The Z-Transform
The z-transform maps discrete-time sequences to functions of a complex variable, turning difference equations into algebra.
The Discrete Laplace Transform
The z-transform is to discrete-time systems what the Laplace transform is to continuous ones. For a sequence x[n], its z-transform is the sum over n of x[n] times z^(-n), where z is a complex variable. Difference equations become algebraic equations in z, giving discrete transfer functions on which digital control design is built.
Key properties
- Linearity: the transform of a sum is the sum of the transforms.
- Time shift: delaying a sequence by one sample multiplies its transform by z^(-1), so z^(-1) is the unit-delay operator.
- Convolution: convolution in time becomes multiplication in the z-domain.
- Final value theorem: the steady-state value follows from a limit of (z - 1) times X(z) as z approaches one.
The unit circle and stability
The stability boundary for discrete systems is the unit circle in the z-plane. A discrete LTI system is stable if and only if all poles of its z-domain transfer function lie strictly inside the unit circle. This mirrors the continuous requirement of left-half-plane poles, connected by the mapping z = e^(s*T), which sends the imaginary axis to the unit circle and the left half-plane to the disk interior.
From difference equation to transfer function
A difference equation such as y[n] = a*y[n-1] + b*u[n] transforms, using the shift property, into Y(z) = a*z^(-1)*Y(z) + b*U(z), giving the transfer function Y(z)/U(z) = b/(1 - a*z^(-1)). The pole at z = a is inside the unit circle, and the system is stable, when the magnitude of a is less than one.
Inverse and implementation
Recovering the time sequence from a z-domain expression is done by partial-fraction expansion, long division, or table lookup. Crucially, a discrete transfer function translates directly into code: it is a recipe for computing each new output from past inputs and outputs, exactly the difference equation a digital controller executes each sample.
The z-transform is thus both the analysis tool for sampled systems and the bridge to implementation, making it indispensable to digital control.