Math 414-501 — Spring 2019

Assignment 1 - Due Wednesday, 1/23/2019.

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

Assignment 2 - Due Wednesday, 1/30/19.

• Read sections 1.3.1-1.3.5.
• Do the following problems.
  1. Chapter 1: 11 (Typo: $f(x)$ is defined on $-\pi \le x \le \pi$.), 12 (skip the plots), 13, 17, 21, 22.
  2. Let $f$ and $g$ have the Fourier series \[ f(x) = a_0 +\sum_{n=1}^\infty a_n \cos(nx)+b_n\sin(nx) \text{ and } g(x) = c_0 +\sum_{n=1}^\infty c_n \cos(nx)+d_n\sin(nx). \]
     1. Show that the Fourier series for $Af(x)+ Bg(x)$, where $A$ and$ B$ are constants is \[ Af(x)+Bg(x) = Aa_0 +Bc_0 + \sum_{n=1}^\infty (Aa_n+Bc_n) \cos(nx)+(Ab_n+Bd_n)\sin(nx). \]
     2. Find a similar form of the identity above for complex Fourier series. (You will need this in problem 13.)

     3. Find the Fourier series for $f(x)=x$ on $[-\pi,\pi]$. Use it, the Fourier for $x^2$ (see exercise 1, pg. 96), and the formulas above to find the Fourier series for $f(x) = x^2-x$ on $[-\pi,\pi]$.

  3. Find the complex form of the Fourier series for $x^2$ on $[-\pi,\pi]$. By breaking the series into its real and imaginary parts, show that this is the same as the real series for $x^2$.

Assignment 3 - Due Friday, 2/8/19.

• Read sections 1.3.1-1.3.5, the Notes on pointwise convergence, and section 0.5 on inner products.
• Do the following problems.
  1. Chapter 1: 23(a,b,c,d) (Hand drawn sketches are fine.), 32(c,d,e,f), 33. (In 32(f), numerically evaluate the integral using a computer.)

  2. In the Notes on pointwise convergence, the Riemann-Lebesgue Lemma was proved for the cosine case. Following the proof for that case, do the proof for the sine case.

  3. 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 $b_n'=-na_n$. (This was essentially done in Theorem 1.30 in the text. The result is also true if $f'$ is only piecewise continuous.) 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 \[ f(x) = \sum_{n=1}^\infty\frac{(-1)^n}{n^3} \sin(nx) \] (Hint: Run the result in part (a) "backwards": If $n\ne 0$, $a_n=-b_n'/n$ and $b_n=a_n'/n$.)

  4. Use Parseval's equation and the series above to find the sum $\sum_{n=1}^\infty \frac{1}{n^6}$.

Assignment 4 - Due Monday, 2/18/2019.

• Read section 0.5 and my notes on Least Squares and Fourier Series
• Do the following problems.
  1. Consider the (real) inner product $\langle f,g\rangle_{L^2[0,\pi]} = \int_{0}^\pi f(x)g(x)dx$ and norm $\|f\|_{L^2[0,\pi]} = \sqrt{\int_{0}^\pi |f(x)|^2dx}$ and let $V_N=\text{span}\{1, \cos(x), \cos(2x), \ldots, \cos(Nx)\}$.
     1. Use the method from Least Squares and Fourier Series to show that the $N^{th}$ partial sum of the cosine series for $f$, $S_N(x) = a_0+\sum_{n=1}^N a_n\cos(nx)$, is the minimizer for $\min_{T_N\in V_N}\|f-T_N\|_{L^2[0,\pi]^2}$.

     2. Assuming that $\lim_{N\to \infty} \|f-S_N\|_{L^2[0,\pi]}=0$, show this version of Parseval's equation: $\int_0^\pi |f(x)|^2dx = \frac{\pi}{2}\big(2a_0^2+\sum_{n=1}^\infty a_n^2\big)$.

     3. Find the cosine series for $f(x)=\frac{1}{8}\big(\pi^2-2\pi x\big)$. Use the version of Parseval's equation above to evaluate the the series $\sum_{n=0}^\infty (2n+1)^{-4}$. (You may use the table of integrals for Test 1.)

  2. Let $W_N=\text{span}\{\sin(nx)\}_{n=1}^N$ and let the inner product and norm be those used above.
     1. Use the method employed in Theorem 0.20, section 0.5.2, to show that $S_N=\sum_{n=1}^N b_n \sin(nx)$, the partial sum for the Fourier sine series for $f$, is the minimizer for $\min_{\, T_N\in W_N}\|f-T_N\|_{L^2[0,\pi]^2}$, where $T_N =\sum_{n=1}^N \beta_n\sin(nx)\in W_N$.

     2. Assuming that $\lim_{N\to \infty} \|f-S_N\|_{L^2[0,\pi]}=0$, show this version of Parseval's equation: $\int_0^\pi |f(x)|^2dx = \frac{\pi}{2}\big(\sum_{n=1}^\infty b_n^2\big)$

     3. Find the sine series for $f(x)=\frac{1}{8}\big(\pi^2x-\pi x^2\big)$. Use the version of Parseval's equation above to evaluate the the series $\sum_{n=1}^\infty (2n-1)^{-6}$. (You may use the table of integrals for Test 1.)

Assignment 5 - Due Wednesday, 2/27/2019.

• 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} $

Assignment 6 - Due Wednesday, 3/6/2019.

• Read sections 2.3 and 2.4
• Problems.
  1. Chapter 2: 5, 6.
  2. Consider the function $f(t)e^{i\alpha t}$, where $\alpha\in \mathbb R$.
     1. Show that $\mathcal F [f(t)e^{i\alpha t}](\lambda) = \hat f(\lambda -\alpha)$. (Hint: this follows from the definition of the FT.)
     2. Using (a) and $e^{i\alpha t}=\cos(\alpha t)+i\sin(\alpha t)$, show that $\mathcal F [f(t)\cos(\alpha t)](\lambda)=\frac12 \big(\hat f(\lambda +\alpha)+ \hat f(\lambda -\alpha)\big)$ and that $\mathcal F [f(t)\sin(\alpha t)](\lambda)=\frac{1}{2i} \big(\hat f(\lambda -\alpha)- \hat f(\lambda +\alpha)\big)$.

     3. If $f(t)=e^{-t}$ for $t\ge 0$, and $0$ otherwise, use the results above to show that $\mathcal F[f(t)\sin(\alpha t)](\lambda)= \frac{\alpha}{\sqrt{2\pi}} ((1+i\lambda)^2+\alpha^2)^{-1}$.

  3. 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 $\sqrt{2\pi}\, \hat\phi(\lambda) \hat\psi(\lambda)$.

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

  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 7 - Due Wednesday, 3/20/2019.

• Read sections 3.1.1 and 3.1.2
• Problems.
  1. Chapter 2: 13

  2. 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.)

  3. Let $a>0$ and $b>0$. Suppose that $f(t)=0$ for all $t>a$ and $t<0$, and $g(t) = 0$ for all $t>b$ and $t<0$. Show that $f\ast g(t) = 0$ for all $t \ge a+b$. (Hint: just look at the definition of the convolution and find the values of $t$ where at least one factor is $0$.)

  4. This is a version of problem 12, chapter 2. Let $h(t):=\begin{cases} 1/d, & 0\le t\le d; \\ 0, &\text{otherwise.}\end{cases}$ be the impulse response for a filter $L$.
     1. Show that if $f(t)=0$ for $t<0$, then $L[f](t) = d^{-1}\begin{cases} 0, & t<0, \\ \int_0^t f(\tau)d\tau, & 0\le t \le d \\ \int_{t-d}^t f(\tau)d\tau, & d\le t .\end{cases}$

     2. Let $f(t) =\begin{cases} 0, & t<0, \\ e^{-t}(\sin(t)+0.05\sin(20t))& 0\le t.\ \ \end{cases}$ Consider the term $0.05e^{-t}\sin(20t)$ to be noise. Use matlab or some other program to experimentally find a value (or values) for $d$ that removes the noise but doesn't change the signal $e^{-t}\sin(t)$ very much.

  5. 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 $h_3=h_2\ast h_1$ is the impulse response for $L[f]=L_2\big[L_1[f]\big]$, and that $L$ is causal. Also, let $h=h_1=h_2$ be 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 8 - Due Wednesday, 4/3/2019.

• Read sections 4.2, 4.3
• Problems.
  1. Chapter 3: 2 (Hint: use problem 2 below.), 16

  2. 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.)

  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 9 - Due Wednesday, 4/10/2019.

• Read sections 0.5.2 (orthogonal projections), 4.4, 5.1.
• Problems.
  1. Chapter 4: 2, 6, 7, 11 (10 pt. bonus.)

  2. Let $f(x)=x$, $0\le x \le 1$, 0 otherwise.
     1. Find $f_3$, the orthogonal projections of $f$ onto the Haar approximation space $V_3$, in terms of the $\{\phi(2^3x-k)\}$ basis for $V_3$.
     2. Use the filter decomposition method given in section 4.3.3 to find the coefficients for $f_2$ and $w_2$ in terms of the bases $\{\phi(2^2x-k)\}$ and $\{\psi(2^2x-k)\}$, for $V_2$ and $W_2$, respectively.

• 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 10- Due Wednesday, 4/17/2019.

• Read sections 5.1 and 5.2
• Problems.
  1. Chapter 5 exercises: 3, 4, 5 (Hint: Use the formula for $p_k$ given in Theorem 5.6 in the text.), 8(c) (Hint: First use example 2.2, p. 95, and the ``quartet'' to show that $\text{sinc}(x)$ is in $V_0$, then use the Sampling Theorem with $f(k/2)=\text{sinc}(k/2)$.)

  2. Do chapter 4, problem 2 using the decomposition filter method. For the coeficients that you get, give the corresponding discrete time. (Refer to the lecture on 4/12/19.)

  3. Do chapter 4, problem 7 using the reconstruction filter method. For the coeficients that you get, give the corresponding discrete time. (Refer to the lecture on 4/12/19.).
• 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