Function Repository Resource:

DPSSWindow

Source Notebook

Generate the zero-order discrete prolate spheroidal sequence window

Contributed by: Mariusz Jankowski

ResourceFunction["DPSSWindow"][n]

gives the order-0 discrete prolate spheroidal sequence (DPSS) window of length n.

ResourceFunction["DPSSWindow"][n,α]

uses the parameter α.

Details and Options

The discrete prolate spheroidal sequence is also known as a Slepian sequence.

The discrete prolate spheroidal sequence of order 0 is a sequence known to have the maximum spectral concentration of any window sequence of the same length, that is based on prolate spheroidal wavefunctions. The very low side lobes of its frequency response make the DPSS useful as a data taper and as a finite impulse response (FIR) filter.

The parameter α is a positive number that defines the width of the main lobe, and typically takes values in the range 0≤α<π/2. It is by default set to π/4.

Examples

Basic Examples (2)

A DPSS window of length 21:

In[1]:=

Out[1]=

Visualize the window:

In[2]:=

Out[2]=

A DPSS window with a specified parameter:

In[3]:=

Out[3]=

Applications (2)

Create a lowpass FIR filter with cut-off frequency of π/5 and length 15:

In[4]:=

$h = LeastSquaresFilterKernel[{"Lowpass", \[Pi]/5}, 15]$

Out[4]=

Apply a DPSS window to the filter to improve stop-band attenuation:

In[5]:=

Out[5]=

Create a Log-magnitude plot of the power spectra of the two filters:

In[6]:=

$LogLinearPlot[ Evaluate[ 20 Log10[Abs@ListFourierSequenceTransform[#, \[Omega]]] & /@ {h, fir}], {\[Omega], 0.1, \[Pi]}, PlotRange -> All, GridLines -> Automatic]$

Out[6]=

Use the DPSS window to diminish the effect of signal partitioning when computing the spectrogram:

In[7]:=

Out[7]=

Properties and Relations (2)

DPSSWindow[n,0] is equivalent to the rectangular window:

In[8]:=

Out[8]=

The spectral concentration ratio of the DPSS window for length n and bandwidth -B<=ω<B:

In[9]:=

$n = 21; B = \[Pi]/10.; w = ResourceFunction["DPSSWindow"][n, B]; W = ListFourierSequenceTransform[w, \[Omega]]; NIntegrate[Abs[W]^2, {\[Omega], -B, B}]/NIntegrate[ Abs[W]^2, {\[Omega], -\[Pi], \[Pi]}]$

Out[12]=

Compare with a Blackman window:

In[13]:=

$w = Array[BlackmanWindow, n, {-0.5, 0.5}]; W = ListFourierSequenceTransform[w, \[Omega]]; NIntegrate[Abs[W]^2, {\[Omega], -B, B}]/NIntegrate[ Abs[W]^2, {\[Omega], -\[Pi], \[Pi]}]$

Out[15]=

Publisher

Mariusz Jankowski

Version History

1.0.0 – 15 February 2023

Source Metadata

Citation:
- Gruenbacher D.M., Hummels D.R., "A simple algorithm for generating discrete prolate spheroidal sequences." IEEE Transactions on Signal Processing, vol. 42, no. 11, 3276-3278, 1994.
  DOI: 10.1109/78.330397
- Slepian D., "Prolate spheroidal wave functions, Fourier analysis, and uncertainty — V: The discrete case." The Bell System Technical Journal, vol. 57, no. 5, 1371-1430, 1978.
  DOI: 10.1002/j.1538-7305.1978.tb02104.x

Related Resources

License Information

This work is licensed under a Creative Commons Attribution 4.0 International License

Wolfram Function Repository

DPSSWindow

Details and Options