ECE 483 Quiz # 2 Solution Problem 1. (10 points) For a discrete-time system with the transfer function Y ( z ) E ( z ) = z +1 ( z-1)( z-0. 5) , (a) (4 pts) Find an equivalent linear di±erence equation model for the input e ( k ) and the output y ( k ), k = 0 , 1 ,... . Solution: Write Y ( z ) E ( z ) = z + 1 ( z-1)( z-0. 5) = z-1 + z-2 1-1 . 5 z-1 + 0 . 5 z-2 ⇒ (1-1 . 5 z-1 + 0 . 5 z-2 ) Y ( z ) = ( z-1 + z-2 ) E ( z ) ⇒ y ( k )-1 . 5 y ( k-1) + 0 . 5 y ( k-2) = e ( k-1) + e ( k-2) . (b) (4 pts) Plot a simulation diagram that implements the above linear di±erence equation. (c) (3 pts) Write a state variable representation corresponding to the simulation diagram in (b), and identify the matrices A , B , C , and D . Solution: Part (b) and (c) have multiple answers; hence omitted. Problem 2. Suppose a discrete-time system with input r ( k ) and output y ( k ) is given by the simulation diagram above.
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