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finals02 - Sample Final Exam Covering Chapters...

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Sample Final Exam Covering Chapters 10-17 (finals02) 1 Sample Final Exam ( finals02) Covering Chapters 10-17 of Fundamentals of Signals & Systems Problem 1 (30 marks) Consider the system depicted below used for discrete-time processing of continuous-time signals. The sampling period is 100 milliseconds ( 0.1 T = s). The discrete-time filter ( ) d H z is given by the following causal difference equation initially at rest: 1 2 0 1 2 [ ] [ 1] [ 2] [ ] [ 1] [ 2] y n a y n a y n b x n b x n b x n + + = + + (a) [8 marks] Find the controllable canonical state-space realization of the filter ( ) d H z , i.e., sketch the block diagram and give the state-space equations. Answer: The state equation: CT/DT Τ DT/CT Τ [ ] d x n x t c ( ) y t c ( ) [ ] d y n ( ) d H z + + + + - - + + [ ] y n z 1 [ ] u n z 1 1 a 2 a 0 b 1 b 2 b
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Sample Final Exam Covering Chapters 10-17 (finals02) 2 N 1 1 2 2 1 2 [ 1] 0 1 [ ] 0 [ ] [ 1] [ ] 1 B A x n x n u n x n a a x n +     = +     +     ±²³²´ . The output equation: [ ] N 2 2 0 1 1 0 0 [ ] [ ] [ ] D C y n b a b b a b x n b u n = + ±²²²³²²²´ (b) [10 marks] The filter ( ) d H z is designed to approximate the unity-gain, second-order, continuous- time, causal LTI filter 2 2 ( ) , Re{ } 1 3 2 H s s s s = > − + + . Find the values of the parameters 1 2 0 1 2 { , , , , } a a b b b of ( ) d H z using the "c2d" transformation. Specify the ROC of ( ) d H z . Answer: The state equation for ( ) H s : N 1 1 2 2 ( ) ( ) 0 1 0 ( ) ( ) ( ) 2 3 1 B A x t x t u t x t x t   = +     µ µ ±²³²´ . The output equation for ( ) H s : [ ] ( ) 2 0 ( ) C y t x t = ±³´ Diagonalize to compute matrix exponential: 1 1 1 0 1 1 , 0 2 1 2 A V AV V = = = 1 1 2 2 2 2 2 2 2 1 2 1 1 2 1 0 1 2 1 1 0 1 1 0.9909 0.0861 2 2 1 2 0.1722 0.7326 2 2 2 3/ 2 1/ 2 2 1 0 T A T AT d T T T T T T T T T T T T T AT d e A e Ve V e e e e e e e e e e e e e e B A e I B = = = = = = + + = = 2 2 2 2 2 2 2 2 0 1 1 2 2 2 1 1 1 3/ 2 1/ 2 0.0045 2 2 1 0 0.0861 2 1 T T T T T T T T T T T T T T T T d e e e e e e e e e e e e e e e C C      + +   + + = = = + = Transfer function:
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Sample Final Exam Covering Chapters 10-17 (finals02) 3 [ ] [ ] 1 1 ( ) ( ) 0.9909 0.0861 0.0045 2 0 0.1722 0.7326 0.0861 0.0045( 0.7326) (0.0861)(0.0861) 1 2 0 ( 0.1722)(0.0045) (0.0861)( 0.9909) ( 0.9909)( 0.7326) 0.0148 0.009( 0.732 d n d d d z C zI A B D z z z z z z z = + = + = + + = H 2 1 2 1 2 6) 2(0.0861)(0.0861) ( 0.9909)( 0.7326) 0.0148 0.009 0.0082 1.7235 0.7407 0.009 0.0082 1 1.7235 0.7407 z z z z z z z z z + + + = + + = + Hence, 0 1 2 1 2 0, 0.009, 0.0082, 1.7235, 0.7407 b b b a a = = = = − = (c) [5 marks] Sketch the pole-zero plot of ( ) d H z . Compute the gain of
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