c12-0 (Numerical Recipes in C), страница 2

To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).−∞499h(t)212.0 IntroductionPh ( f ) (one-sided)0fPh( f )(two-sided)( b)(c)−f0fFigure 12.0.1. Normalizations of one- and two-sided power spectra.

The area under the square of thefunction, (a), equals the area under its one-sided power spectrum at positive frequencies, (b), and alsoequals the area under its two-sided power spectrum at positive and negative frequencies, (c).Be warned that one occasionally sees PSDs defined without this factor two. These,strictly speaking, are called two-sided power spectral densities, but some booksare not careful about stating whether one- or two-sided is to be assumed. Wewill always use the one-sided density given by equation (12.0.14).

Figure 12.0.1contrasts the two conventions.If the function h(t) goes endlessly from −∞ < t < ∞, then its total powerand power spectral density will, in general, be infinite. Of interest then is the (oneor two-sided) power spectral density per unit time. This is computed by taking along, but finite, stretch of the function h(t), computing its PSD [that is, the PSDof a function that equals h(t) in the finite stretch but is zero everywhere else], andthen dividing the resulting PSD by the length of the stretch used.

Parseval’s theoremin this case states that the integral of the one-sided PSD-per-unit-time over positivefrequency is equal to the mean square amplitude of the signal h(t).You might well worry about how the PSD-per-unit-time, which is a functionof frequency f, converges as one evaluates it using longer and longer stretches ofdata.

This interesting question is the content of the subject of “power spectrumestimation,” and will be considered below in §13.4–§13.7. A crude answer forSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited.

To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).t(a)500Chapter 12.Fast Fourier TransformCITED 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 (NewYork: Academic Press).12.1 Fourier Transform of Discretely SampledDataIn the most common situations, function h(t) is sampled (i.e., its value isrecorded) at evenly spaced intervals in time. Let ∆ denote the time interval betweenconsecutive samples, so that the sequence of sampled values ishn = 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 measuredin seconds, for example, then the sampling rate is the number of samples recordedper second.Sampling Theorem and AliasingFor any sampling interval ∆, there is also a special frequency fc , called theNyquist critical frequency, given byfc ≡12∆(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 atthe positive peak again, and so on. Expressed otherwise: Critical sampling of asine wave is two sample points per cycle. One frequently chooses to measure timein units of the sampling interval ∆.

In this case the Nyquist critical frequency isjust 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 remarkableSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use.

Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).now is: The PSD-per-unit-time converges to finite values at all frequencies exceptthose where h(t) has a discrete sine-wave (or cosine-wave) component of finiteamplitude. 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 meansquare 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 thischapter with one exception: In computational work, especially with experimentaldata, we are almost never given a continuous function h(t) to work with, but aregiven, rather, a list of measurements of h(ti ) for a discrete set of ti ’s.

The profoundimplications of this seemingly unimportant fact are the subject of the next section..