ch. 6 z-transform and inverse z-transform, Lecture notes of Digital Signal Processing

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Chapter 6

Z Transform Z Transform

z -Transform

• (^) The DTFT provides a frequency-domain representation of discrete-time signals and LTI discrete-time systems
• (^) Because of the convergence condition, in many cases, the DTFT of a sequence may not exist
• (^) As a result, it is not possible to make use of such frequency-domain characterization in these cases

z -Transform

• (^) Consequently, z -transform has become an im portant tool in the analysis and design of digi tal filters
• (^) For a given sequence g [ n ], its z -transform G ( z ) is defined as where z = Re ( z )+jIm( z ) is a complex variable n n

G z g n z

    

( )  [ ]

z -Transform

• (^) The contour | z |=1 is a circle in the z -plane of unity radius and is called the unit circle
• (^) Like the DTFT, there are conditions on the convergence of the infinite series
• (^) For a given sequence, the set R of values of z for which its z -transform converges is called the region of convergence (ROC) n n

g n z

   

[ ]

z -Transform

• (^) Example - Determine the z -transform X ( z ) of the causal sequence x [ n ]= αnμ [ n ] and its ROC
• (^) Now The above power series converges to
• (^) ROC is the annular region | z | > |α| n n n n n n X z n z z            0 ( )  [ ]  , 1 1 1 ( ) 1 1      for z z X z  

z -Transform

• (^) Example - The z -transform μ ( z ) of the unit st ep sequence μ [ n ] can be obtained from by setting α =1:
• (^) ROC is the annular region 1<│z│≤∞ , 1 1 1 ( ) 1 1      for z z X z   , 1 1 1 ( ) 1 1      for z z  z

z -Transform

• (^) Its z -transform is given by
• (^) ROC is the annular | z | < |α|

1 1 1 1 0 1 1 1  

   ^           

for z z z z z z Y z z z m m m m m n m n n

z -Transform

• (^) Note: The z -transforms of the the two seque nces αnμ [ n ] and - αnμ [- n- 1] are identical even though the two parent sequen ces are different
• (^) Only way a unique sequence can be associa ted with a z -transform is by specifying its R OC

Rational z -Transforms

• (^) In the case of LTI discrete-time systems we are concerned with in this course, all pertin ent z- transforms are rational functions of z -
• (^) That is, they are ratios of two polynomials i n z -1: N N N N M M M M d d z d z d z p p z p z p z D z P z G z                     ( 1 ) 1 1 0 1 ( 1 ) 1 1 0 1 ( ) ( ) ( )  

Rational z -Transforms

• (^) The degree of the numerator polynomial P ( z ) is M and the degree of the denominator polynomial D ( z ) is N
• (^) An alternate representation of a rational z - transform is as a ratio of two polynomials in z: N N N N M M M M N M d z d z d z d p z p z p z p G z z               1 1 0 1 1 1 ( ) 0 1 ( )  

Rational z -Transforms

• (^) At a root z=ξℓ of the numerator pynomial G (ξℓ )=0, and as a result, these values of z are known as the zero s of G ( z )
• (^) At a root z =λ ℓ of the denominator polynom ial G (λℓ)→∞, and as a result, these values of z are known as the poles of G ( z )

Rational z -Transforms

• (^) Consider
• (^) Note G ( z ) has M finite zeros and N finite po les
• (^) If N > M there are additional N - M zeros at z =0 (the origin in the z -plane)
• (^) If N < M there are additional M - N poles at z =         N M N M d z p z G z z 1 0 ( )^01 ( ) ( ) ( )     

Rational z -Transforms

• (^) A physical interpretation of the concepts of poles and zeros can be given by plotting th e log-magnitude 20log 10 │ G ( z )│as shown on next slide for 1 2 1 2 1 0. 8 0. 64 1 2. 4 2. 88 ( )          z z z z G z

Rational z -Transforms