NAME:

Problem	Points	Score
1a	10
1b	10
1c	10
2a	10
2b	10
2c	10
3a	10
3b	10
3c	10
3d	10
Total	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: For the following system:

(a) For c = 1/2, use the Nyquist criterion to determine stability.

(b) Use the Nyquist criterion, and the gain/phase margin, to determine for what range

of values of c the system is stable.

(c) Verify your answer to part (b) using any other technique described in this class.

Problem No. 2: In the following circuit, all resistance, capacitance, and inductance

values have a magnitude of 1.

(a) Define a set of state variables and justify your answer.

The top inductor's current

The capacitor's voltage

The bottom inductor's current

(b) Construct the state equations for this system.

Using nodal analysis:

Using the first 4 equations gives:

the matrices are then

_ _ _ _ _ _

| -19/8 -1 1/8 | B= | 3/8 | C= | 1 0 | D=0

A= | -1 0 1 | |_ 1/8 _| |_ 0 -1 _|

|_ 1/8 -1 -13/8 _|

(c) Find the system transfer function (remember this is a matrix).

Which get nasty real quick.

Problem No. 3: For the signal

(a) What is the minimum sample frequency for which this signal can be sampled

without loss of information? Justify your answer. Be careful!

The Nyquist rate is the minimum frequency. This requires that where fh

is the highest frequency in the signal. In this case it =3300

or fh = 1.65 kHz so

(b) Explain how to reconstruct the original signal from the sampled signal. Be as

specific as possible.

The simplest method is to use an RC low pass filter where the 3 dB frequency

is such that Is true. A more accurate method uses a filter with the transfer function :

convolving h(t) with x( nT ) gives:

(c) Suppose we sample the signal with an 8-bit linear quantizer . What is the signal to

noise ratio of the A/D converter?

D =2A =4 and

(d) Derive the Z-transform of x(t) using the following transform:

sample this at 50 Hz (T=1/50)

= j40 and = j66