Math 414-501 - Spring 2003
Homework
Assignment 1
- Read sections 1.1-1.3.3.
- Problems
- §1.4: 3-7, 21
- Due Thursday, 1/23/03
Assignment 2
- Read sections 1.3.4-1.3.5, 2.1.
- Problems
- §1.4: 9 (In #9, find the complex form first, then get the
real form.), 16, 19, 26, 28
- Due Thursday, 1/30/03
Assignment 3
- Read sections 2.1-2.3.
- Problems
- §1.4: 17(a,b), 18 (series is the same as one in #3), 20, 25,
29
- State and prove a verion of Parseval's Theorem for cosine series.
- Due Thursday, 2/6/03
Assignment 4
- Read sections 2.3-2.4.
- Problems
- §2.6: 1, 2, 4, 6, 7
- Find the Fourier transform of f(t) = e-|t|. In
addition, use this transform and the properties listed in Theorem
2.6 to find the Fourier transforms of the following functions:
- t e-|t|
- e-2|t-3|
- sign(t)e-|t| (Hint : differentiate e-|t|).
- Due Thursday, 2/13/03
Assignment 5
- Read sections 2.5, 3.1.
- Problems
- §2.6: 5, 8, 10
- Use Plancheral's formula (Theorem 2.12) and the Fourier transfrom
of the rectangular function f(t) in Example 2.2 to find
-∞∫∞ sin2(u)
u-2 du.
- Consider the Gaussian function fs(x) in problem 6,
section 2.6. Use the convolution theorem to find
fr*fs , where r and s are positive real numbers.
- Due Thursday, 2/20/03
Assignment 6
- Read sections 3.1, 3.2.
- Problems
- §2.6: 11, 13, 14. (There is an error in the 1st printing in
problem 13. The factor ei(w1+w2)/2 should be
e-it(w1+w2)/2.)
- In problem 11, §2.6, use MATLAB or some other program to
plot the output of the Butterworth filter for A = a = 2, 5, 10, 15,
and 20.
- Condider the dispersion Δaf, where f is
fixed and a is any real number. Show that Δaf is
minimized if
a = ∫ t |f(t)|2 dt / ∫ |f(t)|2 dt, where
the integral is from -∞ to ∞
- Due Thursday, 2/27/03
Assignment 7
- Read sections 3.1, 3.2.
- Problems
- §3.3: 2, 14.
- Due Thursday, 3/6/03
Assignment 8
- Read sections 4.1-4.3
- Problems
- §4.5: 1, 2.
- Do exercises 1-4 on p. 4 in Discrete
Time Signals.
- Due Thursday, 3/27/03
Assignment 9
- Read sections 4.4, 5.1, 5.2
- Project assignment: Groups should write a brief summary of their
plans for a project. Each group should make an appoinment to see me.
- Problems
- §4.5: 5, 6
- This exercise is designed to help you to write a script m-file in
MATLAB code to do a one-level Haar wavelet decomposition.
- The filters for the decomposition are
hi=[-1/2 1/2]
and lo=[1/2 1/2]
.
Write the corresponding discrete-time vectors dt_hi
and
dt_lo
.
- To downsample a signal, recall that you must discard
outputs from a filter corresponding to odd-numbered discrete
times. These are not necessarily the odd-numbered entries in a
row vector. Thus, you must keep track of the discrete times for the
filter outputs
hi*x
and lo*x
and then
discard the appropriate discrete times involved. (Hint, to access
every other entry in a row or column vector z
starting
at entry m
, just use z(m:2:end)
.)
- Try your decomposition on the signal given in Problem 1, pg. 180
of the text. Use
stairs
to plot your result.
- Due Thursday, 4/3/03
Assignment 10
- Read sections 5.2, 5.3
- Work on projects. If you have any questions, be sure to contact
me.
- Problems
- §5.4: 4(c), 5, 8(c,d).
- Due Thursday, 4/17/03