102_1_discussion2

102_1_discussion2 - EE102 Discussion 2 2010-10-05 vasiliy...

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EE102 Discussion 2 2010-10-05 vasiliy karasev 1. Signals Write x ( t ) (shown below) as a superposition of scaled, shifted, tri ( t ) functions. 0 t x(t) 1 -2 2 tri ( t ) = ± 1 - | t | if | t | < 1 0 otherwise. Solution x ( t ) = tri ( t + 1) + tri ( t ) - 1 2 tri (2 t - 1) + tri (2 t - 3) This can be verified by sketching individual components. 2. Linearity, Time Invariance Systems accept functions (signals) as inputs and return functions (signals) as output. Notation (assume continuous time signals).: F - system x - input y - output To denote system acting on an input x to produce y , write: y = F ( x ) or y = Fx . Be careful to distinguish between scalars and signals : F ( α 1 x 1 + α 2 x 2 ) = α 1 F ( x 1 ) + α 2 F ( x 2 ) (condition for linearity) α 1 , α 2 - scalars, x 1 , x 2 - signals. Systems can be specified explicitly: y ( t ) = sin( x ( t )) y ( t ) = x (sin( t )) Are the following systems linear? time invariant? Why or why not? 1. y [ n ] = x [ n + 1] Solution 1
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Verify that the system is linear. Homogeneity Given ˆ x ( n ) = αx ( n ), ˆ y ( n ) = ˆ x ( n + 1). Want to show: ˆ y ( n ) = αy ( n ) if ˆ x ( n ) = αx ( n ) (in words: scaling input results in the same scaling of output). Solution : ˆ y ( n ) = ˆ x ( n + 1) ˆ y ( n ) = αx ( n + 1) ˆ y ( n ) = αy ( n ) Conclusion: homogeneity (scaling) holds.
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102_1_discussion2 - EE102 Discussion 2 2010-10-05 vasiliy...

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