ece306-9 - Solution of Linear Constant-Coefficient...

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1 Solution of Linear Constant-Coefficient Difference Equations °± ²³´µ¶·´¸´¹º³» ¼ ³½¸¾¿ ´¸¶³ ¶ÀÁ¹ÃÄ»¾½¿¼Àº´À½½¿ ´Àº ŽĶ¿¾Ã½À¾ ¶³ ƹ³µ ƹùÀ¶ °±° ²³´ µ¶ °±° ²³´µ¶ · Solution of Linear Constant-Coefficient Difference Equations Example: Determine the response of the system described by the second-order difference equation to the input The homogenous solution is ( ), 0 y n n ( ) 0.7 ( 1) 0.1 ( 2) 2 ( ) ( 2) y n y n y n x n x n = - - - + - - ( ) 4 ( ) n x n u n = ( ) 0.7 ( 1) 0.1 ( 2) 0 y n y n y n - - + - = 1 2 0.7 0.1 0 n n n λ λ λ - - - + = ( ) 2 2 0.7 0.1 0 n λ λ λ + - + = ° and 1 0.5 λ = 2 0.2 λ = ° 1 2 ( ) 0.5 0.2 n n h y n c c = +
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2 °±° ²³´µ¶ ² Solution of Linear Constant-Coefficient Difference Equations n=2 Particular Solution: ( ) 4 ( ) n p y n K u n = 1 2 2 4 ( ) 0.7 4 ( 1) 0.1 4 ( 2) (2)4 ( ) 4 ( 2) n n n n n K u n K u n K u n u n u n - - - - - + - = - - The total solution 2 1 0 2 0 4 0.7 4 0.1 4 2(4) 4 K K K - + = - 16 2.8 0.1 32 1 K K K - + = - 31 2.33 13.3 K = = ( ) 2.33(4) ( ) n p y n u n = 1 2 ( ) 0.5 0.2 2.33(4) ( ) n n n y n c c u n ° ± = + + ² ³ °±° ²³´µ¶ ¸ ¹ Solution of Linear Constant-Coefficient Difference Equations To find c1 and c2 For n=0: , From the total solution, From difference equation, From difference equation, (0) 0.7 (0 1) 0.1 (0 2) 2 (0) (0 2) (0) 2 y y y x x y = - - - + - - = 1 2 (0) 2.33 y c c = + + For n=1: From difference equation, (1) 0.7 (1 1) 0.1 (1 2) 2 (1) (1 2) (1) 1.4 8 9.4 y y y x x y = - - - + - - = + = 1 2 (1) 0.5 0.2 9.32 y c c = + + Therefore, 1 2 2 2.33 c c = + + 1 2 9.4 0.5 0.2 9.32 c c = + + 1 0.466 c = 2 0.807 c = - ( ) 0.466(0.5) 0.807(0.2) 2.33(4) ( ) n n n y n u n ° ± = - + ² ³ Total Solution
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3 °±° ²³´µ¶ º The Impulse Response of a LTI recursive system »¼½¾¼¾¿ÀÁÂÀþ »ÄÅƾǼÈÉÅ ÊÅƾ¼Ë¾ÌÍÅÀǼ The impulse response can be obtained from the linear constant- coefficient difference equation. That is the solution of homogeneous equation and particular solution to the excitation function. In the case where the excitation function is an impulse function. The particular solution is zero , since for n>0. 0
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  • Winter '08
  • Aliyazicioglu
  • LTI system theory, Impulse response, difference equation, total solution, linear constant-coefficient difference

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