# Convolution (Widescreen).pptx - CONVOLUTION A GRAPHICAL...

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CONVOLUTION A GRAPHICAL APPROACH Adrian Gonzalez April 24, 2019 CALIFORNIA STATE POLYTECHNIC UNIVERSITY, POMONA | ELECTRICAL & COMPUTER ENGINEERING DEPARTMENT

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STEPS TO EVALUATE THE CONVOLUTION INTEGRAL: 1. Conversion : Replace with dummy variable for each of the signals. 2. Folding : Take a mirror image of about the ordinate axis to obtain . 3. Displacement : Shift or delay by to obtain . 4. Multiplication : Find the product of and . 5. Integration : For a given time , calculate the area under the product for to get at . CALIFORNIA STATE POLYTECHNIC UNIVERSITY, POMONA | ELECTRICAL & COMPUTER ENGINEERING DEPARTMENT
EXAMPLE 1 Find the convolution of the two signals shown below: 0 1 ? 1 ( ? ) ? 0 1 2 3 ? 2 ( ? ) ? 2 1 CALIFORNIA STATE POLYTECHNIC UNIVERSITY, POMONA | ELECTRICAL & COMPUTER ENGINEERING DEPARTMENT

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CONVERSION Replace with . 0 1 ? 1 ( ? ) ? 2 0 1 2 3 ? 2 ( ? ) ? 1 CALIFORNIA STATE POLYTECHNIC UNIVERSITY, POMONA | ELECTRICAL & COMPUTER ENGINEERING DEPARTMENT
DISPLACEMENT Since the convolution is commutative, we can fold either signal, thus lets fold : 0 1 ? 1 ( ? ) ? 0 1 ? 1 ( ? ) ? -1 2 CALIFORNIA STATE POLYTECHNIC UNIVERSITY, POMONA | ELECTRICAL & COMPUTER ENGINEERING DEPARTMENT

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DISPLACEMENT (CONT.) The signal can now be shifted or delayed by : 0 1 ? 1 ( ? ? ) ? -1 ? 1 ? 2 CALIFORNIA STATE POLYTECHNIC UNIVERSITY, POMONA | ELECTRICAL & COMPUTER ENGINEERING DEPARTMENT
PROCESS: For : No overlap. Note : There will be partial overlap for . ? ( ? ) = ? 1 ( ? ) ∗? 2 ( ? ) = 0 0 1 ? -1 2 3 4 ? 2 ( ? ) 2 1 ? 1 ? ? 1 ( ? ? ) CALIFORNIA STATE POLYTECHNIC UNIVERSITY, POMONA | ELECTRICAL & COMPUTER ENGINEERING DEPARTMENT

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PROCESS (CONT.): For : There’s partial overlap. Note : There will be full overlap for . ? ( ? ) = 1 ? ( 2 ) ( 1 ) ? ? = 2 ? 2 0 1 ? -1 2 3 4 ? 2 ( ? ) 2 1 ? 1 ? ? 1 ( ? ? ) CALIFORNIA STATE POLYTECHNIC UNIVERSITY, POMONA | ELECTRICAL & COMPUTER ENGINEERING DEPARTMENT
PROCESS (CONT.): For : There’s complete overlap. Note : There will partial overlap once again for . ? ( ? ) = ? 1 ? ( 2 ) ( 1 ) ? ? = 2 0 1 ? -1 2 3 4 ? 2 ( ? ) 2 1 ? 1 ? ? 1 ( ? ? ) CALIFORNIA STATE POLYTECHNIC UNIVERSITY, POMONA | ELECTRICAL & COMPUTER ENGINEERING DEPARTMENT

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PROCESS (CONT.): For : There’s partial overlap.

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