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

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1 The One-Side z-Transform °± ²³´µ¶·´¸´¹º³» ¼ ³½¸¾¿ ´¸¶³ ¶ÀÁ¹ÃÄ»¾½¿¼Àº´À½½¿ ´Àº ŽĶ¿¾Ã½À¾ ¶³ ƹ³µ ƹùÀ¶ °±° 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 + +
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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 = + - + - -
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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.
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