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EE 235, Winter 2017, Homework 5: More LTI Systems
Due Friday February 3, 2017 via Canvas Submission
Write down ALL steps for full credit
HW5 Topics:
â€¢ Impulse Response and Step Response
â€¢ LTI System Properties
â€¢ Exponential Response
HW5 Course Learning Goals Satisfied:
â€¢ Goal 1: Describe signals in diï¬€erent domains (time, frequency, and Laplace) and map characteristics
in one domain to those in another
â€¢ Goal 4: Analyze LTI systems given diï¬€erent system representations (including input-output equations
and impulse response).
â€¢ Goal 6: Use and understand standard EE terminology associated with LTI systems (e.g. impulse
response, step response, exponential response).
HW5 References:
â€¢ OWN Sections 2.3, 3.2
HW5 Problems (Total = 74 pts):
1. Review.
(a) Complex Numbers.
1
(2 pts) Let X = Î±âˆ’jÏ‰
, where Î± &gt; 0. Evaluate the magnitude and phase of X. Show that
1
|X| = âˆšÎ±2 +Ï‰2 and âˆ X = arctan( Ï‰Î± ).
(b) Partial Fraction Expansion.
s+8
(2 pts) Let H(s) = s2 +7s+10
. Using the cover-up method for partial fraction expansion (discussed
2
1
in the supplementary notes), write H(s) as a sum of partial fractions. Show that H(s) = s+2
âˆ’ s+5
.
(c) Partial Fraction Expansion.
1
. Using the cover-up method for partial fraction expansion (dis(5 pts) Let H(s) = s(s+2)(s+3)
cussed in the supplementary notes), write H(s) as a sum of partial fractions.
(d) System Properties.
(5 pts) Show whether the following system is linear, time-invariant, causal, stable and invertible?
âˆ« +âˆž
y(t) = âˆ’âˆž Î´(Ï„ âˆ’ x(t + Ï„ ))dÏ„
(e) Convolution.
(5 pts) Let T denote an LTI system with impulse response p(t + 1) âˆ’ 2p(t âˆ’ 1), then find and
sketch y(t) = T [p(âˆ’t/2)].
Hint: The pulse signal p(t) = u(t) âˆ’ u(t âˆ’ 1).
2. LTI System Properties.
(10 pts) For the LTI systems below, determine from the impulse response whether the system is causal
and/or stable:
a) h(t) = eâˆ’2t u(t + 2)
b) h(t) = et u(1 âˆ’ t)
c) h(t) = r(t)p(t/10)
Hint: The ramp signal r(t) = t Â· u(t). The pulse signal p(t) = u(t) âˆ’ u(t âˆ’ 1).
d) h(t) = sin(t)u(t)
e) h(t) = t 1 3. Impulse and Step Response.
Consider the following LTI systems: {
1, âˆ’1 &lt; t &lt; 1
â€¢ T1 : Has impulse response h1 (t) =
0, otherwise
â€¢ T2 : Has step response s2 (t) = 3u(t âˆ’ 1)
â€¢ T3 : Has input-output relationship y3 (t) = x3 (t + 2) (a) (5 pts) What is thestep response s1 (t) of system T1?
0,
t &lt; âˆ’1 Show that s1 (t) = t + 1, âˆ’1 &lt; t &lt; 1 2,
t&gt;1
(b) (2 pts) What is impulse response h2 (t) of system T2? Show that h2 (t) = 3Î´(t âˆ’ 1).
(c) (2 pts) What is the corresponding input-output relationship for system T2? Show that y2 (t) =
3x2 (t âˆ’ 1).
(d) (5 pts) What is the impulse response h3 (t) of system T3? What is the corresponding step response
s3 (t)? Show that h3 (t) = Î´(t + 2) and s3 (t) = u(t + 2).
4. LTI System Interconnections.
Consider the same LTI systems in Problem 3. Answer the following questions.
(a) (2 pts) Is system T2 BIBO stable?âˆ«Using the impulse response test that we discussed in lecture,
âˆž
show that T2 is BIBO stable with âˆ’âˆž |h(t)|dt = 3.
(b) (5 pts) Suppose T1 and T3 are connected in parallel. Is the overall system causal? Use the impulse
response test that we discussed in lecture.
(c) (5 pts) Suppose T2 and T3 are connected in parallel. Find the overall impulse response h(t) and
then find the overall output y(t) when the input is x(t) = cos(t2 ).
(d) (5 pts) Suppose T1 and T3 are connected in series. Is the system causal? Is the system BIBO
stable? Use the impulse response tests.
5. LTI Systems and Exponential Response.
(a) (2 pts) Suppose an LTI system has input-output relationship y(t) = 3x(t + 2). What is its
corresponding transfer function H(s). Show that H(s) = 3e2s .
(b) (2 pts) Using the same system in (a), find the output y(t) using the complex exponential response
method as discussed in lecture for the input x(t) = cos(5t). Show that y(t) = 3 cos(5t + 10).
(c) (5 pts) Repeat (b) for x(t) = ej2t + 2 cos2 (t). Hint: cos2 (Î¸) = 1
2 + 1
2 cos(2Î¸) and 1 = ej0t . (d) (5 pts) The signal x(t) = 2e3t is input to an LTI system. One fact about the output is known:
y(ln(2)) = 8. What is the value of the output at time t = 1, e.g. y(1)? Hint: eln(b) = b.
EXPECTATIONS FROM YOU FOR HW5:
1. Prove whether an LTI system satisfies a particular property (causal or stable) using the impulse response h(t).
2. Understand the relationship between the impulse response and step response.
3. Understand properties of the impulse response when LTI systems are interconnected in series/cascade
or parallel.
4. Understand and apply the definition of the exponential response for LTI systems. 2

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