Linear Shift Invariant

Linear Shift Invariant - Showing a System is Linear and...

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Showing a System is Linear and Shift Invariant Prepared by: Piotr Dollar 1 Showing Linearity To show a system H is linear, we need to show that for all functions f 1 and f 2 , H satisﬁes the following equation: H [ αf 1 ( x ) + βf 2 ( x )] = αH [ f 1 ( x )] + βH [ f 2 ( x )] That is we need to show the left side equals the right side in the above equation. How does one do this? 1. Find H [ αf 1 ( x ) + βf 2 ( x )]. To do this, let r ( x ) = αf 1 ( x ) + βf 2 ( x ), ﬁnd H [ r ( x )] based on the deﬁnition of H , then substitute f 1 ( x ) and f 2 ( x ) back in. We will do some examples below if this is not clear. 2. Find αH [ f 1 ( x )] + βH [ f 2 ( x )]. 3. If H [ αf 1 ( x ) + βf 2 ( x )] = αH [ f 1 ( x )] + βH [ f 2 ( x )] then the system is linear, otherwise it is not. Example 1: H [ f ( x )] = f (2 x ) 1. Let r ( x ) = αf 1 ( x ) + βf 2 ( x ); H [ r ( x )] = r (2 x ) = αf 1 (2 x ) + βf 2 (2 x ). 2.

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Linear Shift Invariant - Showing a System is Linear and...

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