This article is about butterfly diagrams in FFT algorithms. For sunspot diagrams, see Solar cycle.
Signal-flow graph connecting the inputs x (left) to the outputs y that depend on them (right) for a "butterfly" step of a radix-2 Cooley–Tukey FFT. This diagram resembles a butterfly (as in the morpho butterfly shown for comparison), hence the name, although in some countries it is also called the hourglass diagram.
In the context of fast Fourier transform algorithms, a butterfly is a portion of the computation that combines the results of smaller discrete Fourier transforms (DFTs) into a larger DFT, or vice versa (breaking a larger DFT up into subtransforms). The name "butterfly" comes from the shape of the data-flow diagram in the radix-2 case, as described below.[1] The earliest occurrence in print of the term is thought to be in a 1969 MIT technical report.[2][3] The same structure can also be found in the Viterbi algorithm, used for finding the most likely sequence of hidden states.
Most commonly, the term "butterfly" appears in the context of the Cooley–Tukey FFT algorithm, which recursively breaks down a DFT of composite size n = rm into r smaller transforms of size m where r is the "radix" of the transform. These smaller DFTs are then combined via size-r butterflies, which themselves are DFTs of size r (performed m times on corresponding outputs of the sub-transforms) pre-multiplied by roots of unity (known as twiddle factors). (This is the "decimation in time" case; one can also perform the steps in reverse, known as "decimation in frequency", where the butterflies come first and are post-multiplied by twiddle factors. See also the Cooley–Tukey FFT article.)
^Alan V. Oppenheim, Ronald W. Schafer, and John R. Buck, Discrete-Time Signal Processing, 2nd edition (Upper Saddle River, NJ: Prentice Hall, 1989)
^C. J. Weinstein (1969-11-21). Quantization Effects in Digital Filters (Report). MIT Lincoln Laboratory. p. 42. Archived from the original on February 11, 2015. Retrieved 2015-02-10. This computation, referred to as a 'butterfly'
^Cipra, Barry A. (2012-06-04). "FFT and Butterfly Diagram". mathoverflow.net. Retrieved 2015-02-10.
a larger DFT up into subtransforms). The name "butterfly" comes from the shape of the data-flow diagram in the radix-2 case, as described below. The earliest...
states. The butterflydiagram show a data-flow diagram connecting the inputs x (left) to the outputs y that depend on them (right) for a "butterfly" step of...
"Time vs. solar latitude diagram of the radial component of the solar magnetic field (supersynoptic map of "butterfly" diagram". Archived from the original...
solar latitudes in 1793–1796 and a reconstruction of the sunspot butterflydiagram for cycles 3 and 4, although total sunspot numbers only show a single-peaked...
reached. This pattern is best visualized in the form of the so-called butterflydiagram. Images of the Sun are divided into latitudinal strips, and the monthly-averaged...
Maunder. In 1904, he published their results in the form of the "butterfly" diagram. After studying the work of Gustav Spörer, who examined old records...
Tornado diagrams, also called tornado plots, tornado charts or butterfly charts, are a special type of Bar chart, where the data categories are listed...
asymptotically fast multiplication algorithm for large integers Butterflydiagram – a diagram used to describe FFTs Spectral music (involves application of...
Maunder sunspot minimum 1904 — Edward Maunder plots the first sunspot "butterflydiagram" 1906 — Karl Schwarzschild explains solar limb darkening 1908 — George...
A piping and instrumentation diagram (P&ID or PID) is a detailed diagram in the process industry which shows the piping and process equipment together...
progresses. This pattern is best visualized in the form of the so-called butterflydiagram, first constructed by Edward Walter Maunder and Annie Scott Dill Maunder...
interstellar medium. Edward Walter Maunder plots the first sunspot "butterflydiagram". Notable asteroid 522 Helga is discovered by Max Wolf in Heidelberg...
8). Its location conflicted directly with the expectation from the butterflydiagram. In principle new-cycle spots should appear at such a latitude, but...
particular permutation of vectors with 2m entries used in many FFTs. Butterflydiagram Twiddle factor — the trigonometric constant coefficients that are...
disadvantages of a single-loop butterfly. List of knots Entry #532 on page 87 of The Ashley Book of Knots shows a diagram of the butterfly loop under the name harness...
of the Schreier refinement theorem. The 'butterfly' becomes apparent when trying to draw the Hasse diagram of the various groups involved. Zassenhaus'...
In the Time of the Butterflies is a historical fiction novel by Julia Alvarez, relating a fictionalized account of the Mirabal sisters during the time...
of drugs on patients during drug trials. Spaghetti diagrams have been used to study why butterflies are found where they are, and to see how topographic...
that are not butterflies. They were previously classified as suborder Heterocera, but the group is paraphyletic with respect to butterflies (suborder Rhopalocera)...
Entry #532 on page 87 of The Ashley Book of Knots shows a diagram of the alpine butterfly knot under the name harness loop. Ashley appears to have illustrated...
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