330M ASE Linear System Analysis
Unique Number: 12495, Spring 2006
Homework #7
Date given: April 27, 2006
Date Due: May 4, 2006
1. An LTI system has a step response (et cos2 t)u(t). Determine an ordinary dierential equation representation for this system.
2. An LTI system at zero initial conditions is subject to an input signal x(t) = [sin(2t)]u(t). The
resulting output is designated by y (t) = [4et + 2 sin(2t) 4 cos(2t)]u(t).
(a) Determine an ordinary dierential equation representation for this system.
(b) Determine whether the system is BIBO stable.
(c) Depending on your answer in part (b), evaluate (if possible) the frequency response
function H (j ) for this system. Further, obtain expressions for the magnitude and
argument (phase angle) of the frequency response function.
(d) Analytically determine the frequency = max at which the magnitude of the frequency
response function has a maximum value. Also calculate the magnitude of the frequency
response function at frequency max .
(e) Use MATLAB commands impulse and step to plot the impulse and step responses for
this system on the range 0 t 30. Submit both the plots and your source code.
3. An LTI system has transfer function
H (s) =
s2 + 2s
(s2 + 3s + 2)(s2 + s 2)
(a) Determine an ordinary dierential equation representation for this system.
(b) Determine whether the system is BIBO stable.
(c) Depending on your answer in part (b), evaluate (if possible) the frequency response
function for this system. Further, obtain expressions for the magnitude and argument
(phase angle) of the frequency response function.
4. The transfer function of an LTI system is described by
H (s) =
sa
,
s+a
a>0
where is a a real constant.
(a) Determine whether the system is BIBO stable. What happens when a 0?
(b) Sketch the pole-zero diagram for this system.
(c) Do you notice any symmetry in the pole-zero diagram about the imaginary axis?
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(d) Evaluate the frequency response function for this system. Further, prove that the magnitude of the frequency response function is unity for all frequencies.
(e) Comment on the magnitude of the frequency response function for the general case where
any BIBO stable LTI system has n number of zeros and n number of poles having a
symmetry about the imaginary axis. Any system with such properties is known as an allpass system. Remember, since the so called all-pass system is stable, all poles are located
on the left half of the complex plane and accordingly, all the zeros are (symmetrically)
placed on the right-half of the complex plane.
5. A second-order causal LTI system with non-zero initial conditions is subject to an input x1 (t) =
u(t). The resulting output is determined by y1 (t) = 1 + e2t for all t 0 . From the same
initial conditions, when an input x2 (t) = [sin(t)]u(t) is applied, the system produces an
output y2 (t) = 2 cos(t) sin(t) for all t 0 .
(a) Derive the transfer function of this system.
(b) Calculate the non-zero initial conditions of this system.
(c) Now, suppose the same system is set back to zero initial conditions and an input x(t) =
[sin(t)]u(t) is applied [same as x2 (t)!]. Use MATLAB lsim command to generate the
output of the system on 0 t 50. Graphically compare this result with y2 (t) that
was produced as output of the same system. Explain why there is a dierence between
these two quantities.
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