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
degrees of freedom (sample values) in an
-point
window
to obtain a window transform
which maximizes the energy in the
main lobe of the window relative to total energy:
In the continuous-time case, i.e., when




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








![\begin{eqnarray*}
&& [\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
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img468.png)
where
is a rectangular windowing operation which zeros
outside the interval
.
Satisfying (3.37) means that window transform
is an eigenfunction of this sequence of operations; that is, it can be
zeroed outside the interval
, inverse Fourier
transformed, zeroed outside the interval
, and forward
Fourier transformed to yield the original Window transform
multiplied by some scale factor
(the eigenvalue of the
overall operation). We may say that
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
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
, each
associated with a real eigenvalues
.4.9 If
denotes the largest such eigenvalue of (3.37),
then its corresponding eigenfunction,
, 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]:
![]() |
(4.38) |
where




Matlab for the DPSS Window
The function dpss in the Matlab Signal Processing Tool Box4.10 can be used to compute them as follows:
w = dpss(M,alpha,1); % discrete prolate spheroidal sequencewhere







Some examples of DPSS windows and window transforms are given in Fig.3.29.
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Kaiser Window
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Hann-Poisson Window