550 Chapter 13.Fourier and Spectral Applications finite,sampled stretch of it.In this section we'll do roughly the same thing,but with considerably greater attention to details.Our attention will uncover some surprises. The first detail is power spectrum(also called a power spectral density or PSD)normalization.In general there is some relation of proportionality between a measure of the squared amplitude of the function and a measure of the amplitude of the PSD.Unfortunately there are several different conventions for describing the normalization in each domain,and many opportunities for getting wrong the relationship between the two domains.Suppose that our function c(t)is sampled at N points to produce values co...c-1,and that these points span a range of time T,that is T=(N-1)A,where A is the sampling interval.Then here are several different descriptions of the total power: N-1 ≡“sum squared amplitude” (13.4.1) j=0 N-1 lc(t)Pdt≈ ="mean squared amplitude" (13.4.2) j=0 9 N-1 c(t2dt≈△ ∑lsf2≡“time-integral squared amplitude (13.4.3) j=0 PSD estimators,as we shall see,have an even greater variety.In this section, 28 we consider a class of them that give estimates at discrete values of frequency fi, where i will range over integer values.In the next section,we will learn about OF SCIENTIFIC a different class of estimators that produce estimates that are continuous functions 6 of frequency f.Even if it is agreed always to relate the PSD normalization to a particular description of the function normalization(e.g.,13.4.2),there are at least the following possibilities:The PSD is defined for discrete positive,zero,and negative frequencies,and its sum over these is the function mean squared amplitude 10-521 defined for zero and discrete positive frequencies only,and its sum over Numerica these is the function mean squared amplitude 431 defined in the Nyquist interval from-fe to fe,and its integral over this Recipes range is the function mean squared amplitude defined from 0 to fc.and its integral over this range is the function mean squared amplitude North It never makes sense to integrate the PSD of a sampled function outside of the Nyquist interval-fe and fe since,according to the sampling theorem,power there will have been aliased into the Nyquist interval. It is hopeless to define enough notation to distinguish all possible combinations of normalizations.In what follows,we use the notation P(f)to mean any of the above PSDs,stating in each instance how the particular P(f)is normalized.Beware the inconsistent notation in the literature. The method of power spectrum estimation used in the previous section is a simple version of an estimator called,historically,the periodogram.If we take an N-point sample of the function c(t)at equal intervals and use the FFT to compute550 Chapter 13. Fourier and Spectral Applications 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). finite, sampled stretch of it. In this section we’ll do roughly the same thing, but with considerably greater attention to details. Our attention will uncover some surprises. The first detail is power spectrum (also called a power spectral density or PSD) normalization. In general there is some relation of proportionality between a measure of the squared amplitude of the function and a measure of the amplitude of the PSD. Unfortunately there are several different conventions for describing the normalization in each domain, and many opportunities for getting wrong the relationship between the two domains. Suppose that our function c(t) is sampled at N points to produce values c0 ...cN−1, and that these points span a range of time T , that is T = (N − 1)∆, where ∆ is the sampling interval. Then here are several different descriptions of the total power: N
−1 j=0 |cj | 2 ≡ “sum squared amplitude” (13.4.1) 1 T T 0 |c(t)| 2 dt ≈ 1 N N
−1 j=0 |cj | 2 ≡ “mean squared amplitude” (13.4.2) T 0 |c(t)| 2 dt ≈ ∆ N
−1 j=0 |cj | 2 ≡ “time-integral squared amplitude” (13.4.3) PSD estimators, as we shall see, have an even greater variety. In this section, we consider a class of them that give estimates at discrete values of frequency f i, where i will range over integer values. In the next section, we will learn about a different class of estimators that produce estimates that are continuous functions of frequency f. Even if it is agreed always to relate the PSD normalization to a particular description of the function normalization (e.g., 13.4.2), there are at least the following possibilities: The PSD is • defined for discrete positive, zero, and negative frequencies, and its sum over these is the function mean squared amplitude • defined for zero and discrete positive frequencies only, and its sum over these is the function mean squared amplitude • defined in the Nyquist interval from −fc to fc, and its integral over this range is the function mean squared amplitude • defined from 0 to fc, and its integral over this range is the function mean squared amplitude It never makes sense to integrate the PSD of a sampled function outside of the Nyquist interval −fc and fc since, according to the sampling theorem, power there will have been aliased into the Nyquist interval. It is hopeless to define enough notation to distinguish all possible combinations of normalizations. In what follows, we use the notation P(f) to mean any of the above PSDs, stating in each instance how the particular P(f) is normalized. Beware the inconsistent notation in the literature. The method of power spectrum estimation used in the previous section is a simple version of an estimator called, historically, the periodogram. If we take an N-point sample of the function c(t) at equal intervals and use the FFT to compute