Discrete Haar Transform. Let N = 2J where J N and suppose that yk R for
k = 0, 1, . . . , N 1 and that
N 1
f=
yk pJ,k .
k=0
Since f , hj,k = 0 for j J then f may be written as
J 1 2j 1
f , hj,k hj,k .
f = f , p0,0 p0,0 +
j =0 k=0
The following C subroutin
1. [Walnut Exercise 4.44] Prove that if xn is the N -point discrete Fourier transform of
the period-N signal xj then
N 1
j =0
N 1
1
| xj | =
N
2
n=0
| xn | 2 .
By denition
xn =
N 1
xj e2ijn/N .
j =0
Therefore
N 1
n=0
2
| xn | =
=
N 1
N 1
n=0
xn xn =
N 1
j
Math 761 Additional Problems for Homework 2
1. Explicitly compute the 36 entries of the matrix W6 corresponding to the discrete
Fourier transform x = W6 x where x and x are vectors of length 6 given by
x0
x1
x = x2 ,
.
.
.
x5
x0
x1
x = x2
.
.
.
5
a
Math 761 Midterm Version A
1. Fill in the missing blanks in following denitions from from An Introduction to
Wavelet Analysis by David Walnut.
Denition 4.35. (Discrete Fourier Transform) Given a period N signal xn ,
the N -point discrete Fourier transofrm
1. [Walnut Exercise 2.26] Show that if
lim an = a, then lim n = a where n =
n
n
1
n n
ak .
k=1
n
Lemma. If lim bn = 0, then lim
n
n
1
n
bk = 0.
k=1
Suppose > 0. Since bn 0 there is N0 large enough such that n N0 implies |bn | < /2.
N0
Dene M = k=1 |bk | a
Math 761 Quiz 2 Version A
1. Let (t) = e2it where t [0, 1] Use Cauchys integral formula to evaluate the
following integrals:
[ ]
sin
d
2 + 1
[ ]
sin
d
+2
[ ]
sin
d
2 2 + 1
(i)
(ii)
(iii)
Math 761 Quiz 2 Version A
2. Consider the curve given by C = [1
Math 761 Quiz 1 Version A
Formulas and Denitions
Approximate Identity. A collection of functions on R is an approximate identity on R
if the following conditions hold.
K (x) dx = 1.
(a) For every > 0 holds
R
K ( x ) dx M .
(b) There exists M > 0 such that