Finite Fourier Transform, Circulant Matrices, and the Fast Fourier Transform
نویسنده
چکیده
Suppose we have a function s(t) that measures the sound level at time t of an analog audio signal. We assume that s(t) is piecewise-continuous and of finite duration: s(t) = 0 when t is outside some interval a ≤ t ≤ b. Make a change of variable x = (t− a)/(b− a) and set f(x) = s(t). Then 0 ≤ x ≤ 1 when a ≤ t ≤ b, and f(x) is a piecewise continuous function of x. We convert f(x) into a digital signal y ∈ C by sampling at the N equal-spaced values xj = j/N for j = 0, 1, . . . , N − 1:
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