Homework 7, ECE438, Fall 2010, Prof. Boutin

Due in class, Friday October 22, 2010.

The discussion page for this homework is here.

Question 1

Compute the discrete Fourier transform of the following discrete-time signals:

$ x_1[n]= e^{j \frac{2}{3} \pi n}; $

$ x_2[n]= e^{j \frac{2}{\sqrt{3}} \pi n}; $

$ x_3[n]= e^{j \frac{4}{3} \pi n}; $

$ x_4[n]= e^{j \frac{2}{1000} \pi n}; $

$ x_5[n]= e^{-j \frac{2}{1000} \pi n}; $

$ x_6[n]= \cos\left( \frac{2}{1000} \pi n\right) ; $

$ x_7[n]= \cos^2\left( \frac{2}{1000} \pi n\right) ; $.

$ x_8[n]= (-j)^n . $

How do your answers relate to the Fourier series coefficients of x[n]?

Question 2

Obtain the frequency response and the transfer function for each of the following systems:

$ y_1[n]= \frac{x[n]+x[n-1]}{2}; $

$ y_2[n]= \frac{x[n]-x[n-1]}{2}; $

$ y_3[n]= \frac{x[n+1]+x[n]+x[n-1]}{3}; $

$ y_4[n]= \frac{x[n+1]-2 x[n]+x[n-1]}{4}. $

Question 3

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 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 4

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 5

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) $

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 6

Consider a DT LTI system described by the following recursive difference equation

$ y[n]= \frac{1}{8} \left( x[n]-x[n-8] \right) +y[n-1] $

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?

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