Introduction:

The discretization is the process of converting the continuous system into discrete system. There are different discretization methods used for converting the continous system into discrete system. Some methods tend to provide a better frequency-domain match between the original and converted systems, while others provide a better match in the time domain.
Following are the different methods:
1. Zero Order Hold
2. First Order Hold
3. Impulse Invariance method
4. Tustin Approximation
5. Pole-Zero Mapping
The application of z-transform gives an extremely useful mathematical description of a linear time invariant discrete time system. The transfer function of a linear time invariant discrete time system is, by defination, the z-transform of the impulse response of the system. $$ G(z) = \frac{Z[y(k)]}{Z[r(k)]} = \frac {Y(z)}{R(z)} $$ when system is initially relaxed.
Thus, the transfer function of a linear time invariant discrete time sytem is the ratio of the z-transforms of its output and input sequences, assuming that the system is initially relaxed.