Solution_IIR Design1

# Therefore the pole locations are 1230 0921 1 3630

• Notes
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Therefore the pole locations are:

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) 1230 . 0 ( 0921 . 1 ) 3630 . 0 ( 0373 . 1 ) 5847 . 0 ( 9305 . 0 ) 7771 . 0 ( 7771 . 0 ) 9305 . 0 ( 5847 . 0 ) 0373 . 1 ( 3630 . 0 ) 0921 . 1 ( 1230 . 0 13 1 0 099 . 1 14 , 13 12 , 11 10 , 9 8 , 7 6 , 5 4 , 3 2 , 1 28 ) 1 2 ( 2 j s j s j s j s j s j s j s ,..., , k e e s k j j k ± - = ± - = ± - = ± - = ± - = ± - = ± - = = = + π π The transfer function is given by ) 1.2078 2.1842 )( 1.2078 2.0747 )( 1.2078 1.8611 )( 1.2078 1.5542 ( 1 ) 1.2078 1.1694 )( 1.2078 0.7260 )( 1.2078 0.2461 ( 7495 . 3 ) ( ) ( ) ( ) ( 2 2 2 2 2 2 2 13 0 13 0 + + + + + + + + × + + + + + + = - = - = = = s s s s s s s s s s s s s s s s s s G s H k k N c k k Consider the partial fraction expansion of the transfer function, we get . 1230 . 0 0921 . 1 75 . 105 34 . 84 1230 . 0 0921 . 1 75 . 105 34 . 84 3630 . 0 0373 . 1 80 . 46 19 . 97 3630 . 0 0373 . 1 80 . 46 19 . 97 5847 . 0 9305 . 0 j66.08 15.08 - 5847 . 0 9305 . 0 j66.08 15.08 - 7771 . 0 7771 . 0 64 . 32 7771 . 0 7771 . 0 32.64 9305 . 0 5847 . 0 j11.14 2.54 9305 . 0 5847 . 0 j11.14 - 2.54 0373 . 1 3630 . 0 j1.13 2.35 0373 . 1 3630 . 0 j1.13 2.35 0921 . 1 1230 . 0 j0.23 0.18 0921 . 1 1230 . 0 j0.23 0.18 ) ( j s j j s j j s j j s j j s j s j s j s j s j s j s j s j s j s s H + + + + - + - + + + + - + - + - - + + + - + - + + + + + + - + + + + + + - + + + + - + - + + + + + - + - + + = Using the impulse invariance method with Ts=1, the poles of H(z) are given as: j0.0412 0.3330 )) 1230 . 0 ( 0921 . 1 exp( ) 1230 . 0 ( 0921 . 1 j0.1258 0.3313 )) 3630 . 0 ( 0373 . 1 exp( ) 3630 . 0 ( 0373 . 1 j0.2177 0.3288 )) 5847 . 0 ( 9305 . 0 exp( ) 5847 . 0 ( 9305 . 0 j0.3224 0.3278 )) 7771 . 0 ( 7771 . 0 exp( ) 7771 . 0 ( 7771 . 0 j0.4469 0.3329 )) 9305 . 0 ( 5847 . 0 exp( ) 9305 . 0 ( 5847 . 0 0.5989 0.3537 )) 0373 . 1 ( 3630 . 0 exp( ) 0373 . 1 ( 3630 . 0 j0.7848 4703 . 0 )) 0921 . 1 ( 1230 . 0 exp( ) 0921 . 1 ( 1230 . 0 ) ( of Poles ) ( of Poles ± = ± - ± - ± = ± - ± - ± = ± - ± - ± = ± - ± - ± = ± - ± - ± = ± - ± - ± = ± - ± - j j j j j j j j j j j j j j j z H s H The resulting system function is given by
. ) 0412 . 0 3330 . 0 ( 1 75 . 105 34 . 84 ) 0412 . 0 3330 . 0 ( 1 75 . 105 34 . 84 ) 1258 . 0 3313 . 0 ( 1 80 . 46 19 . 97 ) 1258 . 0 3313 . 0 ( 1 80 . 46 19 . 97 ) 2177 . 0 3288 . 0 ( 1 j66.08 15.08 - ) 2177 . 0 3288 . 0 ( 1 j66.08 15.08 - ) 3224 . 0 3278 . 0 ( 1 64 . 32 ) 3224 . 0 3278 . 0 ( 1 32.64 ) 4496 . 0 3329 . 0 ( 1 j11.14 2.54 ) 4496 . 0 3329 . 0 ( 1 j11.14 - 2.54 ) 5989 . 0 3537 . 0 ( 1 j1.13 2.35 ) 5989 . 0 3537 . 0 ( 1 j1.13 2.35 ) 7848 . 0 4703 . 0 ( 1 j0.23 0.18
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