Multitapering

Multitapering is an extension of single taper approaches and consists of dividing the data into overlapping subsets that are each individually tapered, and then Fourier transformed. The individual spectral coefficients of each subset are averaged to reduce the variance. The way in which the data is to be subdivided is defined by a set of tapers indexed by $l=1 \dots L$, the estimated spectral density at frequency bin $f$ is then given by:

$$S(\mathbf{f})=\frac {\sum\limits^{L-1}_{l=0}\lambda_lS_{l}(\mathbf{f})} {\sum\limits^{L-1}_{l=0}\lambda_l}$$ (15)

where $S_l(\mathbf{f})$ is the estimated spectral density using taper $l$ and $\lambda_l$ are weights for each tapered spectral density estimate.

As with the single-window approaches the spectral density is effectively smoothed, but without losing information at the end of the time series. The windows are chosen so that they are orthogonal and reduce leakage as much as possible. Under these requirements the optimal choice is the Discrete Prolate Spheroidal Sequences or Slepian sequences (Percival and Walden, 1993). The Slepian sequence used is determined by the length of the time series $N$, and by a parameter $W$ which corresponds to the half-bandwidth (i.e. $w=2\pi W$ is the half-bandwidth in radians). When using the Slepian sequences to give $S_l(\mathbf{f})$, the weights $\lambda_l$ used in equation 15 could simply be unity or something more complex. In this paper we use the eigenvalues of the Toeplitz matrix associated with the Fourier transformation as the weights (Percival and Walden, 1993). However, even these weights aren't optimal - more elaborate weighting such as Thomsen's non-linear approach Percival and Walden (1993), which adapt to the local variations in the spectral density could be used instead.

As with the single taper approaches the parameter $W$ needs to be chosen to balance the desired reduction in variance with minimising the distortion of the spectral estimate. The variance in the estimation of the spectral density is given by (Percival and Walden, 1993): For Slepian sequence multi-tapering the variance in the estimate is:

$$Var[\hat{\rho}_{xx}(\tau)/\rho_{xx}(\tau)]=2NW$$ (16)

Hence comparing equation 16 with equation 13 a Tukey tapering approach can be compared with multitapering by setting the variances the same. For example, for $N=200$ the recommended Tukey parameter was $M=28$. To give the same variance the requires a multitaper parameter of $NW\approx7$.