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A fast Fourier transform. To compute the Fourier transformh directly from its definition
requires basically N2 add-multiply operations (you should count them!). Fast Fourier transforms (FFTs) are attempts to speed up this computation by more efficiently handling arithmetic. In parts (a), (b), and (c) of this problem, we’ll assume that N can be factored N = p1 p2.
(a) Convince yourself that each of the indices k and v can be expressed in the following forms: