## 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,

*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

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Hann-Poisson Window