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fourier analysis: signals and filters 243aliasing we want no frequencies f>s/2 to be present in our input signal. This isknown as the Nyquist criterion. In practice, some applications avoid the effects ofaliasing by filtering out the high frequencies from the signal and then analyzing theremaining low-frequency part. (The low-frequency sinc filter discussed in §10.7.1is often used for this.) Even though this approach eliminates some high-frequencyinformation, it lessens the distortion of the low-frequency components and so maylead to improved reproduction of the signal.If accurate values for the high frequencies are required, then we will need toincrease the sampling rate s by increasing the number N of samples taken within thefixed sampling time T = Nh. By keeping the sampling time constant and increasingthe number of samples taken, we make the time step h smaller, and this picks up thehigher frequencies. By increasing the number N of frequencies that you compute,you move the higher-frequency components you are interested in closer to themiddle of the spectrum and thus away from the error-prone ends.If we vary the the total time sampling time T = Nh but not the sampling rates = N/T =1/h, we make ω 1 smaller because the discrete frequenciesω n = nω 1 = n 2πT(10.37)are measured in steps of ω 1 . This leads to a smoother frequency spectrum. However,to keep the time step the same and thus not lose high-frequency information, wewould also need to increase the number of N samples. And as we said, this is oftendone, after the fact, by padding the end of the data set with zeros.10.4.3 DFT for Fourier Series (Algorithm)For simplicity let us consider the Fourier cosine series:∞∑y(t)= a n cos(nωt), a k = 2 ∫ Tdt cos(kωt)y(t). (10.38)Tn=0Here T def= 2π/ω is the actual period of the system (not necessarily the period ofthe simple harmonic motion occurring for a small amplitude). We assume that thefunction y(t) is sampled for a discrete set of times0y(t = t k ) ≡ y k , k =0, 1,...,N. (10.39)Because we are analyzing a periodic function, we retain the conventions used inthe DFT and require the function to repeat itself with period T = Nh; that is, weassume that the amplitude is the same at the first and last points:y 0 = y N . (10.40)This means that there are only N independent values of y being used as input.For these N independent y k values, we can determine uniquely only N expansion−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 243