# 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 σ 1 ( x ) * ( g σ 2 ( x ) * I ) = g σ ( x ) * I . Also, find the value of σ in terms of σ 1 and σ 2 . Proof: L = g σ 1 ( x ) * ( g σ 2 ( x ) * I ) ⇒ F ( L ) = F ( g σ 1 ( x ) * ( g σ 2 ( x ) * I )) = F ( g σ 1 ( x )) F (( g σ 2 ( x ) * I )) By Convolution Theorem = F ( g σ 1 ( x )) F ( g σ 2 ( x )) F ( I ) By Convolution Theorem = √ 2 π σ 1 g 1 σ 1 ( u ) √ 2 π σ 2 g 1 σ 2 ( u ) F ( I ) Since F ( g σ ( x )) = √ 2 π σ g 1 σ ( x ) = h √ 2 π σ 1 σ 1 √ 2 π exp ‡- u 2 σ 2 1 2 ·ih √ 2 π σ 2 σ 2 √ 2 π exp ‡- u 2 σ 2 2 2 ·i F ( I ) = exp- u 2 ( σ 2 1 + σ 2 2 ) 2 ¶ F ( I ) = exp ‡- u 2 σ 2 2 · F ( I ) where σ = p σ 2 1 + σ 2 2 = √ 2 π σ h σ √ 2 π exp ‡- u 2 σ 2 2 ·i F ( I ) = √ 2 π σ g 1 σ ( u ) F ( I ) = F ( g σ ( x )) F ( I ) ⇒ F ( L ) = F ( g σ ( x ) * I ) ⇒ L = g σ ( x ) * I 2. g σ ( x,y ) = g σ ( x ) g σ ( y ) . Proof: g σ ( x,y ) = 1 2 πσ 2 exp ‡- x 2 + y 2 2 σ 2 · = ‡ 1 √ 2 πσ ·‡ 1 √ 2 πσ · exp ‡- x 2 2 σ 2 · exp ‡- y 2 2 σ 2 · = h‡ 1 √ 2 πσ · exp ‡- x 2 2 σ 2 ·ih‡ 1 √ 2 πσ · exp ‡- y 2 2 σ 2 ·i = g σ ( x ) g σ ( y ) 1 3. g σ ( x,y ) * I = g σ ( x ) * ( g σ ( y ) * I ) . Proof: Method 1. L = g σ ( x,y ) * I ⇒ F ( L ) = F ( g σ ( x,y ) * I ) = F ( g σ ( x,y )) F ( I ) By Convolution Theorem = 2 π σ 2 g 1 σ ( u,v ) F ( I ) = ‡ √ 2 π σ ·‡ √ 2 π σ · g 1 σ ( u ) g 1 σ ( v ) F ( I ) By Problem 2 = ‡ √ 2 π σ g 1 σ ( u ) ·‡ √ 2 π σ g 1 σ ( v ) · F ( I ) = F...
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Solution2 - Solution for Assignment#2 Khurram...

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