Topic: System Properties

## Question

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!

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)