Question

9.99 Walk-Through: Discrete Fourier Trans- forms. You've measured the following data points for a function f(x):f(0) = 2, /(2) = 3,f(4) =-6, f(6) = 0. (a) Use Equation 9.7.1 to calculate and f2 (b) Find /-, without using Equation 9.7.1. This should lake no more than 20 seconds (c) What are/2 and? Again, more than 20 seconds means you re doing it wrong. (d) What frequencies p are represented by the terms f 1, fo fi and /2? 1J0J1 The Formula for a Discrete Fourier Transform Suppose a function /(a) is known at N discrete points: -/(rAx) for r 0, 1.....N-1. The "dis crete Fourier transform" (DFT)" of f is a set of points/, with-N/2 s n N/2 given by the formula N-I -0 The numbers /, are proportional to the amplitudes of oscillations with frequencies given by: 2πη (9.7.2) Inverse Discrete Fourier Transform The inverse discrete Fourier transform of a set of points /, is N/2 2rin (9.7.3) -N/2+1 (The sum can go over any N consecutive values of n, and it is often written from 0 ¡o N-1, but we prefer this way because it focuses on the most physically meaningful values. Clearly for odd N you would have to adjust the limits of the sum.)

Answer 1

N-1 2-36 rzo - lax1. Scanned wi CamScanner 2.