lect3 - SYSTEMS DEFINED BY RECURSIONS Consider the...

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SYSTEMS DEFINED BY RECURSIONS Consider the difference equation ) 1 ( 4 ) ( 2 ) 2 ( 2 3 ) 1 ( 2 5 ) ( - + = - - - + n x n x n y n y n y . (3.1) We can solve for ) ( n y as, ) 1 ( 4 ) ( 2 ) 2 ( 2 3 ) 1 ( 2 5 ) ( - + + - + - - = n x n x n y n y n y . (3.2) Now suppose that the input, together with the initial output values, ) 1 ( - y and ) 2 ( - y are known. Then we can solve for ) ( n y for all 0 n by recursion on n . That is, substituting n = 0 in (3.2), we can solve for ) 0 ( y as ) 1 ( 4 ) 0 ( 2 ) 2 ( 2 3 ) 1 ( 2 5 ) 0 ( - + + - + - - = x x y y y . Next, with n = 1 in (3.2), we can solve for ) 1 ( y as ) 0 ( 4 ) 1 ( 2 ) 1 ( 2 3 ) 0 ( 2 5 ) 1 ( x x y y y + + - + - = , because ) 0 ( y is known from the previous step. Continuing this recursion we can obtain all the values of ) ( n y for 0 n . In this way, the difference equation (3.1) defines a system with input ) ( n x and output ) ( n y . Notice, however, that the output of the system depends not only on the input, but also on the initial values of the output, ) 1 ( - y and ) 2 ( - y . The general difference equation will be written as
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) ( ) 2 ( ) 1 ( ) ( ) ( ) 2 ( ) 1 ( ) ( 2 1 0 2 1 m n x b n x b n x b n x b N n y a n y a n y a n y M N - + + - + - + = - + + - + - + (3.3) in which the coefficients are assumed known, as are N and M. THE UNILATERAL Z-TRANSFORM We can analyze sytems described by difference equations by employing the unilateral z-transform, defined by = - = 0 ) ( ) ( n n z n f z F . The only modification here, with respect to the bilateral z- transform is that the sum ranges only from zero to infinity. Therefore, the unilateral z-transform contains information only about the signal from zero to infinity. Although the signal need not be zero for n < 0, the unilateral z-transform contains no information about its values for n < 0. Example 3.1 a) ) ( ) ( n U a n f n = a z z z a z F n n n - = = = - 0 ) ( . b) n all a n f n ; ) ( =
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a z z z a z F n n n - = = = - 0 ) ( . Note that the signals in (a) and (b) are quite different for n < 0, but they are equal for 0 n so they have the same unilateral z- transform. The main theorem used in applying the unilateral z-transform to system analysis is the time shifting theorem, which takes a different form than it did in terms of the bilateral transform. Theorem: (Time shifting) Let k be a positive time shift (to the right) of the signal ) ( n f . Then ) ( ) 2 ( ) 1 ( ) ( ) ( ) 2 ( ) 1 ( k f f z f z z F z k n f k k k - + + - + - + - - - - - - Proof: ) ( ) 2 ( ) 1 ( ) ( ) ( ) 2 ( ) 1 ( ) ( ) ( ) 2 ( ) 1 ( ) ( ) ( ) ( ) 2 ( ) 1 ( ) 2 ( ) 1 ( 0 ) ( ) 2 ( ) 1 ( 0 k f f z f z z F z k f f z f z z n f k f f z f z z k n f z k n f k n f k k k k k n k n k k k n n n n - + + - + - + = - + + - + - + = - + + - + - + - = - - - - - - - - - - - = + - - - - - = - = - Q.E.D.
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Example 3.2 Consider the system defined in (3.1) ) 1 ( 4 ) ( 2 ) 2 ( 2 3 ) 1 ( 2 5 ) ( - + = - - - + n x n x n y n y n y where the values ) 1 ( - y and ) 2 ( -
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This note was uploaded on 10/18/2009 for the course EE ee332 taught by Professor Ee during the Spring '07 term at Istanbul Technical University.

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lect3 - SYSTEMS DEFINED BY RECURSIONS Consider the...

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