500 Chapter 12.Fast Fourier Transform PSD-per-unit-time converges to finite values at all frequencies except those where h(t)has a discrete sine-wave (or cosine-wave)component of finite amplitude.At those frequencies,it becomes a delta-function,i.e.,a sharp spike,whose width gets narrower and narrower,but whose area converges to be the mean square amplitude of the discrete sine or cosine component at that frequency. We have by now stated all of the analytical formalism that we will need in this chapter with one exception:In computational work,especially with experimental data,we are almost never given a continuous function h(t)to work with,but are given,rather,a list of measurements of h(ti)for a discrete set ofti's.The profound implications of this seemingly unimportant fact are the subject of the next section. CITED REFERENCES AND FURTHER READING: Champeney,D.C.1973,Fourier Transforms and Their Physical Applications(New York:Academic Press). Elliott,D.F.,and Rao,K.R.1982,Fast Transforms:Algorithms,Analyses,Applications(New York: Academic Press). 12.1 Fourier Transform of Discretely Sampled 西 令 Press. Data In the most common situations,function h(t)is sampled (i.e.,its value is 是8e recorded)at evenly spaced intervals in time.Let A denote the time interval between consecutive samples,so that the sequence of sampled values is 6 hm=h(n△)n=.,-3,-2,-1,0,1,2,3,… (12.1.1) The reciprocal of the time interval A is called the sampling rate;if A is measured in seconds,for example,then the sampling rate is the number of samples recorded per second. Sampling Theorem and Aliasing 4 E喜 Numerical Recipes 10.621 43106 For any sampling interval A,there is also a special frequency fe,called the Nyquist critical frequency,given by (outside f三2△ (12.1.2) If a sine wave of the Nyquist critical frequency is sampled at its positive peak value, then the next sample will be at its negative trough value,the sample after that at the positive peak again,and so on.Expressed otherwise:Critical sampling of a sine wave is two sample points per cycle.One frequently chooses to measure time in units of the sampling interval A.In this case the Nyquist critical frequency is just the constant 1/2. The Nyquist critical frequency is important for two related,but distinct,reasons. One is good news,and the other bad news.First the good news.It is the remarkable500 Chapter 12. Fast Fourier Transform 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). PSD-per-unit-time converges to finite values at all frequencies except those where h(t) has a discrete sine-wave (or cosine-wave) component of finite amplitude. At those frequencies, it becomes a delta-function, i.e., a sharp spike, whose width gets narrower and narrower, but whose area converges to be the mean square amplitude of the discrete sine or cosine component at that frequency. We have by now stated all of the analytical formalism that we will need in this chapter with one exception: In computational work, especially with experimental data, we are almost never given a continuous function h(t) to work with, but are given, rather, a list of measurements of h(t i) for a discrete set of ti’s. The profound implications of this seemingly unimportant fact are the subject of the next section. CITED REFERENCES AND FURTHER READING: Champeney, D.C. 1973, Fourier Transforms and Their Physical Applications (New York: Academic Press). Elliott, D.F., and Rao, K.R. 1982, Fast Transforms: Algorithms, Analyses, Applications (New York: Academic Press). 12.1 Fourier Transform of Discretely Sampled Data In the most common situations, function h(t) is sampled (i.e., its value is recorded) at evenly spaced intervals in time. Let ∆ denote the time interval between consecutive samples, so that the sequence of sampled values is hn = h(n∆) n = ..., −3, −2, −1, 0, 1, 2, 3,... (12.1.1) The reciprocal of the time interval ∆ is called the sampling rate; if ∆ is measured in seconds, for example, then the sampling rate is the number of samples recorded per second. Sampling Theorem and Aliasing For any sampling interval ∆, there is also a special frequency f c, called the Nyquist critical frequency, given by fc ≡ 1 2∆ (12.1.2) If a sine wave of the Nyquist critical frequency is sampled at its positive peak value, then the next sample will be at its negative trough value, the sample after that at the positive peak again, and so on. Expressed otherwise: Critical sampling of a sine wave is two sample points per cycle. One frequently chooses to measure time in units of the sampling interval ∆. In this case the Nyquist critical frequency is just the constant 1/2. The Nyquist critical frequency is important for two related, but distinct, reasons. One is good news, and the other bad news. First the good news. It is the remarkable