Math 658 - Summer II 2002
Homework
July 10, 2002
- Read sections 1.4, 1.5, 1.7.
- Problems
- 1.2, 1.7(a,c), 1.8(b) (pgs. 60-61)
- Due Thursday, 7/11/02
July 11, 2002
- Read section 1.7.
- Problems
- 1.17(a,b) (n=3 in (b)) [Turn in 7/15], 1.19(a) (pgs. 62-63)
- Due Friday, 7/12/02
July 12, 2002
- Read section 1.10.
- Problems
1.13 [Turn in 7/15], 1.34
-
July 15, 2002
- Read section 1.10, 2.1.
- Problem
Prove: Let L be a linear filter with impulse reponse h, where h is
in L1(R) and is continuous. Then, L is
causal if and only if h(t) = 0 for all t < 0.
-
July 16, 2002
- Read sections 2.1-2.3.
- Problems
- 1.14, 1.41 [Turn in 7/18]
July 17, 2002
- Read sections 2.4.
- Problems
- 2.6, 2.9
July 18, 2002
- Read sections 2.7, 3.1
- Problems
- 2.16(a), 2.37 [Turn in 7/22]
July 19, 2002
- Read sections 3.1, 3.3, 3.4
- Problem
- Let {x1, x2, ..., xN} be a set
of distinct points in R. Show that the set of functions
{exp(-ix1t), exp(-ix2t), ...,
exp(-ixNt)}
is linearly independent. (Hint: Let f(t) be any function in
L1 and let F(x) = FT[f].
c1exp(-ix1t) +
c2exp(-ix2t) + ... +
cN exp(-ixNt) = 0 for all t
is equvalent to
c1F(x1) +
c2F(x2) + ... +
cNF(x2) = 0 for all F=FT[f], f in L1.
Choose f and F wisely.
July 22, 2002
- Read sections 3.3, 3.4.
- Problems
- 3.1(a,d)
July 24, 2002
- Read sections 3.4, 3.5.
- Problems
- 3.1(c,e). What periodic functions do the partial sums converge
to? Give an interval on which the convergence is uniform.
July 29, 2002
- Read section 3.5 and my notes on the DFT.
- Problem
- Let f,g be discrete-time signals. Show that if g is in
l1 and f is in l2, then f*g is in l2.
July 30, 2002
- Read my notes on the DFT.
- Problems
- Prove the Convolution Theorem for the DFT. (See pg. 3 of my notes
for the DFT.)
- Write a Matlab M-file that uses
stem3
to plot the
absolute value of the fft for a 1×n row vector y.
July 31, 2002
- Problems
- You are given that the Fourier series for the 2*pi extension of
F(x) = x2/4 - pi*x/2 + pi2/6,
0< x < 2*pi
is
SUMk=1INF k-2 cos(k x).
Use Matlab's FFT to compute the first 10 coefficients in the Fouier
series for F. Compare these with the actual coefficients. Do this for
n=16, 32, and 64. [Turn in Friday, 2 Aug]
August 1, 2002
August 2, 2002
- Read my notes on RBFs. Work
on individual presentations.
August 5, 2002
August 6, 2002
- Find the Radon transform of the function f(x) = 1 if
|x| < 1 and 0 otherwise - that is, f is the characteristic
function of the unit disk.
August 7, 2002
- Find the spherical harmonics corresponding to L=1, m=-1, 0, 1
using the raising and lower operators defined in class.
Return to Narcowich's
homepage