Aside: Linear Time Invariant (LTI) systems

Monday, 22 April 2019

5:34 PM

For any time-invariant, linear system, the relationship between the input 𝑥(𝑡) and the output 𝑦(𝑡) is
𝑦﷐𝑡﷯=ℎ﷐𝑡﷯∗𝑥﷐𝑡﷯

ie. If 𝑥﷐𝑡﷯=𝛿﷐𝑡﷯, then 𝑦﷐𝑡﷯=﷐−∞﷮∞﷮ℎ﷐𝑡−𝜏﷯𝛿﷐𝜏﷯﷯𝑑𝑡=ℎ﷐𝑡﷯ .…𝛿﷐𝑡﷯⇒𝑥﷐𝜏﷯

𝑌﷐𝑠﷯=𝐻﷐𝑠﷯𝑋﷐𝑠﷯
𝐻﷐𝑠﷯=﷐𝑌﷐𝑠﷯﷮𝑋﷐𝑠﷯﷯
(The transfer function is the Laplace transformation of the unit impulse response)

If 𝑥﷐𝑡﷯=δ(𝑡), then 𝑋﷐𝑠﷯=1, and 𝐻﷐𝑠﷯=𝑌﷐𝑠﷯

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