Math 414-501 — Spring 2017

Assignment 1 - Due Wednesday, 1/25/17.

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

Assignment 2 - Due Wednesday, 2/1/17.

• 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, 2/10/17.

• Read sections 1.3.4-1.3.5, 2.1
• Do the following problems.
  1. Chapter 1:
     1. 1. Skip the plots.
     2. 4. Sketch by hand the odd, $2\pi$ periodic extension of $x^2$, $0\le x\le 1$. The FS for the odd extension is the FSS that you are asked to find.
     3. 10. In working this problem, keep this in mind: If a function $f(x)$ is defined on $0\le x \le 1$, then it has three Fourier-type series associated with it: (1) Its FCS, which is the FS for the even, $2$ periodic extension of $f$. (2) Its FSS, which is the FS for the odd, $2$ periodic extension of $f$. (3) Its FS, with $a=1/2$. This is the FS for the $1$ periodic extension of $f$. These are the three series you are asked to find in the problem.
     4. 12. Skip the plots.
     5. 13.
     6. 33.
  2. Let $f(x)=1$ on the interval $0\le x\le \pi$.
     1. Find the FSS for $f(x)$.
     2. Sketch there periods of the function to which the FSS converges pointwise. Is the series uniformly convergent? Justify your answer.
     3. By choosing an appropriate point in the interval $0 < x < \pi$, use the series you found to show that \[ \frac{\pi}{4} = \sum_{k=1}^\infty \frac{(-1)^{k+1}}{2k-1} = 1 - \frac13 + \frac15 -\frac17+ \cdots. \]
  3. 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, 2/24/2017.

• Read sections 2.1 and 2.2.1-2.2.2, 2.2.4
• Problems. You may use the table below.

  "Quartet"

  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 Monday, 3/6/2017.

• 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. 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 $f_3 \ast f_6(t) = \sqrt{\pi} f_{2}(t)=\sqrt{2\pi} e^{-2t^2}$. (Hint: First use the convolution theorem to get $\mathcal F[f_3 \ast f_6(t)](\lambda)$, then find $\mathcal F^{-1}$ of the result to get the answer.)

  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, 3/10/2017.

• Read sections 3.1.1-3.1.4.
• Problems.
  1. Chapter 2: 13

  2. This is a version of problem 12, chapter 2. Let $h(t):=\begin{cases} 1/d, & 0\le t\le d \\ 0, &\text{otherwise}\end{cases}$.
     1. Show that the Filter has the form $L[f] = \frac{1}{d} \int_{ t - d}^t f(\tau)d\tau$.

     2. Find $\hat h(\lambda)$. On the same set of axes, for $d=3,10,30,50$ and $0\le \lambda\le 20$, plot $|\hat h(\lambda)|$. Suppose one wants to filter out frequencies with $\lambda$ above $5$. What would be a good choice for $d$? (It doesn't have to be one of the $d$'s listed.)

     3. Let $f(t) =\begin{cases} e^{-t}(\sin(3t) +\sin(5t) +\sin(40t)), & t\ge 0\\ 0, &\text{otherwise}\end{cases}$.   Filter the signal using the value of $d$ chosen in part (b). On the same set of axes, graph $f$ and $h\ast f$.

  3. Let $h_1$ and $h_2$ be impulse response 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} \ $.

Assignment 7 - Due Friday, 3/31/2017.

• Read sections 4.2, 4.3
• Problems.
  1. 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.)

  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. Start putting together groups for projects.

Assignment 8 - Due Friday, 4/7/2017.

• Read sections 0.5.2, 3.2.1 and 5.1.
• Problems.
  1. Chapter 4: 2, 6, 7, 11 (10 pt. bonus.)
  2. Chapter 0: 15
  3. 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 $\big\{ \frac{1}{\sqrt{2}}, \sqrt{\frac{3}{2}}x, \sqrt{\frac{5}{8}}(3x^2-1)\big\}$.
• Finish groups for projects. Each group leader should send me an email with a list of the group members and a short abstract for the project

Assignment 9 - Due Monday, 4/24/2017.

• Read sections 5.2, 5.3.3, 5.3.4.
• Problems.
  1. Chapter 5: 2, 5, 8, 11
  2. Use the decomposition filter diagram from the text to find $f_{j-1}$ and $w_{j-1}$ for the given $f_j$. (You must use this method to get credit.)
     1. $f_2= 2\phi(4x) -3\phi(4x-1)+ \phi(4x-2) + 3\phi(4x-3)$
     2. $f_3= \phi(8x) +7\phi(8x-1) -\phi(8x-2) +7\phi(8x-3)- 2\phi(8x-4)$
  3. Use the reconstruction filter diagram from the text to find $f_j$ for the given $w_{j-1},f_{j-1}$. (You must use this method to get credit.)
     1. $w_2=-\frac{3}{2} \psi(4x)- \psi(4x-1) + \frac12 \psi(4x-2) - \frac12 \psi(4x-3)$, $f_2= \frac12 \phi(4x) + 2\phi(4x-1)+ \frac52\phi(4x-2) - \frac23\phi(4x-3)$
     2. $w_1=-\psi(2x)- \frac32\psi(2x-1)$, $f_1= \frac32 \phi(2x) -\phi(2x-1)$
• Finish getting your projects together by Thursday, 4/20. If necessary, I'll give some more time -- but only a day or two.