Sample Question Paper for 9210-125
Graduate Diploma in Electronic and Telecommunication
Engineering
Signals and systems
Duration: three hours
You should have the
following for this examination
• one answer book
• non-programmable calculator
• pen, pencil, drawing
instruments
The following data is
attached
• Fourier Transform tables
General instructions
• This paper consists of nine questions.
• Answer any five questions.
• A non-programmable electronic calculator may be used but candidates must show
sufficient steps to justify their answers.
• Drawings should be clear, in good proportion and in pencil. Do not use red ink.
• All questions carry equal marks. The maximum marks for each section within a
question are shown.
1
Consider the continuous time signals
π
x1(t ) = 4cos(5π t), x2(t) = 4cos 5π t – 4 ,
and the discrete time sequences
(
)
x1[n] = 4cos(π n), x2[n] = 4cos (π n – 2).
a)
b)
c)
d)
e)
2
Determine analytically if the following continuous time signals are periodic by
calculating their fundamental period.
i)
x1(t)
(2 marks)
ii)
x2(t)
(2 marks)
iii)
x1(t) + x2(t)
(2 marks)
Determine analytically if the following discrete time sequences are periodic by
calculating their periodic sequence length.
i)
x1[n]
(2 marks)
ii)
x2[n]
(2 marks)
iii)
x1[n] + x2[n]
(2 marks)
Identify a sinusoidal signal x3(t) such that x1(t) + x3(t) is not periodic.
State if the step function u(t) is periodic. Justify your answer.
State if x1(t)u(t) is periodic within any range of t. If it is periodic within some range
of t state the range.
(2 marks)
(3 marks)
(3 marks)
A periodic waveform x(t) is shown in Figure Q2. Its trigonometric Fourier series is given by
⬁
x(t) = a0 + ∑ (an cos(nω 0t) + bn sin(nω 0t))
n=1
where the parameters have their usual meaning.
Figure Q2
a)
b
c)
d)
e)
Estimate the value of a0 in the Trigonometric Fourier series of x(t) giving reasons.
Estimate the values of bn coefficients of the Trigonometric Fourier series of x(t)
giving reasons.
Explain if the coefficients of the complex exponential Fourier series of x(t) are real,
imaginary, or complex giving reasons.
Identify if x(t) has half wave odd symmetry. Justify your answer.
Using the answers in parts a) to d) derive the Fourier series coefficients of x(t).
2
(3 marks)
(3 marks)
(3 marks)
(3 marks)
(8 marks)
3
a)
b)
c)
d)
Verify that if the Fourier transform of x(t) is given by X( ω ), then the Fourier
transform of x(t – t0) is given by e–jω t0 X(ω ).
Compute the Fourier transform of the step function u(t).
Deduce the Fourier transform of the shifted step function u(t – t0).
Using the results of part b) and c) compute the Fourier transform of the waveform
given in Figure Q3.
(2 marks)
(4 marks)
(2 marks)
(12 marks)
Figure Q3
4
Consider the continuous time linear time invariant (LTI) system described by
dy(t)
dt + 4y(t) = 2x(t).
a)
b)
c)
d
Y(ω )
Find the system transfer function H (ω ) = X (ω ) , where X(ω ) and Y(ω ) are the Fourier
transforms of x(t) and y(t) respectively.
Find the impulse response h(t) of the system where H (ω ) is the Fourier transform
of h(t).
Using the result in part b) and the convolution integral, find the response y(t) for
the input signal x(t) = e–2tu(t).
Using the result in part a) and Fourier transform techniques, find the response
y(t) for the input signal x(t) = u(t).
3
(2 marks)
(2 marks)
(8 marks)
(8 marks)
See next page
5
A Linear Time Invariant (LTI) system has the input signal x(t) and an output signal y(t).
The Fourier transform of x(t) and y(t) are X(f ) and Y(f ) respectively where,
X(f ) = Π(f )ejπ f + Π(f )e –jπ f
and
Y(f ) = X(f – f 0 ) + X(f + f 0 ).
Note that Π(f ) is the square window defined by
1, if | f | < 0.5
Π(f ) = 0, otherwise
{
a)
b)
c)
Use Fourier transform theorems to find x(t).
Use Fourier transform theorems to find y(t).
Figure Q5 shows a simple modulation circuit. Design the system by selecting the
parameters A, f1, f2, and W such that the output is y(t).
(5 marks)
(5 marks)
(10 marks)
Figure Q5
6
In the circuit given in Figure Q6, the switch is in the closed position for a long time
before it’s opened at t = 0.
Assume that L = 0.25 H, R1 = 10 Ω, R2 = 5 Ω, and C = 0.1 F.
Figure Q6
a)
b)
c)
d)
Describe the implications of having the switch closed for a long time.
Compute the initial conditions associated with the inductor and capacitor at t = 0.
Construct the Laplace equivalent circuit at t = 0.
Develop an expression for i(t) for t ≥ 0.
4
(2 marks)
(3 marks)
(5 marks)
(10 marks)
7
a)
Use the graphical convolution method to find the output signal
y(t) = x(t) * h(t), where
x(t) = e–5t u(t) and h(t) =Π t .
2
( )
Note that Π t is the square window defined by
2
( )
1, if | t | < 1
t
Π 2 = 0, otherwise
Use the convolution summation to find the response sequence y[n] = x[n] * h[n]
of a discrete time system where
{
( )
b)
h[n] = δ [n] – δ [n – 3] and
1, 0 ≤ n ≤ 3
x[n] =
0, n < 0 or n > 3
{
8
9
(10 marks)
Consider a discrete time LTI system whose input x[n] and output y[n] are related by
y[n] – 0.5y[n–1] = x[n] with y[–1] = 0.
a) Derive the impulse response h[n] of the system.
b) Find the output y[n] for the input x[n] = (0.5)nu[n].
c) Compute the step response sequence s[n] for the system.
d) What would be the steady state value of the system response when n goes
to infinity.
a)
b)
c)
(10 marks)
For a distortionless communication channel the relationship between the input
and the output is given by y(t) = Kx(t – t0), where K and t0 are constant values.
If x(t) = 4 cos(20π t) + 8sin(10π t), sketch the input x(t) and the output y(t) on the
same graph.
Note: the proportions of the graph can be approximate.
Sketch the Bode magnitude and phase diagrams for a distortionless
transmission system.
If the signal x(t) is sent through a channel with the magnitude and phase
characteristics shown in Figure Q9, sketch the output y(t).
(5 marks)
(8 marks)
(4 marks)
(3 marks)
(5 marks)
(5 marks)
(5 marks)
Figure Q9
d)
Compare the results obtained in part a) and c) and comment.
5
(5 marks)
CITY & GUILDS
Data Attachments
Fourier Transform Table
9210-125