Question

A pressure gauge that can be modeled as an LTI system has a time response to a unit step input given by (1 - e^(-rt)e^(-t)). For a certain input x(t), the output is observed to be (2 - 3e^(-r)e^(-3t)). For this observed measurement, determine the true pressure input to the gauge as a function of time.

a) P(t) = x(t) + 3e^(-r)e^(-3t)
b) P(t) = x(t) - 3e^(-r)e^(-3t)
c) P(t) = x(t) / (1 - e^(-rt)e^(-t))
d) P(t) = x(t) * (1 - e^(-rt)e^(-t))

Answer 1

The true pressure input to the gauge as a function of time is P(t) = x(t) - 3e^{-r}e^{-3t}, corresponding to the relationship between the system's step response and the observed output.

Step-by-step explanation:

The observed output of the pressure gauge, which is an LTI system, is given by (2 - 3e-re-3t). If the time response to a unit step input is (1 - e-rte-t), we can deduce the true pressure input by comparing the response of the system to the observed output. Since the system's step response is essentially its impulse response integrated, and the observed output is 2 times the step response minus an additional term, we can infer that the input x(t) must have been a step input of magnitude 2 to get that factor of 2 in the response. The unaccounted term -3e-re-3t in the output suggests that there was an additional impulse that caused this term. Hence, the true pressure input to the gauge as a function of time is P(t) = x(t) - 3e-re-3t, which aligns with option (b).