# ece306-12 - The One-Side z-Transform The One-Side...

• Notes
• 10

This preview shows pages 1–4. Sign up to view the full content.

1 The One-Side z-Transform °± ²³´µ¶·´¸´¹º³» ¼ ³½¸¾¿ ´¸¶³ ¶ÀÁÂ¹ÃÄ»¾½¿¼Àº´À½½¿ ´Àº Å½Ä¶¿¾Ã½À¾ Â¶³ Æ¹³µ Æ¹Ã¹À¶ °±° ²³´ µ¶· °±° ²³´µ¶· · 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 0 ( ) ( ) n n X z x n z + - = = ° ( ) ( ) z x n X z + +

This preview has intentionally blurred sections. Sign up to view the full version.

2 °±° ²³´µ¶· 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 ( ) ( ) ( ) 0 x x n n k X z δ + + = + = 4 4 1 1 ( ) ( ) ( ) 1 x n x n a u n X z az + + - = = - °±° ²³´µ¶· ¸ 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 ( ) ( ) ( ) 0 k x k n n x n k z X z x n z k + - + = ± ² - + - > ³ ´ µ ° ( ) ( ) 0 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 + - - - - - - ± ² - = - + - + - ³ ´ - µ = + - + - -
3 °±° ²³´µ¶· ¹ The One-Side z-Transform Case 2: Time advance Example: if ( ) ( ) x x n X z + + Then 1 0 ( ) ( ) ( ) 0 k x k n n x n k z X z x n z k + - + - = ± ² + - > ³ ´ µ ° 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 + + - - - - ± ² + = + - - ³ ´ - µ = - - - = - - - x n x n a u n z x x z az z x z x z az z z az az ** °±° ²³´µ¶· ´ The One-Side z-Transform Final Value Theorem: This is exits if the ROC of includes the unit circle.

This preview has intentionally blurred sections. Sign up to view the full version.

This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern