Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Applied Math 5600 Bengt Fornberg 1. Homework 3 Due: February 10, 2016 Calculate the finite Fourier transform of order N (even) of the following sequences: (a) uk 1 0 k N 1 (b) uk ( 1) (c) uk k k 0 k N 1 0 k N 1 Simplify your answers as far as you can. Hints: 1. 2. 1 We define the DFT of a sequence uj , j = 0,1,2,..., N-1 as u k N We know how to calculate N 1 k 0 N 1 u e j 2 i k j / N , k 0,1, 2, , N 1. j 0 k t for any t (a finite geometric progression). Can you from that result find a way to calculate the sum N 1 k 0 k kt ? 2. If we start with a set of N complex numbers, apply first the DFT and follow this by an inverse DFT, we get back the very same N complex numbers that we started with (and in the same order). Suppose we again start with N complex numbers but now instead apply the DFT twice in succession (i.e. do not invoke its inverse at either stage). How does the output vector then relate to the input vector? Derive the result, and then explain it in simple words. 3. Consider the period 5 data set x 0 1 2 3 4 y 1 2 3 1 2 . With use of Matlab’s fft and ifft routines (i.e. no explicit use in the code of trigonometric functions), plot the lowest degree trigonometric interpolant to this data (and mark on the plot also the original data entries). 4. The trapezoidal rule for numerical quadrature takes the form b a 1 1 f 0 f1 f 2 f n 1 f n O ( h 2 ) 2 2 f ( x ) dx h where the step size is h [b a ] / n . When the function f(x) is periodic over [a,b] and infinitely differentiable, the accuracy becomes much higher than second order. As an example, consider 2 0 e cos x dx 7.9549265210128453 . One way to understand how well the trapezoidal rule works in this case is to note that the integrand will have a very rapidly convergent Fourier expansion. In the present case e cos x n 0 an cos(nx ) where a0 1 2 2 0 e cos x dx and an 1 2 0 e cos x cos( nx )dx, n 0. By asymptotic analysis (the topic of APPM 5480), one can readily show that limn an 2 n 1 n ! 1. Correctly assuming that this limit result provides good approximations to an also at low values of n, use this to derive an approximation for the trapezoidal error when the original integral is discretized at xi 2 i / 8, i 0,1, ,8. Also tell how many nodes n you would need in order to obtain the very high accuracy of 10-60