Process of sampling
Consider
a continuous time signal
.
Its one sided Laplace transform is defined as :
If
the continuous time signal
is
uniformly sampled with a train of impulses to get a discrete time
signal
,
then it can be represented as:
where
is
the sampling interval.
Now the Laplace transform of the sampled signal (discrete time) is called Star transform and is given by:
It
can be seen that the Laplace
transform of an impulse sampled
signal is the star
transform and is the same as
the Z
transform of the corresponding
sequence when
.
Similar relationship holds when a continuous time system is converted
into a sampled data system by cascading an actual impulse sampler at
the input and a fictitious impulse sampler at the output.[7]
Relationship to Fourier transform
The
Z-transform is a generalization of the discrete-time
Fourier transform (DTFT). The
DTFT can be found by evaluating the Z-transform
at
(where
is
the normalized
frequency) or, in other words,
evaluated on the unit circle. In order to determine the frequency
response of the system the
Z-transform must be evaluated on the unit circle, meaning that the
system's region of convergence must contain the unit circle.
Otherwise, the DTFT of the system does not exist.
Linear constant-coefficient difference equation
The linear constant-coefficient difference (LCCD) equation is a representation for a linear system based on the autoregressive moving-average equation.
Both
sides of the above equation can be divided by
,
if it is not zero, normalizing
and
the LCCD equation can be written
This
form of the LCCD equation is favorable to make it more explicit that
the "current" output
is
a function of past outputs
,
current input
,
and previous inputs
.
Transfer function
Taking the Z-transform of the above equation (using linearity and time-shifting laws) yields
and rearranging results in
Zeros and poles
From the fundamental theorem of algebra the numerator has M roots (corresponding to zeros of H) and the denominator has N roots (corresponding to poles). Rewriting the transfer function in terms of poles and zeros
where
is
the
zero
and
is
the
pole.
The zeros and poles are commonly complex and when plotted on the
complex plane (z-plane) it is called the pole-zero
plot.
In
addition, there may also exist zeros and poles at
and
.
If we take these poles and zeros as well as multiple-order zeros and
poles into consideration, the number of zeros and poles are always
equal.
By factoring the denominator, partial fraction decomposition can be used, which can then be transformed back to the time domain. Doing so would result in the impulse response and the linear constant coefficient difference equation of the system.
Output response
If
such a system
is
driven by a signal
then
the output is
.
By performing partial
fraction decomposition on
and
then taking the inverse Z-transform the output
can
be found. In practice, it is often useful to fractionally decompose
before
multiplying that quantity by
to
generate a form of
which
has terms with easily computable inverse Z-transforms.
See also
Advanced Z-transform
Star transform
Bilinear transform
Finite impulse response
Formal power series
Laplace transform
Laurent series
Probability-generating function
Zeta function regularization
Discrete-time Fourier transform
Difference equation (recurrence relation)
Discrete convolution