BohmanWindow
BohmanWindow[x]
represents a Bohman window function of x.
Details
- BohmanWindow 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.
- BohmanWindow[x] is equal to
![ (-sin(2 pi x)+2 pi x cos(2 pi x)+pi cos(2 pi x))/pi -1/2<=x<0; (sin(2 pi x)-2 pi x cos(2 pi x)+pi cos(2 pi x))/pi 0<=x<=1/2; 0 TemplateBox[{x}, Abs]>1/2;](Files/BohmanWindow.en/2.png " (-sin(2 pi x)+2 pi x cos(2 pi x)+pi cos(2 pi x))/pi -1/2<=x<0; (sin(2 pi x)-2 pi x cos(2 pi x)+pi cos(2 pi x))/pi 0<=x<=1/2; 0 TemplateBox[{x}, Abs]>1/2;")
. - BohmanWindow automatically threads over lists.
Examples
open all close allBasic Examples (3)
Scope (4)
Translated and dilated Bohman window:
Plot[BohmanWindow[(x - 1) / 2], {x, -1, 3}]2D Bohman window with a circular support:
Plot3D[BohmanWindow[Sqrt[x ^ 2 + y ^ 2]], {x, -1, 1}, {y, -1, 1}, PlotRange -> All]BohmanWindow[0.1]Discrete Bohman window of length 15:
ListPlot[Array[BohmanWindow, 15, {-1 / 2, 1 / 2}], Filling -> Axis]Discrete 15×10 2D Bohman window:
ListPointPlot3D[Array[BohmanWindow[#1] BohmanWindow[#2]&, {15, 10}, {{-1 / 2, 1 / 2}}], Filling -> Axis, PlotRange -> All]Applications (3)
Create a moving-average filter of length 21:
h = ConstantArray[1 / 21., 21]Taper the filter using a Bohman window:
w = Array[BohmanWindow, Length[h], {-1 / 2, 1 / 2}];
fir = w h;Log-magnitude plot of the power spectra of the filters:
LogLinearPlot[Evaluate[20Log[10, Abs@ListFourierSequenceTransform[#, ω]]& /@ {h, fir}], {ω, 0.1, Pi}, GridLines -> Automatic]Use a window specification to calculate sample PowerSpectralDensity:
proc = ARMAProcess[1, {.5}, {.3}, 1];
data = RandomFunction[proc, {50}];spec = PowerSpectralDensity[data, w, BohmanWindow];Compare to spectral density calculated without a windowing function:
sd = PowerSpectralDensity[data, w];sd === specThe plot shows that window smooths the spectral density:
Plot[{sd, spec}, {w, -π, π}, PlotRange -> All, PlotLegends -> {"no window", "with window"}]Compare to the theoretical spectral density of the process:
Plot[{spec, Evaluate@PowerSpectralDensity[proc, w]}, {w, -π, π}, PlotLegends -> {"data", "process"}]Use a window specification for time series estimation:
data = RandomFunction[ARMAProcess[1, {.3}, {.4}, 1], {300}];Specify window for spectral estimator:
EstimatedProcess[data, ARMAProcess[1, 1], ProcessEstimator -> {"SpectralEstimator", "Window" -> BohmanWindow}]Properties & Relations (2)
The area under the Bohman window:
area = Integrate[BohmanWindow[x], {x, -∞, ∞}]Normalize to create a window with unit area:
Plot[{BohmanWindow[x], BohmanWindow[x] / area}, {x, -1, 1}, PlotRange -> All]Fourier transform of the Bohman window:
f = FourierTransform[BohmanWindow[x], x, w]Power spectrum of the Bohman window:
LogLinearPlot[20 Log[10, Abs[f]], {w, .1, 80}]Text
Wolfram Research (2012), BohmanWindow, Wolfram Language function, https://reference.wolfram.com/language/ref/BohmanWindow.html.
CMS
Wolfram Language. 2012. "BohmanWindow." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BohmanWindow.html.
APA
Wolfram Language. (2012). BohmanWindow. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BohmanWindow.html