Suppose that $H$ is a linear time invariant system. If the response of $H$ to the input $u$ is $y$, then what is its response to integral of $u$?
LTI system's response to input's integral
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dynamical-systems
1 Answers
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Since $H$ is an LTI system, it has an impulse-response, say $h(t)$ and for $u(t)$ as the input signal: $$y(t)=h(t)*u(t)$$ where $*$ represents the convolution. Note that the above equation is the inherent property of LTI systems. Now since the integral and convolution are both linear operators, if $u_I=\int u$ then $$\begin{align}y_I(t)&=h(t)*u_I(t)\\ &=h(t)*\int_{t_0}^t u(\tau)d\tau\\ &=\int_{t_0}^t h(\tau)*u(\tau)d\tau=\int_{t_0}^t y(\tau)d\tau \end{align}$$ Or in other words $y_I=\int y$
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0I need an explanation on why $h(t)*\int_{t_0}^t u(\tau ) d\tau$ equals $\int_{t_0}^t h(t)*u(\tau ) d\tau$ – 2017-02-26
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0@JohnRailman As I said, both operators are _linear_. So they are interchangeable (they are both integrals and integrals can switch places). And as a side-note, I am just curious that why haven't you accepted any answer for your questions? Not saying that's a big deal, but for some people it is considered as a token of appreciation for their time. – 2017-02-26