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Unformatted text preview: EE 562a Homework Solutions 5 October 27, 2009 1 1. We’ll 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) = αx ( n ) + βx ( 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 [( αx ( n ) + βx ( n 1)) ( by ( n 1) + cy ( n 2))] . The realization follows from this equation. x ( n ) fi fl 1/ a c b z1 z1 z1 z1 = 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 = α T ˜ x + β 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 = α ( T 1 ) ˜ x + β ( 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. Let’s check the superposition property. We want to show that ˜ y 1 = G ˜ x 1 , ˜ y 2 = G ˜ x 2 ⇒ γ 1 ˜ y 1 + γ 2 ˜ y 2 = G ( γ 1 ˜ x 1 + γ 2 ˜ x 2 ) , where ˜ x i is an arbitrary signal in the space and γ i is an arbitrary scalar for i = 1 , 2. Now we’ll claim that G has the superposition property, γ 1 ˜ y 1 + γ 2 ˜ y 2 = G ( γ 1 ˜ x 1 + γ 2 ˜ x 2 ), and show that this is indeed true. G has this superposition property if and only if a T ( γ 1 ˜ y 1 + γ 2 ˜ y 2 ) + b T 1 ( γ 1 ˜ y 1 + γ 2 ˜ y 2 ) + c T 2 ( γ 1 ˜ y 1 + γ 2 ˜ y 2 ) = α T ( γ 1 ˜ x 1 + γ 2 ˜ x 2 ) + β T 1 ( γ 1 ˜ x 1 + γ 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 ( γ 1 ˜ x 1 + γ 2 ˜ x 2 ) = γ 1 T k ˜ x 1 + γ 1 T k ˜ x 2 . Applying this to both sides of the input/output relation yields γ 1 ( a T ˜ y 1 + b T 1 ˜ y 1 + c T 2 ˜ y 1 ) + γ 2 ( a T ˜ y 2 + b T 1 ˜ y 2 + c T 2 ˜ y 2 ) = γ 1 ( α T ˜ x 1 + β T 1 ˜ x 1 ) + γ 2 ( α T ˜ x 2 + β 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 = α T ˜ x i + β T 1 ˜ x i , for i = 1 , 2. It follows that G has the superposition property and therefore is linear....
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This note was uploaded on 04/03/2010 for the course EE 562a taught by Professor Toddbrun during the Fall '07 term at USC.
 Fall '07
 ToddBrun

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