampling is needed in a given application, but I like to have a sampling rate that is at least five times higher than the highest signal frequency of interest. If your signal frequencies are close to the maximum sampling rate of your ADC, you may have to sample closer to the Nyquist rate and then devote more time and money to your reconstruction filter.

Conclusion

We’ve seen that Shannon’s sampling theorem needs to be adapted to the constraints of real-life circuit design. Though perfect reconstruction is mathematically possible when the sampling rate is equal to twice the highest signal frequency, this approach requires an idealized low-pass filter and is, therefore, not directly applicable to engineered systems.

Another important issue that I mentioned is the difference between a reconstruction filter and an anti-aliasing filter. We’ll discuss anti-aliasing filters in the next article.