Download Examiners: Dr SS Singh / Dr DH Lawrence

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EO203 1 / 10
SCHOOL OF ENGINEERING
MODULAR HONOURS DEGREE COURSE
LEVEL 2
SEMESTER 2
2003/2004
SIGNAL THEORY AND APPLICATIONS
Examiners: Dr S S Singh / Dr D H Lawrence
____________________________________________________________________________
Attempt FOUR questions only:
Two from Section A and two from Section B
Time allowed: 3 hours
Total number of questions = 6
All questions carry equal marks.
The figures in brackets indicate the relative weightings of parts of a question.
Special requirement: Mathematical Formulae (B&C).
E0203
2 / 10
SECTION A
1) The circuit depicted in figure Q1 has a constant current source providing current I
in the direction shown. Switch S may be independently operated, causing current
flow to the components at the output R and C. Initially the switch S is in the
position shown forcing current I into the resistor r. Initially the capacitor C has no
charge on it and the switch S is then operated.
a) From the time that switch S is operated, write down the relationship between the
voltage Vs developed by the current source and the voltages VR and Vc across
the components R and C.
(2)
b) From the time that switch S is operated, write down the voltage current
relationship for the resistive component R and comment on the polarity of its
voltage?
(2)
c) From the time that switch S is operated, write down the voltage/current
relationship for the capacitive component C and comment on the polarity and
the rate of change of voltage across it. Comment on the ‘shape’ of the capacitor
voltage profile, giving a reason for its shape
(4)
d) Derive from first principles an expression for the voltage Vs developed across
the current source as a function of time if it is to provide a constant current to
the circuit. Ensure that the expression is a function of the magnitude of the
constant current source and includes the symbols R and C.
(12)
e) Sketch the profile of the current source voltage Vs labelling the axes and
including important features and their values.
Note: Laplace Transform of 1 is 1/s
Laplace Transform of t is 1/s2
Laplace Transform of dx/dt is -x(t=0) + sX(s)
(5)
E0203
3 / 10
S
Vc
C
I
Vs
r
R
VR
FIGURE Q1
I
S
R
C
r
Vc
VR
Vs
constant current
switch
resistor
capacitance
resistor
capacitor voltage
resistor voltage
voltage across current source
E0203
2)
4 / 10
The circuit in figure Q2 has an ideal operational amplifier. The input voltage to
the circuit is that shown in the diagram labelled, ‘Input Voltage Profile against
time’. The input voltage is a series of positive pulses of amplitude Vin.
a) Explain the function of the operational amplifier in the circuit and include in
your explanation reference to the voltages at both inputs to the operational
amplifier as well as its output voltage.
(3)
b) Write down the relationship between the input voltage Vin and the voltage across
the inductor VL.
Justify the relationship you have written with a detailed
explanation of why the relationship you have chosen is correct
(3)
c) Suggest what value the voltage across the resistor r should be? Give a complete
explanation for the value you have suggested. Assuming the value you have
suggested is correct, write down the relationship between the output voltage Vo
and the resistor voltage VR. Justify the relationship you have written with a
detailed explanation that gives the reason why the relationship is correct.
(4)
d) What is the relationship between the current Iin and If and give a full explanation
for the relationship you have put forward?
(2)
e) Derive from first principles, using Laplace transforms the expression for the
output voltage Vo in the circuit of figure Q2 as a function of time. Ensure that
the output voltage is fully defined in the time intervals when the input signal is
at zero volts between input voltage pulses.
(9)
f) Sketch a graph of Vo against time showing clearly the output voltage profile
over a number of successive positive input pulses.
(4)
E0203
Note:
Laplace Transform of 1 is 1/s
Laplace Transform of t is 1/s2
Laplace Transform of dx/dt is -x(t=0) + sX(s)
VR
If
Vin
L
r
R
Vo
VL
VR
Iin
If
A
R
VL
Iin
L
A
L
Vin
Vo
r
0V
Vin
Input Voltage Profile against time
volts

Vin
0
0
time
FIGURE Q2
input voltage
inductor
resistor
resistor
output voltage
inductor voltage
resistor voltage
inductor current
resistor current
operational amplifier
5 / 10
E0203
6 / 10
3) The triangular waveform of figure Q3 is the input signal to a low pass filter. The period of
the triangular waveform is 2T = 20msec and the peak value Vp is 100 mV. The input
signal is symmetrical about the voltage axis.
a) What is the DC output from the filter with such an input signal? Show that the value
you have proposed is correct by evaluating the DC component ao/2 using the
equations below.
(10)
b) Determine what the output amplitude will be of the 2nd harmonic if the filter has
a cut-off frequency that will attenuate the 2nd harmonic by a factor of two. Use
the Fourier equations below to derive and so prove your answer.
(15)
Note: For a function f(t) which has a period 2T the Fourier series is;
α
f t   a o /2   a n cos  b n sin nπ t/T  b n sin nπ t/T 
n 1
Eq.1.
The Fourier coefficients are given by:
T
b n  2/T  f t sin n π tT  dt
Eq.2.
0
T
a n  2/T  f t  cos n π tT  dt
0
Note also:
b
 cost dt = (1/)(sinb11 ’-‘ sina)
a
b
2
 t. cost dt = (1/)(b.sinb ‘-‘ a.sina)’+’ (1/ )(cosb ’-‘ cosa)
a
Eq.3.
E0203
7 / 10
filter input voltage
Vp
-2T
-T
0
Vp
FIGURE
Q3
FIGURE Q4
Time
T
2T
E0203
SECTION B
4) a) State the general mathematical definition for the Fourier transform of a function
f(t).
(1)
b) Sketch the graph of the function f1 (t)  e j t for 0    (2 t). What is the value
of this function when    (2t ) ? Determine the absolute magnitude of f 1 (t ) .
(3)


c) Sketch the graph of the function f 2 (t ) , defined by:
f 2 (t)  u(t)eat sin(  0 t)
where u(t) is the unit-step function, a is a positive constant and  0 is a constant.
(2)

d) Using the definition in part (a), determine the Fourier transform of the function
f 2 (t ) , defined in part (c).
(19)
8 / 10
E0203
5) a) Sketch the graph of the hyperbolic function f1(t)  sinh(  t) .
b) Prove that the function f1(t)  sinh(  t) is an odd function.
9 / 10
(2)
(2)

c) State the general mathematical definition
for the Laplace transform of a function f(t). (1)

d) Using the definition
in part (c), find the Laplace transform of the function
f1(t)  sinh(  t) ;
do not use a table of Laplace transforms. State any
conditions that apply to the validity of your result.

(8)
e) Denoting the Laplace transform of a function f(t) by F(s), state the general
expression for the Laplace transform of the following related function, in terms of
F(s):
f (t )
t
(2)
f) By using your result in part (d) for the Laplace transform of f1(t)  sinh(  t) , and
by exploiting the relationship stated in part (e), determine the Laplace transform
of the following function:

f 2 (t) 
sinh(  t)
t
Note: do not use the formal definition of the Laplace transform to ascertain this

Laplace transform.
(10)
E0203
6)
10 / 10
a) Study the following difference equation:
x[n  2] 2n  x[n 1]  n  1
Determine the order of this equation and hence explain whether it is recursive or
not. In addition,
explain whether it is a linear equation or not.

b) State the general mathematical definition for the ‘z’ transform of a sequence f[k].
(3)
(1)
c) By using the definition in part (b), determine the ‘z’ transform F1(z) for the
sequence defined by:
f 1 [k ]  A

where k ε {0, 1, 2, 3, 4, ...} and A is a constant. Express the transform in closed
form and specify the conditions for its convergence.
(7)
d) A fundamental and general property of the ‘z’ transform is presented below;
study it carefully:
If:
Z { f [k ]} F ( z ),
Then: Z{kf [k ]}   z


d
F( z)
dz
where Z denotes the ‘z’ transform operator, f[k] is the original sequence and F(z)
is the ‘z’ transform of f[k].
Using this property of the ‘z’ transform and the result you obtained in part (c)
for F1(z) , find the ‘z’ transform F2 (z) for the following sequence:
f 2 [k ]  Ak

where k ε {0, 1, 2, 3, 4, ...} and A is a constant. How many poles does

this ‘z’ transform possess?
(9)
e) Sketch the graph of the sequence f 2[k] defined in part (d). Also sketch the
graph of F2 (z) , i.e. the ‘z’ transform of f 2[k], and justify its overall shape by
selecting pertinent values of z and considering the resulting values of F2 (z) .




(5)