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SolSet5 - EE 562a Homework Solutions 5 October 27, 2009 1...

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Unformatted text preview: EE 562a Homework Solutions 5 October 27, 2009 1 1. Well denote a point in the abstract space of deterministic sequences (i.e. discrete time signals) by x = { x ( n ) : n T } and y = { y ( n ) : n T } , where the index set is T = Z , the set of integers. The sequence represented by y is the output of the system G when x is the input (i.e. y = G x ) if and only if ay ( n ) + by ( n- 1) + cy ( n- 2) = x ( n ) + x ( n- 1) . (a) To obtain the block diagram from the difference equation, it is useful to express the output in terms of shifted versions of the input and previous outputs. y ( n ) = 1 a [( x ( n ) + x ( n- 1))- ( by ( n- 1) + cy ( n- 2))] . The realization follows from this equation. x ( n ) fi fl 1/ a c b z-1 z-1 z-1 z-1 = unit delay device realization of system zeros realization of system poles +- +- + + y ( n ) It is possible to realize this system with fewer delay devices - can you see how? (b) & (c) It is actually easier to do (c) first. y = G x a T y + b T- 1 y + c T- 2 y = T x + T- 1 x. Using the fact that T k = ( T 1 ) k ( k unit shifts is the same as one k shift), the solution for (c) is a ( T 1 ) y + b ( T 1 )- 1 y + c ( T 1 )- 2 y = ( T 1 ) x + ( T 1 )- 1 x. Now we can use the linearity of the shift operator to check that G is linear. You can check homogeneity and additivity separately or just check superposition. Lets check the superposition property. We want to show that y 1 = G x 1 , y 2 = G x 2 1 y 1 + 2 y 2 = G ( 1 x 1 + 2 x 2 ) , where x i is an arbitrary signal in the space and i is an arbitrary scalar for i = 1 , 2. Now well claim that G has the superposition property, 1 y 1 + 2 y 2 = G ( 1 x 1 + 2 x 2 ), and show that this is indeed true. G has this superposition property if and only if a T ( 1 y 1 + 2 y 2 ) + b T- 1 ( 1 y 1 + 2 y 2 ) + c T- 2 ( 1 y 1 + 2 y 2 ) = T ( 1 x 1 + 2 x 2 ) + T- 1 ( 1 x 1 + 2 x 2 ) . 2 EE 562a Homework Solutions 5 October 27, 2009 Since T k is linear, we know that it has the superposition property. T k ( 1 x 1 + 2 x 2 ) = 1 T k x 1 + 1 T k x 2 . Applying this to both sides of the input/output relation yields 1 ( a T y 1 + b T- 1 y 1 + c T- 2 y 1 ) + 2 ( a T y 2 + b T- 1 y 2 + c T- 2 y 2 ) = 1 ( T x 1 + T- 1 x 1 ) + 2 ( T x 2 + T- 1 x 2 ) . So G obeys the superposition principle if and only if the above equation is true. This equation is true since we know that, by definition y 1 = G x 1 , y 2 = G x 2 a T y i + b T- 1 y i + c T- 2 y i = T x i + T- 1 x i , for i = 1 , 2. It follows that G has the superposition property and therefore is linear....
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SolSet5 - EE 562a Homework Solutions 5 October 27, 2009 1...

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