Lag-windowing and multiple-data-windowing are roughly equivalent for smooth spectrum estimation

There is no fundamental difference between lag-windowing a correlation sequence and multiple-windowing a data sequence when the objective is to reduce the mean-squared error of a spectrum estimator. By analyzing the approximate low-rank factorization of a bandlimiting Toeplitz operator, we find that lag-windowed (or spectrally smoothed) spectrum estimators have multiple-data-windowed implementations. This makes the Blackman–Tukey–Grenander–Rosenblatt spectrogram equivalent to the Thomson spectrum estimator (and vice-versa), meaning BTGR spectrograms may be implemented in a multichannel filterbank version of the Thomson estimator.