Contents
Homework 9, ECE438, Fall 2016, Prof. Boutin
Hard copy due in class, Wednesday November 16, 2016.
Question 1
Obtain the frequency response and the transfer function for each of the following systems. Sketch the magnitude of the frequency response, and indicate the location of the poles and zeros of the transfer function.
- a. $ y[n]= \frac{x[n]+x[n-2]}{2}; $
- b. $ y[n]= \frac{x[n]-x[n-1]}{2}; $
Question 2
Consider a DT LTI system described by the following equation
$ y[n]=x[n]+2x[n-1]+x[n-2]. $
Find the frequency response of this system using five different approaches.
Question 3
Consider a causal LTI system with transfer function
$ H(z)= \frac{1-\frac{1}{2}z^{-2}} {1-\frac{1}{\sqrt{2}} z^{-1} +\frac{1}{4} z^{-2}} $
- a. Sketch the locations of the poles and zeros.
- b. Determine the magnitude and phase of the frequency response $ H(\omega) $, for
- $ \omega =0,\frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \text{ and }\pi $.
- c. Is the system stable? Explain why or why not?
- d. Find the difference equation for y[n] in terms of x[n], corresponding to this transfer function H(z).
Question 4
Consider a DT LTI system described by the following non-recursive difference equation (moving average filter)
$ y[n]=\frac{1}{8} \left( x[n]+x[n-1]+x[n-2]+x[n-3]+x[n-4]+x[n-5]+x[n-6]+x[n-7]\right) $
i.e.
$ y[n]=\frac{1}{8} \sum_{k=0}^{7}x[n-k] $
- a. Find the impulse response h[n] for this filter. Is it of finite or infinite duration?
- b. Find the transfer function H(z) for this filter.
- c. Sketch the locations of poles and zeros in the complex z-plane.
Hint: To factor H(z), use the geometric series and the fact that the roots of the polynomial $ z^N- p_0 =0 $ are given by
$ z_k =|p_0|^{\frac{1}{N}} e^{j \frac{(\text{arg }p_0+2\pi k)}{N}} ,\quad k=0,\ldots ,N-1 $
Question 5
Consider a DT LTI system described by the following recursive difference equation
$ y[n]= \frac{1}{8} \left( x[n]-x[n-8]+y[n-1] \right) $
- a. Find the transfer function H(z) for this filter.
- b. Sketch the locations of poles and zeros in the complex z-plane.
- Hint: See Part c of the previous problem.
- c. Find the impulse response h[n] for this filter by computing the inverse ZT of H(z). Is it of finite or infinite duration?
Hand in a hard copy of your solutions. Pay attention to rigor!
Presentation Guidelines
- Write only on one side of the paper.
- Use a "clean" sheet of paper (e.g., not torn out of a spiral book).
- Staple the pages together.
- Include a cover page.
- Do not let your dog play with your homework.
Discussion
- Write question/comment here.
- answer will go here
