Suppose we want to approximate the instantaneous power $z(n) = |y(n)|^2$ of the discrete-time signal $y(n)$, where $y(n)$ is the result of filtering $x(n)$ with a given window $h(n)$, $y(n) = x(n) \ast h(n)$, so $z(n) = |x(n) \ast h(n)|^2$. We need to reduce the fluctuation observed in $z(n)$ (i.e. we want to have a smooth $z(n)$), and to achieve this goal we are given the choice of choosing the window function $h(n)$ to convolve with $x(n)$.

Traditional choices for window functions in signal processing are Gaussian, Hanning, Hamming, Kaiser, Triangular, the list goes on. In applications of spectrogram estimation, it is also common to employ the Multitaper approach to reduce the variance of the estimated spectrum, i.e., by averaging out the results of computing $k=1,...,K$ spectrograms, the obtained mean spectrum is smooth.

Could we use the multitaper technique in time domain to obtain smooth approximations of $z(n)$? More specifically, if $h_1(n),...,h_K(n)$ are $K$ orthonormal tapers, shouldn't

$z(n) = \displaystyle \frac{1}{K}\sum_{k=1}^{K}|x(n) \ast h_k(n)|^2$

act like an average filter, yielding to a smoother $z(n)$ than what would be obtained by using a sole window $h(n)$, i.e. $z(n) = |x(n) \ast h(n)|^2$?


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