Advanced Z-transform
Encyclopedia
In mathematics
and signal processing
, the advanced Z-transform is an extension of the Z-transform
, to incorporate ideal delays that are not multiples of the sampling time
. It takes the form
where
It is also known as the modified Z-transform.
The advanced Z-transform is widely applied, for example to accurately model processing delays in digital control
.
.
If
then 
reduces to the Z-transform
which is clearly just the Z-transform of
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and signal processing
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...
, the advanced Z-transform is an extension of the Z-transform
Z-transform
In mathematics and signal processing, the Z-transform converts a discrete time-domain signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation....
, to incorporate ideal delays that are not multiples of the sampling time
Sampling rate
The sampling rate, sample rate, or sampling frequency defines the number of samples per unit of time taken from a continuous signal to make a discrete signal. For time-domain signals, the unit for sampling rate is hertz , sometimes noted as Sa/s...
. It takes the form
where
- T is the sampling period
- m (the "delay parameter") is a fraction of the sampling period
It is also known as the modified Z-transform.
The advanced Z-transform is widely applied, for example to accurately model processing delays in digital control
Digital control
Digital control is a branch of control theory that uses digital computers to act as system controllers.Depending on the requirements, a digital control system can take the form of a microcontroller to an ASIC to a standard desktop computer....
.
Properties
If the delay parameter, m, is considered fixed then all the properties of the Z-transform hold for the advanced Z-transform.Linearity
Damping
Time multiplication
Example
Consider the following example where.If
then reduces to the Z-transformZ-transform
In mathematics and signal processing, the Z-transform converts a discrete time-domain signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation....
which is clearly just the Z-transform of