Homework #1 Due: Tuesday, February 2, 2010
1. (2pt) Compute the following integrals:
1
log( x)dx
0
where log(x) denotes the natural logarithm of x, and 1 x a dx 1 for a fixed parameter a>1. What happ
Homework #11 Due: Thursday, May 6, 2010
Note: Use of Matlab (or any other software) is not permitted. I. (see Exercise 4.11) A 1periodic function on R having the Fourier series:
f ( x) =
k = 
c e
k
Quiz 1, MATH 464, 6 April 2010
Your Name: _
(1) [3 pts] A bandlimited signal is sampled at its critical (Nyquist) sampling rate of 100 KHz. Thus in onesecond interval 100,000 samples are collected.
Quiz 1, MATH 464, 6 April 2010
Your Name: _
(1) [3 pts] A bandlimited signal is sampled at its critical (Nyquist) sampling rate of 100 KHz. Thus in onesecond interval 100,000 samples are collected.
Solutions to Homework 1 1. (a) For 0 < t < 1 use integration by parts in:
1 1 1
log(x)dx = x log(x)1  t
t t
x(log(x) dx = t log(t)
t
1dx = t log(t)1+t
Then
1 1
log(x)dx = lim
0 t
0
log(x)dx = li
R. Balan
Homework #5 Solutions
MATH 464
I. 1. F (0) = 1 2. F (0) = 2i 3. F (1) + F (1) = 0 4.
sF (s)ds = 0

5. F (s) = F (s) , that means F (s) is real valued II. 6. We look for a function f : R C
R. Balan
Homework #6 Solutions
MATH 464
1. Note the function f is the convolution between the box function and the unit Gaussian function , that is:
f (x) =

(u)(x  u)du , (u) =
1 0
, u < , u >
R. Balan
Homework #7 Solutions
MATH 464
1. Method 1 2 Let g(x) = e2ix f (x). Note G(s) = F (s + ) and thus G(s) = 0 for s > . Hence g B (that is, g is band limited). Shannon's sampling formula yie
R. Balan
Homework #8 Solutions
MATH 464
I. The window is g(x) = (x). a) f (x) = e2ix . The windowed Fourier transform is
Vg f (w, t) =

e2iwx f (x)g(x  t)dx =
e2ix(w1) (x  t)dx =
= e2it(w1)
e
R. Balan
Homework #9 Solutions
Note:
MATH 464
I.
a. f (x) = cos(x)(x/).
f (x) =
Its Fourier transform is
1 ix x e + eix 2
F (s) =
1 2
x 1 e2isx (eix +eix )( )dx = 2  y=
x ,
x 1 e2i(s1/2)x ( )
R. Balan
I.
Homework #10 Solutions
MATH 464
(a) Poisson's summation formula implies:
f (x) =
m=
e(xmp) =
2
1 p
e2ikx/p F (k/p)
k=
2
where Thus:
F (s) = F(ex )(s) = es 1 p
2
f (x) =
e2ikx/p ek
k
Homework #10 Due: Thursday, April 29, 2010
Note: Use of Matlab (or any other software) is not permitted. I. (see Exercise 4.2) Use Poisson's relation to find the Fourier series for each of the followi
Homework #9 Due: Thursday, April 22, 2010
Note: Use of Matlab (or any other software) is not permitted. I. (see Exercise 3.9) Find the Fourier transform of each of the following functions: x a. (1pt)
Homework #8 Due: Thursday, April 15, 2010
Note: Use of Matlab (or any other software) is not permitted.
I.
Consider the box function for the window function g=. Compute the windowed Fourier transform
Homework #4 Due: Tuesday, March 3, 2009
Note: Use of Matlab (or any other software) is not permitted. I. Compute the Fourier transform of the following functions (18): 1. 1 , 1 x 2 f ( x) = 0 , other
Homework #5 Due: Tuesday, March 10, 2009
Note: Use of Matlab (or any other software) is not permitted. I. (Exercise 3.23) Let f be a suitably regular function on R with the Fourier transform F. What c
Homework #6 Due: Tuesday, April 7, 2009
Note: Use of Matlab (or any other software) is not permitted. 1. (Exercise 7.4) Use the mesa function from Figure 7.5 of the textbook to construct two Schwartz
Homework #7 Due: Tuesday, April 14, 2009
Note: Use of Matlab (or any other software) is not permitted. (Exercises 7.8, 7.11) Find and simplify the functional fcfw_ , S, that is used to represent the g
Homework #8 Due: Tuesday, April 21, 2009
Note: Use of Matlab (or any other software) is not permitted. I. (1pt) Denote by D the dilation operator that maps a set of function into a set of functions vi
Homework #9 Due: Tuesday, April 28, 2009
Note: Use of Matlab (or any other software) is not permitted. (Exercise 7.27) Find all generalized functions f that satisfy each of the following inhomogeneous
Homework #10 Due: Tursday, May 7, 2009
Note: Use of Matlab (or any other software) is not permitted. I. (Exercises 7.34) Simplify the following expressions. The parameters a,b, are real, and m,n=0,1,2
Homework #2 Due: Tuesday, February 9, 2010
Compute the following integrals. Note: Use of Matlab (or any other software) is not permitted. 1 1. x 2 + 1 dx 0
2.

x
2
1 dx , for a fixed real parameter
Homework #3 Due: Thursday, February 25, 2010
Compute the Fourier coefficients, and expand in Fourier series the following 1periodic functions. Note: Use of Matlab (or any other software) is not permi
Homework #4 Due: Thursday, March 4, 2010
Note: Use of Matlab (or any other software) is not permitted. I. Compute the Fourier transform of the following functions (18): 1. 1 , 1 x 2 f ( x) = 0 , othe
Homework #5 Due: Thursday, March 11, 2010
Note: Use of Matlab (or any other software) is not permitted. I. (Exercise 3.23) Let f be a suitably regular function on R with the Fourier transform F. What
Homework #6 Due: Tuesday, March 23, 2010
1. (2pts) Find the Fourier transform of the function f:RR
1/ 2
f ( x) =
1 / 2
e
 ( x u ) 2
du
2. (2pts) Let f0,f1:RR be functions defined by 2 2 f 0 ( x) =
Homework #7 Due: Thursday, April 8, 2010
Note: Use of Matlab (or any other software) is not permitted. (3pts) Assume f:RR is a squareintegrable function whose Fourier transform is supported in [0,2].