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The document summarizes key concepts about the z-transform, which is the discrete-time counterpart of the Laplace transform. It defines the z-transform, compares it to the discrete-time Fourier transform (DTFT), and discusses its region of convergence (ROC). Examples are provided to illustrate ROCs for different types of sequences, including right-sided, left-sided, and two-sided exponential sequences as well as a finite-length sequence. Common z-transform pairs are also listed.

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Department of Computer Eng. Sharif University of Technology Discrete-time signal processing Chapter 3: THE Z-TRANSFORM Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer and Buck, ©1999-2000 Prentice Hall Inc.
3.1 The Z-Transform • Counterpart of the Laplace transform for discrete-time signals • Generalization of the Fourier Transform Fourier Transform does not exist for all signals • Definition: • Compare to DTFT definition: • z is a complex variable that can be represented as z=r ej • Substituting z=ej will reduce the z-transform to DTFT Chapter 3: The Z-Transform 1           n n z n x z X     n j n j e n x e X        
           j n n z n n re z z n x z X z X n x z X z n x n x                 0 ) ( ) ( ) ( r : ‫اندازه‬ : ‫فاز‬  ‫تبدیل‬ z ‫طرفه‬‫یک‬ ‫تبدیل‬ z ‫طرفه‬‫دو‬ 3.1 The Z-Transform
The z-transform and the DTFT • Convenient to describe on the complex z-plane • If we plot z=ej for =0 to 2 we get the unit circle Chapter 3: The Z-Transform 3 Re Im Unit Circle  r=1 0 2 0 2    j e X
Convergence of the z-Transform • DTFT does not always converge Example: x[n] = anu[n] for |a|>1 does not have a DTFT • Complex variable z can be written as r ej so the z- transform convert to the DTFT of x[n] multiplied with exponential sequence r –n • For certain choices of r the sum maybe made finite Chapter 3: The Z-Transform 4                        n n j n n n j j e n x e n x re X    r r     n j n j e n x e X                 n n x r n -
Region of Convergence (ROC) • ROC: The set of values of z for which the z-transform converges • The region of convergence is made of circles Chapter 3: The Z-Transform 5 Re Im • Example: z-transform converges for values of 0.5<r<2 ROC is shown on the left In this example the ROC includes the unit circle, so DTFT exists
• Example: Doesn't converge for any r. DTFT exists. It has finite energy. DTFT converges in a mean square sense. • Example: Doesn't converge for any r. It doesn’t have even finite energy. But we define a useful DTFT with impulse function.     n n x o  cos    sin c n x n n    Region of Convergence (ROC)
Example 1: Right-Sided Exponential Sequence • For Convergence we require • Hence the ROC is defined as • Inside the ROC series converges to Chapter 3: The Z-Transform 7                        0 n n 1 n n n n az z n u a z X n u a n x       0 n n 1 az a z 1 az n 1         a z z az 1 1 az z X 0 n 1 n 1           Re Im a 1 o x • Region outside the circle of radius a is the ROC • Right-sided sequence ROCs extend outside a circle
( ‫ا‬‫ر‬‫چپگ‬‫دنباله‬ )           a z z az z a z X a z z a z a ROC z a z a z a z n u a z X n n n n n n n n n n n n                                                 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 : 1 1     1     n u a n x n Example 2: Left-Sided Exponential Sequence
Example 3: Two-Sided Exponential Sequence Chapter 3: The Z-Transform 9       1 - n - u 2 1 - n u 3 1 n x n n               1 1 1 0 1 0 1 3 1 1 1 3 1 1 3 1 3 1 3 1                                    z z z z z n n 1 1 0 1 1 1 n n 1 z 2 1 1 1 z 2 1 1 z 2 1 z 2 1 z 2 1                                   z 3 1 1 z 3 1 : ROC 1     z 2 1 1 z 2 1 : ROC 1                                  2 1 z 3 1 z 12 1 z z 2 z 2 1 1 1 z 3 1 1 1 z X 1 1 Im 2 1 oo 12 1 x x 3 1 
Example 4: Finite Length Sequence Chapter 3: The Z-Transform 10          otherwise 0 1 0 N n a n x n N=16 Pole-zero plot         N n u n u a n x n        0 : 1 1 1 ) ( 1 0 1 1 1 1 1 1 0 1 1 0                                 z az az ROC a z a z z az az az z a z X N n n N N N N N n n N n n n
Some common Z-transform pairs Chapter 3: The Z-Transform 11
SEQUENCE TRANSFORM ROC 1  z     0 m if or 0 m if 0 except z All    1  z 1 1 1   z 1 1 1   z m z         m n n u n u n       1 1 z ALL Some common Z-transform pairs
                    1 : cos 2 1 cos 1 cos : 1 1 : 1 : 1 1 1 : 1 1 2 1 0 1 0 0 2 1 1 2 1 1 1 1                                     z ROC z z z n u n a z ROC az az n u na a z ROC az az n u na a z ROC az n u a a z ROC az n u a Z Z n Z n Z n Z n    Some common Z-transform pairs
        0 : 1 1 0 1 0 : cos 2 1 sin sin 1 2 2 1 0 1 0 0                      z ROC az z a otherwise N n a r z ROC z r z r z r n u n r N N Z n Z n                    r z ROC z r z r z r n u n r z ROC z z z n u n Z n Z                  : cos 2 1 cos 1 cos 1 : cos 2 1 sin sin 2 2 1 0 1 0 0 2 1 0 1 0 0       Some common Z-transform pairs
3.2 Properties of The ROC of Z-Transform • The ROC is a ring or disk centered at the origin • DTFT exists if and only if the ROC includes the unit circle • The ROC cannot contain any poles • The ROC for finite-length sequence is the entire z-plane except possibly z=0 and z= • The ROC for a right-handed sequence extends outward from the outermost pole possibly including z=  • The ROC for a left-handed sequence extends inward from the innermost pole possibly including z=0 • The ROC of a two-sided sequence is a ring bounded by poles • The ROC must be a connected region • A z-transform does not uniquely determine a sequence without specifying the ROC Chapter 3: The Z-Transform 15
Stability, Causality, and the ROC • Consider a system with impulse response h[n] • The z-transform H(z) and the pole-zero plot shown below • Without any other information h[n] is not uniquely determined |z|>2 or |z|<½ or ½<|z|<2 • If system stable ROC must include unit-circle: ½<|z|<2 • If system is causal must be right sided: |z|>2 Chapter 3: The Z-Transform 16
3.4 Z-Transform Properties: Linearity • Notation • Linearity – Note that the ROC of combined sequence may be larger than either ROC – This would happen if some pole/zero cancellation occurs – Example: •Both sequences are right-sided •Both sequences have a pole z=a •Both have a ROC defined as |z|>|a| •In the combined sequence the pole at z=a cancels with a zero at z=a •The combined ROC is the entire z plane except z=0 Chapter 3: The Z-Transform 17     x Z R ROC z X n x              2 1 x x 2 1 Z 2 1 R R ROC z bX z aX n bx n ax               N - n u a - n u a n x n n 
Z-Transform Properties: Time Shifting • Here no is an integer – If positive the sequence is shifted right – If negative the sequence is shifted left • The ROC can change – The new term may add or remove poles at z=0 or z= • Example Chapter 3: The Z-Transform 18     x n Z o R ROC z X z n n x o          4 1 z z 4 1 1 1 z z X 1 1                      1 - n u 4 1 n x 1 - n       
Z-Transform Properties: Multiplication by Exponential • ROC is scaled by |zo| • All pole/zero locations are scaled • If zo is a positive real number: z-plane shrinks or expands • If zo is a complex number with unit magnitude it rotates • Example: We know the z-transform pair • Let’s find the z-transform of Chapter 3: The Z-Transform 19     x o o Z n o R z ROC z / z X n x z        1 z : ROC z - 1 1 n u 1 - Z                    n u re 2 1 n u re 2 1 n u n cos r n x n j n j o n o o          r z z re 1 2 / 1 z re 1 2 / 1 z X 1 j 1 j o o          
Z-Transform Properties: Differentiation • Example: We want the inverse z-transform of • Let’s differentiate to obtain rational expression • Making use of z-transform properties and ROC Chapter 3: The Z-Transform 20     x Z R ROC dz z dX z n nx           a z az 1 log z X 1         1 1 1 2 az 1 1 az dz z dX z az 1 az dz z dX                  1 n u a a n nx 1 n           1 n u n a 1 n x n 1 n    
Z-Transform Properties: Conjugation Chapter 3: The Z-Transform 21     x * * Z * R ROC z X n x                            n n n n n n n n n n X z x n z X z x n z x n z X z x n z x n z Z x n                                       
Z-Transform Properties: Time Reversal • ROC is inverted • Example: • Time reversed version of Chapter 3: The Z-Transform 22     x Z R 1 ROC z / 1 X n x           n u a n x n      n u an   1 1 1 - 1 -1 a z z a - 1 z a - az 1 1 z X       
Z-Transform Properties: Convolution • Convolution in time domain is multiplication in z-domain • Example: Let’s calculate the convolution of • Multiplications of z-transforms is • ROC: if |a|<1 ROC is |z|>1 if |a|>1 ROC is |z|>|a| • Partial fractional expansion of Y(z) Chapter 3: The Z-Transform 23         2 x 1 x 2 1 Z 2 1 R R : ROC z X z X n x n x               n u n x and n u a n x 2 n 1     a z : ROC az 1 1 z X 1 1       1 z : ROC z 1 1 z X 1 2              1 1 2 1 z 1 az 1 1 z X z X z Y         1 z : ROC assume 1 1 1 1 1 1 1               az a z a z Y         n u a n u a 1 1 n y 1 n   
Some Z-transform properties Chapter 3: The Z-Transform 24
3.3 The Inverse Z-Transform • Formal inverse z-transform is based on a Cauchy integral • Less formal ways sufficient most of the time – Inspection method – Partial fraction expansion – Power series expansion • Inspection Method Make use of known z-transform pairs such as Example: The inverse z-transform of Chapter 3: The Z-Transform 25   a z az 1 1 n u a 1 Z n              n u 2 1 n x 2 1 z z 2 1 1 1 z X n 1            
Inverse Z-Transform by Partial Fraction Expansion • Assume that a given z-transform can be expressed as • Apply partial fractional expansion • First term exist only if M>N – Br is obtained by long division • Second term represents all first order poles • Third term represents an order s pole – There will be a similar term for every high-order pole • Each term can be inverse transformed by inspection Chapter 3: The Z-Transform 26          N 0 k k k M 0 k k k z a z b z X                     s 1 m m 1 i m N i k , 1 k 1 k k N M 0 r r r z d 1 C z d 1 A z B z X
Inverse Z-Transform by Partial Fraction Expansion • Coefficients are given as • Easier to understand with examples Chapter 3: The Z-Transform 27                     s 1 m m 1 i m N i k , 1 k 1 k k N M 0 r r r z d 1 C z d 1 A z B z X     k d z 1 k k z X z d 1 A               1 i d w 1 s i m s m s m s i m w X w d 1 dw d d ! m s 1 C                
Example 5: 2nd Order Z-Transform Chapter 3: The Z-Transform 28   2 1 z : ROC z 2 1 1 z 4 1 1 1 z X 1 1                                       1 2 1 1 z 2 1 1 A z 4 1 1 A z X   1 4 1 2 1 1 1 z X z 4 1 1 A 1 4 1 z 1 1                                2 2 1 4 1 1 1 z X z 2 1 1 A 1 2 1 z 1 2                            
Example 5 Continued • ROC extends to infinity – Indicates right sided sequence Chapter 3: The Z-Transform 29   2 1 z z 2 1 1 2 z 4 1 1 1 z X 1 1                           n u 4 1 - n u 2 1 2 n x n n             
Example 6 • Long division to obtain Bo Chapter 3: The Z-Transform 30       1 z z 1 z 2 1 1 z 1 z 2 1 z 2 3 1 z z 2 1 z X 1 1 2 1 2 1 2 1                        1 z 5 2 z 3 z 2 1 z 2 z 1 z 2 3 z 2 1 1 1 2 1 2 1 2                   1 1 1 z 1 z 2 1 1 z 5 1 2 z X                  1 2 1 1 z 1 A z 2 1 1 A 2 z X          9 z X z 2 1 1 A 2 1 z 1 1                 8 z X z 1 A 1 z 1 2     
Example 5 Continued • ROC extends to infinity – Indicates right-sided sequence Chapter 3: The Z-Transform 31   1 z z 1 8 z 2 1 1 9 2 z X 1 1                 n 8u - n u 2 1 9 n 2 n x n         
Inverse Z-Transform by Power Series Expansion • The z-transform is power series • In expanded form • Z-transforms of this form can generally be inversed easily • Especially useful for finite-length series Chapter 3: The Z-Transform 32           n n z n x z X                          2 1 1 2 2 1 0 1 2 z x z x x z x z x z X
     1 2 1 1 1 2 z 2 1 1 z 2 1 z z 1 z 1 z 2 1 1 z z X                             1 n 2 1 n 1 n 2 1 2 n n x                                 2 n 0 1 n 2 1 0 n 1 1 n 2 1 2 n 1 n x Example 6

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Signals and systems3 ppt

  • 1. Department of Computer Eng. Sharif University of Technology Discrete-time signal processing Chapter 3: THE Z-TRANSFORM Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer and Buck, ©1999-2000 Prentice Hall Inc.
  • 2. 3.1 The Z-Transform • Counterpart of the Laplace transform for discrete-time signals • Generalization of the Fourier Transform Fourier Transform does not exist for all signals • Definition: • Compare to DTFT definition: • z is a complex variable that can be represented as z=r ej • Substituting z=ej will reduce the z-transform to DTFT Chapter 3: The Z-Transform 1           n n z n x z X     n j n j e n x e X        
  • 3.            j n n z n n re z z n x z X z X n x z X z n x n x                 0 ) ( ) ( ) ( r : ‫اندازه‬ : ‫فاز‬  ‫تبدیل‬ z ‫طرفه‬‫یک‬ ‫تبدیل‬ z ‫طرفه‬‫دو‬ 3.1 The Z-Transform
  • 4. The z-transform and the DTFT • Convenient to describe on the complex z-plane • If we plot z=ej for =0 to 2 we get the unit circle Chapter 3: The Z-Transform 3 Re Im Unit Circle  r=1 0 2 0 2    j e X
  • 5. Convergence of the z-Transform • DTFT does not always converge Example: x[n] = anu[n] for |a|>1 does not have a DTFT • Complex variable z can be written as r ej so the z- transform convert to the DTFT of x[n] multiplied with exponential sequence r –n • For certain choices of r the sum maybe made finite Chapter 3: The Z-Transform 4                        n n j n n n j j e n x e n x re X    r r     n j n j e n x e X                 n n x r n -
  • 6. Region of Convergence (ROC) • ROC: The set of values of z for which the z-transform converges • The region of convergence is made of circles Chapter 3: The Z-Transform 5 Re Im • Example: z-transform converges for values of 0.5<r<2 ROC is shown on the left In this example the ROC includes the unit circle, so DTFT exists
  • 7. • Example: Doesn't converge for any r. DTFT exists. It has finite energy. DTFT converges in a mean square sense. • Example: Doesn't converge for any r. It doesn’t have even finite energy. But we define a useful DTFT with impulse function.     n n x o  cos    sin c n x n n    Region of Convergence (ROC)
  • 8. Example 1: Right-Sided Exponential Sequence • For Convergence we require • Hence the ROC is defined as • Inside the ROC series converges to Chapter 3: The Z-Transform 7                        0 n n 1 n n n n az z n u a z X n u a n x       0 n n 1 az a z 1 az n 1         a z z az 1 1 az z X 0 n 1 n 1           Re Im a 1 o x • Region outside the circle of radius a is the ROC • Right-sided sequence ROCs extend outside a circle
  • 9. ( ‫ا‬‫ر‬‫چپگ‬‫دنباله‬ )           a z z az z a z X a z z a z a ROC z a z a z a z n u a z X n n n n n n n n n n n n                                                 1 1 1 0 1 1 0 1 1 1 1 1 1 1 1 1 : 1 1     1     n u a n x n Example 2: Left-Sided Exponential Sequence
  • 10. Example 3: Two-Sided Exponential Sequence Chapter 3: The Z-Transform 9       1 - n - u 2 1 - n u 3 1 n x n n               1 1 1 0 1 0 1 3 1 1 1 3 1 1 3 1 3 1 3 1                                    z z z z z n n 1 1 0 1 1 1 n n 1 z 2 1 1 1 z 2 1 1 z 2 1 z 2 1 z 2 1                                   z 3 1 1 z 3 1 : ROC 1     z 2 1 1 z 2 1 : ROC 1                                  2 1 z 3 1 z 12 1 z z 2 z 2 1 1 1 z 3 1 1 1 z X 1 1 Im 2 1 oo 12 1 x x 3 1 
  • 11. Example 4: Finite Length Sequence Chapter 3: The Z-Transform 10          otherwise 0 1 0 N n a n x n N=16 Pole-zero plot         N n u n u a n x n        0 : 1 1 1 ) ( 1 0 1 1 1 1 1 1 0 1 1 0                                 z az az ROC a z a z z az az az z a z X N n n N N N N N n n N n n n
  • 12. Some common Z-transform pairs Chapter 3: The Z-Transform 11
  • 13. SEQUENCE TRANSFORM ROC 1  z     0 m if or 0 m if 0 except z All    1  z 1 1 1   z 1 1 1   z m z         m n n u n u n       1 1 z ALL Some common Z-transform pairs
  • 14.                     1 : cos 2 1 cos 1 cos : 1 1 : 1 : 1 1 1 : 1 1 2 1 0 1 0 0 2 1 1 2 1 1 1 1                                     z ROC z z z n u n a z ROC az az n u na a z ROC az az n u na a z ROC az n u a a z ROC az n u a Z Z n Z n Z n Z n    Some common Z-transform pairs
  • 15.         0 : 1 1 0 1 0 : cos 2 1 sin sin 1 2 2 1 0 1 0 0                      z ROC az z a otherwise N n a r z ROC z r z r z r n u n r N N Z n Z n                    r z ROC z r z r z r n u n r z ROC z z z n u n Z n Z                  : cos 2 1 cos 1 cos 1 : cos 2 1 sin sin 2 2 1 0 1 0 0 2 1 0 1 0 0       Some common Z-transform pairs
  • 16. 3.2 Properties of The ROC of Z-Transform • The ROC is a ring or disk centered at the origin • DTFT exists if and only if the ROC includes the unit circle • The ROC cannot contain any poles • The ROC for finite-length sequence is the entire z-plane except possibly z=0 and z= • The ROC for a right-handed sequence extends outward from the outermost pole possibly including z=  • The ROC for a left-handed sequence extends inward from the innermost pole possibly including z=0 • The ROC of a two-sided sequence is a ring bounded by poles • The ROC must be a connected region • A z-transform does not uniquely determine a sequence without specifying the ROC Chapter 3: The Z-Transform 15
  • 17. Stability, Causality, and the ROC • Consider a system with impulse response h[n] • The z-transform H(z) and the pole-zero plot shown below • Without any other information h[n] is not uniquely determined |z|>2 or |z|<½ or ½<|z|<2 • If system stable ROC must include unit-circle: ½<|z|<2 • If system is causal must be right sided: |z|>2 Chapter 3: The Z-Transform 16
  • 18. 3.4 Z-Transform Properties: Linearity • Notation • Linearity – Note that the ROC of combined sequence may be larger than either ROC – This would happen if some pole/zero cancellation occurs – Example: •Both sequences are right-sided •Both sequences have a pole z=a •Both have a ROC defined as |z|>|a| •In the combined sequence the pole at z=a cancels with a zero at z=a •The combined ROC is the entire z plane except z=0 Chapter 3: The Z-Transform 17     x Z R ROC z X n x              2 1 x x 2 1 Z 2 1 R R ROC z bX z aX n bx n ax               N - n u a - n u a n x n n 
  • 19. Z-Transform Properties: Time Shifting • Here no is an integer – If positive the sequence is shifted right – If negative the sequence is shifted left • The ROC can change – The new term may add or remove poles at z=0 or z= • Example Chapter 3: The Z-Transform 18     x n Z o R ROC z X z n n x o          4 1 z z 4 1 1 1 z z X 1 1                      1 - n u 4 1 n x 1 - n       
  • 20. Z-Transform Properties: Multiplication by Exponential • ROC is scaled by |zo| • All pole/zero locations are scaled • If zo is a positive real number: z-plane shrinks or expands • If zo is a complex number with unit magnitude it rotates • Example: We know the z-transform pair • Let’s find the z-transform of Chapter 3: The Z-Transform 19     x o o Z n o R z ROC z / z X n x z        1 z : ROC z - 1 1 n u 1 - Z                    n u re 2 1 n u re 2 1 n u n cos r n x n j n j o n o o          r z z re 1 2 / 1 z re 1 2 / 1 z X 1 j 1 j o o          
  • 21. Z-Transform Properties: Differentiation • Example: We want the inverse z-transform of • Let’s differentiate to obtain rational expression • Making use of z-transform properties and ROC Chapter 3: The Z-Transform 20     x Z R ROC dz z dX z n nx           a z az 1 log z X 1         1 1 1 2 az 1 1 az dz z dX z az 1 az dz z dX                  1 n u a a n nx 1 n           1 n u n a 1 n x n 1 n    
  • 22. Z-Transform Properties: Conjugation Chapter 3: The Z-Transform 21     x * * Z * R ROC z X n x                            n n n n n n n n n n X z x n z X z x n z x n z X z x n z x n z Z x n                                       
  • 23. Z-Transform Properties: Time Reversal • ROC is inverted • Example: • Time reversed version of Chapter 3: The Z-Transform 22     x Z R 1 ROC z / 1 X n x           n u a n x n      n u an   1 1 1 - 1 -1 a z z a - 1 z a - az 1 1 z X       
  • 24. Z-Transform Properties: Convolution • Convolution in time domain is multiplication in z-domain • Example: Let’s calculate the convolution of • Multiplications of z-transforms is • ROC: if |a|<1 ROC is |z|>1 if |a|>1 ROC is |z|>|a| • Partial fractional expansion of Y(z) Chapter 3: The Z-Transform 23         2 x 1 x 2 1 Z 2 1 R R : ROC z X z X n x n x               n u n x and n u a n x 2 n 1     a z : ROC az 1 1 z X 1 1       1 z : ROC z 1 1 z X 1 2              1 1 2 1 z 1 az 1 1 z X z X z Y         1 z : ROC assume 1 1 1 1 1 1 1               az a z a z Y         n u a n u a 1 1 n y 1 n   
  • 25. Some Z-transform properties Chapter 3: The Z-Transform 24
  • 26. 3.3 The Inverse Z-Transform • Formal inverse z-transform is based on a Cauchy integral • Less formal ways sufficient most of the time – Inspection method – Partial fraction expansion – Power series expansion • Inspection Method Make use of known z-transform pairs such as Example: The inverse z-transform of Chapter 3: The Z-Transform 25   a z az 1 1 n u a 1 Z n              n u 2 1 n x 2 1 z z 2 1 1 1 z X n 1            
  • 27. Inverse Z-Transform by Partial Fraction Expansion • Assume that a given z-transform can be expressed as • Apply partial fractional expansion • First term exist only if M>N – Br is obtained by long division • Second term represents all first order poles • Third term represents an order s pole – There will be a similar term for every high-order pole • Each term can be inverse transformed by inspection Chapter 3: The Z-Transform 26          N 0 k k k M 0 k k k z a z b z X                     s 1 m m 1 i m N i k , 1 k 1 k k N M 0 r r r z d 1 C z d 1 A z B z X
  • 28. Inverse Z-Transform by Partial Fraction Expansion • Coefficients are given as • Easier to understand with examples Chapter 3: The Z-Transform 27                     s 1 m m 1 i m N i k , 1 k 1 k k N M 0 r r r z d 1 C z d 1 A z B z X     k d z 1 k k z X z d 1 A               1 i d w 1 s i m s m s m s i m w X w d 1 dw d d ! m s 1 C                
  • 29. Example 5: 2nd Order Z-Transform Chapter 3: The Z-Transform 28   2 1 z : ROC z 2 1 1 z 4 1 1 1 z X 1 1                                       1 2 1 1 z 2 1 1 A z 4 1 1 A z X   1 4 1 2 1 1 1 z X z 4 1 1 A 1 4 1 z 1 1                                2 2 1 4 1 1 1 z X z 2 1 1 A 1 2 1 z 1 2                            
  • 30. Example 5 Continued • ROC extends to infinity – Indicates right sided sequence Chapter 3: The Z-Transform 29   2 1 z z 2 1 1 2 z 4 1 1 1 z X 1 1                           n u 4 1 - n u 2 1 2 n x n n             
  • 31. Example 6 • Long division to obtain Bo Chapter 3: The Z-Transform 30       1 z z 1 z 2 1 1 z 1 z 2 1 z 2 3 1 z z 2 1 z X 1 1 2 1 2 1 2 1                        1 z 5 2 z 3 z 2 1 z 2 z 1 z 2 3 z 2 1 1 1 2 1 2 1 2                   1 1 1 z 1 z 2 1 1 z 5 1 2 z X                  1 2 1 1 z 1 A z 2 1 1 A 2 z X          9 z X z 2 1 1 A 2 1 z 1 1                 8 z X z 1 A 1 z 1 2     
  • 32. Example 5 Continued • ROC extends to infinity – Indicates right-sided sequence Chapter 3: The Z-Transform 31   1 z z 1 8 z 2 1 1 9 2 z X 1 1                 n 8u - n u 2 1 9 n 2 n x n         
  • 33. Inverse Z-Transform by Power Series Expansion • The z-transform is power series • In expanded form • Z-transforms of this form can generally be inversed easily • Especially useful for finite-length series Chapter 3: The Z-Transform 32           n n z n x z X                          2 1 1 2 2 1 0 1 2 z x z x x z x z x z X
  • 34.      1 2 1 1 1 2 z 2 1 1 z 2 1 z z 1 z 1 z 2 1 1 z z X                             1 n 2 1 n 1 n 2 1 2 n n x                                 2 n 0 1 n 2 1 0 n 1 1 n 2 1 2 n 1 n x Example 6