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242 chapter 10102 4 6-1Figure 10.2 A plot of the functions sin(πt/2) and sin(2πt ). If the sampling rate is not highenough, these signals will appear indistinguishable. If both are present in a signal (e.g., as asignal that is the sum of the two) and if the signal is not sampled at a high enough rate, thededuced low-frequency component will be contaminated by the higher-frequencycomponent.and may contaminate the low-frequency components that we deduce. This effectis called aliasing and is the cause of the moiré pattern distortion in digital images.As an example, consider Figure 10.2 showing the two functions sin(πt/2) andsin(2πt) for 0 ≤ t ≤ 8, with their points of overlap in bold. If we were unfortunateenough to sample a signal containing these functions at the times t =0, 2, 4, 6, 8,then we would measure y ≡ 0 and assume that there was no signal at all. However,if we were unfortunate enough to measure the signal at the filled dots inFigure 10.2 where sin(πt/2) = sin(2πt), specifically, t =0, 1210 , 4 3,..., then our Fourieranalysis would completely miss the high-frequency component. In DFT jargon,we would say that the high-frequency component has been aliased by thelow-frequency component. In other cases, some high-frequency values may beincluded in our sampling of the signal, but our sampling rate may not behigh enough to include enough of them to separate the high-frequency componentproperly. In this case some high-frequency signals would be includedspuriously as part of the low-frequency spectrum, and this would lead to spuriouslow-frequency oscillations when the signal is synthesized from its Fouriercomponents.More precisely, aliasing occurs when a signal containing frequency f is sampledat a rate of s = N/T measurements per unit time, with s ≤ f/2. In this case,the frequencies f and f − 2s yield the same DFT, and we would not be able todetermine that there are two frequencies present. That being the case, to avoid−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 242