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 happens for a=1 ?
2. (5pt) Consider the following function:
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
2 ikx
(with F[k] replaced by ck) takes the values: 1 1
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. If only 1000 samples are used to compute the signal at
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. If only 1000 samples are used to compute the signal at
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 = lim (t log(t) + t)  1 = 1
t t 0
Note:
t
lim t log(t) =
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 that satisfies the given equation. Since we have the fr
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 >
1 2 1 2
, (u) = eu
2
Thus the Fourier transform is t
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 yields g(x) =
n
g(nT )sinc(
x  nT T
But g(nT ) = ein f (
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)
e2iy(w1) (y)dy = e2it(w1) sinc(w  1)
b) f (x) = e2ia
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 ( )dx+ 2 
x e2i(s+1/2)x ( )dx 
Change the integration v
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=
2
/p2
which is the Fourier series expansion of f (x)
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 following pperiodic functions on R: a. (1pt) f ( x) =
m = 
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) f ( x) = cos( x) b. (1pt) f ( x) = (2 x + 1) (2 x  1)
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 Vgf of the following functions: a. (2pts) f ( x) = e 2i
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 , otherwise 0 , x < 3 ( x + 3) / 2 , 3 x < 1 f ( x) = 1 , 1 x
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 can you deduce about F if you know that:
1.
f ( x)dx
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 functions (one for case (a) below, and another one for
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 generalized function f when: 1. (1pt) f ( x) = x 2. (1pt
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 via D:fg, where g(x)=af(ax), for some a0, and by T the tr
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 equations. 1. (1pt) ( x 4 1) f ( x) = ( x) ; 2. (1pt)
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, You can freely use rules from Fourier transform calcu
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 a>0 + a2 1 dx , for a fixed real parameter a>0 + 1)( x
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 permitted. 1.
2
(1  2 x )
, for 0 < x < 1
2.
2
6
(1  6 x +
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 , otherwise 0 , x < 3 ( x  3) / 2 ,  3 x < 1 f ( x) = 1 ,
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 can you deduce about F if you know that:
1.

f ( x)d
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) = e  x , f 1 ( x) = xe  x Compute the following convolu
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]. Assume we know the samples cfw_f(nT) , n=.,2,1,0,1,2