**Assignment 1** - Due Friday, 1/29/16.

- Read sections 1.2.1-1.2.5.
- Do the following problems.
- Chapter 1: 1, 3, 4, 7, 8, 9, 10, 14, 25

- Point distribution
- Exercise 1: 25 pts, 9 for series, 16 for plots;
- Exercise 7: 15 pts;
- Exercise 8: 30 pts, 18 for series, 12 for plots;
- Exercise 10 (c): 20 pts;
- Exercise 25: 10 pts.

**Assignment 2** - Due Friday, 2/5/16.

- Read sections 1.3.1-1.3.5.
- Do the following problems.
- Chapter 1: 11, 18, 21, 22, 23(a,e,f),
- Suppose that $f$ and $f'$ are continuous 2π-periodic
functions, and that the Fourier series for $f$ and $f'$ are $
f(x) = a_0 +\sum_{n=1}^\infty a_n \cos(nx)+b_n\sin(nx)$ and $f'(x) =
a'_0 +\sum_{n=1}^\infty a'_n \cos(nx)+b'_n\sin(nx)$, respectively.
- Use integration by parts to show that the coefficients of the two
series are related this way for n ≥ 1: $a'_n = nb_n$ and
$b_n'=-na_n$. If f is k times continuously differentiable, use
induction to derive a similar formula for the the Fourier coefficients
of the $k^{th}$ derivative of $f$, $f^{(k)}$.
- 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 \[ f(x) = \sum_{n=1}^\infty\frac{(-1)^n}{n^3} \sin(nx) \]

- Use integration by parts to show that the coefficients of the two
series are related this way for n ≥ 1: $a'_n = nb_n$ and
$b_n'=-na_n$. If f is k times continuously differentiable, use
induction to derive a similar formula for the the Fourier coefficients
of the $k^{th}$ derivative of $f$, $f^{(k)}$.

- Chapter 1: 11, 18, 21, 22, 23(a,e,f),

**Assignment 3** - Due Friday, 2/12/16.

- Read sections 1.3.4-1.3.5, 2.1
- Do the following problems.
- Chapter 1: 12, 13, 32
- In the previous assignment, you showed that $f(x) =
\frac{1}{12}(x^3 - \pi^2x) =\sum_{n=1}^\infty\frac{(-1)^n}{n^3}
\sin(nx)\,$ on the interval $-\pi\le x \le \pi$.
- Use the fact that this Fourier series converges pointwise to find the value of the series $\sum_{k=1}^\infty \frac{(-1)^k}{(2k-1)^3}$.
- In addition, use this FS and Parceval's theorem to find the value of the series $\sum_{n=1}^\infty \frac{1}{n^6}$.

**Assignment 4** - Due Friday, 2/26/2016.

- Read sections 2.1 and 2.2.1-2.2.2, 2.2.4
- Problems. The table below, which was discussed in class on
2/24/16, may be used to do the assignment.
**"Quartet"**Function Fourier Transform $f(t)$ $\hat f(\lambda)$ $\hat f(t)$ $f(-\lambda)$ - 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)

- $t e^{-|t|}$ (Use #2.)
- Find the Fourier transforms of these functions.
- $g(t)=\begin{cases}1 & -1 \le t \le 2, \\ 0 & \text{otherwise}.
\end{cases}$
- $h(t) = \begin{cases} 0, & t<0 \\
Ae^{-\alpha t},& t\ge 0. \end{cases} $, where $A>0$ and $\alpha>0$.
- $F(t) = \begin{cases} 1 & 0 \le t \le 1,\\ -1 & -1 \le t < 0 \\ 0 & \text{otherwise.} \end{cases} $

- $g(t)=\begin{cases}1 & -1 \le t \le 2, \\ 0 & \text{otherwise}.
\end{cases}$
- Use Example 2.5, pg. 99, to find the the value of the integral $I = \int_0^\infty \frac{\sin^2(t)}{t^2}dt$. (Hint: Use Theorem 2.1, pg. 94.)

**Assignment 5** - Due Friday, 3/4/2016.

- 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 $\phi\ast \psi(t)$.
- Find $\hat\phi(\lambda)$ and $\hat\psi(\lambda)$.
- Find $\phi\ast \psi(t)$.
- Verify the convolution theorem by directly finding ${\mathcal F}(\phi\ast \psi)$ and comparing it with $\sqrt{2\pi}\, \hat\phi(\lambda) \hat\psi(\lambda)$.

- Let $h(t) = \begin{cases} \pi + t & -\pi \le t \le 0 , \\
\pi-t & 0\le t \le \pi \\
0 & \text{otherwise}. \end{cases} \ $ Recall that $\hat h(\lambda) =
\sqrt{\frac{8}{\pi}} \frac{\sin^2(\pi \lambda/2)}{\lambda^2} $. Use
Plancheral's theorem to find $I=\int_0^\infty
\frac{\sin^4(t)}{t^4}dt$.
- 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
\[ L[f] = Ae^{-\alpha t} \int_0^{\min(1,t)} e^{\alpha \tau} f(\tau)d\tau, \ \text{if }t\ge 0, \ \text{and } L[f] = 0 \ \text{if } t<0. \]

**Assignment 6** - Due Friday, 3/11/2016.

- Read sections 3.1.1-3.1.4.
- Problems.
- Chapter 2: 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.

- Show that the Filter has the form $L[f] = \frac{1}{d} \int_{ t -
d}^t f(\tau)d\tau$.
- 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. If $h_1=h_2=h$, where $h$ is the impulse response function
for the Butterworth filter, show that $h*h(t) = \begin{cases} A^2
te^{-\alpha t},& t\ge 0\\ 0,& t \le 0 \end{cases} \ $.
- 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 $f_3 \ast f_6(t) = \sqrt{\pi} f_{2}(t)=\sqrt{2\pi} e^{-2t^2}$.

- Chapter 2: 13

**Assignment 7** - Due Monday, 4/4/2016.

- Read sections 4.2, 4.3
- Problems.
- Suppose that
**x**is an n-periodic sequence (i.e.,**x**∈**S**_{n}). 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.)
- Chapter 4: 2, 6, 7

- Suppose that

**Assignment 8** - Due Friday, 4/15/2016.

- Read sections 0.5.2, 3.2.1 and 5.1.
- Problems.
- Chapter 0: 15
- Using the inner product $\langle f,g\rangle = \int_{-1}^1 f(x)\overline{g(x)}dx$, find the projection of $\sin(\pi x)$ onto the span of $\big\{ \frac{1}{\sqrt{2}}, \sqrt{\frac{3}{2}}x, \sqrt{\frac{5}{8}}(3x^2-1)\big\}$.
- Re-do problem 4.7 using the coefficient method.
- Chapter 4: 11 (10 pt. bonus.)

- Start putting together groups for projects.

**Assignment 9** - Due Friday, 4/22/2016.

- Read sections 5.2, 5.3.3, 5.3.4.
- Problems.
- Chapter 5: 2, 5, 8, 11, 17

- Finish getting your projects together by Wednesday, 4/19. If necessary, I'll give some more time -- but only a day or two.

Updated 4/15/2016