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Unformatted text preview: TL/H/5620 AnIntroductiontotheSamplingTheoremAN-236 National Semiconductor Application Note 236 January 1980 An Introduction to the Sampling Theorem An Introduction to the Sampling Theorem With rapid advancement in data acquistion technology (i.e. analog-to-digital and digital-to-analog converters) and the explosive introduction of micro-computers, selected com- plex linear and nonlinear functions currently implemented with analog circuitry are being alternately implemented with sample data systems. Though more costly than their analog counterpart, these sampled data systems feature programmability. Additionally, many of the algorithms employed are a result of develop- ments 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 these 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 hardware effects that one might en- counter when applying the sampled theorem. With this . . . let us begin. I. An Intuitive Development The sampling theorem by C.E. Shannon in 1949 places re- strictions on the frequency content of the time function sig- nal, 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 rate greater than twice its highest frequency component. Practically speaking for example, to sample an analog sig- nal 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 frequency component results in a phenomenon known as aliasing . This concept results in a frequency mis- takenly taking on the identity of an entirely different frequen- cy when recovered. In an attempt to clarify this, envision the ideal sampler of Figure 1(a) , with a sample period of T shown in (b), sampling the waveform f(t) as pictured in (c). The sampled data points of f(t) are shown in (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 points as f(t) in (d). Because of this it is difficult to determine which frequency a(t), is truly being observed. This effect is similar to that observed in western movies when watching the TL/H/56201 FIGURE 1. When sampling, many signals may be found to have the same set of data points. These are called aliases of each other....
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- Fall '08