ece306-9 - 1 Solution of Linear Constant-Coefficient...

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Unformatted text preview: 1 Solution of Linear Constant-Coefficient Difference Equations &¡ ¢£¤¥¦§¤¨¤©ª£« ¬ £­¨®¯ ¤¨¦£ ¦°±²©³´«®­¯¬°ª¤°­­¯ ¤°ª µ­´¦¯®³­°® ²¦£ ¶©£¥ ¶©³©°¦ &¡& ¢£¤ ¥¦ &¡& ¢£¤¥¦ § 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 ( ), 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) y n y n y n-- +- = 1 2 0.7 0.1 n n n λ λ λ--- + = ( ) 2 2 0.7 0.1 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 = + 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 2 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 & ¡ = + + ¢ £ &¡& ¢£¤¥¦ § ¨ 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 & ¡ =- + ¢ £ Total Solution 3 &¡& ¢£¤¥¦ § The Impulse Response of a LTI recursive system ¨©ª«©«¬­®¯­°« ¨±²³«´©µ¶² ·²³«©¸«¹º²­´© 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....
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This note was uploaded on 03/11/2008 for the course ECE 306 taught by Professor Aliyazicioglu during the Winter '08 term at Cal Poly Pomona.

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ece306-9 - 1 Solution of Linear Constant-Coefficient...

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