9.16. A causal LTI system S with impulse response h(t) has its input x(t) and output y(t) related through a linear constant-coefficient differential equation of the form dy(t) dt d³ y(t) d13 + (1 + α) d² y(t) dt² + a(a + 1). + a² y(t) = x(t). (a) If g(t) = dh(t) + h(t), dt how many poles does G(s) have? (b) For what real values of the parameter a is S guaranteed to be stable?

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This is a practice problem from the Signals and Systems course in my Electrical Engineering program.

Could you please walk me through the steps of solving it so that I can solve more complicated ones in the future?

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9.16. A causal LTI system S with impulse response h(t) has its input x(t) and output y(t)
related through a linear constant-coefficient differential equation of the form
dy(t)
dt
d³ y(t)
d13
+ (1 + α) d² y(t)
dt²
+ a(a + 1). + a² y(t) = x(t).
(a) If
g(t)
=
dh(t)
+ h(t),
dt
how many poles does G(s) have?
(b) For what real values of the parameter a is S guaranteed to be stable?
Transcribed Image Text:9.16. A causal LTI system S with impulse response h(t) has its input x(t) and output y(t) related through a linear constant-coefficient differential equation of the form dy(t) dt d³ y(t) d13 + (1 + α) d² y(t) dt² + a(a + 1). + a² y(t) = x(t). (a) If g(t) = dh(t) + h(t), dt how many poles does G(s) have? (b) For what real values of the parameter a is S guaranteed to be stable?
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