Name:	Jim Rutland

ProblemPointsScore

1a10
1b10
2a10
2b10
2c10
3a10
3b10
3c10
3d10
3e10
Total100

Notes:

1.	The exam is closed books/closed notes - except for one page of notes.

2.	Please show ALL work. Incorrect answers with no supporting explanations or work will be given no partial credit.

3.	Please indicate clearly your answer to the problem. If I can't read it (and I am the judge of legibility), it is wrong. If I can't follow your solution (and I get lost easily), it is wrong. All things being equal, neat and legible work will get the higher grade:)

Problem No. 1: Show the following properties of the Fourier transform for real signals:

(a) |X(f)| = |X(-f)|

X(f) can be written as follows:

(b) (f) = - (f)

Using A( ) and B( ) from the above proof, it can be shown that:

Therefore,

Problem No. 2: For the following two signals:

(a)	Plot the magnitude spectrum of x(t):

(b)	Plot the magnitude spectrum of y(t):

(

(C)	Compare and contrast these plots (note that you can answer this question correctly without getting the correct answers for (a) and (b).

x(t) has a unit pulse width of 1s and y(t) has a unit pulse width of 2s. Signals which are

wider in the in the time domain are more narrow in the frequency domain. This is seen

when you compare the two plots of the specta in the frequency domain. x(t) is also

shifted in time 0.5s. It is proven in the Fourier time delay theorem that a shift in time

does not effect the magnitude of the x(f) but it does effect the phase. The Fourier time

delay theorem is given below:

The scale change theorem is given below:

This theorem can be applied which reveals the sinc function in part b) is ½ the width of

the sinc function in part a).

Problem No. 3: For the following system,

(a) Find X(s):

(b) Find Y(s)

(c) For the circuit shown, construct the Bode magnitude plot.

(d) How many poles and zeros does the system have?

This system has 2 poles and 2 zeros.

(e) Using concepts introduced in Chaps. 4 - 6, determine the stability of the system.

This system is marginally stable because it is a simple system with one inductor and one

capacitor. The inductor and capacitor balance each other out and will not allow the system to

ever have an infinite magnitude. There are no poles in the right half or left half plane. The poles

are located on the imaginary axis.