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)**