On the spectral window of Thomson's estimator

Thomson's multi-window estimator is one of the most widely used techniques in the estimation of the spectrum of a time series with bandwidth W from N observations in the time domain. The success of the method is due to variance reduction and low bias (low spectral leakage resulting from convolving with the spectral window). While the reduction of variance is achieved by an averaging process, the control on the bias is based on the fact that the average of the square of the first K=2NW discrete prolate wave functions becomes closer, as K grows, to the ideal averaging kernel, given by the normalized characteristic function of the target bandwidth region. In this technical report, we derive an analytic estimate supporting and quantifying this fact: we bound the L1 distance between the spectral window and the ideal averaging kernel.