{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The Bartlett window is defined as" } ] }, { "__type": "Math", "__tag": 4058, "value": "w(n) = \\frac{2}{M-1} \\left(\\frac{M-1}{2} - \\left|n - \\frac{M-1}{2}\\right|\n\\right)" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Most references to the Bartlett window come from the signal processing literature, where it is used as one of many windowing functions for smoothing values. Note that convolution with this window produces linear interpolation. It is also known as an apodization (which means \"removing the foot\", i.e. smoothing discontinuities at the beginning and end of the sampled signal) or tapering function. The Fourier transform of the Bartlett window is the product of two sinc functions. Note the excellent discussion in Kanasewich " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "2" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ], "title": [], "level": 0, "target": null }, "Warns": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Raises": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Yields": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Methods": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Returns": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "out", "annotation": "array", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The triangular window, with the maximum value normalized to one (the value one appears only if the number of samples is odd), with the first and last samples equal to zero." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Return the Bartlett window." } ] } ], "title": [], "level": 0, "target": null }, "Receives": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Warnings": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Attributes": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Parameters": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "M", "annotation": "int", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Number of points in the output window. If zero or less, an empty array is returned." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Extended Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The Bartlett window is very similar to a triangular window, except that the end points are at zero. It is often used in signal processing for tapering a signal, without generating too much ripple in the frequency domain." } ] } ], "title": [], "level": 0, "target": null }, "Other Parameters": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null } }, "_ordered_sections": [ "Summary", "Extended Summary", "Parameters", "Attributes", "Methods", "Returns", "Yields", "Receives", "Other Parameters", "Raises", "Warns", "Warnings", "Notes" ], "item_file": "/numpy/lib/_function_base_impl.py", "item_line": 3136, "item_type": "function", "aliases": [ "numpy.bartlett" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Code", "__tag": 4050, "value": "import numpy as np\nimport matplotlib.pyplot as plt\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "np.bartlett(12)\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nPlot the window and its frequency response (requires SciPy and matplotlib).\n\n.. plot::\n :include-source:\n\n import matplotlib.pyplot as plt\n from numpy.fft import fft, fftshift\n window = np.bartlett(51)\n plt.plot(window)\n plt.title(\"Bartlett window\")\n plt.ylabel(\"Amplitude\")\n plt.xlabel(\"Sample\")\n plt.show()\n plt.figure()\n A = fft(window, 2048) / 25.5\n mag = np.abs(fftshift(A))\n freq = np.linspace(-0.5, 0.5, len(A))\n with np.errstate(divide='ignore', invalid='ignore'):\n response = 20 * np.log10(mag)\n response = np.clip(response, -100, 100)\n plt.plot(freq, response)\n plt.title(\"Frequency response of Bartlett window\")\n plt.ylabel(\"Magnitude [dB]\")\n plt.xlabel(\"Normalized frequency [cycles per sample]\")\n plt.axis('tight')\n plt.show()" } ], "title": [], "level": 0, "target": null }, "see_also": [ { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "blackman", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "current-module", "version": "current-version", "kind": "to-resolve", "path": "blackman" }, "kind": "module" }, "descriptions": [], "type": null }, { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "hamming", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "current-module", "version": "current-version", "kind": "to-resolve", "path": "hamming" }, "kind": "module" }, "descriptions": [], "type": null }, { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "hanning", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "current-module", "version": "current-version", "kind": "to-resolve", "path": "hanning" }, "kind": "module" }, "descriptions": [], "type": null }, { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "kaiser", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "current-module", "version": "current-version", "kind": "to-resolve", "path": "kaiser" }, "kind": "module" }, "descriptions": [], "type": null } ], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "M", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "bartlett" }, "references": [ ".. [1] M.S. Bartlett, \"Periodogram Analysis and Continuous Spectra\",", " Biometrika 37, 1-16, 1950.", ".. [2] E.R. Kanasewich, \"Time Sequence Analysis in Geophysics\",", " The University of Alberta Press, 1975, pp. 109-110.", ".. [3] A.V. Oppenheim and R.W. Schafer, \"Discrete-Time Signal", " Processing\", Prentice-Hall, 1999, pp. 468-471.", ".. [4] Wikipedia, \"Window function\",", " https://en.wikipedia.org/wiki/Window_function", ".. [5] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling,", " \"Numerical Recipes\", Cambridge University Press, 1986, page 429." ], "qa": "numpy:bartlett", "arbitrary": [], "local_refs": [ "M", "out" ] }