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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 () 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 * Z Transform Examples * * 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: