华侨大学：《信号与系统》课程教学资源（PPT课件讲稿，机电版）证明3-7 终值定理证明

终值定理 若:因果信号x()+X(z), 则imx(n)=lim[(二-1)X(=) 证明::z{x(n+1)-x(m)}=zX(z)-x(0)-X(z) =(=-1)X(=)-2x(0 lim[(z-DX(z]=lime[(n+1)x(n)]+zx(0) z→1 x(0)+lim 2[x(n+1)x(n)]= x(0)+[x(1)-x(O)+[x(2)-x(1)]+…+[x(m)-x(n-1)]+ lim[(z-1)X(zl X(oo n

X 第 1 终值定理 页 若: 因果信号x(n) → X(z), 1 lim ( ) lim[( 1) ( )] n z x n z X z → → = − Z x n x n zX z zx X z { ( 1) ( )} ( ) (0) ( ) + − = − − 1 1 lim[( 1) ( )] lim{ [ ( 1) ( ) } ] (0) z Z z X z Z x n x n zx → →  − = + − + = − − ( 1) ( ) (0) z X z zx 则： 证明: 1 lim[( 1) ( )] ( ) z z X z x → − =  1 0 (0) lim [ ( 1) ( )] n z n x x n x n z  − → = = + +  − = + − + − + + − − + x( ) 0 [ (1) (0)] [ (2) (1)] [ ( ) ( 1 x x x x x n x n )] n→∞