OpenAI. An explicit power saving for the exact discrete Fourier transform.

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8.3 The middle transform and the final permutation The factor QFt in (35) can be computed from a cyclic convolution and two pointwise phase multiplications. The chirp identity below performs the frequency change in Q without applying Q as a separate permutation of the array. The second identity shows how the retained source permutation R cancels when computing a convolution. Here ∗ denotes cyclic convolution on the product of the coordinate groups and a dot denotes coordinatewise multiplication.