Minimum bias multiple taper spectral estimation

Two families of orthonormal tapers are proposed for multitaper spectral analysis: minimum bias tapers, and sinusoidal tapers {/spl upsisup (k/)}, where /spl upsisub nsup (k/)=/spl radic/(2/(N+1))sin(/spl pi/kn/N+1), and N is the number of points. The resulting sinusoidal multitaper spectral estimate is S/spl circ/(f)=(1/2K(N+1))/spl Sigmasub j=1sup K/|y(f+j/(2N+2))-y(f-j/(2N+2))|/sup 2/, where y(f) is the Fourier transform of the stationary time series, S(f) is the spectral density, and K is the number of tapers. For fixed j, the sinusoidal tapers converge to the minimum bias tapers like 1/N. Since the sinusoidal tapers have analytic expressions, no numerical eigenvalue decomposition is necessary. Both the minimum bias and sinusoidal tapers have no additional parameter for the spectral bandwidth. The bandwidth of the jth taper is simply 1/N centered about the frequencies (/spl plusmn/j)/(2N+2). Thus, the bandwidth of the multitaper spectral estimate can be adjusted locally by simply adding or deleting tapers. The band limited spectral concentration, /spl intsub -wsup w/|V(f)|/sup 2/df of both the minimum bias and sinusoidal tapers is very close to the optimal concentration achieved by the Slepian (1978) tapers. In contrast, the Slepian tapers can have the local bias, /spl intsub - 1/2 sup 1/2 /f/sup 2/|V(f)|/sup 2/df, much larger than of the minimum bias tapers and the sinusoidal tapers. >