ExactBlackmanWindow
represents an exact Blackman window function of x.
Details
- ExactBlackmanWindow is a window function typically used for finite impulse response (FIR) filter design and spectral analysis.
- Window functions are used in applications where data is processed in short segments and have a smoothing effect by gradually tapering data values to zero at the ends of each segment.
- ExactBlackmanWindow[x] is equal to
![ (4620 cos(2 pi x)+715 cos(4 pi x)+3969)/(9304) -1/2<=x<=1/2; 0 TemplateBox[{x}, Abs]>1/2;](Files/ExactBlackmanWindow.en/2.png " (4620 cos(2 pi x)+715 cos(4 pi x)+3969)/(9304) -1/2<=x<=1/2; 0 TemplateBox[{x}, Abs]>1/2;")
- ExactBlackmanWindow automatically threads over lists.
Examples
open all close allBasic Examples (3)
Shape of a 1D exact Blackman window:
Plot[ExactBlackmanWindow[x], {x, -1, 1}]Shape of a 2D exact Blackman window:
Plot3D[ExactBlackmanWindow[x]ExactBlackmanWindow[y], {x, -1, 1}, {y, -1, 1}, PlotRange -> All]Extract the continuous function representing the exact Blackman window:
FunctionExpand[ExactBlackmanWindow[x]]Scope (4)
ExactBlackmanWindow[0.1]Translated and dilated exact Blackman window:
Plot[ExactBlackmanWindow[(x - 1) / 2], {x, -1, 3}]2D exact Blackman window with a circular support:
Plot3D[ExactBlackmanWindow[Sqrt[x ^ 2 + y ^ 2]], {x, -1, 1}, {y, -1, 1}, PlotRange -> All]Discrete exact Blackman window of length 15:
ListPlot[Array[ExactBlackmanWindow, 15, {-1 / 2, 1 / 2}], Filling -> Axis]Discrete 15×10 2D exact Blackman window:
ListPointPlot3D[Array[ExactBlackmanWindow[#1] ExactBlackmanWindow[#2]&, {15, 10}, {{-1 / 2, 1 / 2}}], Filling -> Axis]Applications (3)
Create a moving average filter of length 11:
a = ConstantArray[1 / 11, 11]Smooth the filter using a exact Blackman window:
a2 = # / Total[#]&[a Array[ExactBlackmanWindow, 11, {-1 / 2, 1 / 2}]];Log-magnitude plot of the frequency spectrum of the filters:
LogLinearPlot[Evaluate[20Log[10, Abs@ListFourierSequenceTransform[#, ω]]& /@ {a, a2}], {ω, 0.0314, Pi}, PlotRange -> All, GridLines -> Automatic, ImageSize -> 300]Use a window specification to calculate sample PowerSpectralDensity:
proc = ARMAProcess[1, {.5}, {.3}, 1];
data = RandomFunction[proc, {50}];spec = PowerSpectralDensity[data, w, ExactBlackmanWindow];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" -> ExactBlackmanWindow}]Properties & Relations (2)
The area under the exact Blackman window:
area = Integrate[ExactBlackmanWindow[x], {x, -∞, ∞}]Normalize to create a window with unit area:
Plot[{ExactBlackmanWindow[x], ExactBlackmanWindow[x] / area}, {x, -1, 1}, PlotRange -> All]Fourier transform of the exact Blackman window:
f = FourierTransform[ExactBlackmanWindow[x], x, w]Power spectrum of the exact Blackman window:
LogLinearPlot[20 Log[10, Abs[f]], {w, .1, 80}]Text
Wolfram Research (2012), ExactBlackmanWindow, Wolfram Language function, https://reference.wolfram.com/language/ref/ExactBlackmanWindow.html.
CMS
Wolfram Language. 2012. "ExactBlackmanWindow." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ExactBlackmanWindow.html.
APA
Wolfram Language. (2012). ExactBlackmanWindow. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ExactBlackmanWindow.html