ECSE-4760
Real-Time Applications in Control & Communications
Spring 2009
Exam #2
Tuesday 5/5 11:30 a.m. Low-3051
Name ______________________________
Section: MR 1:30___
MR 6:30___
(Grades will be posted on SIS when they are available.)
This exam is open book, open notes, etc., calculators but NO laptops, Internet connections, …
Answer questions in only 5 out of the 8 sets corresponding to each of the experiments in the lab. You do not need
to choose the experiments you performed but you will probably wish to do so. If you answer more than 5 sets, the
first 5 will be graded. Each set is worth 20 points (max) with the breakdown shown by the questions.
If you are missing any grades on RPILMS, make sure you bring the reports to me (CII-6219) before the end of the
day today to get your records corrected.
I Digital Logic ______
II Voice Processing ______
III Binary Communications ______
IV Digital Filter ______
V Interactive Graphics ______
VI Hybrid Control ______
VII DC Motor ______
VIII Optimal Control ______
TOTAL ______
‘09
I Digital Logic
Name _________________________
001
1
0/0
0/0
010
2
1
000
0
0/1
0/1
1/0
0/0
011
3
1/0
1/0
1/0
100
4
1/0
1/1
1/1
110
6
101
5
0/0
0/0
1. (10pt) Given the above state machine, with arrows labeled INPUT/OUTPUT, find the Next State Table and
Output Table.
Input
0
1
P.S.
000
001
010
011
100
101
110
111
Next State Table
Input 0
1
P.S.
000
001
010
011
100
101
110
111
Output Table
2. (3pt) For a T flip-flop implementation of the state machine, find the expression for T0 the low bit in the
implementation and simplify it using a Carnough map.
‘09
I Digital Logic
Name _________________________
3. (2pt) Find the logic expression for the output of the state machine and simplify.
The 11 state counter design (0 – 10) in the lab procedure found the equation for the D-state T flip-flop by using a
K-map to find the inverse (NOT) of the change. This flip-flop should change every time (T4 = 1) except when
ABCD = 1010. The K-map with the don’t cares and the inverse logic equation will be:
CD
AB
00
01
11
10
00
1
1
1
1
01
1
1
1
d = don’t care
11
d
d
1
d
d
T1 = AC
10
1
1
d
0
T1 = AC
4. (3pt) Using the standard method to cover the K-map, what is the reduced expression for T1?
5. (2pt) What can be used to show that the answer to 4. is equivalent to the expression to the right of the K-map
above?
II Voice Processing
‘09
Name _________________________
1. (10pt) The zero order hold reconstruction has the same effect as convolving the impulse function samples with
a square pulse or multiplying the sampled spectrum with a sinc function. If the reconstruction is just a Z.O.H.,
what effect does this have on the spectrum in the 0 – 40kHz band of a 5kHz input sine wave? (Use equations or
appropriately scaled plots to explain results.)
Antialias
Filter
fco =10kHz
Ideal
Sampler
fs = 20kHz
Z.O.H.
Reconstruction
2. (2pt) Given a sine, triangle, and sawtooth wave with equal peak-peak voltages, which has the lowest crest factor?
II Voice Processing
‘09
Name _________________________
3. (2pt) A 12-bit ±2.5V 2’s complement analog converter is used in a delta-modulator system. What is the
smallest value A can have for a dead-band modulator with this hardware?
4. (5pt) For a delta modulator with A = 0.1V, B = 0.2V, C = 0.05V, and D = 0.15V, what is the peak-peak voltage
of the output for an input 2.2Vp-p square wave that is sampled 16 times in its period T?
5. (1pt) Give an example of a speech sound that has a wide frequency band and little to none regular repeating
waveform patterns.
III Binary Communications
‘09
Name _________________________
1. (2pt) For the following codewords, what is the minimum Hamming distance?
00000000, 01010101, 10100101, 10101010, 11111111
2. (3pt) What is the maximum number of bit errors that can be corrected?
3. (3pt) What is the maximum number of bit errors that can be detected?
4. (3pt) With a single bit error auto correction, how many additional bit errors may be detected?
5. (3pt) Find an additional 8-bit codeword that could be used with the code above that maintains the same
minimum hamming distance.
6. (2pt) For a code with 16 unique codewords and 12 bits per word, what is the maximum efficiency of this code?
7. (4pt) A channel has an approximate relationship Pe 
104
where Pe is the probability of bit errors and SNR is
SNR
signal power 
-6
the channel signal-to-noise ratio 
. For 0.5V NZ PCM signal pulses Pe = 2 x 10 . How should the
noise
power


pulse voltage be changed to improve Pe to4 x 10-7?

IV Digital Filter
‘09
Name _________________________
1. Below is the pole-zero diagram of a digital filter with fs = 36kHz.
1a. (3pt) What is the approximate cutoff frequency of the filter?
1b. (4pt) What type of filter is it (Butterworth, Elliptic,… & LPF, BPF,…)?
1c. (2pt) What is the order of the filter?
1d. (4pt) Sketch |H(ej)| vs. for 0 ≤  ≤ 2. Scale the x-axis in kHz.
IV Digital Filter
‘09
Name _________________________
1e. (3pt) Sketch the pole-zero diagram of the inverse filter where the inverse is the filter obtained by switching the
pass and stop bands around. The cutoff frequencies are not changed but frequencies that were previously passed
are blocked and those that were previously blocked are passed.
2. (4pt) For a 7th order Butterworth LPF with fs = 50kHz and a 2kHz cutoff, if |H(4kHz)| = -43dB, what is the
approximate attenuation at 8kHz (find |H(8kHz)|)?
V Interactive Graphics
‘09
Name _________________________
5000
, find the ideal impulse time response y(t)
s2
if the system is configured with a purely proportional feedback gain of -2.
1a. (5pt) Using the transfer function of just the DC motor G(s) 

1b. (2pt) What is the oscillation frequency of the feedback system in 1a?
1

1
exp  x 2  y 2 , and a second non-Gaussian uniform pdf f 2 (x, y)
 2

2
st
that is a flat-topped square region in the 1 quadrant with corners (0,0). (3,0), (0,3), and (3,3) shown below for
clarity. The height of the box is 1/9 and C1P1 = C2P2. Draw the decision boundary between the two probability
functions, looking down on the x-y plane. Scale the axes.


2. (7pt) Given a Gaussian pdf f1 (x, y) 
1
2
3 point (1.5,0.5)
3. (6pt) Given 3 Gaussian pdfs with parameters below, determine to which of the 3 functions the
0
most likely belongs and to which function it least likely belongs.
.
x0
y0
C
P
x y 
8
f1(x,y)
1
1
0
0
0
1
1/3
4
f2(x,y)
0.5 2
0
0
3
1
1/3
8
f3(x,y)
2.5 1.5 0.1 2
2
2
1/3
VI Hybrid Control
‘09
Name _________________________
ANSWER ANY COMBINATION OF QUESTIONS THAT ADD UP TO 20 POINTS.
1. (5pt) Which parameter or factor in a PID controller tends to dampen the response on a system whose pure
proportional response is very under damped?
i) KP
ii) KI
iii) KD
iv) the presence of a free integrator in the plant Gp(s)
2a. (3pt) TRUE or FALSE: A ripple free (FST) controller for a system with a very slow step response may be
slightly improved by introducing a small delay in the controller’s output to the plant.
2b. (2pt) TRUE or FALSE: Using a digital approximation to a PID controller maintains step and impulse
responses that are effectively identical to the step and impulse response of a purely analog controller.
3. (10pt) A PI Ziegler-Nichols controller for a process has Kp = 0.1 and Ki = 0.01. If the process is modified by
the addition of a pure delay that doubles L but leaves R unchanged, what will the new values of Kp and Ki be?
5
, find one equivalent expression for G(s) that is appropriate for a cascaded
(s  2)(s  3)
implementation on an analog computer and another appropriate for a parallel implementation.
4. (5pt) Given G(s) 

VI Hybrid Control
‘09
Name _________________________
5. (10pt) Find the FST controllers process pulse transfer function G p (z) for G p (s) 
0.05
when the
(s  0.5)(s  0.1)
sampling period is 0.5 s.


6. (10pt) Find the PD gains for a controller that minimizes the integrated time absolute error, given
1
G p (s) 
and setting the natural frequency response of the closed loop system to 5 rad/s.
(s  3)(s  4)

‘09
VII DC Motor Control
Name _________________________
1. (3pt) Which of the following can be phase lead compensators? Circle all that are correct. (p, z, &  > 0)
(s  z)
(s  z)
(s  2 p)
(10  s)
(s  z)
9 (s  3) 2
i)
ii) 
iii) 
iv) 0.5
v) 2
vi) 2
2
(s /  z)
(s  z)
(s  p)
(20  s)
(s  2z)
100 (s  10)





50s 12
s 2
2. (4pt) For the plant Gp (s)  2
and compensator Gc (s)  5
, what is the steady state error to an input
s  3s
s 10
r(t) = 2t (t ≥ 0)?


3. (5pt) What is the difference equation for an integrator obtained from the Tustin approximation with T = 10ms?
VII DC Motor Control
‘09
Name _________________________
4. (8pt) Find the gain K of a pure proportional controller that will place the closed loop poles of the plant
15
G p (s) 
on the  = 0.5 line in the s-plane.
(s  1)(s  2)
VIII Optimal Control
‘09
Name _________________________
1. (4pt) For a 2nd order system similar to that in the experiment with 5V initial conditions on x1(t), find the
equivalent impulse (height and pulse width with shortest possible duration) for ideal control, assuming the
controller output has a range of 15V.
2. (2pt) How does the value of the initial conditions (ICs) of a system affect the optimal Riccati LQR feedback
gains?
5
is a solution to the Riccati equation for the system with:
2
Ac  1, Bc  5, Cc  3, Dc  2, Q  5 13 , & R  3
3a. (5pt) Determine if P 

3b. (3pt) Simplify (but do not solve) the performance index for the system in 3a when tf = 100 and H = 1, but
there is no steady state error on x(t).
3c. (3pt) What is needed before the actual equation for the performance index can be evaluated numerically?
4. (3pt) TRUE or FALSE: The Riccati equations can be used to find the optimal gains for a LQR controlling a
time-varying continuous system but not a time-varying discrete system.