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###### 日期：2022-05-09 09:16

SIGNALS & SYSTEMS

5CCE2SAS

COURSEWORK 2

There are 4 Questions, answer all.

Detailed answers and careful sketches are required for

each one of the questions.

the deadline, as indicated on Keats.

Question one (total: 30 marks):

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

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).

Question two (total: 20 marks):

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

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

Question three (total: 25 marks):

The impulse response of a linear time invariant (LTI) system is given as ?() 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 ?(), as given in Figure Q3, sketch with careful labelling ?1():

c) Is the system with impulse response ?1() that you found in part (b) causal?

Why? (5 marks)

d) Let the impulse response ?2() of a LTI system be given as

?2() = ?1( ? 10), where ?1() 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(),

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 ?().

a) Find and sketch carefully with labelling the output signal (), when the

impulse response of this LTI system is given as:

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)