LanczosWindow
represents a Lanczos window function of x.
Details
- LanczosWindow, also known as the sinc window, is typically used for antialiasing and resampling.
- 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.
- LanczosWindow[x] is equal to
![ (sin(2 pi x))/(2 pi x) -1/2<=x<=1/2; 0 TemplateBox[{x}, Abs]>1/2;](Files/LanczosWindow.en/2.png " (sin(2 pi x))/(2 pi x) -1/2<=x<=1/2; 0 TemplateBox[{x}, Abs]>1/2;")
. - LanczosWindow automatically threads over lists.
Examples
open all close allBasic Examples (3)
Plot[LanczosWindow[x], {x, -1, 1}]Plot3D[LanczosWindow[x]LanczosWindow[y], {x, -1, 1}, {y, -1, 1}, PlotRange -> All, Exclusions -> None]Extract the continuous function representing the Lanczos window:
FunctionExpand[LanczosWindow[x]]Scope (4)
LanczosWindow[0.1]Translated and dilated Lanczos window:
Plot[LanczosWindow[(x - 1) / 2], {x, -1, 3}]2D Lanczos window with a circular support:
Plot3D[LanczosWindow[Sqrt[x ^ 2 + y ^ 2]], {x, -1, 1}, {y, -1, 1}, PlotRange -> All, Exclusions -> None]Discrete Lanczos window of length 15:
ListPlot[Array[LanczosWindow, 15, {-1 / 2, 1 / 2}], Filling -> Axis]Discrete 15×10 2D Lanczos window:
ListPointPlot3D[Array[LanczosWindow[#1] LanczosWindow[#2]&, {15, 10}, {{-1 / 2, 1 / 2}}], Filling -> Axis]Applications (4)
Decrease the size of an image by a factor of 5 using the Lanczos method:
Image[ImageResize[[image], Scaled[1 / 5], Resampling -> "Lanczos"], Magnification -> 5]By default, "Lanczos" resampling is used when resampling audio signals:
a = ExampleData[{"Sound", "RollingCoin"}, "Audio"];
AudioSampleRate[a]ar = AudioResample[a, 8000];
{AudioSampleRate[ar], ar}Use a window specification to calculate sample PowerSpectralDensity:
proc = ARMAProcess[1, {.5}, {.3}, 1];
data = RandomFunction[proc, {50}];spec = PowerSpectralDensity[data, w, LanczosWindow];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" -> LanczosWindow}]Properties & Relations (2)
The area under the Lanczos window:
area = Integrate[LanczosWindow[x], {x, -∞, ∞}]Normalize to create a window with unit area:
Plot[{LanczosWindow[x], LanczosWindow[x] / area}, {x, -1, 1}, PlotRange -> All]Fourier transform of the Lanczos window:
f = FourierTransform[LanczosWindow[x], x, w]Power spectrum of the Lanczos window:
LogLinearPlot[20 Log[10, Abs[f]], {w, .1, 80}]Text
Wolfram Research (2012), LanczosWindow, Wolfram Language function, https://reference.wolfram.com/language/ref/LanczosWindow.html.
CMS
Wolfram Language. 2012. "LanczosWindow." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/LanczosWindow.html.
APA
Wolfram Language. (2012). LanczosWindow. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LanczosWindow.html