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### Thomson's method of spectral estimation

Thomson's multiple window method of spectral estimationthomson provides a very good estimate of the power spectrum by measuring the energy contained within a collection of rectangular shaped frequency intervals. The spectral estimate is formulated by averaging together, with appropriately chosen weightings (the eigenvalues), multiple power spectral estimates, each computed with a different window.

The windows comprise a family of discrete prolate spheroidal sequences (DPSS), have been studied extensively (Landau, Pollack, Slepiandps1,dps2,dps3,dps4,dps5) and are commonly referred to as prolates or Slepians.

The remarkable property of this family of windows is that their energy contributions add up in a very special way that collectively defines an ideal (ideal in the sense of the total in-bin versus out-of-bin energy concentration) rectangular frequency bin. Furthermore, for a time series of a given length, the power spectrum may be estimated at various resolutions (e.g. we can choose the frequency bin size). While it might at first seem unclear why one would want anything other than the highest resolution, the Thomson method allows us to trade resolution for improved statistical properties (reduced variance of the spectral estimate). Often, much of the fine structure of a spectral estimate is due to noise. It should be stressed that while other methods of spectral estimation (such as the Welchoppenheim method) exist, the Thomson method is particularly noteworthy for its precisely defined rectangular frequency bins.

Generally, the Thomson method is thought of as a multiple window method, but another way of thinking of the Thomson method is by the way that the energy in each frequency bin is calculated. To determine the quantity of energy inside the bin centered at , we frequency-shift each of the windows to , and sum the energy contributions from each of the frequency-shifted windows:

S(f_c) = _i |

C_0,f_c,0,0,0 g_i \: s

|^2

Writing the Thomson method in this way, we can generalize it further by replacing the one-parameter operator, , with multi-parameter operators.

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Steve Mann
Thu Jan 8 19:50:27 EST 1998