Mathematical proof for non linear system

  • #1

Homework Statement


Consider the continuous-time processing system in figure 1, which has two inputs and one output. The linear sub-system H is characterised by the impulse response h(t) = e −2t , where t denotes time.

Block diagram is the product of x1(t) and x2(t) going through a block step of h(t) (exp-2t)

Give a mathematical proof that this is a nonlinear system.

Question attached in picture.

Homework Equations


α1y1(t) + α2y2(t) = H {α1x1(t) + α2x2(t)} , condition for linear stability

where x1(t) and x2(t) are two input signals and y1(t) and y2(t) are the corresponding output signals for system H

The Attempt at a Solution



Would my equations become

y1(t) = x1(t)e(-2t)

y2(t) = x2(t)e(-2t)

then calculating one way

y(t) = α1x1(t)e(-2t) + α2x2(t)e(-2t)

then the other

y(t) = e(-2t)[α1x1(t) + α2x2(t)] = α1x1(t)e(-2t) + α2x2(t)e(-2t)

But that would suggest linearity... Is there something I'm missing?
 

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Answers and Replies

  • #2
epenguin
Homework Helper
Gold Member
3,898
926

Homework Statement


Consider the continuous-time processing system in figure 1, which has two inputs and one output. The linear sub-system H is characterised by the impulse response h(t) = e −2t , where t denotes time.

Block diagram is the product of x1(t) and x2(t) going through a block step of h(t) (exp-2t)

Give a mathematical proof that this is a nonlinear system.

Question attached in picture.

Homework Equations


α1y1(t) + α2y2(t) = H {α1x1(t) + α2x2(t)} , condition for linear stability

where x1(t) and x2(t) are two input signals and y1(t) and y2(t) are the corresponding output signals for system H

The Attempt at a Solution



Would my equations become

y1(t) = x1(t)e(-2t)

y2(t) = x2(t)e(-2t)

then calculating one way

y(t) = α1x1(t)e(-2t) + α2x2(t)e(-2t)

then the other

y(t) = e(-2t)[α1x1(t) + α2x2(t)] = α1x1(t)e(-2t) + α2x2(t)e(-2t)

But that would suggest linearity... Is there something I'm missing?

You seem to be missing totally that the input to H is g(t) and inventing two outputs y when there is only one.

Seem. Thisnis not my field but it sure seems.
 
  • #3
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,722

Homework Statement


Consider the continuous-time processing system in figure 1, which has two inputs and one output. The linear sub-system H is characterised by the impulse response h(t) = e −2t , where t denotes time.

Block diagram is the product of x1(t) and x2(t) going through a block step of h(t) (exp-2t)

Give a mathematical proof that this is a nonlinear system.

Question attached in picture.

Homework Equations


α1y1(t) + α2y2(t) = H {α1x1(t) + α2x2(t)} , condition for linear stability

where x1(t) and x2(t) are two input signals and y1(t) and y2(t) are the corresponding output signals for system H

The Attempt at a Solution



Would my equations become

y1(t) = x1(t)e(-2t)

y2(t) = x2(t)e(-2t)

then calculating one way

y(t) = α1x1(t)e(-2t) + α2x2(t)e(-2t)

then the other

y(t) = e(-2t)[α1x1(t) + α2x2(t)] = α1x1(t)e(-2t) + α2x2(t)e(-2t)

But that would suggest linearity... Is there something I'm missing?

Yes, you are missing the effects of "g".
 

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