Homework 4: Systems and Signals

Homework 4
ECE 102: Systems and Signals

1. Linear Differential Equations with Laplace Transforms
A system S has the following IPOP:
3
d
2y(t)
dt2
+ 19
dy(t)
dt + 20y(t) = 2dx(t)
dt − x(t), t ≥ 0
y
′
(0) = y(0) = 0, x(0) = 0
(a) Find the transfer function of S. Is the system stable?
(b) Find the output given input: x(t) = e
1
2
(t−3)u(t − 3)
2. Block diagram representation of LTI systems
Consider the system S characterized by the differential equation
d
3y(t)
dt3
+ 6
d
2y(t)
dt2
+ 11
dy(t)
dt + 6y(t) = 2d
2x(t)
dt2
− 14dx(t)
dt − 16x(t).
(a) Draw a block diagram realisation of the system using integration and differentiation blocks.
(b) Draw the pole-zero constellation of system S and comment on its stability.
(c) Find the output when x(t) = u(t + 2) − u(t − 2) is applied as input.
3. Transfer function from system bock diagram
Consider the system S whose input output relation are shown by the block diagram below:
(a) Find the transfer function H(s) for the system.
(b) Express the system using a differential equation in x(t) and y(t).
(c) Compute the inverse Laplace transform of e
−4sH(
s
3 − 4).
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4. Fourier series representation
Fourier proposed to represent a periodic signal as a sum of sinusoids, perhaps an infinite number
of them. For instance, consider the representation of a periodic signal x(t) as a sum of cosines of
different frequencies
x(t) = X∞
k=0
Ak cos (Ωkt + θk)
(a) If x(t) is periodic of period T0, what should the frequencies Ωk be?
(b) Consider x(t) = 2 + cos(2πt) − 3 cos(6πt + π/4). Is this signal periodic? If so, what is
its period T0 ? Determine its trigonometric Fourier series as given above by specifying the
values of Ak and θk for all values of k = 0, 1, . . .
(c) Let the signal x1(t) = 2 + cos(2πt)−3 cos(20t+π/4) (this signal is almost like x(t) given
above, except that the frequency 6πrad/sec of the second cosine has been approximated by
20rad/sec ). Is this signal periodic? Can you determine its Fourler series as given above by
specifying the values of Ak and θk, for all values of k = 0, 1, . . . ? Explain.
5. Matlab assignment:
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