Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 DiscreteDolphChebyshevWindow

Source Notebook

Return a discrete Dolph-Chebyshev window sequence

Contributed by: Mariusz Jankowski

ResourceFunction["DiscreteDolphChebyshevWindow"][n]

gives the discrete Dolph-Chebyshev window of length n.

ResourceFunction["DiscreteDolphChebyshevWindow"][n,α]

gives the discrete window with spectral side lobe attenuation of -20α dB.

Details and Options

The argument n should be an Integer defining the length of the window.
The argument α is a positive number, defining the spectral side lobe attenuation of the window. The default value is α=2.

Examples

Basic Examples (2) 

A Dolph-Chebyshev window of length 21:

In[1]:=
ResourceFunction["DiscreteDolphChebyshevWindow"][21]
Out[1]=

Plot the window:

In[2]:=
ListPlot[%, Filling -> 0]
Out[2]=

A Dolph-Chebyshev window with a specified parameter:

In[3]:=
ResourceFunction["DiscreteDolphChebyshevWindow"][21, 3]
Out[3]=

Applications (2) 

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

In[4]:=
h = LeastSquaresFilterKernel[{"Lowpass", \[Pi]/5}, 15]
Out[4]=

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

In[5]:=
fir = #/Total[#] &[ h * ResourceFunction["DiscreteDolphChebyshevWindow"][Length[h]]];

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

In[6]:=
LogLinearPlot[ Evaluate[ 20 Log[10, Abs@ListFourierSequenceTransform[#, \[Omega]]] & /@ {h, fir}], {\[Omega], 0.1, \[Pi]}, PlotRange -> All, GridLines -> Automatic]
Out[6]=

Use the Dolph-Chebyshev window to diminish the effect of signal partitioning when computing a spectrogram:

In[7]:=
{Spectrogram[\!\(\* TagBox[ DynamicModuleBox[{Audio`AudioObjects`audio$$ = HoldComplete[ Audio[CompressedData[" 1:eJztm4k7ltv3xpuUiiKaSdFAURpMeZ+9Io2aNaeUNEpKNA9kqsgQRSFRhpIh UuF99iIRKk6SoUgjColKSfTb+/tv/JzrOudcnVN4n2fvte57rfszxtp+xfYe 3bp1OyDL/rFiyyETB4ctRyx6819s27Jn+rQ9/H92Z3/PUuv2v7+KHt6H28tr YE3ZQXB9v4reNCW4f9xn6lexQ3rgeV8a4TYbN8j5YGzMX3p04nqw8/4GbtEv YcqYCNKR7Yf5Os9Qq9AR7dfeNF766jH5dfGecGCBJVW3kyVR74+CnVYalFuv gJNLNVCn/AtGHq9B3SlD0NvBHAa1x0DrCgsYdylNaP4liqkml0ivst4wVGWB 9NDOc0isXuPhjQE4ffBdUqhRClsaamG32Ty4m/aNqqkcRO3zgzHuq5soL9SL yzXH4z1Ta7Q0WUt3eHnC/Vvf4LXuAwiz6S2u00vFHi4VuCTbEqPfmZMh64bC 9Plu5J1VlOg8dDWx7LSHMsNwEN50hzwbW6wNa8VlJcW4efB5OnmFN1g8kEKs rAVU5XpKjfWP0U0hDcL2j7Fk9VtDepj4obVuHp5avBorRk2B57cb4HFiNfzp Lwsh9iZ49kUg7lgyEeHlejFpoJI4/HgHLZo8AWVmKAj7ll6G2rG14DIhEJZv zaHJpAQNKisxov90LC6rJilvF8GuBc/IBZsDM58d20IWLF8N40+6wIc9FmSW 7wXcu+c3Gpy/h0aCm+DtkgTvrQrA0sQEIn8Y0XevX1Pf4YfEEyXaZOjoALrU 8Dxu2pGArko9MTLsKMj0+wNFWc9gbM/95JiMB8bo3MKIn1NQY22S5O83KuQ3 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Initialization:>Audio`CheckID[Audio`AudioObjects`audioID$$], UnsavedVariables:>{Audio`AudioObjects`audioID$$}], Audio`AudioBox["AudioClass" -> "AudioData"], Editable->False, Selectable->False]\), 512, Automatic, None]}
Out[7]=

Properties and Relations (2) 

The Dolph-Chebyshev window side lobes are equiripple:

In[8]:=
Plot[Evaluate[ 20 Log10[ Abs@ListFourierSequenceTransform[ ResourceFunction["DiscreteDolphChebyshevWindow"][ 17], \[Omega]]]], {\[Omega], 0, \[Pi]}, GridLines -> {None, {-60}}]
Out[8]=

The Dolph-Chebyshev window side lobe attenuation is equal to -20α dB:

In[9]:=
Labeled[Plot[ Evaluate[ 20 Log10[ Abs@ListFourierSequenceTransform[ ResourceFunction["DiscreteDolphChebyshevWindow"][ 17, #], \[Omega]]]], {\[Omega], 0, \[Pi]}, GridLines -> {None, {-20 #}}, PlotRange -> {5, -80}], StringJoin["\[Alpha] = ", ToString[#]]] & /@ {1, 2, 3}
Out[9]=

Possible Issues (1) 

For small α, the window end-points may be larger than other sample values. The magnitude of the end-point impulses is on the order of the size of the spectral side lobes:

In[10]:=
ResourceFunction["DiscreteDolphChebyshevWindow"][11, 0.5]
Out[10]=
In[11]:=
ListPlot[%, Filling -> 0]
Out[11]=

Publisher

Mariusz Jankowski

Version History

  • 1.0.0 – 08 July 2020

Related Resources

Author Notes

NA

License Information