Name: Trae Rushing

Problem	Points	Score
1a	10	10
1b	10	10
2a	10	10
2b	10	10
2c	10	10
3a	10	10
3b	10	10
3c	10	10
3d	10	10
3e	10	10
Total	100	100

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)|

Using Euler's theorem, we get the following for all reals:

Assume the following to be true,

Substituting (- ) for ( )

The cosine of a negative number is the same as the cosine of a positive number with the same magnitude.

The sine of a negative number is the negative of the value of a positive number of the same magnitude.

Since

Then

Therefore,

(b) (f) = - (f)

From the proof above, the following is true:

Hence,

Since the tangent of a negative number is the same as the negative tangent of a positive number with the same magnitude:

Problem No. 2: For the following two signals:

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

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

(

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

x(t) is a function that has an amplitude of one and has been shifted by .5. y(t) is a function that has been scaled by two. x(t) magnitude spectrum is twice as wide as y(t) because its width in the time domain is half of y(t). The magnitude spectrum of x(t) is not affected by the time shift. This is shown through the time-delay theorem .

A shift in time does not affect the magnitude spectrum but it does affect the phase spectrum.

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.

This bode plot was done in matlab using the following code:>> bode([1 0 0)],[1 0 1])

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

- The squared term in the numerator accounts for two zeros at = 0 and the positive 40dB/decade slope of the line.

- The squared term in the denominator accounts for two poles at = 1, which essentially turns the slope off.

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

With the Routh Array shown below, we can see that the system is marginally stable. Stability requires that there will be no right half plane poles, and since there are no changes in column one, this characteristic seems to be true of this system. The system itself is made up of two linear components that cancel each other out with their equal magnitudes. This allows for a steady-state response when an ac or sinusoidal input is applied. The characteristics of the components will force the system to become unstable when a D.C. input is supplied. The capacitor acts as an open circuit and the inductor acts as a short circuit. The Routh array is shown below:

s2	1	1
s1	0
s0	1