{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The defaults are chosen such that " }, { "__type": "InlineCode", "__tag": 4051, "value": "f(x)" }, { "__type": "Text", "__tag": 4046, "value": " is equivalent to " }, { "__type": "InlineCode", "__tag": 4051, "value": "fft.fft(x)" }, { "__type": "Text", "__tag": 4046, "value": " and, if " }, { "__type": "InlineCode", "__tag": 4051, "value": "m > len(x)" }, { "__type": "Text", "__tag": 4046, "value": ", that " }, { "__type": "InlineCode", "__tag": 4051, "value": "f(x, m)" }, { "__type": "Text", "__tag": 4046, "value": " is equivalent to " }, { "__type": "InlineCode", "__tag": 4051, "value": "fft.fft(x, m)" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If " }, { "__type": "ParamRef", "__tag": 4071, "name": "w" }, { "__type": "Text", "__tag": 4046, "value": " does not lie on the unit circle, then the transform will be around a spiral with exponentially-increasing radius. Regardless, angle will increase linearly." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "For transforms that do lie on the unit circle, accuracy is better when using " }, { "__type": "CrossRef", "__tag": 4002, "value": "ZoomFFT", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.signal._czt:ZoomFFT" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": ", since any numerical error in " }, { "__type": "ParamRef", "__tag": 4071, "name": "w" }, { "__type": "Text", "__tag": 4046, "value": " is accumulated for long data lengths, drifting away from the unit circle." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The chirp z-transform can be faster than an equivalent FFT with zero padding. Try it with your own array sizes to see." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "However, the chirp z-transform is considerably less precise than the equivalent zero-padded FFT." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "As this CZT is implemented using the Bluestein algorithm, it can compute large prime-length Fourier transforms in O(N log N) time, rather than the O(N**2) time required by the direct DFT calculation. (" }, { "__type": "CrossRef", "__tag": 4002, "value": "scipy.fft", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.fft" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": " also uses Bluestein's algorithm'.)" } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "(The name \"chirp z-transform\" comes from the use of a chirp in the Bluestein algorithm. 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[1] Leo I. Bluestein, \"A linear filtering approach to the computation", " of the discrete Fourier transform,\" Northeast Electronics Research", " and Engineering Meeting Record 10, 218-219 (1968).", ".. [2] Rabiner, Schafer, and Rader, \"The chirp z-transform algorithm and", " its application,\" Bell Syst. Tech. J. 48, 1249-1292 (1969)." ], "qa": "scipy.signal._czt:CZT", "arbitrary": [], "local_refs": [ "a", "f", "m", "n", "w" ] }