DFT and FFT By using the complex roots

DFT and FFT • By using the complex roots of unity, we can evaluate and interpolate a polynomial in O(n lg n) • An example, here are the solutions to 8 = 1 • The principal root is • We often write these values in trigonometric form

Properties of Roots of Unity • The halving lemma is essential since it guarantees the recursive solutions of subproblems are only half as large

Discrete Fourier Transform • We wish to evaluate at the points • We can assume n to be a power of 2 by adding high-order zero coefficients as necessary • Assuming a = (a 0, a 1, …, an-1) we define • This calculation is O(n 2), but by using the complex roots of unity, it can be done in O(n lg n)

Fast Fourier Transform • We separate the even and odd index coefficients • Our desired result is • So to evaluate A(x) at the complex roots of unity

FFT - continued • Due to the halving lemma, there are not n distinct values, rather only n/2 complex (n/2)th roots • Since the solutions of the subproblems are of the same form as the original problem, these recursive solutions are also only half in size • So we can solve an n-element DFTn computation by solving two n/2 -element DFTn/2 computations • In the following recursive FFT algorithm, we compute the DFT for the vector where n is a power of 2

FFT Algorithm - 1 • Lines 2 -3 handle the base case • Lines 4, 5, 13 update so that at lines 11 -12 we have • Lines 6 -7 define the coefficient vectors for the subpolynomials

FFT Algorithm - 2 • Lines 8 -9 solve the subproblems recursively • The algorithm complexity is

Interpolation • The DFT can be written as the matrix product y = Vna • The DFTn-1 can be computed in O(n lg n) time too