## Pages

### 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)
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)