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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