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248 chapter 5 Nonparametric Power Spectrum Estimation N 1 0 Data record N 1 0 Taper 1 N 1 0 Periodogram 1 N 1 0 Periodogram 2 N 1 0 Periodogram M Averaging PSD Estimate N 1 0 Taper 2 N 1 0 Taper M FIGURE 5.30 A pictorial description of the multitaper approach to power spectrum estimation. The function ¯ R w (e ) is the spectral window of the averaged multitaper estimator, which is obtained by averaging spectra of the individual tapers. Hence, for ¯ R w (e ) to produce a good leakage-free estimate ˆ R ( MT ) x (e ) , all K spectral windows must provide good protection against leakage. Therefore, each taper must have low sidelobe levels. Furthermore, the averaging of K individual periodograms also reduces the overall variance of ˆ R ( MT ) x (e ) . Thereductioninvarianceispossibleifthe ˆ R k,x (e ) arepairwiseuncorrelatedwithcommon variance, in which case the variance reduces by a factor of 1 /K . Thus, we need K orthonormal data tapers such that each one provides a good pro- tection against leakage and such that the resulting individual spectral estimates are nearly uncorrelated. One such set is obtained by using DPSS with parameter W and of orders k = 0 , . . . , K 1, where K is chosen to be less than or equal to the number 2 W (called the Shannon number , which is also a fi xed-resolution bandwidth). The design of these se- quences is discussed in detail in Thomson (1982) and in Percival and Walden (1993). In Matlab these tapers are generated by using the [w]=dpss(L,W) function, where L is the length of 2 W tapers computed in matrix w . The fi rst four 21-point DPSS tapers with W = 4 and their Fourier transforms are shown in Figure 5.31 while the next four DPSS tapers are shown in Figure 5.32. It can be seen that higher-order tapers assume both positive and negative values. The zeroth-order taper (like other windows) heavily attenuates data values near n = 0 and n = L . The higher- order tapers successively give greater weights to these values to the point that tapers for k K have very poor bias properties and hence are not used. This behavior is quite evident in the frequency domain where as the taper order increases, mainlobe width and sidelobe attenuation decrease. The multitapering approach can be interpreted as a technique in which higher-order tapers capture information that is lost when only the fi rst taper is used. In Matlab the function [Pxx,Pxxc,F]=PMTM(x,W,Nfft,Fs) estimates the power spectrum of the data vector x in the array Pxx , using the multitaper approach. The function uses DPSS tapers with parameter W and adaptive weighted averaging Manolakis, Dimitris, et al. Statistical and Adaptive Signal Processing : Spectral Estimation, Signal Modeling, Adaptive Filtering and Array Processing, Artech House, 2005.

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