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# CTSS Stability Debate Problem

Module by: John Buck. E-mail the author

Summary: Undergraduate signals and systems module examining students understanding of the stability criteria for LTI systems

Two Signals and Systems students, Ty Minvariant and Lynn Near are arguing about whether the LTI system with impulse response h(t)=u(t-3)h(t)=u(t-3) is stable. You must settle the argument. Ty claims that the system is stable. He says

An LTI system is stable if |h(t)|<|h(t)|<, i.e., the magnitude of the impulse response is bounded. Since |u(t-3)|1|u(t-3)|1 for all tt, the system satisfies the conditions for stability.

Lynn claims that the system is not stable. She says

If the input to a stable system is bounded, i.e., |x(t)|B|x(t)|B, the output must also be bounded, i.e., |y(t)|C|y(t)|C for some constant CC. If I set x(t)x(t) to be a unit step u(t)u(t), I get the convolution
y ( t ) = - h ( τ ) x ( t - τ ) d τ = - u ( τ - 3 ) u ( t - τ ) d τ If t < 3 , this product is 0, so y ( t ) = 0 . For t > 3 = 3 t u ( τ - 3 ) u ( t - τ ) d τ = 3 t 1 d τ = t - 3 y ( t ) = - h ( τ ) x ( t - τ ) d τ = - u ( τ - 3 ) u ( t - τ ) d τ If t < 3 , this product is 0, so y ( t ) = 0 . For t > 3 = 3 t u ( τ - 3 ) u ( t - τ ) d τ = 3 t 1 d τ = t - 3
This is a case where a bounded input produces an unbounded output. Since y(t)y(t) grows forever, it cannot be bounded by a constant CC. Since a bounded input produces an unbounded output, the system cannot be stable.

1. Which student is right, Ty or Lynn?
2. In three or fewer sentences, explain what the mistake is in the argument made by the student who is not correct.

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