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(ELEC211)00fallmid.pdf
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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)

(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)

(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)

(d)
The magnitude of H ( jw) does not contain any information about the group delay. Is this statement true or false? Why? (1 mark)


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).

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

(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 jt
(d) Sketch the Fourier Transform of the signals e p and e p for w = 0 to 5p. (2 marks)
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)
Question 4: (5 marks) A causal LTI system is described by the differential equation dy(t) dx(t)
+ay(t) =-+ bx(t) where a and b are two different non-zeroes real numbers.
dt dt
(a)
Find the relationship between a and b such that the frequency response of the system satisfies | H ( jw)|= 1for allw . ( 1 mark)

(b)
Sketch | H ( jw) | . (1 mark)

(c) Drive the impulse response (1 mark)

(d)
Find and plot the output of this system when the input is x(t) = e -tu(t) . (2 marks)


Question 5: (4 marks) Use the results in Tables 3.1, 3.2, 4.1, 4.2, 5.1 and 5.2 to do the following tasks. Anyone who uses the basic definition of FS, FT, Inverse FS and IFT will result in zero mark.
sin(3pn / 4)
(a) Compute y[n] , where y[n] = x1[n]* x2[n], x1[n] = and
pn sin(pn / 2)
x2[n] = . (2 marks)
pn
sin(Wt)
(b) Compute the total energy of x(t), where x(t) =cos(w0t) and w0 >> 2W. (2
pt
marks)
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