Assignment 1 - Due Friday, 1/24/2020.

• Read sections 0.1-0.5.
• Do the following problems.
  1. Chapter 0 (p. 34): 2, 3.
  2. Consider the space $L^2[a,b]$. This exercise is designed to show that $L^2[a,b]$ is a vector space.
     1. Let $\alpha=a_1+ia_2$ and $\beta=b_1+ib_2$ be complex numbers, where $a_1,a_2$ and $b_1,b_2$ are real. (Recall that the complex conjugate of $\alpha$ is $\bar \alpha=a_1-ia_2$, and the length $\alpha$ is $|\alpha| =\sqrt{a_1^2+a_2^2}=\sqrt{\alpha \bar \alpha}\ $.) Show that $|\alpha+\beta|^2 \le 2(|\alpha|^2 +|\beta|^2)$.

     2. Let $f,g$ be in $L^2[a,b]$. Apply part (a) to show that $\|f+g\|^2\le 2(\|f\|^2+ \|g\|^2)$.

     3. Let $f\in L^2[a,b]$. Show that for any complex scalar $c$, $\|cf\|^2=|c|^2\|f\|^2$. Conclude that $L^2[a,b]$ is a vector space. Thus signals of finite energy obey the "principle of superposition."

  3. For a real (real scalars) inner product space $V$, the angle between two vectors $f,g$ is defined to be $\theta=\arccos\big(\frac{\langle f,g\rangle}{\|f\|\|g\|}\big)$. Find the angle between these real-valued functions (vectors) in $L^2[-\pi,\pi]$, with real scalars.
     1. $f(t)=e^t\sin(t)$ and $g(t)=\cos(3t)$.
     2. $f(t)=\sin(2t)$ and $g(t)=\sin(4t)$.
     3. $f(t)= \begin{cases}1&-\pi \le t< 0, \\ -1 & 0 \le t\le \pi \end{cases}$ and $g(t)= 2t^2+1$
     4. $f$ even and $g$ odd.

  4. For the pair $f,g$ in part (b) above, find the distance $\|f-g\|$ between $f$ and $g$.

  5. Suppose that $f$ is in $L^2[0,1]$, and that $c,d$ satisfy $0 < c,d <1/2$. Is it true that, in general, $f(c+d)=f(c)+f(d)$? Prove your answer.

Assignment 2 - Due Friday, 1/31/2020.

• Read sections 0.7.1, 1.1.1 - 1.1.3, 1.2.1
• Problems.
  1. Chapter 0, exercises: 5, 9, 10, 12 (Only do $1,x,x^2$.) 13 (Only do the space spanned by $1,x,x^2$; use ex. 12 to do the problem.), 17.

  2. Consider the Haar scaling function, $\phi(x):=\begin{cases} 1 & 0 \le x<1,\\ 0 & \text{otherwise}\end{cases}$ and $\psi(x):=\begin{cases}1&0\le x < 1/2,\\ -1 & 1/2\le x <1,\\0 & \text{otherwise}\end{cases}$. Sketch (by hand is okay) $\psi(2x)$ and $\psi(2x-1)$. Show these functions form an orthogonal set. Find the corresponding orthonormal set.

  3. Let V be a vector space with a complex inner product $\langle \cdot,\cdot\rangle>$. Suppose that the set $S = \{u_1,u_2,\cdots,u_n\}$ is an orthonormal basis for $V$. Show that if $v=\sum_{j=1}^n a_j u_j$ and $w=\sum_{k=1}^n b_k u_k$, then $\langle v,w\rangle= \sum_{j=1}^n a_j\bar b_j=\bar b^Ta$, where $a=(a_1 \cdots a_n)^T$ and $b=(b_1\cdots b_n)^T$.

Assignment 3 - Due Friday, 2/7/2020.

• Read sections 1.2.1-1.2.4
• Problems.
  1. Chapter 0, exercise: 0.15. (The o.n set to be used for the projection is $\{\phi(x), \psi(x), 2^{1/2}\psi(2x), 2^{1/2}\psi(2x-1)\}$

  2. Chapter 1, exercises: 1, 15, 25.

  3. Find the Fourier series for $f(x)=\pi -|x|$, $-\pi\le x\le \pi$

  4. Sketch (by hand is okay) the even and odd $2\pi$ periodic extensions of the functions defined on the interval $[0, \pi]$: $x$, $\pi -x$, $\sin(x)$, and $e^x$.

  5. Consider the vector space $L^2[0,\pi]$.
     1. Show that $\{\frac{1}{\sqrt{\pi}}, \sqrt{\frac{2}{\pi}}\cos(x), \sqrt{\frac{2}{\pi}}\cos(2x), \ldots, \sqrt{\frac{2}{\pi}}\cos(nx)\ \ldots\}$ is an othonormal set for $L^2[0,\pi]$.
     2. Show that $\{\sqrt{\frac{2}{\pi}}\sin(x), \sqrt{\frac{2}{\pi}}\sin(2x), \ldots, \sqrt{\frac{2}{\pi}}\sin(nx)\ \ldots\}$ is also an orthonormal set for $L^2[0,\pi]$.

Assignment 4 - Due Friday, 2/14/2020.

• Read sections 1.2.4-1.2.5, 1.3.1
• Problems.
  1. Chapter 1, exercises: 3, 4, 7, 10(a,b),

  2. Find the Fourier series for $f(x)=x$, $0\le x \le 2\pi$. Sketch (by hand is okay) the $2\pi$ periodic extension of $f$. Use your favorite software to plot $x$ and the partial sum $S_N$, for $N=5,10$, on the interval $[-2\pi,2\pi]$.

  3. In class we showed that "the Fourier series of a sum or difference is the sum or difference of the Fourier series." This means that the Fourier series for $f\pm g$ is the Fourier series for $f$ plus (or minus) the Fourier series for $g$.
     1. Find the complex form of the Fourier series for $f(x)=e^{2x}$ on the interval $[-\pi,\pi]$.

     2. Find the complex form of the Fourier series for $g(x)=e^{-2x}$ on the interval $[-\pi,\pi]$.

     3. Use the rule above to find the Fourier series of $\cosh(2x)$ and $\sinh(2x)$.

Assignment 5 - Due Friday, 2/21/2020.

• Read sections 1.3.1, 1.3.3-1.3.5, and the Notes on pointwise convergence.
• Problems.
  1. Chapter 1, exercises: 23(a,b,c,d) (Hand drawn sketches are fine.), 32(c,d,e,f) (I'll explain this in class.), 33.

  2. In the proof of the Riemann-Lebesgue lemma we assumed that $f$ was continuously differentiable on $[a,b]$. Show that the result holds for a piecewise smooth funtion which has a jump disconinuity at $x=c $, where $a < c < b$. (Hint: split the integral into two pieces, $a\le x\le c$ and $c\le x \le b$ and use the fact that $f$ is continuously differentiable on each piece.)

Assignment 6 - Due Friday, 3/6/2020.

• Read sections 2.1 and 2.2.
• Problems.
  1. Chapter 2 exercises: 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. $te^{-|t|}$ (Use #2.)
     2. $e^{-2|t-3|}$ (#6 and #7)
     3. ${\rm sign}(t)e^{-|t|}$ (Hint: differentiate $e^{-|t|}$; use #4.)
     4. $(1+(t-2)^2)^{-1}$  (Hint: How is this function related to $\hat f(\lambda)$, where $f(t)=e^{-|t|}$? Once you've gotten this, use #6.)

  3. Find the Fouirer transforms of these functions.
     1. $g(t) = \left\{\begin{array}{cl} 1 & \text{if }-1 \le t \le 2 \\ 0 & \text{otherwise}. \end{array} \right.$

     2. $h(t) = \left\{\begin{array}{cl} -1 & \text{if }-3 \le t \le 0 \\ 1 & \text{if }\ 0 < t \le 3 \\ 0 & \text{otherwise}. \end{array} \right.$

Assignment 7 - Due Monday, 4/6/2020.

• Read section 4.2, 4.3.1 and 4.3.2
• Problems.
  1. Chapter 4: 2, 6, 7

  2. Let $d>1$. Consider the running average of a function: $L[f](t) =\frac{1}{d}\int_{t-d}^t f(\tau)d\tau$.
     1. Using the definition of a linear time invariant filter, show that $L$ is LTI; that is, show directly from the formula for $L[f]$ that $L[f_a]=(L[f])_a$.
     2. Use $L[\delta](t)=h(t)$ to find the impuse response function.

  3. Let f(t) be a signal that is 0 when t < 0 or t > 1. Show that, for the Butterworth filter, one has

     \[ L[f] = Ae^{-\alpha t} \int_0^{\min(1,t)} e^{\alpha \tau} f(\tau)d\tau, \ \text{if }t\ge 0, \ \text{and } L[f] = 0 \ \text{if } t<0. \]

  4. Consider a function $f=\begin{cases}\cos(\pi x/4) &|x| \le 2\\ 0&\text{otherwise}\end{cases}$. Find the projection of $f$ onto $V_0$, $W_0$ and $V_1$. Do the coefficients satisfy the formulas invoving a's and b's in Theorem 4.12?

Updated 3/28/2020