# Math 414-501 — Spring 2016

## Assignments

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

• Read sections 1.2.1-1.2.5.
• Do the following problems.
1. 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.
1. Chapter 1: 11, 18, 21, 22, 23(a,e,f),

2. 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.

1. 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 $f$, $f^{(k)}$.

2. 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 $f(x) = \sum_{n=1}^\infty\frac{(-1)^n}{n^3} \sin(nx)$

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

• Read sections 1.3.4-1.3.5, 2.1
• Do the following problems.
1. Chapter 1: 12, 13, 32
2. 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 $-\pi\le x \le \pi$.
1. Use the fact that this Fourier series converges pointwise to find the value of the series $\sum_{k=1}^\infty \frac{(-1)^k}{(2k-1)^3}$.
2. 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.

 Function Fourier Transform $f(t)$ $\hat f(\lambda)$ $\hat f(t)$ $f(-\lambda)$

1. Chapter 2: 1, 2, 4.
2. 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:

1. $t e^{-|t|}$ (Use #2.)

2. $e^{-2|t-3|}$ (#6 and #7)

3. $\text{sgn}(t)e^{-|t|}$ (Hint: differentiate $e^{-|t|}$; use #4.) Here, $\text{sgn}(t) = \begin{cases} 1 & t > 0,\\ 0 & t = 0, \\ -1 & t < 0.\end{cases}$

4. $\frac{1}{1+(t-2)^2}$  (Hint: How are Fourier transforms and inverse Fourier transforms related? Use the answer to this and #6)

3. Find the Fourier transforms of these functions.
1. $g(t)=\begin{cases}1 & -1 \le t \le 2, \\ 0 & \text{otherwise}. \end{cases}$

2. $h(t) = \begin{cases} 0, & t<0 \\ Ae^{-\alpha t},& t\ge 0. \end{cases}$, where $A>0$ and $\alpha>0$.

3. $F(t) = \begin{cases} 1 & 0 \le t \le 1,\\ -1 & -1 \le t < 0 \\ 0 & \text{otherwise.} \end{cases}$
4. 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.
1. Chapter 2: 5, 6.
2. 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 $\phi\ast \psi(t)$.
1. Find $\hat\phi(\lambda)$ and $\hat\psi(\lambda)$.
2. Find $\phi\ast \psi(t)$.
3. Verify the convolution theorem by directly finding ${\mathcal F}(\phi\ast \psi)$ and comparing it with $\sqrt{2\pi}\, \hat\phi(\lambda) \hat\psi(\lambda)$.

3. 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 $I=\int_0^\infty \frac{\sin^4(t)}{t^4}dt$.

4. 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 $|t| \ge a+b$.

5. 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.
1. Chapter 2: 13

2. This is a version of problem 12, chapter 2. Take $h(t)$ to be the function defined in that problem.
1. Show that the Filter has the form $L[f] = \frac{1}{d} \int_{ t - d}^t f(\tau)d\tau$.

2. 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 $\lambda$.)

3. 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.)

4. 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.

3. 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} \$.

4. 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 $f_3 \ast f_6(t) = \sqrt{\pi} f_{2}(t)=\sqrt{2\pi} e^{-2t^2}$.

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

• Read sections 4.2, 4.3
• Problems.
1. Suppose that x is an n-periodic sequence (i.e., xSn). 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.)

2. Chapter 3: 2 (Hint: use the previous problem.)
3. 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.)
4. Chapter 4: 2, 6, 7

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

• Read sections 0.5.2, 3.2.1 and 5.1.
• Problems.
1. Chapter 0: 15
2. 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 $\big\{ \frac{1}{\sqrt{2}}, \sqrt{\frac{3}{2}}x, \sqrt{\frac{5}{8}}(3x^2-1)\big\}$.
3. Re-do problem 4.7 using the coefficient method.
4. 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.
1. 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