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Unformatted text preview: Inverse Z Transforms and Solving Difference Equations EE 341 Discrete Time Linear Systems Fall 2007 Howard Jay Chizeck University of Washington Seattle, Washington Ref: Text 11.6 ; Inclass Ch 11 notes pp. 1722 19,2223 Oct 2007 2 For LTI systems, described by linear constant coefficient difference equations ] [ ... ] 1 [ ] [ ] [ ... ] 1 [ ] [ 1 1 m k x b k x b k x b n k y a k y a k y m n − + + − + = − + + − + current and past outputs Using Z Transforms To Solve Difference Equations current and past inputs 19,2223 Oct 2007 3 Take Z transform of both sides (or one sided Z transform assuming zero initial conditions ): For LTI systems, described by linear constant coefficient difference equations ] ... )[ ( ] ... 1 )[ ( 1 1 1 1 m m n n z b z b b z X z a z a z Y − − − − + + + = + + + Using Z Transforms To Solve Difference Equations Delays become powers of z1 ] [ ... ] 1 [ ] [ ] [ ... ] 1 [ ] [ 1 1 m k x b k x b k x b n k y a k y a k y m n − + + − + = − + + − + 19,2223 Oct 2007 4 ] ... 1 [ ] ... [ ) ( ) ( ) ( 1 1 1 1 n n m m z a z a z b z b b z X z Y z H − − − − + + + + + + = = ] ... [ ] ... [ 1 1 1 1 n n n m n m n n a z a z z b z b z b + + + + + + = − − − for n ≥ m Solve for the Transfer Function H(z) by dividing: ] ... )[ ( ] ... 1 )[ ( 1 1 1 1 m m n n z b z b b z X z a z a z Y − − − − + + + = + + + 19,2223 Oct 2007 5 Poles and Zeros • Poles of H(z): roots of denominator polynomial • Zeros of H(z): roots of numerator polynomial note: find these after canceling any common factors—and do this for polynomials in z (not z1 ) 19,2223 Oct 2007 6 • Find the output of an LTI system in the Z domain, Y(z), by multiplying the z transform of the input, X(z) with H(z) = the Z transform of the impulse response • Then you can use the Inverse Z Transform to get the output signal y[k] from its Z transform, Y(z) Using Z Transforms To Solve Difference Equations 19,2223 Oct 2007 7 u(kT ) (right sided) 19,2223 Oct 2007...
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This note was uploaded on 04/07/2008 for the course EE EE 341 taught by Professor Chezick during the Spring '08 term at University of Washington.
 Spring '08
 Chezick

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