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EE211 Signals and Systems

Middle Term Exam

Nov 11, 2000 (Sat., Week 10) Requirements: This exam is worth of 25% of the total marks of this subject. Do all questions. This is an open book exam, but the students only need the textbook, the lecture notes, matlab notes and tutorial notes. No any electronic device will be allowed.
Duration: exam time ( 2:00 pm - 4:00 pm )
Students shall come 5 minutes early to write down their name and student ID number.
Question 1 (4 marks) Judge the following statements to see whether it is true or false and
give the reason.
(a)Any periodic CT signal can be represented as sum of a set of sine waveforms. Is this
statement true or false? Why?(1 mark)

(a) False, convergence conditions
(b)Fourier transform of any real signal is real. So we can build up a device (spectrum analyser) to display the spectrum of any real signal easily. Is this statement true or false? Why? (1 mark)
False, FT[real signal]=complex spectrum in many cases. What showed on spectrum analyser is the magnitude of the spectrum.
(c) The impulse response of any cable which can pass a CT signal comprising of any frequency component without any loss is d (t) . Is this statement true or false? Why? (1 mark)
Yes, FT[d (t) ]=1 an all-pass filter.
(d) The magnitude of H ( jw) does not contain any information about the group delay. Is this
statement true or false? Why? (1 mark) Yes, delay is in the phase of H ( jw).
Question 2 (8 marks) Do the following simple tasks.
(a) Sketch the waveform sin(2pt) for t=0 to 2 and determine the period of this signal. Please label the axes clearly. (2 marks).

Period = 1

(b) Sketch the waveform sin(2pn / 5) for n=0 to 10 and determine the sampling frequency. Please label the axes clearly. (2 marks)

sin(2pn / 5) = sin(2ptt)
==> the sampling period is 1/5
t=n /5

==> sampling frequency = 10p rad/s
(c) Sketch the waveform ejpn for n = -3 to 3. Determine and sketch the Fourier series coefficients, ak, of this signal for k = 0 to 7. Please label the axes clearly (2 marks).
jn

e p
The waveform :


Fourier series coefficients of this signal:

jn jt

(d) Sketch the Fourier Transform of the signals e p and e p for w = 0 to 5p. (2 marks) Fourier Transform of the signals ejpn :


Fourier Transform of the signals ejpt :


Question 3: (4 marks) Using equations (5.8) and (5.9) in the textbook to prove that
FT (x[n]* h[n]) = FT (x[n]) FT (h[n]) , where FT(.) denotes Fourier Transform, * denotes the convolutional sum operation. (4 marks)
DTFT jw DTFT jw DTFT jw
Denote y[n]....Y (e ), x[n].... X (e ) and h[n].... H (e ) Consider the convolution sum
y[n] = x[n]* h[n] = . x[k]h[n -k]
k =-

We desire Y (ejw ), which is

jw . .-jwn
Y (e )=. . x[k]h[n -k] e
..
n=- ok =- .
. -jwn .
=. x[k]..h[n -k]e
.
k=- on=- .
jw -jwk
= . x[k]H (e )e
k=-
jw -jwk
= H (e ) . x[k]e
k =-
jw jw

= X (e )H (e )

Quest