San Jose State University

Department of Electrical Engineering

Exam 1

EE 210 Spring 2008

Closed Book, Closed Notes,

Instructions: There are 4 problems. Interpretation of questions will not be given during the exam. If you are unsure about the meaning of a question, make an assumption, state what it is and continue.

Problem Score Possible

1 25

2 25

3 25

4 25

Total 100

Problems are weighted as shown. Please write your name and Student ID at the bottom of this page and work all 4 problems. Show all your work. No credit will be given for work not shown. Use back of page if necessary but label clearly.

Last Name: First Name: ID: .

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PROBLEM 1: ( 25 points) A continuous-time signal ( )tx is shown in below figure. Sketch and label carefully each of the following signals. 1) ( )1tx (10 points)

( )x t

t1

1

1

23

2

1 2

2) ( )24 tx (15 points)

3

PROBLEM 2: (25 points)

Determine and sketch the convolution of the following two signals:

( )

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PROBLEM 3: (25 points): Consider the following system:

+

0.6

1z

( )X z

( )Y z

a) Find the pulse response of the system ( )H z .(5 points)

b) Find the power transfer functions for the system. (5 points)

c) State ROC and plot their pole zero diagram. (5 points)

ROC of ( )T z

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d) If the input to the system is assumed white noise with ( ) 2xxS z = , use residue theory to find the mean-squared output fluctuations

[ ] ( ) ( )2 1102yy yy

y n R S z z dzj

= = % at the output (10 points)

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PROBLEM 4: (25 points) In the input to system of problem 3 has [ ] ( )10 0.6 nxxR n = a) Find the power spectral density of the output noise. (10 points)

b) The cross-spectral density of the input and output noise ( )xyS z . (5 points)

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c) Find the signal to noise ratio at the input and output for 0n if an input signal of [ ] [ ]6f n u n= is added to the input noise. (10 points)