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