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To recover it you must specify " }, { "__type": "InlineCode", "__tag": 4051, "value": "orthogonalize=False" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "For " }, { "__type": "InlineCode", "__tag": 4051, "value": "norm=\"ortho\"" }, { "__type": "Text", "__tag": 4046, "value": " both the " }, { "__type": "CrossRef", "__tag": 4002, "value": "dst", "reference": { "__type": "RefInfo", "__tag": 4000, "module": null, "version": null, "kind": "local", "path": "dst" }, "kind": "local" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineRole", "__tag": 4003, "value": "idst", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " are scaled by the same overall factor in both directions. By default, the transform is also orthogonalized which for types 2 and 3 means the transform definition is modified to give orthogonality of the DST matrix (see below)." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "For " }, { "__type": "InlineCode", "__tag": 4051, "value": "norm=\"backward\"" }, { "__type": "Text", "__tag": 4046, "value": ", there is no scaling on the " }, { "__type": "CrossRef", "__tag": 4002, "value": "dst", "reference": { "__type": "RefInfo", "__tag": 4000, "module": null, "version": null, "kind": "local", "path": "dst" }, "kind": "local" }, { "__type": "Text", "__tag": 4046, "value": " and the " }, { "__type": "InlineRole", "__tag": 4003, "value": "idst", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " is scaled by " }, { "__type": "InlineCode", "__tag": 4051, "value": "1/N" }, { "__type": "Text", "__tag": 4046, "value": " where " }, { "__type": "InlineCode", "__tag": 4051, "value": "N" }, { "__type": "Text", "__tag": 4046, "value": " is the \"logical\" size of the DST." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "There are, theoretically, 8 types of the DST for different combinations of even/odd boundary conditions and boundary off sets " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "1" }, { "__type": "Text", "__tag": 4046, "value": ", only the first 4 types are implemented in SciPy." } ] }, { "__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 DST-I; we use the following for " }, { "__type": "InlineCode", "__tag": 4051, "value": "norm=\"backward\"" }, { "__type": "Text", "__tag": 4046, "value": ". DST-I assumes the input is odd around " }, { "__type": "InlineMath", "__tag": 4057, "value": "n=-1" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "n=N" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Math", "__tag": 4058, "value": "y_k = 2 \\sum_{n=0}^{N-1} x_n \\sin\\left(\\frac{\\pi(k+1)(n+1)}{N+1}\\right)" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Note that the DST-I is only supported for input size > 1. The (unnormalized) DST-I is its own inverse, up to a factor " }, { "__type": "InlineMath", "__tag": 4057, "value": "2(N+1)" }, { "__type": "Text", "__tag": 4046, "value": ". The orthonormalized DST-I is exactly its own inverse." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineCode", "__tag": 4051, "value": "orthogonalize" }, { "__type": "Text", "__tag": 4046, "value": " has no effect here, as the DST-I matrix is already orthogonal up to a scale factor of " }, { "__type": "InlineCode", "__tag": 4051, "value": "2N" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__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 DST-II; we use the following for " }, { "__type": "InlineCode", "__tag": 4051, "value": "norm=\"backward\"" }, { "__type": "Text", "__tag": 4046, "value": ". DST-II assumes the input is odd around " }, { "__type": "InlineMath", "__tag": 4057, "value": "n=-1/2" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "n=N-1/2" }, { "__type": "Text", "__tag": 4046, "value": "; the output is odd around " }, { "__type": "InlineMath", "__tag": 4057, "value": "k=-1" }, { "__type": "Text", "__tag": 4046, "value": " and even around " }, { "__type": "InlineMath", "__tag": 4057, "value": "k=N-1" } ] }, { "__type": "Math", "__tag": 4058, "value": "y_k = 2 \\sum_{n=0}^{N-1} x_n \\sin\\left(\\frac{\\pi(k+1)(2n+1)}{2N}\\right)" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If " }, { "__type": "InlineCode", "__tag": 4051, "value": "orthogonalize=True" }, { "__type": "Text", "__tag": 4046, "value": ", " }, { "__type": "InlineCode", "__tag": 4051, "value": "y[-1]" }, { "__type": "Text", "__tag": 4046, "value": " is divided " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\sqrt{2}" }, { "__type": "Text", "__tag": 4046, "value": " which, when combined with " }, { "__type": "InlineCode", "__tag": 4051, "value": "norm=\"ortho\"" }, { "__type": "Text", "__tag": 4046, "value": ", 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 of the DST-III, we use the following (for " }, { "__type": "InlineCode", "__tag": 4051, "value": "norm=\"backward\"" }, { "__type": "Text", "__tag": 4046, "value": "). DST-III assumes the input is odd around " }, { "__type": "InlineMath", "__tag": 4057, "value": "n=-1" }, { "__type": "Text", "__tag": 4046, "value": " and even around " }, { "__type": "InlineMath", "__tag": 4057, "value": "n=N-1" } ] }, { "__type": "Math", "__tag": 4058, "value": "y_k = (-1)^k x_{N-1} + 2 \\sum_{n=0}^{N-2} x_n \\sin\\left(\n\\frac{\\pi(2k+1)(n+1)}{2N}\\right)" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If " }, { "__type": "InlineCode", "__tag": 4051, "value": "orthogonalize=True" }, { "__type": "Text", "__tag": 4046, "value": ", " }, { "__type": "InlineCode", "__tag": 4051, "value": "x[-1]" }, { "__type": "Text", "__tag": 4046, "value": " is multiplied by " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\sqrt{2}" }, { "__type": "Text", "__tag": 4046, "value": " which, when combined with " }, { "__type": "InlineCode", "__tag": 4051, "value": "norm=\"ortho\"" }, { "__type": "Text", "__tag": 4046, "value": ", 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": "Text", "__tag": 4046, "value": "The (unnormalized) DST-III is the inverse of the (unnormalized) DST-II, up to a factor " }, { "__type": "InlineMath", "__tag": 4057, "value": "2N" }, { "__type": "Text", "__tag": 4046, "value": ". 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