{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "For a single dimension array " }, { "__type": "InlineCode", "__tag": 4051, "value": "x" }, { "__type": "Text", "__tag": 4046, "value": ", " }, { "__type": "InlineCode", "__tag": 4051, "value": "dct(x, norm='ortho')" }, { "__type": "Text", "__tag": 4046, "value": " is equal to MATLAB " }, { "__type": "InlineCode", "__tag": 4051, "value": "dct(x)" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "There are, theoretically, 8 types of the DCT, only the first 4 types are implemented in scipy. 'The' DCT generally refers to DCT type 2, and 'the' Inverse DCT generally refers to DCT type 3." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Strong", "__tag": 4048, "children": [ { "__type": "Text", "__tag": 4046, "value": "Type I" } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "There are several definitions of the DCT-I; we use the following (for " }, { "__type": "InlineCode", "__tag": 4051, "value": "norm=None" }, { "__type": "Text", "__tag": 4046, "value": ")" } ] }, { "__type": "Math", "__tag": 4058, "value": "y_k = x_0 + (-1)^k x_{N-1} + 2 \\sum_{n=1}^{N-2} x_n \\cos\\left(\n\\frac{\\pi k n}{N-1} \\right)" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If " }, { "__type": "InlineCode", "__tag": 4051, "value": "norm='ortho'" }, { "__type": "Text", "__tag": 4046, "value": ", " }, { "__type": "InlineCode", "__tag": 4051, "value": "x[0]" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineCode", "__tag": 4051, "value": "x[N-1]" }, { "__type": "Text", "__tag": 4046, "value": " are multiplied by a scaling factor of " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\sqrt{2}" }, { "__type": "Text", "__tag": 4046, "value": ", and " }, { "__type": "InlineCode", "__tag": 4051, "value": "y[k]" }, { "__type": "Text", "__tag": 4046, "value": " is multiplied by a scaling factor " }, { "__type": "InlineCode", "__tag": 4051, "value": "f" } ] }, { "__type": "Math", "__tag": 4058, "value": "f = \\begin{cases}\n \\frac{1}{2}\\sqrt{\\frac{1}{N-1}} & \\text{if }k=0\\text{ or }N-1, \\\\\n \\frac{1}{2}\\sqrt{\\frac{2}{N-1}} & \\text{otherwise} \\end{cases}" }, { "__type": "Admonition", "__tag": 4056, "kind": "versionadded", "base_type": "neutral", "children": [ { "__type": "AdmonitionTitle", "__tag": 4055, "children": [ { "__type": "Text", "__tag": 4046, "value": "versionadded 1.2.0" } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Orthonormalization in DCT-I." } ] } ] }, { "__type": "Admonition", "__tag": 4056, "kind": "note", "base_type": "note", "children": [ { "__type": "AdmonitionTitle", "__tag": 4055, "children": [ { "__type": "Text", "__tag": 4046, "value": "note " } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The DCT-I is only supported for input size > 1." } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Strong", "__tag": 4048, "children": [ { "__type": "Text", "__tag": 4046, "value": "Type II" } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "There are several definitions of the DCT-II; we use the following (for " }, { "__type": "InlineCode", "__tag": 4051, "value": "norm=None" }, { "__type": "Text", "__tag": 4046, "value": ")" } ] }, { "__type": "Math", "__tag": 4058, "value": "y_k = 2 \\sum_{n=0}^{N-1} x_n \\cos\\left(\\frac{\\pi k(2n+1)}{2N} \\right)" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If " }, { "__type": "InlineCode", "__tag": 4051, "value": "norm='ortho'" }, { "__type": "Text", "__tag": 4046, "value": ", " }, { "__type": "InlineCode", "__tag": 4051, "value": "y[k]" }, { "__type": "Text", "__tag": 4046, "value": " is multiplied by a scaling factor " }, { "__type": "InlineCode", "__tag": 4051, "value": "f" } ] }, { "__type": "Math", "__tag": 4058, "value": "f = \\begin{cases}\n\\sqrt{\\frac{1}{4N}} & \\text{if }k=0, \\\\\n\\sqrt{\\frac{1}{2N}} & \\text{otherwise} \\end{cases}" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "which makes the corresponding matrix of coefficients orthonormal (" }, { "__type": "InlineCode", "__tag": 4051, "value": "O @ O.T = np.eye(N)" }, { "__type": "Text", "__tag": 4046, "value": ")." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Strong", "__tag": 4048, "children": [ { "__type": "Text", "__tag": 4046, "value": "Type III" } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "There are several definitions, we use the following (for " }, { "__type": "InlineCode", "__tag": 4051, "value": "norm=None" }, { "__type": "Text", "__tag": 4046, "value": ")" } ] }, { "__type": "Math", "__tag": 4058, "value": "y_k = x_0 + 2 \\sum_{n=1}^{N-1} x_n \\cos\\left(\\frac{\\pi(2k+1)n}{2N}\\right)" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "or, for " }, { "__type": "InlineCode", "__tag": 4051, "value": "norm='ortho'" } ] }, { "__type": "Math", "__tag": 4058, "value": "y_k = \\frac{x_0}{\\sqrt{N}} + \\sqrt{\\frac{2}{N}} \\sum_{n=1}^{N-1} x_n\n\\cos\\left(\\frac{\\pi(2k+1)n}{2N}\\right)" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The (unnormalized) DCT-III is the inverse of the (unnormalized) DCT-II, up to a factor " }, { "__type": "InlineCode", "__tag": 4051, "value": "2N" }, { "__type": "Text", "__tag": 4046, "value": ". The orthonormalized DCT-III is exactly the inverse of the orthonormalized DCT-II." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Strong", "__tag": 4048, "children": [ { "__type": "Text", "__tag": 4046, "value": "Type IV" } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "There are several definitions of the DCT-IV; we use the following (for " }, { "__type": "InlineCode", "__tag": 4051, "value": "norm=None" }, { "__type": "Text", "__tag": 4046, "value": ")" } ] }, { "__type": "Math", "__tag": 4058, "value": "y_k = 2 \\sum_{n=0}^{N-1} x_n \\cos\\left(\\frac{\\pi(2k+1)(2n+1)}{4N} \\right)" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If " }, { "__type": "InlineCode", "__tag": 4051, "value": "norm='ortho'" }, { "__type": "Text", "__tag": 4046, "value": ", " }, { "__type": "InlineCode", "__tag": 4051, "value": "y[k]" }, { "__type": "Text", "__tag": 4046, "value": " is multiplied by a scaling factor " }, { "__type": "InlineCode", "__tag": 4051, "value": "f" } ] }, { "__type": "Math", "__tag": 4058, "value": "f = \\frac{1}{\\sqrt{2N}}" }, { "__type": "Admonition", "__tag": 4056, "kind": "versionadded", "base_type": "neutral", "children": [ { "__type": "AdmonitionTitle", "__tag": 4055, "children": [ { "__type": "Text", "__tag": 4046, "value": "versionadded 1.2.0" } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Support for DCT-IV." } ] } ] } ], "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": "y", "annotation": "ndarray of real", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The transformed input array." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Return the Discrete Cosine Transform of arbitrary type sequence x." } ] } ], "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": "x", "annotation": "array_like", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The input array." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "type", "annotation": "{1, 2, 3, 4}, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Type of the DCT (see Notes). 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Default is None." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "overwrite_x", "annotation": "bool, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If True, the contents of " }, { "__type": "ParamRef", "__tag": 4071, "name": "x" }, { "__type": "Text", "__tag": 4046, "value": " can be destroyed; the default is False." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Extended Summary": { "__type": "Section", "__tag": 4015, "children": [], "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": "/scipy/fftpack/_realtransforms.py", "item_line": 235, "item_type": "function", "aliases": [ "scipy.fftpack.dct" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Text", "__tag": 4046, "value": "The Type 1 DCT is equivalent to the FFT (though faster) for real,\neven-symmetrical inputs. The output is also real and even-symmetrical.\nHalf of the FFT input is used to generate half of the FFT output:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "from scipy.fftpack import fft, dct\nimport numpy as np\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "fft(np.array([4., 3., 5., 10., 5., 3.])).real\ndct(np.array([4., 3., 5., 10.]), 1)\n", "execution_status": "failure" } ], "title": [], "level": 0, "target": null }, "see_also": [ { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "idct", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "current-module", "version": "current-version", "kind": "to-resolve", "path": "idct" }, "kind": "module" }, "descriptions": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Inverse DCT" } ] } ], "type": null } ], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "x", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "type", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "2" }, { "__type": "SigParam", "__tag": 4030, "name": "n", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "axis", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "-1" }, { "__type": "SigParam", "__tag": 4030, "name": "norm", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "overwrite_x", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "False" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "dct" }, "references": [ ".. [1] 'A Fast Cosine Transform in One and Two Dimensions', by J.", " Makhoul, `IEEE Transactions on acoustics, speech and signal", " processing` vol. 28(1), pp. 27-34,", " :doi:`10.1109/TASSP.1980.1163351` (1980).", ".. [2] Wikipedia, \"Discrete cosine transform\",", " https://en.wikipedia.org/wiki/Discrete_cosine_transform" ], "qa": "scipy.fftpack._realtransforms:dct", "arbitrary": [], "local_refs": [ "axis", "n", "norm", "overwrite_x", "type", "x", "y" ] }