Math 414 - Spring 2001
Homework
Assignment 1
- Read sections 0.1-0.5, 1.1-1.2.
- Problems
- §0.8: 1, 3, 4, 18, 23
- §1.4: 1, 3, 13.
- Due Thursday, 1/25/01
Assignment 2
- Read sections 0.3.2, 0.5.2, 0.7.1, 0.7.2, 1.2.4, 1.2.5, 1.3.
- Problems
- §0.8: 6, 7, 10, 15.
- §1.4: 5, 8, 11, 24.
- Due Thursday, 1 February.
Assignment 3
- Read sections 0.3.2, 0.7.1, 0.7.2, 1.3
- Problems
- §0.8: 27.
- §1.4: 28(a,c,d,e), 29.
-
Find the complex form of the Fourier series for f(x) = exp(r*x), -pi
< x < pi.
- Due Thursday, 8 February.
Assignment 4
- Read sections 2.1 and 2.2.
- Problems
- §0.8: 24.
- §1.4: 16, 17, 19(a,d,f), 20(a,d,f), 28(f,g), 32.
-
- Due Thursday, 15 February.
Assignment 5
- Read sections 2.3.
- Problems
- § 2.6: 4-6.
- Find the Fourier transform of the following functions:
- f(t) = e-| t | cos(t)
- f*f (f convolved with itself).
- tf'(t-1)
- Consider the function f(t) in example 2.2, pg. 94. Use this
function and the Plancheral/Parseval Theorem 2.12 to find || sin(x)/x
||, which is the L2 norm of sin(x)/x.
- Suppose that h(x) is 2*pi periodic in x, with only a jump
discontinity at x=0 (and multiples of 2*pi, of course). Let A =
½[h(0+)+h(0-)] (midpoint of the right and
left limits of h at x=0) and let B=
½[(h(0+)-h(0-)] (½ the
jump). Finally, let f(x) be the function defined in problem 28,
§1.4. Show that for g(x) := h(x) - A - B*[f(x)/pi], we have
lim x ->0 g(x) = 0. Thus g(x) is continuous at x=0,
provided we take g(0)=0. (Harder to state than do!)
- Due Thursday, 22 February.
Assignment 6
- Read sections 2.4, 2.5, 3.1
- Problems
- §2.6: 11-14 (Note: Problem 13, §2.6. The factor
ei(w1+w2)/2) should be e-ti
(w1+w2)/2)).
- Let H and K be causal filters with impulse response functions
h(t) and k(t). Show that the filter with impulse response h*k is
causal.
- Refer to problem 4 on Assignment 5 for notation. Show that if
h(x) is piecewise smooth, then g(x) is differentiable everywhere,
except for "corners" at even multiples of . Use this fact to show that for every > 0 and for all x there is an
N0 such that if N>N0
Here, hN(x) is the Nth partial sum of the Fourier series
for h.
- Due Thursday, 1 March.
Assignment 7
- Read sections 3.1, 3.2.
- Problems
- §3.3: 2, 14(e)
- Refer to problem 4 on Assignment 5 and problem 3 on Assignment
6. Show that, for the function h, the Gibbs' phenomenon is exhibited
as an overshoot of about 0.18B (18% of ½ the total jump), as
long as N is large enough. This is the universality of the Gibbs'
phenomenon.
- Think about projects!
- Due Thursday, 22 March.
Assignment 8
- Read sections 4.1 and 4.2.
- Problems (Useful M-files can be
found under MATLAB mfiles
on my home page.)
- §3.3: 4, 5, 10, 13, 14(f)
- Consider the time series xk=exp(-k/2).
- Make a stem plot for the time series xk=exp(-k/2), for
k=-20, 20.
- Let p = 3 and let x be the time series above. Do a stem plot for
yk = Tp[x]k for k=-20, 20.
- Let F be a filter with FIR f = (...,0,½,0,0,
¼,0,...). (Here, f0=½,
f3=¼.) Make a stem plot for (f*x)k
- Projects!
- Due Thursday, 29 March.
Assignment 9
- Read sections 4.3 and 4.4.
- Problems
- §4.5: 1, 2, 4, 5
- Consider the function f(x)=exp(-|x|).
- For all j, find f j, the orthogonal projection of f
onto the Haar scaling space Vj.
- For j=-1, 0, 1, and 16, plot the corresponding projection over
the interval x=-8 to 8. (For best effect, use MATLAB's
subplot
command along with stairs
.)
- Plot the wavelet part w0 for f. Again, use x=-8 to 8.
- Due Thursday, 5 April.
Assignment 10
- Read sections 5.1 and 5.2.
- Problems
- §4.5: 6,7, 9, 10, 11. (You may use the Wavelet Toolbox in
MATLAB to do the numerical parts of problems.)
- Hand in a title and outline for your project.
- Due Thursday, 12 April.
Assignment 11
- Read sections 5.3 and 6.1.
- Problems
- §5.4: 2(b), 4(c), 5, 8
- Due Thursday, 19 April.