EE 350 Exam # 2 16 October 2014

Last Name (Print):

First Name (Print):

ID number (Last 4 digits):

Section:

DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO

Problem Weight Score

1 25

2 25

3 25

4 25

Total 100

INSTRUCTIONS

1. You have 2 hours to complete this exam.

2. This is a closed book exam. You may use one 8.5”× 11” note sheet.

3. Calculators are not allowed.

4. Solve each part of the problem in the space following the question. If you need more space, continue your solutionon the reverse side labeling the page with the question number; for example, Problem 1.2 Continued. NO

credit will be given to solutions that do not meet this requirement.

5. DO NOT REMOVE ANY PAGES FROM THIS EXAM. Loose papers will not be accepted and agrade of ZERO will be assigned.

6. The quality of your analysis and evaluation is as important as your answers. Your reasoning must be preciseand clear; your complete English sentences should convey what you are doing. To receive credit, you must

show your work.

7. Any student caught cheating on an exam will receive a grade of zero for the exam. Additional sanctions,including assigning an XF grade, will be pursued following university guidelines.

1

Problem 1: (25 Points)

1. (10 points) The network in Figure 1, with input f(t) and output y(t), is represented by the ordinary differentialequation

y +1

LCy(t) =

1

LCf(t).

Determine the impulse response representation h(t) of the network.

Figure 1: Passive LC network.

2

3

2. (7 points) A LTI system, different from the one considered in part 1, has the impulse response representation

h(t) = δ(t + 1) + e−tu(t).

(a) (2 points) Is the system causal or noncausal? In order to receive credit, you must justify your answerusing a short sentence.

(b) (5 points) Determine if the system is bounded-input bounded-output stable. Justify you answer.

4

3. (8 points) Figure 2 shows the block diagram of a LTI system with input f(t) and output y(t). Each blockrepresents a LTI system, and the impulse response representations of these systems are

h1(t) = 4u(t)

h2(t) =1

2δ(t).

Using the properties of convolution, represent the system by the ordinary differential equation

y + a0y(t) = bof(t)

by providing the numeric values of ao and bo.

Figure 2: Block diagram representation of a LTI system.

5

Problem 2: (25 points)

1. (15 points) A linear time-invariant system with input f(t) and output y(t) is represented by the impulseresponse

h(t) = e−2tu(t).

Determine the zero-state response of this system to the input

f(t) = e−t [u(t) − u(t − 1)]

using the graphical convolution approach. Do not sketch y(t). In order to receive credit, clearly specify theregions of integration and, for each region, provide a sketch of f and h.

6

7

2. (10 points) The steady-state output of a LTI system is the same form as its input when its input is eωt. In

mathematical terms, eωt is said to be an eigenfunction of the system. Show that the zero-state response of aLTI system with impulse response h(t) to the complex exponential input

f(t) = eωt

isy(t) = ejωtH(ω),

where

H(ω) =

∫

∞

−∞

h(t)e−ωtdt.

8

Problem 3: (25 points)

1. (12 points) Consider the RC network in Figure 3 that has input f(t) and output y(t). Using phasor analysis,determine the frequency response function of the system and express your answer in the standard form

H(ω) =bm(ω)m + bm−1(ω)m−1 + · · ·+ b1(ω) + b0

(ω)n + an−1(ω)n−1 + · · ·+ a1(ω) + a0

.

Figure 3: Passive RC network.

9

2. (13 points) A LTI system, different from the one considered in part 1, has the frequency response functionrepresentation

H(ω) =(ω)2

(ω)2 + 2(ω) + 100.

(a) (3 points) Represent the system as an ODE in the standard form

dny

dtn+ an−1

dn−1y

dtn−1+ · · ·+ aoy = bm

dmf

dtm+ bm−1

dm−1f

dtm−1+ · · ·+ bof.

(b) (2 points) Determine the DC gain of the system.

(c) (2 points) Determine the high frequency AC gain of the system.

(d) (6 points) Determine the sinusoidal steady-state response of the system to the input

f(t) = cos(10t)

10

Problem 4: (25 points)

1. (8 points) Determine if the signals

φ1(t) =√

2e−tu(t)

φ2(t) =[

3e−2t − 2e−t]

u(t),

defined over the interval t ≥ 0, form an orthonormal set. Justify your answer.

11

2. (9 points) Approximate a signal f(t) as a weighted sum of functions x1(t) and x2(t),

f(t) ≈ c1x1(t) + c2x2(t),

where the approximation error signal is

e(t) = f(t) − c1x1(t) − c2x2(t).

The signals f(t), e(t), x1(t), and x2(t) are real-valued and defined on the interval [t1, t2]. Table 1 lists therelevant inner products. For example, 〈f, x1〉 = 6. Determine the value of the weighting coefficients c1 and c2

that minimizes the energy

Ee =

∫ t2

t1

e2(t)dt

of the approximation error signal.

〈·, ·〉 x1(t) x2(t)x1(t) 1 2

x2(t) 2 1f(t) 6 9

Table 1: Table of inner products.

12

3. (8 points) A periodic triangle waveform is approximated as

f(t) = 2 +

N∑

n=1

an cos(10nt)

where

an =16

π2n2

and N represents the integer number of sinusoids used to generate the approximation. As the value of N

increases, the energy of the approximation error signal decreases. A partially complete MATLAB function fornumerically generating the signal f(t) appears in Figure 4. The function accepts a vector t containing thetime instants at which to evaluate the function f(t) and an integer N specifying the number of cosine termsto include in the approximation. The function returns a vector f containing the values of f(t) at the timeinstants specified in the vector t. Complete the code in Figure 4.

% EE 350 Fall 2014

% Exam 2

% Problem 4 Part 3

%

function [ f ] = find_f(t, N)

% Function find_f approximates f(t) using N terms

%

Figure 4: MTALB m-file for realizing the user defined function find f.

13

EE 350 Exam # 2 16 October 2014

Last Name (Print):

First Name (Print):

ID number (Last 4 digits):

Section:

DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO

Problem Weight Score

1 25

2 25

3 25

4 25

Total 100

INSTRUCTIONS

1. You have 2 hours to complete this exam.

2. This is a closed book exam. You may use one 8.5”× 11” note sheet.

3. Calculators are not allowed.

4. Solve each part of the problem in the space following the question. If you need more space, continue your solutionon the reverse side labeling the page with the question number; for example, Problem 1.2 Continued. NO

credit will be given to solutions that do not meet this requirement.

5. DO NOT REMOVE ANY PAGES FROM THIS EXAM. Loose papers will not be accepted and agrade of ZERO will be assigned.

6. The quality of your analysis and evaluation is as important as your answers. Your reasoning must be preciseand clear; your complete English sentences should convey what you are doing. To receive credit, you must

show your work.

7. Any student caught cheating on an exam will receive a grade of zero for the exam. Additional sanctions,including assigning an XF grade, will be pursued following university guidelines.

1

Problem 1: (25 Points)

1. (10 points) The network in Figure 1, with input f(t) and output y(t), is represented by the ordinary differentialequation

y +1

LCy(t) =

1

LCf(t).

Determine the impulse response representation h(t) of the network.

Figure 1: Passive LC network.

2

3

2. (7 points) A LTI system, different from the one considered in part 1, has the impulse response representation

h(t) = δ(t + 1) + e−tu(t).

(a) (2 points) Is the system causal or noncausal? In order to receive credit, you must justify your answerusing a short sentence.

(b) (5 points) Determine if the system is bounded-input bounded-output stable. Justify you answer.

4

3. (8 points) Figure 2 shows the block diagram of a LTI system with input f(t) and output y(t). Each blockrepresents a LTI system, and the impulse response representations of these systems are

h1(t) = 4u(t)

h2(t) =1

2δ(t).

Using the properties of convolution, represent the system by the ordinary differential equation

y + a0y(t) = bof(t)

by providing the numeric values of ao and bo.

Figure 2: Block diagram representation of a LTI system.

5

Problem 2: (25 points)

1. (15 points) A linear time-invariant system with input f(t) and output y(t) is represented by the impulseresponse

h(t) = e−2tu(t).

Determine the zero-state response of this system to the input

f(t) = e−t [u(t) − u(t − 1)]

using the graphical convolution approach. Do not sketch y(t). In order to receive credit, clearly specify theregions of integration and, for each region, provide a sketch of f and h.

6

7

2. (10 points) The steady-state output of a LTI system is the same form as its input when its input is eωt. In

mathematical terms, eωt is said to be an eigenfunction of the system. Show that the zero-state response of aLTI system with impulse response h(t) to the complex exponential input

f(t) = eωt

isy(t) = ejωtH(ω),

where

H(ω) =

∫

∞

−∞

h(t)e−ωtdt.

8

Problem 3: (25 points)

1. (12 points) Consider the RC network in Figure 3 that has input f(t) and output y(t). Using phasor analysis,determine the frequency response function of the system and express your answer in the standard form

H(ω) =bm(ω)m + bm−1(ω)m−1 + · · ·+ b1(ω) + b0

(ω)n + an−1(ω)n−1 + · · ·+ a1(ω) + a0

.

Figure 3: Passive RC network.

9

2. (13 points) A LTI system, different from the one considered in part 1, has the frequency response functionrepresentation

H(ω) =(ω)2

(ω)2 + 2(ω) + 100.

(a) (3 points) Represent the system as an ODE in the standard form

dny

dtn+ an−1

dn−1y

dtn−1+ · · ·+ aoy = bm

dmf

dtm+ bm−1

dm−1f

dtm−1+ · · ·+ bof.

(b) (2 points) Determine the DC gain of the system.

(c) (2 points) Determine the high frequency AC gain of the system.

(d) (6 points) Determine the sinusoidal steady-state response of the system to the input

f(t) = cos(10t)

10

Problem 4: (25 points)

1. (8 points) Determine if the signals

φ1(t) =√

2e−tu(t)

φ2(t) =[

3e−2t − 2e−t]

u(t),

defined over the interval t ≥ 0, form an orthonormal set. Justify your answer.

11

2. (9 points) Approximate a signal f(t) as a weighted sum of functions x1(t) and x2(t),

f(t) ≈ c1x1(t) + c2x2(t),

where the approximation error signal is

e(t) = f(t) − c1x1(t) − c2x2(t).

The signals f(t), e(t), x1(t), and x2(t) are real-valued and defined on the interval [t1, t2]. Table 1 lists therelevant inner products. For example, 〈f, x1〉 = 6. Determine the value of the weighting coefficients c1 and c2

that minimizes the energy

Ee =

∫ t2

t1

e2(t)dt

of the approximation error signal.

〈·, ·〉 x1(t) x2(t)x1(t) 1 2

x2(t) 2 1f(t) 6 9

Table 1: Table of inner products.

12

3. (8 points) A periodic triangle waveform is approximated as

f(t) = 2 +

N∑

n=1

an cos(10nt)

where

an =16

π2n2

and N represents the integer number of sinusoids used to generate the approximation. As the value of N

increases, the energy of the approximation error signal decreases. A partially complete MATLAB function fornumerically generating the signal f(t) appears in Figure 4. The function accepts a vector t containing thetime instants at which to evaluate the function f(t) and an integer N specifying the number of cosine termsto include in the approximation. The function returns a vector f containing the values of f(t) at the timeinstants specified in the vector t. Complete the code in Figure 4.

% EE 350 Fall 2014

% Exam 2

% Problem 4 Part 3

%

function [ f ] = find_f(t, N)

% Function find_f approximates f(t) using N terms

%

Figure 4: MTALB m-file for realizing the user defined function find f.

13