12.1 Fourier Transform of Discretely Sampled Data 501 fact known as the sampling theorem:If a continuous function h(t).sampled at an interval A,happens to be bandwidth limited to frequencies smaller in magnitude than fe,i.e.,if H(f)=0 for all |f>fe,then the function h(t)is completely determined by its samples hn.In fact,h(t)is given explicitly by the formula sin[2rfe(t-n△)] h(t)=△ hn (12.1.3) π(t-n△) n=- This is a remarkable theorem for many reasons,among them that it shows that the "information content"of a bandwidth limited function is,in some sense,infinitely smaller than that of a general continuous function.Fairly often,one is dealing with a signal that is known on physical grounds to be bandwidth limited(or at least approximately bandwidth limited).For example,the signal may have passed through an amplifier with a known,finite frequency response.In this case,the sampling theorem tells us that the entire information content of the signal can be recorded by sampling it at a rate A-equal to twice the maximum frequency passed by the amplifier (cf.12.1.2). Now the bad news.The bad news concerns the effect of sampling a continuous function that is not bandwidth limited to less than the Nyquist critical frequency. 9 In that case,it turns out that all of the power spectral density that lies outside of the frequency range -fe<f fe is spuriously moved into that range.This phenomenon is called aliasing.Any frequency component outside of the frequency range (-fe,fe)is aliased (falsely translated)into that range by the very act of discrete sampling.You can readily convince yourself that two waves exp(2mifit) a%29 9 and exp(2mif2t)give the same samples at an interval A if and only if fi and f2 differ by a multiple of 1/A,which is just the width in frequency of the range (-fe,fe).There is little that you can do to remove aliased power once you have a冰 OF SCIENTIFIC 6 discretely sampled a signal.The way to overcome aliasing is to (1)know the natural bandwidth limit of the signal-or else enforce a known limit by analog filtering of the continuous signal,and then (ii)sample at a rate sufficiently rapid to give at least two points per cycle of the highest frequency present.Figure 12.1.1 illustrates these considerations. To put the best face on this,we can take the alternative point of view:If a Numerica 10621 continuous function has been competently sampled,then,when we come to estimate its Fourier transform from the discrete samples,we can assume(or rather we might 431 as well assume)that its Fourier transform is equal to zero outside of the frequency Recipes range in between-fe and fe.Then we look to the Fourier transform to tell whether 腿 the continuous function has been competently sampled(aliasing effects minimized). We do this by looking to see whether the Fourier transform is already approaching North zero as the frequency approaches fe from below,or -fe from above.If,on the contrary,the transform is going towards some finite value,then chances are that components outside of the range have been folded back over onto the critical range. Discrete Fourier Transform We now estimate the Fourier transform of a function from a finite number of its sampled points.Suppose that we have N consecutive sampled values hk≡h(tk),tk≡k△，k=0,1,2,..,N-1 (12.1.4)12.1 Fourier Transform of Discretely Sampled Data 501 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). fact known as the sampling theorem: If a continuous function h(t), sampled at an interval ∆, happens to be bandwidth limited to frequencies smaller in magnitude than fc, i.e., if H(f)=0 for all |f| ≥ fc, then the function h(t) is completely determined by its samples hn. In fact, h(t) is given explicitly by the formula h(t)=∆
+∞ n=−∞ hn sin[2πfc(t − n∆)] π(t − n∆) (12.1.3) This is a remarkable theorem for many reasons, among them that it shows that the “information content” of a bandwidth limited function is, in some sense, infinitely smaller than that of a general continuous function. Fairly often, one is dealing with a signal that is known on physical grounds to be bandwidth limited (or at least approximately bandwidth limited). For example, the signal may have passed through an amplifier with a known, finite frequency response. In this case, the sampling theorem tells us that the entire information content of the signal can be recorded by sampling it at a rate ∆−1 equal to twice the maximum frequency passed by the amplifier (cf. 12.1.2). Now the bad news. The bad news concerns the effect of sampling a continuous function that is not bandwidth limited to less than the Nyquist critical frequency. In that case, it turns out that all of the power spectral density that lies outside of the frequency range −fc <f<fc is spuriously moved into that range. This phenomenon is called aliasing. Any frequency component outside of the frequency range (−fc, fc) is aliased (falsely translated) into that range by the very act of discrete sampling. You can readily convince yourself that two waves exp(2πif 1t) and exp(2πif2t) give the same samples at an interval ∆ if and only if f1 and f2 differ by a multiple of 1/∆, which is just the width in frequency of the range (−fc, fc). There is little that you can do to remove aliased power once you have discretely sampled a signal. The way to overcome aliasing is to (i) know the natural bandwidth limit of the signal — or else enforce a known limit by analog filtering of the continuous signal, and then (ii) sample at a rate sufficiently rapid to give at least two points per cycle of the highest frequency present. Figure 12.1.1 illustrates these considerations. To put the best face on this, we can take the alternative point of view: If a continuous function has been competently sampled, then, when we come to estimate its Fourier transform from the discrete samples, we can assume (or rather we might as well assume) that its Fourier transform is equal to zero outside of the frequency range in between −fc and fc. Then we look to the Fourier transform to tell whether the continuous function has been competently sampled (aliasing effects minimized). We do this by looking to see whether the Fourier transform is already approaching zero as the frequency approaches f c from below, or −fc from above. If, on the contrary, the transform is going towards some finite value, then chances are that components outside of the range have been folded back over onto the critical range. Discrete Fourier Transform We now estimate the Fourier transform of a function from a finite number of its sampled points. Suppose that we have N consecutive sampled values hk ≡ h(tk), tk ≡ k∆, k = 0, 1, 2,...,N − 1 (12.1.4)