554 Chapter 13.Fourier and Spectral Applications and f&is given by (13.4.6).The more general form of (13.4.7)can now be written in terms of the window function wj as W-1 12 W(s)= isk/N wk k=0 (13.4.12) N/2 cos(2nsk/N)w(k-N/2)dk N/2 Here the approximate equality is useful for practical estimates,and holds for any window that is left-right symmetric (the usual case).and for sN (the case of interest for estimating leakage into nearby bins).The continuous function w(k-N/2) in the integral is meant to be some smooth function that passes through the points wk. 海 There is a lot of perhaps unnecessary lore about choice ofa window function,and practically every function that rises from zero to a peak and then falls again has been named after someone.A few of the more common(also shown in Figure 13.4.1)are: RECIPES 方-N 05=1- ≡Bartlett window (13.4.13) The“Parzen window”is very similar to this.) A入 9 Hann window” 2 (13.4.14) (The"Hamming window"is similar but does not go exactly to zero at the ends.) 61 w5=1- -N NΓ ≡Welch window (13.4.15) We are inclined to follow Welch in recommending that you use either(13.4.13) or (13.4.15)in practical work.However,at the level of this book,there is Recipes Numerica 10521 effectively no difference between any of these(or similar)window functions.Their 431 difference lies in subtle trade-offs among the various figures of merit that can be Recipes used to describe the narrowness or peakedness of the spectral leakage functions computed by (13.4.12).These figures of merit have such names as:highest sidelobe level(dB,sidelobe fall--of(dB per octave，以equivalent noise bandwidth (bins，以3-dB bandwidth (bins),scallop loss (dB),worst case process loss (dB).Roughly speaking, the principal trade-off is between making the central peak as narrow as possible versus making the tails of the distribution fall off as rapidly as possible.For details,see(e.g.)[21.Figure 13.4.2 plots the leakage amplitudes for several windows already discussed. There is particularly a lore about window functions that rise smoothly from zero to unity in the first small fraction(say 10 percent)of the data,then stay at unity until the last small fraction(again say 10 percent)of the data,during which the window function falls smoothly back to zero.These windows will squeeze a little bit of extra narrowness out of the main lobe of the leakage function(never as554 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). and fk is given by (13.4.6). The more general form of (13.4.7) can now be written in terms of the window function wj as W(s) = 1 Wss      N
−1 k=0 e2πisk/N wk      2 ≈ 1 Wss      N/2 −N/2 cos(2πsk/N)w(k − N/2) dk      2 (13.4.12) Here the approximate equality is useful for practical estimates, and holds for any window that is left-right symmetric (the usual case), and for s  N (the case of interest for estimating leakage into nearby bins). The continuous function w(k−N/2) in the integral is meant to be some smooth function that passes through the points w k. There is a lot of perhaps unnecessary lore about choice of a window function, and practically every function that rises from zero to a peak and then falls again has been named after someone. A few of the more common (also shown in Figure 13.4.1) are: wj = 1 −     j − 1 2N 1 2N     ≡ “Bartlett window” (13.4.13) (The “Parzen window” is very similar to this.) wj = 1 2  1 − cos 2πj N  ≡ “Hann window” (13.4.14) (The “Hamming window” is similar but does not go exactly to zero at the ends.) wj = 1 − j − 1 2N 1 2N 2 ≡ “Welch window” (13.4.15) We are inclined to follow Welch in recommending that you use either (13.4.13) or (13.4.15) in practical work. However, at the level of this book, there is effectively no difference between any of these (or similar) window functions. Their difference lies in subtle trade-offs among the various figures of merit that can be used to describe the narrowness or peakedness of the spectral leakage functions computed by (13.4.12). These figures of merit have such names as: highest sidelobe level (dB), sidelobe fall-off (dB per octave), equivalent noise bandwidth (bins), 3-dB bandwidth (bins), scallop loss (dB), worst case process loss (dB). Roughly speaking, the principal trade-off is between making the central peak as narrow as possible versus making the tails of the distribution fall off as rapidly as possible. For details, see (e.g.) [2]. Figure 13.4.2 plots the leakage amplitudes for several windows already discussed. There is particularly a lore about window functions that rise smoothly from zero to unity in the first small fraction (say 10 percent) of the data, then stay at unity until the last small fraction (again say 10 percent) of the data, during which the window function falls smoothly back to zero. These windows will squeeze a little bit of extra narrowness out of the main lobe of the leakage function (never as