Aside: Convolution

Monday, 22 April 2019

4:47 PM

 

Convolution means: take the mirror image of  to obtain

Then shift (delay)  by  to obtain

Multiply  with

Then integrate over the

 

 

Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Mirror Image Ink Drawings Ink Drawings Ink Drawings Shift by 𝑡 Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings      Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings  Ink Drawings Ink Drawings Ink Drawings   
𝑡≤1 No overlap between ℎ(𝑡−𝜏) and 𝑥(𝑡) 𝑦﷐𝑡﷯=0 1<𝑡≤2 ℎ(𝑡−τ) and 𝑥(𝑡) overlap between 1 and 𝑡 𝑦﷐𝑡﷯=﷐1﷮𝑡﷮ℎ𝑥﷯𝑑𝜏 𝑦﷐𝑡﷯=﷐1﷮𝑡﷮2﷯𝑑𝜏 𝑦﷐𝑡﷯=﷐﷐2τ﷯﷮1﷮𝑡﷯=2𝑡−2 2<𝑡≤3 ℎ(𝑡−τ) and 𝑥(𝑡) completely overlap between 𝑡−1 and 𝑡 𝑦﷐𝑡﷯=﷐𝑡−1﷮𝑡﷮(2)(1)﷯𝑑𝜏 𝑦﷐𝑡﷯=﷐﷐2τ﷯﷮𝑡−1﷮𝑡﷯ 𝑦(𝑡)=2 3<𝑡≤4 ℎ﷐𝑡−𝜏﷯ and 𝑥(𝑡) overlap between 𝑡−1 and 3 𝑦﷐𝑡﷯=﷐𝑡−1﷮3﷮﷐2﷯﷐1﷯﷯𝑑𝜏 𝑦﷐𝑡﷯=﷐﷐2τ﷯﷮𝑡−1﷮3﷯ 𝑦(𝑡)=8−2𝑡 𝑡>4 No overlap 𝑦﷐𝑡﷯=0 Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings      Ink Drawings Ink Drawings Ink Drawings Ink Drawings 

 

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