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EL 611 HW_ZT Soln 2011

# EL 611 HW_ZT Soln 2011 - Z Transform Solutions EL611 Fall...

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EL611 Z Transform Solutions Fall 2008 1.) 2 2 ) 2 / 1 )( 3 ( ) 4 / 15 2 / 15 2 ( ) ( z z z z z z H . a) ) ( z H has zeros at 0 z , 197 . 4 z and 447 . 0 z . It has a pole at 3 z and a double pole at 2 / 1 z . b) The ROC for stability must be 3 2 / 1 z . We then expand in partial fractions as 2 2 2 ) 2 / 1 ( 25 / 70 2 / 1 25 / 83 3 25 / 33 ) 2 / 1 )( 3 ( 4 / 15 2 / 15 2 ) ( z z z z z z z z z H 2 ) 2 / 1 ( 25 / 70 2 / 1 25 / 83 3 25 / 33 ) ( z z z z z z z H . Now we separate this into causal and anticausal parts, both stable, as follows 4 / 1 14 . 1 32 . 3 3 25 / 33 ) ( 2 2 z z z z z z z H . The two systems are: 2 1 1 2 2 1 4 / 1 1 14 . 1 32 . 3 4 / 1 14 . 1 32 . 3 ) ( z z z z z z z z H ; Causal 3 25 / 33 ) ( 2 z z z H ; Anticausal The recursions are: Recursion 1: ) 1 ( 14 . 1 ) ( 32 . 3 ) 2 ( 4 / 1 ) 1 ( ) ( 1 1 1 n x n x n y n y n y Recursion 2: ) 1 ( 25 / 33 ) ( 3 ) 1 ( 2 2 n x n y n y

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c) Use the recursions to find the impulse response ) ( n h of the system for 3 3 n , i.e. find ) 3 ( ), 2 ( ), 1 ( ), 0 ( ), 1 ( ), 2 ( ), 3 ( h h h h h h h . We now use the input ) ( ) ( n n x and run both recursions to get ] ) 3 ( ) 2 ( ) 1 ( ) 0 ( ) 1 ( ) 2 ( ) 3 ( [ 1 1 1 1 1 1 1 y y y y y y y [0 0 0 -3.3200 -4.4600 -3.6300 -2.5150] and ] ) 3 ( ) 2 ( ) 1 ( ) 0 ( ) 1 ( ) 2 ( ) 3 ( [ 2 2 2 2 2 2 2 y y y y y y y [-0.0489 -0.1467 -0.4400 0 0 0 0 ] Adding these up gives the impulse response ] ) 3 ( ) 2 ( ) 1 ( ) 0 ( ) 1 ( ) 2 ( ) 3 ( [ h h h h h h h [-0.0489 -0.1467 -0.4400 -3.3200 -4.4600 -3.6300 -2.5150 ] d) Find ) ( n h by inverse transforming ) ( z H and check that it agrees with part (c). Recall that 2 ) 2 / 1 ( 25 / 70 2 / 1 25 / 83 3 25 / 33 ) ( z z z z z z z H . The stable inverse to this is ) ( 2 1 25 / 70 ) ( 2 1 25 / 83 ) 1 ( ) 3 ( 25 / 33 ) ( 1 n U n n U n U n h n n n . This does check with the recursion results. 2.) a)   ) 2 )( 3 / 1 ( 3 / 5 2 3 / 1 ) ( ) 1 ( 2 ) ( 3 1 ) ( z z z z z z z z X n U n U n x n n .
ROC for ) ( z X is 2 3 / 1 z ) 3 / 2 )( 3 / 1 ( 3 / 5 3 / 2 5 3 / 1 5 ) ( ) ( 3 2 5 ) ( 3 1 5 ) ( z z z z z z z z Y n U n U n y n n .

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EL 611 HW_ZT Soln 2011 - Z Transform Solutions EL611 Fall...

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