2.5: Causal Systems
- Page ID
- 47230
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All physical systems respond to input only after the input is applied. In math terms, this means \(h(t) = 0\) for all \(t < 0\). For convenience, we also usually consider input signals to be zero before time zero. The convolution is adapted in a very reasonable way:
\begin{align} y(t) &= \int \limits_{-\infty}^{\infty} u(\xi) h(t - \xi) \, d\xi \\[4pt] &= \int \limits_{0}^{\infty} u(\xi) h(t - \xi) \, d\xi \\[4pt] &= \int \limits_{0}^{t} u(\xi) h(t - \xi) \, d\xi. \end{align}
The lower integration limit is set by the assumption that \(u(t) = 0\) for \(t < 0\), and the upper limit is set by the causality of the impulse response. The complementary form with integrand \(u(t − \xi) h(\xi) \) also holds.