ece306-12 - 1 The One-Side z-Transform&...

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Unformatted text preview: 1 The One-Side z-Transform &¡ ¢£¤¥¦§¤¨¤©ª£« ¬ £­¨®¯ ¤¨¦£ ¦°±²©³´«®­¯¬°ª¤°­­¯ ¤°ª µ­´¦¯®³­°® ²¦£ ¶©£¥ ¶©³©°¦ &¡& ¢£¤ ¥¦§ &¡& ¢£¤¥¦§ § The One-Side z-Transform The one-sided z-transform of a signal x(n) is defined as The one-sided z-transform has the following characteristics: 1. It does not contain information about the signal x(n) for negative values of time (i.e., for n<0) 2. It is unique only for causal signals, because only these signals are zero for n<0. 3. The one-sided z-transform X + (z) of x(n) is identical to the two-sided z-transform of the signal x(n)u(n). 4. ROC of X + (z) is always the exterior of the circle. So it is not necessary to refer to their ROC ( ) ( ) n n X z x n z ∞ +- = = & ( ) ( ) z x n X z + + ↔ 2 &¡& ¢£¤¥¦§ ¢ The One-Side z-Transform Example: 1 2 3 5 1 1 ( ) {1,2,5,7,0,1} ( ) 1 2 5 7 x x n X z z z z z + +---- = ↔ = + + + + ↑ 1 3 2 2 ( ) {1,2,5,7,0,1} ( ) 5 7 z x n X z z z + +-- = ↔ = + + ↑ 2 3 4 5 7 3 3 ( ) {0,0,1,2,5,7,0,1} ( ) 1 2 5 7 x x n X z z z z z z + +----- = ↔ = + + + + ↑ , 4 4 ( ) ( ) ( ) 1 x x n n X z δ + + = ↔ = 5 5 ( ) ( ) ( ) x k x n n k X z z δ + +- =- ↔ = 6 6 ( ) ( ) ( ) x x n n k X z δ + + = + ↔ = 4 4 1 1 ( ) ( ) ( ) 1 x n x n a u n X z az + +- = ↔ =- &¡& ¢£¤¥¦§ ¨ The One-Side z-Transform if Shifting Property Case 1:Time Delay In case x(n) is causal, then Example: The z-transform of ,. … ( ) ( ) x x n X z + + ↔ Then 1 ( ) ( ) ( ) k x k n n x n k z X z x n z k +- + = & ¡- ↔ +- > ¢ £ ¤ ¥ ¦ ( ) ( ) x k x n k z X z k +- +- ↔ > 1 1 ( ) ( ) ( ) 1 x n x n a u n X z az + +- = ↔ =- 1 ( ) ( 2) x n x n =- 2 2 2 1 2 1 1 1 ( 2) ( 2) ( 1) ( 2) 1 ( 1) ( 2) 1 x n x n a u n z x z x z az z x z x az +------ & ¡- =- ↔ +- +- ¢ £- ¤ ¥ = +- +-- 3 &¡& ¢£¤¥¦§ ¨ The One-Side z-Transform Case 2: Time advance Example: if ( ) ( ) x x n X z + + ↔ Then 1 ( ) ( ) ( ) k x k n n x n k z X z x n z k +- +- = & ¡ + ↔- > ¢ £ ¤ ¥ ¦ 1 1 ( ) ( ) ( ) 1 x n x n a u n X z az + +- = ↔ =- 1 ( ) ( 2) x n x n = + The z-transform of 2 2 1 1 2 2 1 2 2 1 1 ( 2) ( 2) (0) (1) 1 (0) (1) 1 1 + +---- & ¡ + = + ↔-- ¢ £- ¤ ¥ =--- =--- x n x n a u n z x x z az z x z x z az z z az az ** &¡& ¢£¤¥¦§ ¤ The One-Side z-Transform Final Value Theorem: This is exits if the ROC of includes the unit circle....
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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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ece306-12 - 1 The One-Side z-Transform&...

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