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Practice Question on Linearity of a System
The input x(t) and the output y(t) of a system are related by the equation
$ y(t)=\int_{-\infty}^t x(\tau) d\tau . \ $
Is the system linear (yes/no)? Justify your answer.
You will receive feedback from your instructor and TA directly on this page. Other students are welcome to comment/discuss/point out mistakes/ask questions too!
Answer 1
Yes, this system is linear.
If
$ x_1(t) \to \Bigg[ system \Bigg] \to y_1(t)= \int_{-\infty}^{t} x_1(\tau) d\tau $
and
$ x_2(t) \to \Bigg[ system \Bigg] \to y_2(t)= \int_{-\infty}^{t} x_2(\tau) d\tau $
Then
$ ax_1(t)+bx_2(t) \to \Bigg[ system \Bigg] \to y(t)= \int_{-\infty}^{t} ax_1(\tau)+bx_2(\tau) d\tau = a\int_{-\infty}^{t} x_1(\tau) d\tau\ +\ b\int_{-\infty}^{t} x_2(\tau) d\tau = ay_1(t)+by_2(t) $
--Cmcmican 19:20, 26 January 2011 (UTC)
- TA's comment: Excellent!
--Ahmadi 17:27, 27 January 2011 (UTC)
Answer 2
Write it here.
Answer 3
Write it here.
