Integer Reversal

Digit-reversal permutation, and in particular bit-reversal permutation, is a reordering technique used in some fast Fourier transform algorithms.

Define a function that generates the reversal permutation of degree .

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reversalperm[k_, b_] := IntegerReverse[Range[0, b^k - 1], b, k] + 1;

Generate the bit-reversal permutation for lists of length .

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Table[reversalperm[k, 2], {k, 0, 5}]

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Generate the base-3 digit-reversal permutation for lists of 9 elements.

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reversalperm[2, 3]

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Represent those swaps.

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segment[{p1_, p2_}] := {PointSize[Large], Point[{p1, p2}], Line[{p1, p2}], Text[Last[p1], p1 - {1/2, 0}], Text[Last[p2], p2 + {1/2, 0}]};

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With[{k = 2, b = 3}, Graphics[ segment /@ Thread[{Thread[{0, Range[b^k]}], Thread[{b^k, reversalperm[k, b]}]}] ]]

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