Properties

Saturday, 20 April 2019

2:49 PM

Linearity
𝐿﷐﷐𝑎﷮1﷯﷐𝑓﷮1﷯﷐𝑡﷯+﷐𝑎﷮2﷯﷐𝑓﷮2﷯﷐𝑡﷯﷯=﷐𝑎﷮1﷯﷐𝐹﷮1﷯﷐𝑠﷯+﷐𝑎﷮2﷯﷐𝐹﷮2﷯﷐𝑠﷯

Scaling
𝐿﷐𝑓﷐𝑎𝑡﷯﷯=﷐1﷮𝑎﷯𝐹﷐﷐𝑠﷮𝑎﷯﷯

Time Shift
𝐿﷐𝑓﷐𝑡−𝑎﷯𝑢﷐𝑡−𝑎﷯﷯=﷐𝑒﷮−𝑎𝑠﷯𝐹﷐𝑠﷯

Frequency Shift
𝐿﷐﷐𝑒﷮−𝑎𝑡﷯𝑓﷐𝑡﷯𝑢﷐𝑡﷯﷯=𝐹﷐𝑠+𝑎﷯

Time Differentiation
𝐿﷐﷐𝑓﷮′﷯﷐𝑡﷯﷯=﷐﷐0﷮−﷯﷮∞﷮﷐𝑓﷮′﷯﷐𝑡﷯﷐𝑒﷮−𝑠𝑡﷯﷯𝑑𝑡

𝐿﷐﷐𝑓﷮′′﷯﷐𝑡﷯﷯=﷐𝑠﷮2﷯𝐹﷐𝑠﷯−𝑠𝑓﷐﷐0﷮−﷯﷯−﷐𝑓﷮′﷯﷐0﷯

Time Integration
𝐿﷐﷐0﷮𝑡﷮𝑓﷐τ﷯𝑑𝜏﷯﷯=﷐﷐0﷮−﷯﷮∞﷮﷐﷐0﷮𝑡﷮𝑓﷐τ﷯﷯𝑑𝜏﷯﷐𝑒﷮−𝑠𝑡﷯﷯𝑑𝑡
=﷐﷐𝑓﷐𝑡﷯﷐𝑒﷮−𝑠𝑡﷯﷯﷮﷐0﷮−﷯﷮∞﷯−﷐﷐0﷮−﷯﷮∞﷮𝑓﷐𝑡﷯﷐−𝑠﷐𝑒﷮−𝑠𝑡﷯﷯﷯𝑑𝑡
=0−𝑓﷐﷐0﷮−﷯﷯+𝑠﷐﷐0﷮−﷯﷮∞﷮𝑓﷐𝑡﷯﷐𝑒﷮−𝑠𝑡﷯﷯𝑑𝑡
=𝑠𝐹﷐𝑠﷯−𝑓﷐﷐0﷮−﷯﷯

Frequency Differentiation

Initial Value Theorem

We know that

Final Value Theorem

Let , then

So

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