7/29/2019 Midterm Quiz

1/2

Midterm Quiz Signals, Spectra, and Signal Processing

Name: Section:

General Instructions: Answer each of the following questions completely, writing your complete solutions on separate

sheets of short bond paper for items requiring solutions and writing your final answers in this answer sheet. Items with

erasures in this answer sheet will render that answer invalid, hence will be considered incorrect. Credit points for each

item are indicated opposite the item. Good luck!

I. Answer the following questions as precise as possible. Write your answers on separate sheet of paper. The

maximum score for each item five (5) points, and the points credited in each item depend on how precise the answers

are.

1. Discuss the relationship of the region of convergence of the z-transform of the signal and its length and causality.2. How does the placement of the poles of the z-transform of the signal affect its time-domain behavior?3. Enumerate and briefly discuss the properties of two-sided z-transform. What are the fundamental differences of theproperties of the two-sided z-transform with that of the one-sided z-transform?4. Does the stability and causality of the system depend on the poles of the system function? Why or why not?

5. In analyzing discrete-time systems in z-domain, when should one use the bilateral z-transform and when should oneuse the unilateral

z-transform?

II. Solve each of the problems completely, showing your solutions on separate answer sheets. For computer-aided

solutions, write the MATLAB command/s used to generate the answers. The maximum score for each problem is

fifteen (15) points. Write your final answers in separate sheet of paper.

1. Using the power series definition of the two-sided z-transform, obtain the z-transform of the following signals andspecify their ROCs.

a) xn = 3 0 0 0 0 6 1 4b) xn = , n 70,n < 7c)

xn = a

+ a

un, where

ais a real number.

d) xn = , n 0 , n < 0e) xn = 2 un

2. Using the table ofz-transform pairs and the properties ofz-transforms, obtain the z-transform of the following signalsand sketch their pole-zero plots and their ROCs.

a) xn = 1 + nunb) xn = 1 2unc) xn = un un10d)

xn = n + n un 1

e) xn = nun3. Determine the causal signal xn if its z-transform Xz is given as follows:

a) Xz = b) Xz = c) Xz = .d) Xz = e) The pole-zero plot is sketched below, with a gain constant of

G =.

7/29/2019 Midterm Quiz

2/2

4. Use the one-sided z-transform to determine the closed-form of the output yn for n 0. Write out and plot thesamples of the output yn for 0 n 10.

a) yn 1.5yn 1 + 0.5yn 2 = 0, with y1 = 1 and y2 = 0

b) yn =

yn 2 + xn with xn = un, y1 = 0 and y2 = 1

5. The system is described by the difference equation

yn 3

5yn 1 +

3

25yn 2 = xn 1 +

1

2xn 2

a) Assuming zero initial conditions, what is the systems transfer function? Specify the ROC ofHz if the system

is to be causal.

b) With the initial conditions zero, compute the poles and zeros of the system and sketch the pole-zero pattern

including the systems ROC. From this, draw conclusions on the stability of the system.

c) Determine the closed-form impulse response of the system, and plot the first eleven samples of hn, i.e0 n 10. From this verify if the conclusion about the stability is the same as the one drawn in part (b).

d) Determine the step response of the system assuming zero initial conditions. Plot the first eleven samples of

the step response.

e) Determine the step response of the system with y1 = 1 and y2 = 2. Plot the first eleven samples of

the step response.

Notes: Print this questionnaire. All answers and solutions should be LEGIBLY HANDWRITTEN on separate LONG BOND

paper sheets (see format of submission below), including plots and graphs. Papers must be stapled at the TOP LEFT

corner of the paper. Do not plagiarize, it is to be dealt with accordingly. Hand in your papers by 3:00 PM tomorrow (Feb.

12, 2013). Deduction of 10 points per minute beyond 3:00 PM will be given.

Format of submission:

First and second pages: This questionnaire with your name and section written on the spaces provided.

Third page, front: Your responses to part I. Limit your answer to part I to a SINGLE, ONE SIDE long bond paper ONLY. No

erasures allowed. Erasures = NO CREDIT.

Third page, back: Your responses (only the answers, no solutions) to part II. No erasures allowed. Erasures = NO CREDIT

Label your answers accordingly (i.e. (1a), (2b), and so on). Graphs and plots must be labeled clearly or else no credit will

be given to the graph or plot.

Fourth up to the sixth pages: Your solutions to part II. Box the final answers. Answers should tally the one written on the

second page. In case of discrepancy, the answer will be invalidated and no credit will be given even if one of the two is

correct. No erasures allowed. Erasures = NO CREDIT.

Limit your submissions up to SIX pages only. Deduction of 10 points per excess page beyond six will be given. Any

infraction on the rules stated above entails a deduction of FIVE points PER INFRACTION.