ELEC270 SIGNALS AND SYSTEMS FIRST SEMESTER EXAMINATIONS 2019/20
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ELEC270
FIRST SEMESTER EXAMINATIONS 2019/20
SIGNALS AND SYSTEMS
1. Consider the periodic square wave signal f(t) shown in Figure Q1 below .
Figure Q1
a) Express f(t) as a Fourier Series expansion.
b) Sketch the magnitude of the first six spectral components , i.e . a! , b! and |c! | = +a!(2) + b!(2) for n = 0 to 6 .
c) Find the average power off(t) .
d) How many harmonics would be required to retain at least 90% of the signal’s power .
e) Design a suitable filter that would block the fifth and higher harmonics (i.e n ≥ 5). Your design should include the type of filter, appropriate cut-off
frequency (or frequencies) and a frequency domain plot.
Total 25
2. The potential difference v(t) across a capacitor as it charges to Vo Volts and subsequently discharges through an RC circuit is given by:
vc (t) = Vo 41 − e__
vd (t) = Vo e τd
where vc (t) and vd (t) are the potential differences during charging and
discharging, respectively, and τc and τd represent the respective charging and
discharging time constants .
An experiment is carried out in which the capacitor begins to charge at t = 0 to a voltage Vo = 10 V, and is then made to discharge at t = 4 s; it is left to discharge for a long time. Both time constants are set to 0.5 s .
a) Write an expression for ?(?), and plot the signal from ? = 0 to ? = 8 s.
b) Find V(幼), the Fourier Transform of v(t).
c) Find the energy of the signal v(t).
d) Find 
凯 v(t) δ(t − 5) dt and comment on what this value represents .
e) State Parseval’s Theorem and verify it for the signal f(t) = e__atu(t) given that e__atu(t) and a +1j幼 form a Fourier pair (a > 0).
Hint: ∫ 
dx = 
tan__1 
Total 25
|
3. |
The input x(t) and impulse response ℎ(t) of a continuous time LTI system are given by x(t) = u(t) ℎ(t) = e%atu(t), a > 0 a) Find the output of the system, y(t). b) Find the system step response s(t). c) Find the system transfer function H(s). d) Another (different) system has the following transfer function |
|
G(s) = 
+ s2 
49
Find the impulse response of this system .
e) x(t) and x(幼) shown below form a Fourier pair. The signal x(t) is sampled 5
at regular time intervals , at a sampling rate greater than the Nyquist rate .
Sketch the spectrum of the sampled signal , and comment on this phenomenon .
Figure Q3-2
Total 25
4. A discrete-time system has the following difference equation:
y[n] = 2x[n] + x[n − 2] − 0.5y[n − 1]
a) Draw a block diagram for the above system.
b) Find the transfer function H(z).
c) Plot the poles and zeros on a pole-zero diagram, and comment on the system stability.
d) The input to the above system is given as
x [n] = {1, 2, 0, 3, 0, 0, ⋯ }
Find the output Y(z) of the above system to the input given.
e) Find the inverse z-transform of G(z) given below and suggest a suitable region of convergence.
G(z) = − 3z
Total 25
2023-09-04