Time shifting property of z transform

Before we learn about time shifting property first let we learn about what is z transform system in digital signal processing subject.

Statement : 

              Z
If  x(n) ↔ X (Z)

                      Z
Then x (n-k) ↔ Z -k  X (Z)

Proof :
                                 -∞
So here Z{x(n)} = X  (Z) = ∑     x (n)  Z-n
                                 +∞


Z { x(n-k) }  can be written as,


                    -∞
Z{x(n-k)} = ∑     x (n-k)  Z-n
                     +∞

Now  Z-n can be  written as  Z-n  =   Z-(n-k) Z-k  thus the equation 

                    -∞
Z{x(n-k)} = ∑     x (n-k)  Z-(n-k) Z-k
                     +∞

Since the limits of summation are in terms of  'n'  we can take Z-k outside the summation sign.

                             -∞
so Z{x(n-k)} = Z-k  ∑     x (n-k)  Z-(n-k) 
                             +∞

Now put n-k =m on R.H.S 

At  n = -∞, -∞-k = m → m =  -∞
At n = ∞, ∞-k = m  → m =  ∞

                              ∞
Z{x(n-k)} = Z-k  ∑     x (m)  Z-(m) 
                          m = -∞ 
                  
Comparing Equation : 
                           
Z{x(n-k)} = Z-k   X (Z)


= x (n-k) ↔ Z-k   X (Z)

Similarly, we can write 

= x (n+k) ↔ Z+k   X (Z)

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