13.4 Power Spectrum Estimation Using the FFT 555 square window 8 Welch window Bartlett window .6 Permission is .com or call (including this one) Hann window 111800-672 (North 0 1988-1992 by Cambridge University Press. 50 100 150 200 250 bin number America tusers to make one paper from NUMERICAL RECIPES IN C: THE Figure 13.4.1.Window functions commonly used in FFT power spectral estimation.The data segment, ART here of length 256,is multiplied (bin by bin)by the window function before the FFT is computed.The square window,which is equivalent to no windowing,is least recommended.The Welch and Bartlett 9 Programs windows are good choices. much as a factor of two,however),but trade this off by widening the leakage tail by a significant factor (e.g.,the reciprocal of 10 percent,a factor of ten).If we 兰。 可 distinguish between the width of a window(number of samples for which it is at its maximum value)and its rise/fall time (number of samples during which it rises and falls):and if we distinguish between the FWAM(full width to half maximum OF SCIENTIFIC COMPUTING (ISBN value)of the leakage function's main lobe and the leakage width(full width that contains half of the spectral power that is not contained in the main lobe);then these quantities are related roughly by 10-6211 N Fuurggoglrion (FWHM in bins)≈ (13.4.16) Numerical Recipes 43106 (window width) W (leakage width in bins) (13.4.17) (outside (window rise/fall time) North For the windows given above in (13.4.13)-(13.4.15),the effective window widths and the effective window rise/fall times are both of order N.Generally speaking,we feel that the advantages of windows whose rise and fall times are only small fractions of the data length are minor or nonexistent,and we avoid using them.One sometimes hears it said that flat-topped windows "throw away less of the data,"but we will now show you a better way of dealing with that problem by use of overlapping data segments. Let us now suppose that we have chosen a window function,and that we are ready to segment the data into K segments of N=2M points.Each segment will be FFT'd,and the resulting K periodograms will be averaged together to obtain a13.4 Power Spectrum Estimation Using the FFT 555 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). amplitude 0 .2 .4 .6 .8 1 0 50 100 150 200 250 bin number Bartlett window Welch window square window Hann window Figure 13.4.1. Window functions commonly used in FFT power spectral estimation. The data segment, here of length 256, is multiplied (bin by bin) by the window function before the FFT is computed. The square window, which is equivalent to no windowing, is least recommended. The Welch and Bartlett windows are good choices. much as a factor of two, however), but trade this off by widening the leakage tail by a significant factor (e.g., the reciprocal of 10 percent, a factor of ten). If we distinguish between the width of a window (number of samples for which it is at its maximum value) and its rise/fall time (number of samples during which it rises and falls); and if we distinguish between the FWHM (full width to half maximum value) of the leakage function’s main lobe and the leakage width (full width that contains half of the spectral power that is not contained in the main lobe); then these quantities are related roughly by (FWHM in bins) ≈ N (window width) (13.4.16) (leakage width in bins) ≈ N (window rise/fall time) (13.4.17) For the windows given above in (13.4.13)–(13.4.15), the effective window widths and the effective window rise/fall times are both of order 1 2N. Generally speaking, we feel that the advantages of windows whose rise and fall times are only small fractions of the data length are minor or nonexistent, and we avoid using them. One sometimes hears it said that flat-topped windows “throw away less of the data,” but we will now show you a better way of dealing with that problem by use of overlapping data segments. Let us now suppose that we have chosen a window function, and that we are ready to segment the data into K segments of N = 2M points. Each segment will be FFT’d, and the resulting K periodograms will be averaged together to obtain a