Integrals, Unit Step, etc.

Signals and Systems

–

Final Exam

Date Due: April 22, 2013, Monday @4:00 PM

Total = 100 points

Name:

Each question has three parts and each student will be answering the part

of each question

assigned to him

as indicated

below:

Part a

Part b

Part c

David

Marquidris

Jirreaubi

Qu

estion 1

: Express

the following complex numbers in polar form:

(10 points)

a.

3 + j4

b.

-

100 + j46

c.

-

23 + j7

Question 2

: Let Z1 = 7 + j5 and Z2 =

-

3 + j4.

Determine the following in both Cartesian and Polar form:

(10 po

ints)

a.

Z1/Z2

b. Z1*Z2

c. (Z1

-

Z2)/(Z1+Z2)

Question 3

: Classify the signals belo

w as periodic or aperiodic. If periodic, then identify the period.

(1

5

points)

a.

x(t) = cos(4t) + 2sin(8t)

b. x(t) = 3cos(4t) + sin(πt)

c. x(t) = cos(3πt) + 2cos(4πt)

Question 4

: Determine if the following systems are time

-

invariant, linear, causal, and/or memoryless?

(1

5

po

ints)

a) dy/dt + 6 y(t) = 4 x(t)

b) dy/dt + 4 y(t) =

2 x(t)

c. y(t) = sin(x(t))

Question 5

: Solve the following difference equations using recursion first by hand (for n=0 to n=4

) and then plot.

C

heck your solution using

Maple or

MATLAB (for n=0 to n=30).

Attach plots to your solution.

(

20

points)

a) y[n] + 0.5y[n

-

1] = 2x[n

-

1]; x[n] = δ[n], y[

-

1] = 0

b.

y[n] + 2y[n

-

1] = 2x[n

-

1]; x[n] = δ[n], y[

-

1] = 0

c) y[n] + 1.2y[n

-

1] + 0.32y[n

-

2] = x[n]

-

x[n

-

1]; x[n] = u[n], y[

-

2] = 1, y[

-

1]=2

Question 6

.

Solve the di

?

erential equation

s

:

(15 points)

a.

x

’

’

+

4

x

’

+ 13x = 0

;

x(0) = 3, x

’

(0) = 0

b.

x

’

’ +

6

x

’

+ 9

x =

50 sin(t)

;

x(0) = 1

, x

’(0) =

4

c.

x

’

’ +

4

x

’

-

3x =

4e

t

;

x(0) =

1

, x

’(0) =

-

2

Question 7

:

Find the Fourier series of the function

: (15 points)

a.

b.

c.

Signals and Systems – Final Exam

Date Due: April 22, 2013, Monday @4:00 PM

Total = 100 points

Name:

Each question has three parts and each student will be answering the part of each question assigned to him

as indicated below:

Part a Part b Part c

David Marquidris Jirreaubi

Question 1: Express the following complex numbers in polar form: (10 points)

a. 3 + j4 b. -100 + j46 c. -23 + j7

Question 2: Let Z1 = 7 + j5 and Z2 = -3 + j4.

Determine the following in both Cartesian and Polar form: (10 points)

a. Z1/Z2 b. Z1*Z2 c. (Z1-Z2)/(Z1+Z2)

Question 3: Classify the signals below as periodic or aperiodic. If periodic, then identify the period. (15 points)

a. x(t) = cos(4t) + 2sin(8t) b. x(t) = 3cos(4t) + sin(πt) c. x(t) = cos(3πt) + 2cos(4πt)

Question 4: Determine if the following systems are time-invariant, linear, causal, and/or memoryless? (15 points)

a) dy/dt + 6 y(t) = 4 x(t) b) dy/dt + 4 y(t) = 2 x(t) c. y(t) = sin(x(t))

Question 5: Solve the following difference equations using recursion first by hand (for n=0 to n=4) and then plot.

Check your solution using Maple or MATLAB (for n=0 to n=30). Attach plots to your solution. (20 points)

a) y[n] + 0.5y[n-1] = 2x[n-1]; x[n] = δ[n], y[-1] = 0

b. y[n] + 2y[n-1] = 2x[n-1]; x[n] = δ[n], y[-1] = 0

c) y[n] + 1.2y[n-1] + 0.32y[n-2] = x[n]-x[n-1]; x[n] = u[n], y[-2] = 1, y[-1]=2

Question 6. Solve the di?erential equations: (15 points)

a. x’’ + 4x’ + 13x = 0; x(0) = 3, x’(0) = 0

b. x’’ + 6x’ + 9x = 50 sin(t); x(0) = 1, x’(0) = 4

c. x’’ + 4x’ - 3x =

4e

t

; x(0) = 1, x’(0) = -2

Question 7: Find the Fourier series of the function: (15 points)

a.

b.

c.