z – Transform

The z-transform is a very powerful tool for the analysis of discrete time signals and LTI systems. The z-transform of a discrete-time signal x(n) is defined as the power series

Z[x(n)] = X(z) = SIGMA x(n)z(power)-n …..(i)

where z is a complex variable. This expression is generally referred to as two-sided z-transform.

If x(n) is a causal sequence, x(n) = 0 for n<0, then its z-transform is

X(z) = SIGMA x(n)z(power)-n

This expression is also called a one-sided z-transform.

The equation (i) is sometimes called the direct z-transform because it transform the time-domain signal x(n) into its complex plalne representation X(z).

There are following uses of z-transform as given below —

(i) Stability of discrete time LTI system.

(ii) Causality of discrete time LTI system.

(iii) System behaviour related to the system function of a discrete time LTI system.