{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The Kaiser window is defined as" } ] }, { "__type": "Math", "__tag": 4058, "value": "w(n) = I_0\\left( \\beta \\sqrt{1-\\frac{4n^2}{(M-1)^2}}\\right)/I_0(\\beta)" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "with" } ] }, { "__type": "Math", "__tag": 4058, "value": "\\quad -\\frac{M-1}{2} \\leq n \\leq \\frac{M-1}{2}," }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where " }, { "__type": "InlineMath", "__tag": 4057, "value": "I_0" }, { "__type": "Text", "__tag": 4046, "value": " is the modified zeroth-order Bessel function." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The Kaiser was named for Jim Kaiser, who discovered a simple approximation to the DPSS window based on Bessel functions. The Kaiser window is a very good approximation to the discrete prolate spheroidal sequence, or Slepian window, which is the transform which maximizes the energy in the main lobe of the window relative to total energy." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The Kaiser can approximate other windows by varying the beta parameter. (Some literature uses alpha = beta/pi.) " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "4" } ] }, { "__type": "Code", "__tag": 4050, "value": "==== =======================\nbeta Window shape\n==== =======================\n0 Rectangular\n5 Similar to a Hamming\n6 Similar to a Hann\n8.6 Similar to a Blackman\n==== =======================", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "A beta value of 14 is probably a good starting point. Note that as beta gets large, the window narrows, and so the number of samples needs to be large enough to sample the increasingly narrow spike, otherwise NaNs will be returned." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Most references to the Kaiser window come from the signal processing literature, where it is used as one of many windowing functions for smoothing values. 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[1] J. F. Kaiser, \"Digital Filters\" - Ch 7 in \"Systems analysis by", " digital computer\", Editors: F.F. Kuo and J.F. Kaiser, p 218-285.", " John Wiley and Sons, New York, (1966).", ".. [2] E.R. Kanasewich, \"Time Sequence Analysis in Geophysics\", The", " University of Alberta Press, 1975, pp. 177-178.", ".. [3] Wikipedia, \"Window function\",", " https://en.wikipedia.org/wiki/Window_function", ".. [4] F. J. Harris, \"On the use of windows for harmonic analysis with the", " discrete Fourier transform,\" Proceedings of the IEEE, vol. 66,", " no. 1, pp. 51-83, Jan. 1978. :doi:`10.1109/PROC.1978.10837`." ], "qa": "scipy.signal.windows._windows:kaiser", "arbitrary": [], "local_refs": [ "M", "beta", "device: any", "sym", "w", "xp" ] }