Homework 6, ECE438, Fall 2011, Prof. Boutin

Due Wednesday October 26, 2011 (in class)

Questions 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 response of this system to the input

$ x[n]=\left\{ \begin{array}{rl} -2, & \text{ if }n=-2,\\ 1, & \text{ if }n=0,\\ -2 & \text{ if }n=2,\\ 0, & \text{ else. } \end{array} \right. $

by the following approaches:

a. Directly substitute x[n] into the difference equation describing the system;

b. Find the impulse response h[n] and convolve it with x[n];

c. Find the frequency response by the following two approaches:

i. apply the input $ e^{ j \omega_0 n} $ to the difference equation describing the system,

ii. find the DTFT of the impulse response.

(verify that both methods lead to the same result) then find the DTFT of the input, multiply it by the frequency response of the system to yield the DTFT of the output, and finally calculate the inverse DTFT y[n].

d. Verify that all three approaches for finding y[n] lead to the same result.

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?

Discussion

Write your questions/comments here

Yimin:

1.what does Q2-c-i mean?

i. apply the input $ e^{ j n} $ to the difference equation describing the system..

My understanding is that, first DTFT the difference equation to get the equation contains Y(w) and X(w) then develop H(w), second approach is to DTFT the h[n] obtained in b. part, then compare?

Yes, but there are two ways to get the frequency response. One method is to compute the DTFT of h[n]. Another method, which I used in class, is to compute the system's response to a complex exponential. You need to use both methods. -pm

Oops, I see what you mean. There was a typo in the question (a missing $ \omega_0 $). -pm

2.I changed the equation on Q4. That's too long to view on portable devices.

3.in the last question, the initial values of y is not given, so the LTI system is not uniquely defined, so the IZF of H(z) is not uniquely defined. Do you want us to discuss both situation?

Excellent! You should say something like: "we must assume the system is XXX, otherwise it is not uniquely defined. Now assuming XXX, we have ... (answer the questions)". -pm

Professor, could you give some hint for Q3,d? I used factorization, but it seemed too complicated. Should just I used the property of causal system, like bringing n=0 in, and the parts with negative index are all 0? Does that make sense? Thanks.

Sure. Basically, you have to retrace the steps for deriving H(z) backward. Start from the fact that y(z)=H(z)X(z), then replace H(z) by its expression in terms of z. After that, multiply both sides of the equation by the denominator of H(z). Can you figure out how to continue? -pm

So should our solution for 3d be recursive or do we need to factor everything into terms of x[n]...?

It should be recursive. -pm

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