EE200_Weber_10-28

# EE200_Weber_10-28 - EE 200 Frequency Response of Composite...

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40 EE 200 Frequency Response of Composite Systems A system S that is the cascade of two LTI systems, S 1 and S 2 , is also an LTI system. The cascade of S 1 and S 2 is time invariant S o D " = S 2 o S 1 o D = S 2 o D o S 1 = D o S 2 o S 1 = D o S x S 1 D ! S(x) S 2 x S 1 S(x) S 2 D ! x S 1 S(x) S 2 D ! x S 1 S(x) S 2 D ! S S S S

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41 EE 200 Frequency Response of Composite Systems If S 1 and S 2 are homogeneous, so is the cascade. S ( " x ) = ( S 2 o S 1 )( x ) = S 2 ( S 1 ( x )) = S 2 ( S 1 ( x )) = S 2 ( S 1 ( x )) = ( S 2 o S 1 )( x ) = S ( x ) " x S 1 S(x) S 2 " x S 1 S(x) S 2 " x S 1 S(x) S 2 " x S 1 S(x) S 2 " x S 1 S(x) S 2
42 EE 200 Frequency Response of Composite Systems If S 1 and S 2 are additive, so is the cascade. S ( x 1 + x 2 ) = ( S 2 o S 1 )( x 1 + x 2 ) = S 2 ( S 1 ( x 1 + x 2 )) = S 2 ( S 1 ( x 1 ) + S 1 ( x 2 )) + S 2 ( S 1 ( x 1 )) + S 2 ( S 1 ( x 2 )) = ( S 2 o S 1 )( x 1 ) + ( S 2 o S 1 )( x 2 ) = S ( x 1 ) + S ( x 2 ) x 1 S 1 S(x 1 +x 2 ) S 2 S 1 S(x 1 +x 2 ) S 2 S 1 + + x 2 x 1 x 2 S 1 S(x 1 +x 2 ) S 2 S 1 + x 1 x 2 S 2 S S S

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43 EE 200 Frequency Response of Composite Systems If the frequency response of S 1 is H 1 ( # ), the output of S 1 when the input is a complex exponential is a scaled version of x(t): y(t) = H 1 ( # )x(t). If the frequency response of S 2 is H 2 ( # ), the output of S 2 is then a scaled version of y(t): z(t) = H 2 ( # )y(t). The frequency response of the cascade system is the product of the responses of the individual systems z(t) = H 2 ( # )H 1 ( # )x(t) = H( # )x(t) x(t) S 1 S(x) S 2 S y(t) z(t)
44 EE 200 Frequency Response of Composite Systems Since H 2 ( # )H 1 ( # ) = H 1 ( # )H 2 ( # ), this says that it doesn’t make any difference which order the systems are composed in a cascade system. S 1 S 2 S S 2 S 1 S = Cascade systems with discrete input and output values work the same way. H( # ) = H 1 ( # )H 2 ( # )

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45 EE 200 Frequency Response of Composite Systems A feedback system with component system S 1 and S 2 is LTI if the the component systems are LTI. S 1 S 2 + x(t) y(t) y ( t ) = S 1 ( x ( t ) + S 2 ( y ( t )) = S 1 ( x ( t ) + S 1 ( S 2 ( y ( t ))) y ( t ) " S 1 ( S 2 ( y ( t ))) = S 1 ( x ( t ))
46 EE 200 Frequency Response of Composite Systems If the frequency response of S 1 is H 1 ( # ), and of S 2 is H 2 ( # ), then for an input of a complex exponential the response of the feedback system S is given by H( # ) y ( t ) " S 1 ( S 2 ( y ( t ))) = S 1 ( x ( t )) H ( # ) x " S 1 ( S 2 ( H ( ) x )) = S 1 ( x ) H ( )( x " S 1 ( S 2 ( x )) = S 1 ( x ) H ( )( x " H 1 ( ) H 2 ( ) x ) = H 1 ( ) x H ( )(1 " H 1 ( ) H 2 ( )) = H 1 ( ) H ( ) = H 1 ( ) 1 " H 1 ( ) H 2 ( )

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1 EE 200 Filtering Linear time invariant systems are often called filters because of their property that all frequency components in the output are scaled versions of input frequencies.
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EE200_Weber_10-28 - EE 200 Frequency Response of Composite...

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