## 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 is a continuous function of , the function which maximize this ratio is the first

*prolate spheroidal wave function*for the given main-lobe bandwidth [101], [202, p. 205].

^{4.8}

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.

### Matlab for the DPSS Window

The function `dpss` in the Matlab Signal Processing Tool
Box^{4.10} can be used to compute them as follows:

w = dpss(M,alpha,1); % discrete prolate spheroidal sequencewhere is the desired window length, and can be interpreted as half of the window's time-bandwidth product in ``cycles''. Alternatively, can be interpreted as the highest bin number inside the main lobe of the window transform , when the DFT length is equal to the window length . (See the next section on the Kaiser window for more on this point.)

Some examples of DPSS windows and window transforms are given in Fig.3.29.

**Next Section:**

Kaiser Window

**Previous Section:**

Hann-Poisson Window