In this section, we will study the properties of the Z transform. We know that x (n) and X (Z) are Z transform pair and denoted as,
Z
If x (n) ↔ X(Z)
1. Linearity :
In this property
In this property
Z Z
If x1 (n) is ↔ X1(Z) and x2 (n) ↔ X2(Z) then
DTFT
a1 x1 (n) + a2 x2 (n) ↔ a1 X1 (Z) + a2 X2 (Z)
2. Time shifting :
Z
x (n) is ↔ X(Z) then
Z
x(n-k) ↔ Z –k X(Z)
3. Scaling in the Z domain :
If
Z Z
x (n) is ↔ X(Z) then a n x(n) ↔ X ( Z / a)
4. Time reversal :
If
Z
x (n) is ↔ X(Z) then
Z
x (-n) ↔ X(Z-1)
5. Differentiation :
If
Z
x (n) is ↔ X(Z) then
Z
n x (n) ↔ -Z dX(Z) / dZ
6. Convolution theorem :
Z Z
If x1 (n) is ↔ X1(Z) and x2 (n) ↔ X2(Z) then
Z
x1 (n) * x2 (n) is ↔ X1(Z) . X2(Z)