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,

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

where is the nonzero duration of in seconds. This integral equation can be understood as ``cropping'' to zero outside its main lobe (note that the integral goes from to , followed by a convolution of with a sinc function which ``time limits'' the window to a duration of seconds centered at time 0 in the time domain. In operator notation,

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 denotes the desired window length in samples, is the desired main-lobe cut-off frequency in radians per second, and is the sampling period in seconds. The main-lobe bandwidth is thus 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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