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1、An Introduction to the Sampling TheoremAn Introduction to the Sampling TheoremWith rapid advancement in data acquistion technology (i.e. analog-to-digital and digital-to-analog converters) and the explosive introduction of micro-computers, selected complex linear and nonlinear functions currently im
2、plemented with analog circuitry are being alternately implemented with sample data systems.介绍了采样定理 介绍了采样定理 日新月异的数据采集技术(即模拟到数字和数字到模拟转换器)和爆炸物引进微型计算机， 选择复杂的线性和非线性职能目前实施模拟电路正在轮流实施样本数据系统。Though more costly than their analog counterpart, these sampled data systems feature programmability. Additionally, ma
3、ny of the algorithms employed are a result of developments made in the area of signal processing and are in some cases capable of functions unrealizable by current analog techniques.With increased usage a proportional demand has evolved to understand the theoretical basis required in interfacing the
4、se sampled data-systems to the analog world.This article attempts to address the demand by presenting the concepts of aliasing and the sampling theorem in a manner, hopefully, easily understood by those making their first attempt at signal processing. Additionally discussed are some of the unobvious
5、 hardware effects that one might encounter when applying the sampled theorem. With this. . . let us begin. 虽然比自己更昂贵的模拟对应，这些采样数据系统的功能可编程。此外，许多算法是由于就 业的发展领域取得的信号处理，并在某些情况下无法实现的功能能够由目前的模拟技术。 BR随着需求的使用比例已演变理解的理论基础需要在这些接口采样数据系统的模拟世 界。BR本文试图解决的需求，提出的概念混叠和采样定理的方式，希望容易理解那些使 他们第一次尝试信号处理。此外讨论的一些硬件效果不明显的一个申请时可
6、能遇到的采样定 理。与此有关。 。 。让我们开始。An Intuitive DevelopmentThe sampling theorem by C.E. Shannon in 1949 places restrictions on the frequency content of the time function signal, f(t), and can be simply stated as follows: In order to recover the signal function f(t) exactly, it is necessary to sample f(t) at a
7、 rate greater than twice its highest frequency component.Practically speaking for example, to sample an analog signal having a maximum frequency of 2Kc requires sampling at greater than 4Kc to preserve and recover the waveform exactly.The consequences of sampling a signal at a rate below its highest
8、 frequency component results in a phenomenon known as aliasing. This concept results in a frequency mistakenly taking on the identity of an entirely different frequency when recovered. In an attempt to clarify this, envision the ideal sampler of Figure 1(a), with a sample period of T shown in Figure
9、 1(b), sampling the waveform f(t) as pictured in Figure 1(c). The sampled data points of f(t) are shown in Figure 1(d) and can be defined as the sample set of the continuous function f(t). Note in Figure 1(e) that another frequency component, a(t), can be found that has the same sample set of data p
10、oints as f(t) in Figure 1(d). Because of this it is difficult to determine which frequiesntcryulyabe（itn）g,observed. This effect is similar to that observed in western movies when watching the spoked wheels of a rapidly moving stagecoach rotate backwards at a slow rate. The effect is a result of eac
11、h individual frame of film resembling a discrete strobed sampling operation flashing at a rate slightly faster than that of the rotating wheel. Each observed sample point or frame catches the spoked wheel slightly displaced from its previous position giving the effective appearance of a wheel rotati
12、ng backwards. Again, aliasing is evidenced and in this example it becomes difficult to determine which is the true rotational frequency being observed.一个直观的发展BR采样定理的申农限制在194 9年的频率内容的时间函数信号，f（t）, 和可简单说明如下：BR为了恢复信号函数f（t）完全相同，有必要样本函数f （ t） 在速度大于两倍BR频率最高的组成部分。BR实际上例如，样本有一个模拟信号的最 大频率为2KC需要取样大于4KC在维护和恢复波形
13、完全相同。BR造成的后果采样信号 的速度在低于其最高频率分量结果的现象称为走样。这一概念的结果在频率上采取错误的身 份，是完全不同的频率时收回。为了澄清这一点，设想的理想采样的图1 （ a ）项的样本 期间的T如图1（ b ）款，取样的波形函数f （吨）为图图1（ C ）项。采样数据点的F （吨）列于图1（ d ）和可界定为样本集的连续函数f （ T ）类。注意：在图1 （ e ）款，另一频率部分，（t ）款，可以发现，具有相同的样本集的数据点为F （ t ）的图 1（ d ）项。由于这一原因，很难确定哪些频率1 （ t ）款， BR真正得到遵守。这样的效果相似，观察到西方电影的时候看轮辐式轮
14、毂，快速旋转向后移动 马车在缓慢。大意是由于每个帧的电影类似离散strobed采样作业闪动的速度稍快于旋转 车轮。每个观测采样点或框架的轮辐车轮渔获量略有流离失所，其先前的立场给予有效的外 观车轮旋转倒退。再次，别名证明和在这个例子中有困难，以确定哪些是真正的旋转频率得 到遵守。On the surfaCe it is easily said that anti-aliasing designs Can be aChievedbysamplingatarategreaterthantwiCethemaximumfrequenCy found within the signal to be sa
15、mpled. In the real world, however, most signals Contain the entire speCtrum of frequenCy Components; from the desired to those present in white noise. To reCover suCh information aCCuratelythesystemwouldrequireanunrealizablyhighsamplerate. This diffiCulty Can be easily overCome by preConditioning th
16、e input signal, the means of whiCh would be a band-limiting or frequenCy filtering funCtion performed prior to the sample data input. The prefilter, typiCally Called anti-aliasing filter guarantees, for example in the low pass filter Case, that the sampled data system reCeivesanalogsignalshavingaspeCtralContentnogreaterthanthose frequenCiesallowedbythefilter.AsillustratedinFigure2,itthus beComes a simple matter to sample at greater than
