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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

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HW 3: SolutionsHW 3: Solutions

))(()())(()(:prove toneed we,invariance- timeprove To

txftytxfty

invariant- timeis system

)(

txdtdtx

tddty

txdtdtxfty

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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

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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)