Math 414-501 Spring 2015
Assignments
Assignment 1 - Due Friday, 1/30/2015.
- Read sections 0.1-0.5, 0.7.1
- Problems.
- Chapter 0: 2 (Do Example 0.3, p. 4.), 3, 10, 11, 12, 13, 15, 28
- Use the inner product in Example 0.3, pg. 4. Find the projection
of w onto v, where the vectors are displayed
below. Verify that if p is the required projection,
then q := w − p is orthogonal
to v.
- Point distribution:
1. Text problems, chapter 0:
3 -- 15 pts. (If they only do positivity, give them full credit.)
12 -- 25 pts.
15 -- 20 pts.
28 -- 20 pts.
2. 20 pts.
Total = 100 pts.
Assignment 2 - Due Friday, 2/6/2015.
- Read sections 1,1-1.1.3, 1.2.1-1.2.5
- Problems.
- Chapter 0: 14 1, 17, 23
- Chapter 1: 1
- Let $V$ be an inner product space and let $V_0$ be a finite
dimensional subspace of $V$. Show that if $v\in V$ has $v_0 =
\text{proj}_{V_0}(v)$, then
\[
\| v - v_0\|^2 = \|v\|^2 - \|v_0\|^2.
\]
- Let V be a vector space with a complex inner product <
·,· >. Suppose that the set B =
{u1, u2,
..., un} is an orthonormal basis for V.
- Re-do the proof we did in class on 1/30/15. That is, show that
if
v = a1u1 +
a2u2 + ... +
anun and w =
b1u1 +
b2u2 + ... +
bnun,
then <v, w > = ∑j aj
bj
= bT a. (Hint: Put the expression
for v in the inner product and then use activity and
homogeneity. Finally, identify the coefficients that multiply the a's in the
resulting sum.)
- Verify this identity in $V=L^2[-1,1]$, where $B$ is the set of
orthonormal Legendre polynomials,
\[
p_0(x)=\frac{1}{\sqrt{2}}, \quad p_1(x) = \sqrt{\frac {3}{2}}\, x,
\quad p_2(x)=\sqrt{\frac{5}{8}}(3x^2-1)
\]
and v, w are replaced by $x-x^2$ and $12+x-3x^2$,
respectively.
- Point distribution:
1. Text problems, chapter 0.
14 -- 25 pts. (Give 9 pts. for the plots.)
23 -- 15 pts.
2. Text problem, chapter 1.
1 -- 25 pts. (Give 12 pts. for the plots.)
3. 5 pts.
4. (a) 15 pts. (b) 15 pts.
Total = 100 pts.
1 The space $V_n$ isn't clear
in the problem. It should be $V_n = \{\frac{1}{\sqrt{2\pi}},
\frac{\cos(x)}{\sqrt{\pi}}, \frac{\sin(x)}{\sqrt{\pi}}, \cdots,
\frac{\cos(nx)}{\sqrt{\pi}}, \frac{\sin(nx)}{\sqrt{\pi}}\}$. The functions
given in the set are orthonormal; you do not need to show this.
Assignment 3 - Due Friday, 2/13/2015.
- Read sections 1,3.1-1.3.4
- Problems.
- Chapter 1: 4, 7, 8 (do only n=5 and n = 10), 10
- Suppose that f and f′ are continuous 2π-periodic
functions. If the Fourier series for f and f′ are
f(x) = a0 + ∑n ancos(nx) +
bnsin(nx) and f′(x) = a′0 +
∑n a′ncos(nx) +
b′nsin(nx),
then show that the coefficients of the two series are related this way
for n ≥ 1:
a′n = n bn, b′n = −
n an and a′0 = 0.
(Hint: Integrate by parts.) If f is k times continuously differentiable, use
induction to derive a similar formula for the the Fourier coefficients
of f(k).
- Let $f(x) =
\frac{1}{12}(x^3 - \pi^2x)$, $-\pi \le x \le \pi$.
In the text (cf. Example 1.9), we derived the Fourier series for $g(x)
= x$ on $-\pi \le x <\pi$. Use the series for $g$ and the results from
the previous problem to show that the Fourier series for $f(x) =
\frac{1}{12}(x^3 - \pi^2x)$ is given by
\[
f(x) = \sum_{n=1}^\infty\frac{(-1)^n}{n^3} \sin(nx)
\]
- Point distribution:
1. Text problems, chapter 1.
7 -- 10 pts.
10 -- (a) 20 pts. (b) 20 pts. (c) 20 pts.
2. 10 pts.
3. 20 pts.
Total = 100 pts.
Assignment 4 - Due Wednesday, 2/18/2015.
- Read sections 1.3.4-1.3.5
- Problems.
- Chapter 1: 111 (Plot=sketch
by hand.), 21, 23(a,b,c) (Plot=sketch by hand; note that the functions
have period = 2 and are defined on $-1 \le x \le 1$.), 33
- In the previous assignment you showed that
\[
\frac{1}{12}(x^3 - \pi^2x) = \sum_{n=1}^\infty\frac{(-1)^n}{n^3}
\sin(nx), \ -\pi \le x \le \pi.
\]
Use this formula and Parseval's equation to find the sum of the series
$\sum_{n=1}^\infty \frac{1}{n^6}$.
- Point distribution: 1. Text problems, chapter 1.
11 -- (a) 20 pts. (b) 20 pts.
21 -- (a) 5 pts. (b) 20 pts.
23 -- (a) 5 pts. (b) 5 pts. (c) 5 pts.
33 -- 10 pts.
2. 10 pts.
Total = 100 pts.
1 There is a typo in the
problem. The function $f(x)$ is defined on $-\pi \le x \le \pi$; $x$
was omitted from the interval.
Assignment 5 - Due Friday, 2/27/2015.
- Read sections 1.3.1, 1.3.2, 2.1 and 2.2.
- Problems.
- Chapter 1: 32(a,c,d,e,f)
- Chapter 2: 1, 2, 4.
- 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:
- $t e^{-|t|}$ (Use #2.)
- $e^{-2|t-3|}$ (#6 and #7)
- $\text{sgn}(t)e^{-|t|}$ (Hint: differentiate $e^{-|t|}$;
use #4.) Here, $\text{sgn}(t) = \begin{cases} 1 & t > 0,\\ 0 & t = 0,
\\ -1 & t < 0.\end{cases} $
- $\frac{1}{1+(t-2)^2}$ (Hint: How are Fourier transforms and
inverse Fourier transforms related? Use the answer to this and #6)
- Find the Fourirer transforms of these functions.
- $g(t)=\begin{cases}1 & -1 \le t \le 2, \\ 0 & \text{otherwise}.
\end{cases}$
- $h(t) = \begin{cases} 1 & 0 \le t \le 1,\\ -1 & -1 \le t < 0 \\ 0
& \text{otherwise.} \end{cases} $
- Point distribution: 1. Chapter 1.
32 -- (a) 10 pts. (c) 10 pts. (f) 10 pts.
2. Chapter 2.
2 -- 15 pts.
3. (a) 15 pts. (c) 10 pts. (d) 15 pts.
4. (b) 15 pts.
Total = 100 pts.
Assignment 6 - Due Monday, 3/9/2015.
- Read sections 2.3 and 2.4.
- Problems.
- Chapter 2: 5, 6.
- Let $\phi(t) := \begin{cases} 1 & 0 \le t < 1, \\ 0 &
\text{otherwise},\end{cases}$ and $\psi(t) := \begin{cases} 1 &
0 \le t < 1/2, \\ -1 & 1/2 \le t <1, \\0 &
\text{otherwise.}\end{cases} \ $ Find $\phi\ast \psi(t)$.
- Let $h(t) = \begin{cases} 1 & -\pi \le t < \pi , \\ 0 &
\text{otherwise}. \end{cases} \ $ Recall that $\hat h(\lambda) =
\sqrt{\frac{2}{\pi}} \frac{\sin(\pi \lambda)}{\lambda} $. Use
Plancheral's theorem to find $\int_0^\infty
\frac{\sin^2(x)}{x^2}dx$.
- Suppose that $f(t)=0$ for all $|t| \ge a >0$ and $g(t) = 0 $ for
all $|t| \ge b > 0$. Show that $f\ast g(t) = 0$ for all $|t| \ge
a+b$.
- Let f(t) be a signal that is 0 when t < 0 or t > 1. Show
that, for the Butterworth filter, one has
L[f] = A e− α
t ∫ 0min(1,t)
eατ f(τ)dτ,
if t ≥ 0, and that L[f] = 0 if t < 0.
- Point distribution: 1. Chapter 2, problem 6 -- 20 pts.
2. 20 pts.
3. 20 pts.
4. 20 pts.
5. 20 pts.
Total = 100 pts.
Assignment 7 - Due Wednesday, 3/25/2015.
- Read section 3.1.
- Problems.
- Chapter 2: 8, 13.
- This is a version of problem 12, chapter 2. Take $h(t)$ to be the
function defined in that problem.
- Show that the Filter has the form $L[f] = \frac{1}{d} \int_{ t -
d}^t f(\tau)d\tau$.
- Find $\hat h(\lambda)$. Make the plots required in problem 12,
but use $|\hat h(\lambda)|$ rather than $\hat
h(\lambda)$. (Interpret cycles/$2\pi$ as a $\lambda$.)
- For $t \ge 0$, let $g_\beta(t) = e^{-t} \sin(\beta t)$, where
$\beta$ is a real number, and for $ t < 0$, let $g_\beta(t) =
0$. Find $h\ast g_\beta(t)$ for all $t \ge 0$. (Be aware that the
cases $0 \le t < d$ and $d \le t$ have to be treated differently.)
- Let $f$ be as in problem 12. Use your answer to the previous part
to write $f$ as a sum of the $g_\beta$'s, and then find $h\ast
f$. Make the plots required for $h\ast f$ in problem 12.
- Let $h_1$ and $h_2$ be impulse resonse functions for causal
filters $L_1[f] = h_1\ast f$ and $L_2[f]=h_2\ast f$. Show that if
$h=h_1\ast h_2$ is the impulse response for $L[f]=h\ast f$, then $L$
is causal.
- Recall that we have defined the Gaussian $f_s$ by $f_s(t) =
\sqrt{s} e^{-s t^2}$ and shown that $\hat f_s(\lambda) =
\frac{1}{\sqrt{2}} e^{-\lambda^2/(4s)}$. (Chapter 2, problem 6.)
Consider the two Gaussians $f_3(t) = \sqrt{3}e^{-3 t^2}$ and $f_6(t)
= \sqrt{6}e^{-6t^2}$. Show that $f_3 \ast f_6(t) = \sqrt{\pi}
f_{2}(t)=\sqrt{2\pi} e^{-2t^2}$.
- Point distribution:
1. Chapter 2:
8 -- 10 pts.; 13 -- 20.
2. (c) 25 pts. (d) 25 pts.
4. 20 pts.
Total = 100 pts.
Assignment 8 - Due Wednesday, 4/1/2015.
- Read sections 3.2.1, 4.2.1, 4.2.2.
- Problems.
- Suppose that x is an n-periodic sequence (i.e., x
∈ Sn). Show that $ \sum_{j=m}^{m+n-1}{\mathbf
x}_j = \sum_{j=0}^{n-1}{\mathbf x}_j $. (This is the DFT analogue of
Lemma 1.3, p. 44.)
- Chapter 3: 2 (Hint: use the previous problem.)
- Consider the Gaussian function $f_1(t) = e^{-t^2}$. The Fourier
transform of this function is $\hat f_1(\lambda) =
\frac{1}{\sqrt{2}} e^{-\lambda^2/4}$. Numerically approximate $\hat
f_1(\lambda)$ using the FFT, with $f_1$ being sampled over the
interval $[-5,5]$ for n = 256, 512, and 1024. Graph $\hat
f_1$ and its FFT approximation $\hat f_{ap}$ for these three values of
$n$. (For an example of this type of problem, see
Approximating the FT with the FFT.)
- Point distribution: 1. 20 pts. 2. Chapter 3: 2 (Four
parts: (i) shifts -- 10 pts. (ii) convolution -- 20 pts. (iii)
convolution theorem -- 20 pts. (iv) relation for "y hats" -- 10 pts.)
3. 20 pts. (The graphs in this problem should be
labeled. Each plot should have n, FT_actual and FT_approx. Take off
two points if there were no labels.)
Total = 100 pts.
Assignment 9 - Due Friday, 4/10/2015.
- Read sections 4.2-4.4.
- Problems.
- Chapter 3: 16.
- Let $x_j = \begin{cases} 1, & 0 \le j \le N-1, \\ 0, &
\text{otherwise}. \end{cases}$ Show that
$
\hat x(\phi) =
\frac{e^{-i\frac{N-1}{2}\phi}\sin\big(\frac{N}{2}\phi\big) }{\sin\big
(\frac12 \phi\big)}
$.
- Let $x = (\cdots x_{-2}\ \ x_{-1}\ \ x_0 \ \ x_1 \ \ x_2 \ \ x_3 \cdots )$
be in $\ell^2$ and let $y = T_p(x)$. Show that the Z-transforms of
$x$ and $y$ are related by $\hat y(\phi) = e^{-ip\phi}\hat x(\phi)$.
- Start putting together groups for projects.
- Point distribution: 1. Chapter 3: 16, 30 pts.
2. 35 pts.
3. 35 pts.
Total=100 pts.
Assignment 10 - Due Wednesday, 4/15/2015.
- Read sections 4.2-4.4.
- Problems.
- Chapter 4: 1. Do this problem by the method in Example
4.11. Sketch this by hand.
- Chapter 4: 2. Do this problem by using the formulas in Theorem
4.12 to find the coefficients, and then the component
parts. Sketch this by hand.
- Chapter 4: 6, 7. Do the sketches by hand.
- Show that $a^j_k = 2^{\frac{j}{2}}\int_{-\infty}^\infty
f(x)\phi_{j,k}(x)dx =2^{j}\int_{-\infty}^{\infty} f(x)\phi(2^jx -
k)dx$. (This is an easy problem. Use Theorem 0.21, p. 17, along with
the fact that $\{\phi_{j,k}\}_{k\in \mathbb Z}$ is an orthonormal
basis for $V_j$.)
- Start putting together groups for projects.
Assignment 11 - Due Monday, 4/27/2015.
- Read sections 5.1 and 5.2.
- Form a group (2-4 is OK) and choose someone to represent the
group. Pick a topic for
the project. The reprentative should
email me a list of the members and a short explanation of
the topic.
Updated 4/22/2015