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Although approximated
differentials are often used to convert continuous time systems
to discrete ones, by far the the most commonly used
tool for the analysis of sampled data systems is the z-transform.
Let us discover what this transformation is, and why it is better
to use this transform instead of the Backward
Difference Method or Bilinear
Transform.
Suppose we sample
a continuous time variable f(t). Because the sampled
signal exists only at the sampling instants, the sequence
of pulses can be represented mathematically as:
To simplify notation,
let the sampled sequence be denoted by  .
Recall that the Laplace
transform of f(t) is defined as:
Since the sampled
signal
is a subset of f(t), we can also apply the Laplace transform
to it, that is:
As
only exists at sampling instants, this means that we can replace
the integral with a summation, that is,
Defining:
then
This is the definition of the
z-transform of a continuous time signal f(t) sampled
with a sampling interval of Ts, i.e.
Thus, the
z-transform is merely the Laplace Transform of a sampled data
sequence and as such, inherits many of the properties
of the Laplace Transform.
Properties
Some of the more
important properties of the z-transforms are as follows:
Linearity
The z-transform
is a linear transform. That is, given constants a and
b and time variables f(t) and g(t):
z-transforms
of time delays
If f(t-kTs)
is f(t) delayed by k sampling intervals
(k is an integer), and f(t) = 0 for
t < 0, then the z-transform of f(t-kTs)
is given by:
Final
Value Theorem
This theorem
allows the calculation of the final value of a z-transformed
function and is stated as:
Relationship
with the s-plane
The mapping from
the s-plane to the z-plane is accomplished through the relationship
This function
maps the whole of the left side of the s-plane to a unit circle
on the z-plane:
In the case of Laplace transfer
functions, systems are stable if they do no possess poles on
the right half of the s-plane. In the case of sampled data systems,
they are stable if they do not possess poles that lie
outside the unit-circle in the z-plane.
Example
z-transforms
a) z-transform of a
unit step function
The time domain representation
of a unit step function is:
Thus 
That is,
which is an infinite series,
illustrating that the z-transforms operates on an
infinite sequence. Fortunately, there is also a 'closed-form'
equivalent, as it can be shown that
and this is the form that is
always presented in z-transform tables.
b) z-transform of an
exponential decay
The time function in this case
is:
Thus
which is another infinite series.
Again the closed-form solution is available and can be verified
by long division to be:
Inversion
of z-transforms
Like Laplace
transforms, z-transforms can be inverted back into the time
domain. Given a transfer function, we can either apply long
division to obtain the series form of the sampled signal or
make use of the tables. The first is simple but tedious and
the result may not be suitable for further analysis. Thus tables
are often used. In this case, the transfer function is factored
into lower order components using partial fraction expansion,
and tables are used to look up the corresponding time functions
of each component. The final result is obtained by adding up
these individual time function
However, due
to the nature of the problem, it is not often that z-transform
functions need to be converted back to the time domain. Given
the range of simulation tools available nowadays, it is usually
simpler to simulate the response of the discrete system to enable
visualisation of response characteristics.
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