HannPoissonWindow

✖
`HannPoissonWindow`

✖

HannPoissonWindow[x]

represents a Hann–Poisson window function of x.

✖

HannPoissonWindow[x,α]

uses the parameter α.

Details

HannPoissonWindow is a window function typically used in signal processing applications where data needs to be processed in short segments.
Window functions have a smoothing effect by gradually tapering data values to zero at the ends of each segment.

HannPoissonWindow[x,α] is equal to $ (cos(2 pi x)+1)/(2 ⅇ^(2 alpha x)) 0<=x<=1/2; 1/2 ⅇ^(2 alpha x) (cos(2 pi x)+1) -1/2<=x<0; 0 TemplateBox[{x}, Abs]>1/2;$ .
HannPoissonWindow[x] is equivalent to HannPoissonWindow[x,1].
HannPoissonWindow automatically threads over lists.

Examples

open allclose all

Basic Examples (3)Summary of the most common use cases

Shape of a 1D Hann–Poisson window:

In[1]:=1

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-55nvy5

Out[1]=1

Shape of a 2D Hann–Poisson window:

In[1]:=1

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-yyuiog

Out[1]=1

Extract the continuous function representing the Hann–Poisson window:

In[1]:=1

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-i5bqnh

Out[1]=1

Parameterized Hann–Poisson window:

In[2]:=2

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-zwjlmu

Out[2]=2

Scope (6)Survey of the scope of standard use cases

Evaluate numerically:

In[1]:=1

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-rfdjbb

Out[1]=1

Shape of a 1D Hann–Poisson window using a specified parameter:

In[1]:=1

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-y9d2xu

Out[1]=1

Variation of the shape as a function of the parameter α:

In[1]:=1

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-eu37ay

Out[1]=1

Translated and dilated Hann–Poisson window:

In[1]:=1

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-fzqpy6

Out[1]=1

2D Hann–Poisson window with a circular support:

In[1]:=1

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-h4a5s3

Out[1]=1

Discrete Hann–Poisson window of length 15:

In[1]:=1

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-5yhlc9

Out[1]=1

Discrete 15×10 2D Hann–Poisson window:

In[2]:=2

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-ivcfcj

Out[2]=2

Applications (3)Sample problems that can be solved with this function

Create a moving average filter of length 21:

In[1]:=1

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-b4svbb

Out[1]=1

Taper the filter using a Hamming window:

In[8]:=8

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-1k06dy

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

In[10]:=10

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-65dzhv

Out[10]=10

Use a window specification to calculate sample PowerSpectralDensity:

In[1]:=1

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-la7hx0

Calculate the spectrum:

In[2]:=2

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-xvpxo9

Compare to spectral density calculated without a windowing function:

In[3]:=3

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-012m6s

In[4]:=4

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-phnp33

Out[4]=4

The plot shows that window smooths the spectral density:

In[5]:=5

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-z97d2x

Out[5]=5

Compare to the theoretical spectral density of the process:

In[6]:=6

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-2bqb4v

Out[6]=6

Use a window specification for time series estimation:

In[1]:=1

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-8tceex

Specify window for spectral estimator:

In[2]:=2

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-nkbyb7

Out[2]=2

Properties & Relations (3)Properties of the function, and connections to other functions

HannPoissonWindow[x,0] is equivalent to a Hann window:

In[1]:=1

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-vfnwv2

Out[1]=1

The area under the Hann–Poisson window:

In[1]:=1

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-bvk48s

Out[1]=1

Normalize to create a window with unit area:

In[2]:=2

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-uvswua

Out[2]=2

Fourier transform of the Hann–Poisson window:

In[1]:=1

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-hw628m

Out[1]=1

Power spectrum of the Hann–Poisson window:

In[2]:=2

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-6mml6t

Out[2]=2

Possible Issues (1)Common pitfalls and unexpected behavior

2D sampling of Hann–Poisson window will use a different parameter for each row of samples when passed as a symbol to Array:

In[1]:=1

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-0oh59a

Out[1]=1

Use a pure function instead:

In[2]:=2

✖

https://wolfram.com/xid/0rkt7ijm1lluqhe-iqrxnt

Out[2]=2

Top