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HW 3: Solutions. The output of a particular system S is the time derivative of its input. Prove that system S is linear time-invariant (LTI). Solution:. HW 3: Solutions. HW 3: Solutions. What is the unit impulse response of this system? Solution:. d/dt ( (t)). (t). 1/. 1/ 2. - PowerPoint PPT Presentation
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HW 3: SolutionsHW 3: Solutions1.1. The output of a particular system S is the time The output of a particular system S is the time
derivative of its input.derivative of its input.
a)a) Prove that system S is linear time-invariant (LTI).Prove that system S is linear time-invariant (LTI).
Solution:Solution:
linear is system
Then,
Let
21
21
2121
txBftxAf
txdtdBtx
dtdA
tBxtAxdtdtBxtAxf
txdtdtxfty
2
HW 3: SolutionsHW 3: Solutions
))(()())(()(:prove toneed we,invariance- timeprove To
txftytxfty
invariant- timeis system
)(
txdtdtx
tddty
txdtdtxfty
3
HW 3: SolutionsHW 3: Solutionsb)b) What is the unit impulse response of this system?What is the unit impulse response of this system?
Solution:Solution:
tt
(t)(t)1/1/
tt
d/dt (d/dt ((t))(t))1/1/22
-1/-1/22
Limit as Limit as tends to tends to 00
unit impulseunit impulse unit impulseunit impulseresponseresponse
4
HW 3: SolutionsHW 3: Solutions2.2. Prove Property 5. Prove Property 5. That is, prove that, for an arbitrary LTI That is, prove that, for an arbitrary LTI
system, for a given input waveform system, for a given input waveform x(t)x(t), the time , the time derivative of its output is identical to the output of that derivative of its output is identical to the output of that system when subjected to the time derivative of its inputsystem when subjected to the time derivative of its input. . In other words, differentiation on the input and output In other words, differentiation on the input and output sides are equivalent.sides are equivalent.
Solution:Solution: Follows from Problem 1, and commutativity of Follows from Problem 1, and commutativity of convolution.convolution.
Arbitrary Arbitrary
LTI LTI systemsystem
d/dtd/dtx(t)x(t) y(t)y(t) y’(t)y’(t) Arbitrary Arbitrary
LTI LTI systemsystem
d/dtd/dtx(t)x(t) x’(t)x’(t) y’(t)y’(t)
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