The Discrete Fourier Transform∗

1 Motivation We want to numerically approximate coefficients in a Fourier series. The first step is to see how the trapezoidal rule applies when numerically computing the integral (2π) −1 2π 0 F (t)dt, where F (t) is a continuous, 2π-periodic function. Applying the trapezoidal rule with the stepsize taken to be h = 2π/n for some integer n ≥ 1 results in (2π) −1 2π 0 F (t)dt ≈ 1 n n−1 j=0 Y j , where Y j := F (hj) = F (2πj/n), j = 1. .. n − 1. We remark that we made use of Y n = F (2π) = F (0) = Y 0 in employing the trapezoidal rule to arrive at the right hand side of the equation above. Recall that the coefficients in a Fourier series expansion for a continuous, 2π-periodic function f (t) have the form c k = 1 2π 2π 0 f (t) exp(−ikt)dt. We can apply the version of the trapezoidal rule derived above to approximately calculate the c k 's, since f (t) exp(−ikt) is 2π-periodic. Doing so yields c k ≈ 1 n n−1 j=0 f (2πij/n) exp(−2πijk/n) = 1 n n−1 j=0 y j w jk , where y j = f (2πij/n) and w = exp(2πi/n). If we replace k by k + n, the right hand side of the last equation is unchanged, for w n = exp(−2πi) = 1. * These notes are based on [1, Chapter 3].