[<< wikibooks] Electronics Communication/Signal Processing/Signal Transformation/Z Transform
== Z Transform Properties ==
Initial value theorem
  
    
      
        x
        [
        0
        ]
        =
        
          lim
          
            z
            →
            ∞
          
        
        X
        (
        z
        )
         
      
    
    {\displaystyle x[0]=\lim _{z\rightarrow \infty }X(z)\ }
  , If 
  
    
      
        x
        [
        n
        ]
        
      
    
    {\displaystyle x[n]\,}
   causalFinal value theorem
  
    
      
        x
        [
        ∞
        ]
        =
        
          lim
          
            z
            →
            1
          
        
        (
        z
        −
        1
        )
        X
        (
        z
        )
         
      
    
    {\displaystyle x[\infty ]=\lim _{z\rightarrow 1}(z-1)X(z)\ }
  , Only if poles of 
  
    
      
        (
        z
        −
        1
        )
        X
        (
        z
        )
         
      
    
    {\displaystyle (z-1)X(z)\ }
  	 are inside unit circle


== Z Transform Table ==
Here:

  
    
      
        u
        [
        n
        ]
        =
        1
      
    
    {\displaystyle u[n]=1}
   for 
  
    
      
        n
        >=
        0
      
    
    {\displaystyle n>=0}
  , 
  
    
      
        u
        [
        n
        ]
        =
        0
      
    
    {\displaystyle u[n]=0}
   for 
  
    
      
        n
        <
        0
      
    
    {\displaystyle n<0}
  

  
    
      
        δ
        [
        n
        ]
        =
        1
      
    
    {\displaystyle \delta [n]=1}
   for 
  
    
      
        n
        =
        0
      
    
    {\displaystyle n=0}
  , 
  
    
      
        δ
        [
        n
        ]
        =
        0
      
    
    {\displaystyle \delta [n]=0}
   otherwise