Here we derive the bandwidth measure of Equation (355b) (Green hall, XXXXXXXXXXAssume that we have a time series that is a portion X 0 , X 1 , . . . , X N−1 of a zero mean Gaussian stationary process...

Here we derive the bandwidth measure of Equation (355b) (Green hall, 2006). Assume that we have a time series that is a portion X₀, X₁, . . . , X_N−1
of a zero mean Gaussian stationary process {Xt} with sampling interval ∆t and SDF S(·). Use this time series to form the weighted multitaper SDF estimator

  of Equation (352b), and assume that the tapers have unit energy and are pairwise orthogonal in the sense of Equation (354b).

(a) Assuming that
  are such that we can use the approximation of Equation (436b), show that

Where

Argue also that R^(WMT)(0) = 1.

(c) Consider now the special case  for which
  reduces to the basic multitaper SDF estimator
  k = 1/K, for which Sˆ (WMT)(·) reduces to the basic multitaper SDF estimator Sˆ (MT)(·) of Equation (352a). Show that the right-hand side of Equation (355b) reduces to the right-hand side of Equation (355a), and hence width
 becomes width_e