Math 414-501 — Spring 2020

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?