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**EECS 20N: Structure and Interpretation of Signals and Systems
**Department of Electrical Engineering and Computer Sciences
UNIVERSITY OF CALIFORNIA BERKELEY

MIDTERM 2 03 April 2008

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” sheet
of handwritten notes having no appendage. Computing, communication,
and other electronic devices (except dedicated timekeepers) must be turned
off. Noncompliance 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 paper 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 ex-
plain your responses (and reasoning) succinctly, but clearly and convincingly.

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

1

**MT2.1 (35 Points) **The non-zero impulse response values of the discrete-time LTI
system F

appear below:

(a) (7 Points) Determine a reasonably simple expression for *F *(*ω*), where *F *is the
frequency response of the filter F.

(b) (8 Points) Determine a reasonably simple linear, constant-coefficient differ-
ence equation governing the input-output behavior of the LTI system. Ex-
plain (briefly) why it is apt to call the system F a *three-point, center-weighted,
moving average filter.
*

2

(c) (12 Points) Determine the output of the system if the input is

(i) the unit-step function: *x*(*n*) = *u*(*n*).

(ii) a constant function: *x*(*n*) = 1.

(iii) the sign-alternating signal *x*(*n*) = cos(*πn*).

Be sure to provide a well-labeled sketch of all the signals involved.

(d) (8 Points) Provide well-labeled sketches of *|F *(*ω*)*| *and ∠*F *(*ω*), the magnitude
and phase responses, respectively, of the filter. Is your magnitude response
consistent with your last two results in part (c)?

3

**MT2.2 (30 Points) **Consider a discrete-time LTI filter

The two parts of this problem refer to two different, and mutually independent, filters, which we generically name F. Filter properties disclosed (or inferred) in one part may not be carried over to the other. You may tackle the parts in either order.

(a) (15 Points) Assume F is an ideal high-pass LTI filter with the frequency re- sponse

*F *(*ω*) =

{
0*, |ω| ≤ π/*4
*e−iω/*2*, π/*4 *< |ω| ≤ π .
*

Determine a simple expression for the output of the system, if the input is
*x*(*n*) = *A *+ *B *cos

(*π
*3
*n
*)

*, *where *A *and *B *are constants.

(b) (15 Points) The impulse response of the filter is *f*(*n*) = *a|n|*, where 0 *< |a| <
*1. Determine an expression for *F *(*ω*), the frequency response of the filter;
simplify as much as you can.

4

**MT2.3 (15 Points) **Consider a continuous-time LTI system F for which one input-
output behavior is depicted below:

The input and output values are zero in the ranges of *t *not shown in the figure.

Determine the output of the system in response to the input *x*(*t*) = 1, *∀t*.

You may use the blank space below for scratch work. Nothing written below this line on this page will be considered in evaluating your work.

5

**MT2.4 (25 Points) **The continuous-time *unit doublet *is defined as the ”derivative”
of the Dirac delta:

*δ̇(t)
4
*=

*dδ*(*t*)

*dt
.
*

(a) (15 Points) Let *x *be a continuous-time function. Show that

(*x ∗ δ̇)(t) *= *ẋ(t) 4*= *dx*(*t*)
*dt
*

for every value of *t *at which *x *is ”well-behaved.” In particular, note that

(*x ∗ δ̇)(t) *=
∫ +*∞
−∞
*

*x*(*τ*) *δ̇(t− τ*) *dτ ,
*

and employ the method of integration by parts to show that the integral eval-
uates to *ẋ(t). *An LTI system whose impulse response is *δ̇ *is called an *ideal
differentiator*.

(b) (10 Points) Consider a continuous-time LTI system F whose impulse response
is *f*(*t*) = *δ̇(t). *Use the result of part (a) to determine the output of the system
in response to the input shown in the figure below. Briefly explain your work.

6

You may use this page for scratch work only.
Without exception, subject matter on this page will *not *be graded.

7

LAST Name FIRST Name

Lab Time

Problem Points Your Score Name 10

1 35

2 30

3 15

4 25

**Total 115
**

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