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EECS 20N: Structure and Interpretation of Signals and Systems

Department of Electrical Engineering and Computer Sciences

University of California Berkeley

MAKEUP

MIDTERM 2

15 March 2012

LAST Name FIRST Name

Lab Time

• (10 Points) Print your name and lab time in legible, block lettering above AND on the last page where the grading table appears.

• This exam should take up to 70 minutes to complete. You will be given at least 70 minutes, up to a maximum of 80 minutes, to work on the exam.

• This exam is closed book. Collaboration is not permitted. You may not use or access, or cause to be used or accessed, any reference in print or electronic

form at any time during the exam, except two double-sided 8.5”×11” sheets of handwritten notes having no appendage. Computing, communication, and other

electronic devices (except dedicated timekeepers) must be turned off. Noncom-

pliance with these or other instructions from the teaching staff—including, for

example, commencing work prematurely or continuing beyond the announced

stop time—is a serious violation of the Code of Student Conduct. Scratch pa-

per will be provided to you; ask for more if you run out. You may not use your

own scratch paper.

• The exam printout consists of pages numbered 1 through 8. When you are prompted by the teaching staff to begin work, verify that your copy of the exam

is free of printing anomalies and contains all of the eight numbered pages. If

you find a defect in your copy, notify the staff immediately.

• Please write neatly and legibly, because if we can’t read it, we can’t grade it.

• For each problem, limit your work to the space provided specifically for that problem. No other work will be considered in grading your exam. No exceptions.

• Unless explicitly waived by the specific wording of a problem, you must explain your responses (and reasoning) succinctly, but clearly and convincingly.

• We hope you do a fantastic job on this exam.

1

MT2.1 (30 Points) The impulse response h of a discrete-time LTI system H is

∀n ∈ Z, h(n) = +∞∑ `=0

α` g(n − `N),

where 0 < α < 1; N is a positive integer greater than 2; and g is the impulse response

of the two-point moving-average filter G. That is, g(n) = δ(n) + δ(n − 1)

2 , ∀n ∈ Z.

(a) (5 Points) Provide a well-labeled plot of h(n).

(b) (10 Points) Determine a reasonably simple expression for the frequency response

H(ω) of the filter H; your expression must be in terms of the parameters α and N. You may receive partial credit if you express H(ω) in terms of the frequency

response G(ω) of the moving-average filter G.

(c) (15 Points) Provide a well-labeled plot of the magnitude response |H(ω)| of the filter H. Assume α = 1/2 and N = 3.

2

MT2.2 (35 Points) The impulse response h of a continuous-time system H is shown below.

We apply to the system the input signal described by x(t) =

+∞∑ `=−∞

(−1)` δ(t − 2`).

(a) (15 Points) Provide a well-labeled plot of input x(t) as well as y(t), the corre-

sponding output.

3

(b) (20 Points) Consider a related LTI system G whose impulse response g is shown below:

Express H(ω), the frequency response of the system H, in terms of G(ω), the frequency response of the system G. Provide a succinct, but clear and convinc- ing, explanation.

Note that H(ω) 4 =

∫ +∞ −∞

h(t) e−iωt dt, and G(ω) is defined similarly.

4

MT2.3 (35 Points) Consider a linear (but possibly time-varying) discrete-time sys-

tem G. If the input to the system is x(n) = δ(n−m), an impulse shifted by m samples, then the corresponding output is y(n) = gm(n). In general, the discrete-time function

gm depends on m.

The output of the system is given by

y(n) =

+∞∑ m=−∞

x(m) gm(n), for all integers n.

(a) (10 Points) Each of the figures below depicts—by a combination of circles, stars,

and triangles—the non-zero values gm(n) of a possible function gm. Determine

which one corresponds to a time-invariant system.

All the stars correspond to the same numerical value A of gm(n), all the circles

correspond to the same numerical value B of gm(n), and all the triangles corre-

spond to the same numerical value C of gm(n).

0 1

1

0 1

1

5

(b) (25 Points) Prove that the system G is BIBO stable if, and only if,

+∞∑ m=−∞

|gm(n)| <∞, for every integer n.

Hint: Show the necessity and sufficiency separately, just as you saw it done for

LTI systems in class. In particular, show that if ∑+∞ m=−∞ |gm(n)| <∞, then every

bounded input produces a bounded output. This is the sufficiency condition, and

it’s easier to prove than the necessity condition.

Afterward, show that if ∑+∞ m=−∞ |gm(n)| = ∞ for even a single value of n, a

bounded input signal x exists such that a bound for the output y does not

exist. You must find such a bounded input and show how the output becomes

unbounded.

6

MT2.4 (5 Points) Which Exam Should We Grade?

Select one, and only one, option below, and indicate your choice on the back cover.

Definitions:

Original Midterm 2 (OrigMT2): The midterm administered in EE 20 on Tuesday,

13 March 2012.

SOrig Your score on the Original Midterm 2, normalized to be out of 115 points.

Makeup Midterm 2 (MkupMT2): The midterm administered in EE 20 on Thursday,

15 March 2012.

SMkup Your score on the Makeup Midterm 2, out of 115 points.

S2: Your overall Midterm 2 score (out of 115 points) counted toward your final course

grade.

(A) Grade only my MkupMT2 (i.e., let it count for 100% of my overall Midterm 2

score). That is, disregard my OrigMT2 so that S2 = SMkup.

(B) Grade only my OrigMT2 (i.e., let it count for 100% of my Midterm 2 score).

That is, disregard my MkupMT2 so that S2 = SOrig.

(C) Grade bothmy OrigMT2 and MkupMT2. Compute my overall score for Midterm

2 according to the following weighted linear combination:

S2 = 0.25SOrig + 0.75SMkup.

That is, let my OrigMT2 score count for 25% and my MkupMT2 score for 75%

of my overall Midterm 2 score.

(D) Grade bothmy OrigMT2 and MkupMT2. Compute my overall score for Midterm

2 according to the following weighted linear combination:

S2 = 0.75SOrig + 0.25SMkup.

That is, let my OrigMT2 score count for 75% and my MkupMT2 score for 25%

of my overall Midterm 2 score.

(E) Grade both my OrigMT2 and MkupMT2, and count them equally toward my

overall Midterm 2 score. That is,

S2 = SOrig + SMkup

2 .

7

LAST Name FIRST Name

Lab Time

The Grading Option You’ve Chosen for Your Midterm 2 (See Prob. 4 on p. 7):

Problem Points Your Score

Name 10

1 30

2 35

3 35

4 5

Total 115

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