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Slepian or DPSS Window

A window having maximal energy concentration in the main lobe is given by the digital prolate spheroidal sequence (DPSS) of order 0 [256,136]. It is obtained by using all $ M$ degrees of freedom (sample values) in an $ M$ -point window $ w(n)$ to obtain a window transform $ W(\omega)\approx\delta(\omega)$ which maximizes the energy in the main lobe of the window relative to total energy:

$\displaystyle \max_w \left[ \frac{ \mbox{main lobe energy} } { \mbox{total energy} } \right] \protect$ (4.36)

In the continuous-time case, i.e., when $ W(\omega)$ is a continuous function of $ \omega\in(-\infty,\infty)$ , the function $ W(\omega)$ which maximize this ratio is the first prolate spheroidal wave function for the given main-lobe bandwidth $ 2\omega_c$ [101], [202, p. 205].4.8

A prolate spheroidal wave function is defined as an eigenfunction of the integral equation

$\displaystyle \int_{-\omega_c}^{\omega_c} W(\nu) \frac{\sin[\pi D\cdot(\omega-\nu)]}{\pi(\omega-\nu)}\, d\omega = \lambda W(\omega) \protect$ (4.37)

where $ D$ is the nonzero duration of $ w(t)$ in seconds. This integral equation can be understood as ``cropping'' $ W(\omega)$ to zero outside its main lobe (note that the integral goes from $ -\omega _c$ to $ \omega_c$ , followed by a convolution of $ W(\omega)$ with a sinc function which ``time limits'' the window $ w(t)$ to a duration of $ D$ seconds centered at time 0 in the time domain. In operator notation,

&& [\hbox{\sc Chop}_{2\omega_c}(W)]*[D\,\mbox{sinc}(D\omega)]\\
&=& \hbox{\sc FT}(\hbox{\sc Chop}_D(\hbox{\sc IFT}(\hbox{\sc Chop}_{2\omega_c}(W)))) \eqsp \lambda W

where $ \hbox{\sc Chop}_D(w)$ is a rectangular windowing operation which zeros $ w(t)$ outside the interval $ t\in[-D/2,D/2]$ .

Satisfying (3.37) means that window transform $ W(\omega)$ is an eigenfunction of this sequence of operations; that is, it can be zeroed outside the interval $ [-\omega_c,\omega_c]$ , inverse Fourier transformed, zeroed outside the interval $ [-D/2,D/2]$ , and forward Fourier transformed to yield the original Window transform $ W(\omega)$ multiplied by some scale factor $ \lambda$ (the eigenvalue of the overall operation). We may say that $ W$ is the bandlimited extrapolation of its main lobe.

The sinc function in (3.37) can be regarded as a symmetric Toeplitz operator kernel), and the integral of $ W$ multiplied by this kernel can be called a symmetric Toeplitz operator. This is a special case of a Hermitian operator, and by the general theory of Hermitian operators, there exists an infinite set of mutually orthogonal functions $ W_m(\omega)$ , each associated with a real eigenvalues $ \lambda_m$ .4.9 If $ \lambda_0$ denotes the largest such eigenvalue of (3.37), then its corresponding eigenfunction, $ W_0(\omega)\;\leftrightarrow\;
w_0(t)$ , is what we want as our Slepian window, or prolate spheroidal window in the continuous-time case. It is optimal in the sense of having maximum main-lobe energy as a fraction of total energy.

The discrete-time counterpart is Digital Prolate Spheroidal Sequences (DPSS), which may be defined as the eigenvectors of the following symmetric Toeplitz matrix constructed from a sampled sinc function [13]:

$\displaystyle S[k,l] = \frac{\sin[\omega_c T(k-l)]}{k-l}, \quad k,l=0,1,2,\ldots,M-1$ (4.38)

where $ M$ denotes the desired window length in samples, $ \omega_c$ is the desired main-lobe cut-off frequency in radians per second, and $ T$ is the sampling period in seconds. The main-lobe bandwidth is thus $ 2\omega_c$ rad/sec, counting both positive and negative frequencies.) The digital Slepian window (or DPSS window) is then given by the eigenvector corresponding to the largest eigenvalue. A simple matlab program is given in §F.1.2 for computing these windows, and facilities in Matlab and Octave are summarized in the next subsection.

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2020-07-26 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University