This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 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 = 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....
View Full
Document
 Fall '07
 ToddBrun

Click to edit the document details