Solution2 - Solution for Assignment #2 Khurram...

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Unformatted text preview: Solution for Assignment #2 Khurram Hassan-Shafique Prove the following results 1 g 1 ( x ) * ( g 2 ( x ) * I ) = g ( x ) * I . Also, find the value of in terms of 1 and 2 . Proof: L = g 1 ( x ) * ( g 2 ( x ) * I ) F ( L ) = F ( g 1 ( x ) * ( g 2 ( x ) * I )) = F ( g 1 ( x )) F (( g 2 ( x ) * I )) By Convolution Theorem = F ( g 1 ( x )) F ( g 2 ( x )) F ( I ) By Convolution Theorem = 2 1 g 1 1 ( u ) 2 2 g 1 2 ( u ) F ( I ) Since F ( g ( x )) = 2 g 1 ( x ) = h 2 1 1 2 exp - u 2 2 1 2 ih 2 2 2 2 exp - u 2 2 2 2 i F ( I ) = exp- u 2 ( 2 1 + 2 2 ) 2 F ( I ) = exp - u 2 2 2 F ( I ) where = p 2 1 + 2 2 = 2 h 2 exp - u 2 2 2 i F ( I ) = 2 g 1 ( u ) F ( I ) = F ( g ( x )) F ( I ) F ( L ) = F ( g ( x ) * I ) L = g ( x ) * I 2. g ( x,y ) = g ( x ) g ( y ) . Proof: g ( x,y ) = 1 2 2 exp - x 2 + y 2 2 2 = 1 2 1 2 exp - x 2 2 2 exp - y 2 2 2 = h 1 2 exp - x 2 2 2 ih 1 2 exp - y 2 2 2 i = g ( x ) g ( y ) 1 3. g ( x,y ) * I = g ( x ) * ( g ( y ) * I ) . Proof: Method 1. L = g ( x,y ) * I F ( L ) = F ( g ( x,y ) * I ) = F ( g ( x,y )) F ( I ) By Convolution Theorem = 2 2 g 1 ( u,v ) F ( I ) = 2 2 g 1 ( u ) g 1 ( v ) F ( I ) By Problem 2 = 2 g 1 ( u ) 2 g 1 ( v ) F ( I ) = F...
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Solution2 - Solution for Assignment #2 Khurram...

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