奈奎斯特速率

在信号处理中，奈奎斯特速率（Nyquist rate，以哈里·奈奎斯特命名）等于某一函数或信号的最高频率（带宽）的两倍，即 $f_{\mathrm {NR} }=2f_{m}$ ，是不发生混叠失真的最低采样率。其单位是单位时间的采样次数，通常为每秒采样次数（Sps）或赫兹（Hz）^[1]。奈奎斯特速率对应的最大抽样间隔，称为奈奎斯特间隔。需要注意的是，“奈奎斯特速率”是连续时间信号的属性，而“奈奎斯特频率”是离散时间系统的属性。

“奈奎斯特速率”一词也用于不同的语境，以符号每秒为单位，表示带宽受限的基带信道（如电报线路）^[2]或通带信道（如受限的无线电频段或频分多路复用信道）中符号速率的上限。

与采样的关系

当以恒定采样率 $f_{s}$ sample/second对连续函数 $x(t)$ 进行采样时，总会有无限多的其他连续函数可以符合这些采样点。但其中只有一个函数的带宽限制为 ${\tfrac {1}{2}}f_{s}$ Hz，也就是 $|f|\geq {\tfrac {1}{2}}f_{s}$ 时它的傅里叶变换 $X(f)$ 为零。通常，用于从采样点重建连续函数的数学算法可以无限接近这一理论上的、但无限长的函数。因此，如果原始函数 $x(t)$ 的带限为 ${\tfrac {1}{2}}f_{s}$ （即奈奎斯特准则），插值算法将重建出唯一的该函数。

按照函数自身的带宽 $B$ 来表示（如图所示），奈奎斯特准则通常被表述为 $f_{s}>2B$ 。 $2B$ 被称为带宽为 $B$ 的函数的奈奎斯特速率。当不满足奈奎斯特准则时（例如 $B>{\tfrac {1}{2}}f_{s}$ ）时，就会发生混叠失真。

有意的混叠

图3描绘了一类被称为基带或低通的函数，其正频范围的主要能量集中在 $[0,B)$ 。如果函数的频率范围是 $(A,A+B)$ ，且 $A>B$ ，则称其为带通函数。在某些情况下，人们希望将带通函数转换为基带。这可以通过混频（外差（英语：Heterodyne））的方式，将带通函数下移到 $(0,B)$ 的频率范围内。这么做的一个常见原因是为了降低奈奎斯特速率，从而更高效地存储数据。

事实上，可以通过直接以低于奈奎斯特速率的方式对带通函数进行采样（称为欠采样）来实现类似的结果。具体而言，只需以一个最小的整数子倍频率对带通信号采样，这个频率满足 $f_{s}>2B$ 的基带奈奎斯特准则即可。

参考资料

^ Oppenheim, Alan V.; Schafer, Ronald W.; Buck, John R. Discrete-time signal processing 2nd. Upper Saddle River, N.J.: Prentice Hall. 1999: 140. ISBN 0-13-754920-2. T is the sampling period, and its reciprocal, f_s=1/T, is the sampling frequency, in samples per second.
^ Roger L. Freeman. Telecommunication System Engineering. John Wiley & Sons. 2004: 399. ISBN 0-471-45133-9.

查论编數位訊號處理
理論	信号检测理论離散訊號估計理論取樣定理
子領域	音频信号处理影像處理語音處理統計訊號處理（英语：Statistical signal processing）
技術	Z轉換高级Z变换匹配Z变换雙線性轉換常數Q轉換傅里叶变换離散傅立葉轉換（DFT）離散分數傅立葉轉換（DFFT）离散時間傅立葉轉換（DTFT）冲激不變法積分變換拉普拉斯變換拉普拉斯逆變換星標變換札克变换
取樣	混疊抗混叠濾波器奈奎斯特速率 / 頻率升取樣降取樣（英语：Undersampling）過取樣欠取樣（英语：Undersampling）取樣率量化

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