Digital Control Systems
The z-Transform
The z Transform
Definition of z-Transform
The z transform method is an operational method that is very powerful when working with discrete time
systems.
In considering the z-transform of a time function x(t), we consider only the sampled values of x(t), that is:
x(0), x(T),x(2T),x(3T)….
The z-Transform of a time function x(t), where t is nonnegative, or of a sequence of values x(kT), where k
takes zero or positive integers and T is sampling period is defined by the following equation:
For a sequence of numbers x(k), the z-transform is defined by
In the one sided z transform we assume x(t)=0 for t<0 or x(k)=0 for k<0. Note that z is a complex variable
The z Transform
Definition of z-Transform
Note that, when dealing with a time sequence x(kT) obtained by sampling a time signal x(t), the z transform
X(z) involves T explicitly. However for a number sequence x(k), the z transform X(z) does not involve T
explicitly.
The z transform of x(t), where −∞ <t<∞ or of x(k), where k takes integer values (k=0,±1, ±2, . .) is defined by
In the two sided z transform, the time function x(t) is assumed to be nonzero values for k<0 we assume
x(t)=0 for t<0 or x(k)=0 for k<0. Note that z is a complex variable
Z Transforms of Elementary Functions
By definition
Step Function
𝛿(𝑘) Impulse Function
Z Transforms of Elementary Functions
Decaying Exponential Function
Damped Cosine Wave
Damped Cosine Wave
Z Transforms of Elementary Functions
Unit Ramp Function
Z Transforms of Elementary Functions
Polynomial Function ak
Table of Z Transforms
Z-Transform Examples
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Z Transform Examples
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Properties of the z-Transform
Linearity
Time Shift
Multiplication by an Exponential Sequence
Differentiation in z-Domain
Properties of the z-Transform
Time Reversal
Initial Value Theorem
Convolution
Properties of the z-Transform
Example:
Properties of the z-Transform
Example
Find the Z-transform of the causal sine sequence.
Properties of the z-Transform
Inverse z-Transform
The inverse Z-transform can yield the
corresponding time sequence f(kt) uniquely.
However, it says nothing about f(t). There might
be numerous f(t) for a given f(kT).
Other than referring to z transform tables, four methods for obtaining the inverse z transform are available:
• Direct Division Method
• Computational Method
• Partial Fraction Expansion Method
• Inversion Integral Method
Inverse z-Transform
Direct Division Method
Express X(z) in powers of z−1
Inverse z-Transform
Computational Method-Difference Equation Approach
Example:
Inverse z-Transform
Computational Method-Difference Equation Approach
Example:
Inverse z-Transform
Computational Method-Difference Equation Approach
Example(cntd.):
Finding difference equation from TF.
By substituting k=-2 into eqn., we find y(0)=0
By substituting k=-1 into eqn., we find y(1)=0.4673
By substituting k=0 into eqn., we find y(2)=0.3769
. ……………………………….
………
Inverse z-Transform
Computational Method-Matlab Approach
Example:
Inverse z-Transform
Computational Method-Matlab Approach
Example (cntd.):
Inverse z-Transform
Partial Fraction Expansion
Inverse z-Transform
Example:
z-Transform Method for Solving Difference Equation
Time shift Property of Z-transform
z-Transform Method for Solving Difference Equation
Example:
z-Transform Method for Solving Difference Equation
Example: