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LINEARITY
ADDITIVE
X Y
X=f(x+y)
Y=f(x)+f(y)
LINEAR
SYSTEM
MULTIPLICATION
X Y
X=f(ax)
Y=af(x)
LINEAR
SYSTEM
LINEAR SYSTEM
X Y
X=f(a1x+a2x)
Y=a1f(x)+a2f(x)
LINEAR SYSTEM
EE-2027 SaS 06-07, L11 4/12
Linearity of the Fourier Transform
• The Fourier transform is a linear function of
x(t)
1 1
2 2
1 2 1 2
( ) ( )
( ) ( )
( ) ( ) ( ) ( )
F
F
F
x t X j
x t X j
ax t bx t aX j bX j


 


  
Linearity of Fourier Transform
Example: Linearity
4 2( ) ( ) ( )x t p t p t 
2
( ) 4sinc 2sincX
 

 
   
    
   
Example

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Linearity