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Digital signals and sampled systems

Discrete-time systems

In discrete-time systems variables are sampled and quantized (we will not consider sample-data signals that can take any value into a range). Then, variables will change every T seconds. T is called sample time. A typical response of a discrete time system will look like in the figure:

The sample and hold process of a continuous signal is represented in the following figure:

If we consider the values of the resulting signal of the sampling process of x(t), x(kT):

x(0*T), x(1*T), x(2*T),...

The Z-transform of x(kT) is defined as
X(z)=Z(x(kT))=x(0*T) z^0+x(1*T) z^(-1)+…=∑n=0∞ [x(nt)z^(-n)]
where z is a complex-variable. 

Relation between s and z plane

All continuous functions that have Laplace Transform, also have Z transform. The relation between both planes is given by teh expression  :

 

Inverse transform 

The inverse transform of a variable can be done from the definition of the Z-transform. However, if the signal is a polynomial fraction, that is common in control engineering, the inverse transform can be obtained using fraction decomposition and then uusing the Z transform tables for the resulting single terms.

Properties of the Z-transfer function

Some of the properties of the Z-trasform are presented. These have special relevance from the modelling and control design perspective.

Linearity


Multiplying by a constant Multiplicación por una constante:

Time shift: