EE200_Weber_9-16

# EE200_Weber_9-16 - EE 200 Linear Functions A linear...

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1 EE 200 Linear Functions A linear function possesses the property of “superposition”: If the input to a function is a weighted sum of signals, then the output is the weighted sum of the responses of the individual signals. Input = u = α 1 u 1 + α 2 u 2 + … + α n u n Output = v = f(u) = f( α 1 u 1 + α 2 u 2 + … + α n u n ) = α 1 v 1 + α 2 v 2 + … + α n v n where v 1 = f(u 1 ), v 2 = f(u 2 ), … , v n = f(u n )

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2 EE 200 Linear Functions Superposition is the result of two properties: Homogeneity: f( α u) = α f(u) Additivity: f(u 1 + u 2 ) = f(u 1 ) + f(u 2 ) v = f(u) = f( α 1 u 1 + α 2 u 2 + … + α n u n ) = f( α 1 u 1 ) + f( α 2 u 2 ) + … + f( α n u n ) by additivity = α 1 f(u 1 ) + α 2 f(u 2 ) + … + α n f(u n ) by homogeneity = α 1 v 1 + α 2 v 2 + … + α n v n
3 EE 200 Linear Functions A linear function is of the form y = f(x) = ax f(u) = f( α 1 u 1 + α 2 u 2 + α 3 u 3 ) = a( α 1 u 1 + α 2 u 2 + α 3 u 3 ) = a α 1 u 1 + a α 2 u 2 + a α 3 u 3 = α 1 au 1 + α 2 au 2 + α 3 au 3 = α 1 f(u 1 ) + α 2 f(u 2 ) + α 3 f(u 3 ) Functions of the form “y = mx + b” are not linear functions by this definition.

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4 EE 200 Linear Functions If graphed, a linear function is a line passing through the the origin of the coordinate system. x y = ax x 1 x 2 y = a 1 x 1 + a 2 x 2 Same is true for higher number of dimensions, but tough to draw.
5 EE 200 Linear Functions Examples of functions that are not linear since superposition does not hold. f(x) = x 2 f(x) = ax + b f ( x 1 ) = x 1 2 f ( x 2 ) = x 2 2 f ( " 1 x 1 + 2 x 2 ) = ( 1 x 1 + 2 x 2 ) 2 = 1 x 1 ( ) 2 + 2 x 2 ( ) 2 + 2 1 2 x 1 x 2 = 1 2 x 1 2 + 2 2 x 2 2 + 2 1 2 x 1 x 2 = 1 2 f ( x 1 ) + 2 2 f ( x 2 ) + 2 1 2 x 1 x 2 f ( x 1 ) = ax 1 + b f ( x 2 ) = ax 2 + b f ( 1 x 1 + 2 x 2 ) = a ( 1 x 1 + 2 x 2 ) + b = 1 ax 1 + b ( ) + 2 ax 2 + b ( ) + b (1 # ( 1 + 2 )) = 1 f ( x 1 ) + 2 f ( x 2 ) + b (1 # ( 1 + 2 ))

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6 EE 200 Linear Functions Expand the definition of linear functions to handle tuples on input and output. Let x 1 , x 2 , … be N-tuples, Real N a 1 , a 2 , … be scalars, Real y 1 , y 2 , … are M-tuple, Real M f: Real N Real M Superposition still holds y = f(x) = f(a 1 x 1 + a 2 x 2 + … + a n x n ) = a 1 y 1 + a 2 y 2 + … + a n y n where y 1 = f(x 1 ), y 2 = f(x 2 ), … , y n = f(x n )
7 EE 200 Linear Functions Every linear function can be represented by the multiplication of a matrix and a vector. If A is a M

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## This note was uploaded on 12/03/2008 for the course EE 200 taught by Professor Zadeh during the Fall '08 term at USC.

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EE200_Weber_9-16 - EE 200 Linear Functions A linear...

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