SIGNALS & SYSTEMS
5CCE2SAS
COURSEWORK 2

Dr. M. R. Nakhai, Department of Engineering,
March 2022

There are 4 Questions, answer all.
Detailed answers and careful sketches are required for
each one of the questions.

Upload clearly scanned copies of your written answers by
the deadline, as indicated on Keats.



1
Question one (total: 30 marks):

The Fourier transform of a continuous-time signal (), i.e., (), is given as

() = ( −

3
) + 2 ( −

7
)

a) Find the corresponding time-domain signal ().
(10 marks)

b) Given that the signal () in part (a) is a periodic function of time, find
the fundamental (radian) frequency and the Fourier series coefficients of
the signal ().
(10 marks)

c) Determine how (), i.e., the Fourier transform of (), given above,
can be derived using its fundamental (radian) frequency and the Fourier
series coefficients you have found in part (b).
(10 marks)





Question two (total: 20 marks):

The impulse response of a linear time invariant (LTI) system is given as

ℎ() =
sin(32)

−
sin(16)



a) Sketch (with careful labelling) the frequency response, i.e., (), of this
system.
(10 marks)


b) Find the output () of this system, in terms of cosine functions, to the
input
() = −35 + −30 + −20 + −15 + 1 + 15 + 20 + 30 + 35

(10 marks)





2
Question three (total: 25 marks):

The impulse response of a linear time invariant (LTI) system is given as ℎ() in
Figure Q3:

Figure Q3: The impulse response

a) Find and sketch with careful labelling the output signal () when the input
signal () is: (6 marks)
() = ( − 3) + ()

b) For ℎ(), as given in Figure Q3, sketch with careful labelling ℎ1():
(6 marks)
ℎ1() = ℎ(−
1
2
− 1)

c) Is the system with impulse response ℎ1() that you found in part (b) causal?
Why? (5 marks)

d) Let the impulse response ℎ2() of a LTI system be given as
ℎ2() = ℎ1( − 10), where ℎ1() is the same signal you found in part (b).
Find and sketch with careful labelling the output 2() of this system to the
input 2(), given by
2() =
1
2
[( − 2) − ()]
(8 marks)













3
Question four (total: 25 marks):

The signal () as shown in Figure Q4(a) is applied to a linear time invariant (LTI)
system with impulse response ℎ().









a) Find and sketch carefully with labelling the output signal (), when the
impulse response of this LTI system is given as:

ℎ() = () − ( − 1)
(10 marks)

b) Determine and sketch carefully with labelling the output signal (), when
the impulse response of this LTI system is as shown in Figure Q4(b):

(10 marks)










c) Is this system as described in part (b) memoryless? Why?
(2 marks)

d) Is this system as described in part (b) stable? Why?
(3 marks)




()
t
1 2
1
0
Figure Q4(a): Input signal
ℎ()
t
1
1
0
Figure Q4(b): Impulse response

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