Z-Transforms and Discrete-Time Systems

Introduction

Modern mechatronic systems rarely operate exclusively in continuous time. Sensors sample position, velocity, temperature, pressure, and current; digital controllers calculate commands; and actuators are driven through digitally generated signals. A continuous-time plant therefore becomes part of a sampled-data system containing analog-to-digital conversion, computation, zero-order holding, and digital-to-analog or pulse-width-modulation interfaces.

The Z-transform is the principal mathematical tool for analyzing linear discrete-time systems. It converts difference equations into algebraic equations, provides a systematic method for solving initial-value problems, and supports the analysis of poles, zeros, frequency response, causality, and stability. In engineering mathematics, it plays a role analogous to that of the Laplace transform for continuous-time systems.

A discrete-time signal is a sequence indexed by an integer, normally written x[k], where k is the sample number. If the sampling period is T_s seconds, the corresponding physical sampling instant is t=kT_s. The sequence may represent a measured signal, a control command, or a computed state.

Learning objectives

By the end of this chapter, you should be able to:

• define the unilateral and bilateral Z-transforms and use their properties;
• calculate inverse Z-transforms using tables, long division, partial fractions, and power-series methods;
• formulate and solve linear constant-coefficient difference equations;
• interpret discrete convolution as the input–output operation of an LTI system;
• identify poles and zeros and relate them to transient response and stability;
• analyze sampled-data control and embedded mechatronic applications;
• recognize implementation limitations, numerical issues, and safety implications.

1. Discrete-time systems and notation

A discrete-time system maps an input sequence x[k] to an output sequence y[k]. It is linear if superposition holds:

for arbitrary constants a and b. It is time invariant if delaying the input by k₀ delays the output by the same amount. A system is causal if y[k] depends only on present and past inputs and states, not future input values. It is BIBO stable if every bounded input produces a bounded output.

The unit impulse, or Kronecker delta, is

The unit step is u[k]=1 for k≥0 and zero otherwise. The discrete-time impulse is especially important because the impulse response completely characterizes an LTI system.

For a sampled physical signal, the notation x[k]=x(kT_s) is often used. Sampling introduces the possibility of aliasing: frequency components above half the sampling frequency cannot be uniquely distinguished after sampling. The Nyquist frequency is f_s/2, where f_s=1/T_s. An anti-aliasing low-pass filter should therefore be used before the ADC.

2. The Z-transform

The bilateral, or two-sided, Z-transform of x[k] is

where z is a complex variable. The set of values of z for which the sum converges is the region of convergence (ROC). The transform is not fully specified by an algebraic expression alone; the ROC is also needed, particularly when determining causality and stability.

The unilateral, or one-sided, Z-transform is

It is useful for causal systems with specified initial conditions. In this chapter, X(z) denotes the bilateral transform unless the unilateral form is explicitly stated.

Common transform pairs

For example, the geometric-series identity gives

The ROC condition is the convergence condition. It is not an optional annotation.

Important properties

Linearity:

Time shifting, for a bilateral transform:

Convolution:

where

Multiplication by the index is represented by differentiation:

For the unilateral transform, the delay property includes initial terms. For example,

and

These terms are essential when solving initial-value problems.

3. Inverse Z-transforms

The inverse transform recovers a sequence from X(z). The result depends on the ROC. Three practical methods are particularly important.

Power-series method

Write X(z) as a series in z^-1:

The coefficient of z^-k is x[k]. This method is useful for numerical or asymptotic inspection.

Partial fractions

Suppose

Assume a causal ROC, |z|>0.8. Write

Equating coefficients gives A+B=1 and 0.2A+0.8B=0.5. Hence A=0.5 and B=0.5. Therefore

The sequence is causal and decays because both pole magnitudes are less than one.

Long division

For a causal sequence, expand in descending powers of z or ascending powers of z^-1. The first coefficients directly provide x[0], x[1], …. This method is often convenient for checking a partial-fraction result.

4. Difference equations

A linear constant-coefficient difference equation has the form

where the coefficients are constants. The system function, under zero initial conditions, is

The corresponding implementation equation is

provided a₀≠0. In embedded software, this equation is executed once per sampling interval. Previous input and output values must be stored accurately and updated in the correct order.

Worked example: solving a recurrence

Consider

with input x[k]=u[k]. Taking the unilateral transform gives

Since y[-1]=0 and X⁺(z)=1/(1-z^-1),

Partial fractions yield

Thus

At k=0, this gives y[0]=1, as required by the recurrence. As k→∞, 0.6^k→0, so the output approaches 1, the steady-state response to a unit step.

5. Discrete convolution and system response

For an LTI system with impulse response h[k], the output is

For causal sequences this reduces to

Convolution means that every input sample produces a shifted and scaled copy of the impulse response, and the output is their sum. In digital signal processing, finite-length convolution can be computed directly, while fast convolution commonly uses the FFT for long signals.

For a finite impulse response (FIR) filter,

There is no feedback, so an FIR filter is always BIBO stable when its coefficients are finite. An infinite impulse response (IIR) filter uses previous outputs and can achieve sharp filtering with fewer coefficients, but its stability and numerical sensitivity require analysis.

6. Poles, zeros, and frequency response

A rational system function can be written as

where z_i are zeros, p_j are poles, and K is a gain constant. A zero makes the system response vanish at that location; a pole produces large response near its location and determines natural modes.

The pole-radius interpretation is central. A causal mode p^k u[k] decays if |p|<1, remains constant in magnitude if |p|=1, and grows if |p|>1. For a causal rational LTI system, BIBO stability requires every pole to lie strictly inside the unit circle:

Poles exactly on the unit circle generally violate BIBO stability, even when a particular input produces a bounded output. Pole-zero cancellation must be treated cautiously: exact algebraic cancellation may hide an internal unstable mode, and finite-precision arithmetic prevents perfect cancellation.

The frequency response is obtained by evaluating the system function on the unit circle:

where ω is digital angular frequency in radians per sample, normally -π≤ω≤π. If the sampling frequency is f_s, the physical frequency is f=ωf_s/(2π). The magnitude |H(e^(jω))| describes gain and the argument describes phase.

For a real second-order system with denominator

the poles are r e^(±jθ). The radius r controls decay rate, while θ controls oscillation frequency. Such pole pairs commonly model flexible mechanical modes and resonances.

7. Stability and design criteria

For a causal discrete-time system, stability may be checked in several complementary ways:

1. Pole test: all poles must satisfy |p|<1.
2. Impulse-response test: Σ|h[k]|<∞ is sufficient for BIBO stability and is necessary for standard causal LTI rational systems.
3. State-space test: all eigenvalues of the state-transition matrix must be inside the unit circle for asymptotic stability.
4. Jury test: an algebraic test for polynomial roots inside the unit circle, useful when numerical root calculation is inconvenient.

The pole test is usually the most transparent for low-order systems. Stability is not the same as good performance. A pole close to the unit circle may technically be stable but produce slow settling, large sensitivity, and substantial quantization effects.

Sampling period affects the discrete model. A high sampling rate generally improves the approximation of rapid dynamics, but increases processor load, communication traffic, and sensitivity to measurement noise. A low sampling rate reduces computational demand but can cause phase lag, aliasing, and instability.

8. Sampled-data control

A typical digital control loop contains a reference, error calculation, digital controller, actuator interface, continuous plant, sensor, and sampler. The controller may be represented by a difference equation. For example, a proportional–integral controller in discrete form can be implemented as

where e[k] is the error, K_p and K_i are controller gains, T_s is the sampling period, and u[k] is the command. This incremental form is computationally efficient, but actuator saturation can cause integral windup. Anti-windup logic is therefore needed in practical systems.

A zero-order hold keeps a computed command constant between samples. The plant, however, remains continuous-time. Accurate design may require discretizing a continuous model using zero-order-hold equivalence, forward Euler, backward Euler, or the bilinear (Tustin) transformation. For a continuous pole s=p, exact sampling maps it to

Thus a stable continuous pole with negative real part maps inside the unit circle. Approximate discretization methods do not all preserve this relationship equally well; the selected method should reflect bandwidth, phase requirements, and implementation constraints.

9. Illustrative case: digital speed control of a conveyor motor

Background

Consider a small conveyor driven by a DC motor. An encoder supplies speed measurements at f_s=1 kHz, so T_s=0.001 s. The controller is implemented on a microcontroller and must reject load disturbances while avoiding oscillation and excessive actuator activity.

Problem

A first-order discrete approximation of the motor speed response is selected as

where ω[k] is normalized speed and u[k] is normalized motor command. The coefficient 0.85 represents a stable sampled motor mode. The engineering team must determine the response to a step command, assess stability, and identify implementation risks.

Analysis

The transfer function from command to speed is

There is one pole at z=0.85, which lies inside the unit circle. Therefore the model is asymptotically and BIBO stable. For a unit-step command, the impulse response is h[k]=0.15(0.85^k)u[k], and convolution gives

At k=0, ω[0]=0.15. At k=9, ω[9]=1-0.85¹0≈0.803. The response approaches 1 without overshoot because the pole is real and positive.

The approximate time constant in samples can be estimated from 0.85^k≈e^(-k(0.1625)), giving about 6.15 samples, or approximately 6.15 ms. This is an approximate interpretation, not a replacement for the exact discrete response.

Approach and solution

The firmware should:

• sample the encoder at a fixed, verified rate;
• filter measurement noise without adding unacceptable delay;
• calculate the command using bounded arithmetic;
• enforce actuator limits and anti-windup if integral control is added;
• monitor missed deadlines and sensor faults;
• test pole locations after every controller-parameter change.

Outcome

The first-order model predicts a stable, monotonic response. In practice, unmodeled sampling delay, quantization, friction, saturation, and flexible belt dynamics may introduce additional poles or zeros. Hardware-in-the-loop testing should therefore precede deployment.

Learning points

The case demonstrates that a simple recurrence can be converted directly into a transfer function, that pole location predicts qualitative response, and that mathematical stability does not by itself guarantee safe or satisfactory mechatronic performance.

10. Implementation, limitations, and professional practice

Finite precision

Microcontrollers represent numbers with finite word length. Rounding changes coefficients and pole locations. Direct-form IIR structures can be sensitive when poles are close to the unit circle. Second-order sections, coefficient scaling, saturation arithmetic, and numerical testing are preferred for higher-order filters.

Delays and scheduling

A one-sample computational delay contributes a factor z^-1, adding phase lag. Interrupt jitter makes the effective sampling period variable and can degrade control performance. Real-time scheduling must guarantee that computation finishes before the next sample.

Noise and quantization

ADC quantization introduces error, while differentiation amplifies high-frequency noise. Filtering improves measurement quality but adds delay. Designers must balance noise rejection against phase margin and responsiveness.

Verification and safety

A controller should be verified through simulation, code review, processor-in-the-loop or hardware-in-the-loop tests, fault injection, and bounded operating tests. Safety-related systems require traceability, access control, secure firmware updates, and compliance with the applicable industry standards. A mathematically stable controller can still cause harm if a sensor fails, an actuator saturates, or communication data are corrupted.

Contemporary developments

Embedded control increasingly uses model-based design, automatic code generation, fixed-point optimization, and edge machine learning. Machine-learning components may estimate disturbances or identify plant parameters, but their outputs should be monitored by deterministic bounds and fallback control logic. Z-transform models remain valuable because they provide interpretable timing, stability, and frequency-response guarantees around such components.

Chapter Summary

The Z-transform represents a discrete-time sequence as an algebraic function of the complex variable z. The ROC is essential because it distinguishes causal, anticausal, and stability-related interpretations. Linearity, shifting, differentiation, and convolution properties make the transform useful for system analysis.

Difference equations describe executable digital algorithms. Under zero initial conditions, their Z-transforms produce a system function whose numerator identifies zeros and whose denominator identifies poles. Inverse transforms can be obtained through power series, long division, or partial fractions. Convolution explains how an LTI system combines input samples with its impulse response.

For causal rational systems, all poles must lie strictly inside the unit circle for BIBO stability. Pole radius controls decay, pole angle controls oscillation, and zeros shape frequency response. In sampled-data mechatronic systems, sampling, computation delay, quantization, saturation, scheduling, and sensor noise must be considered alongside mathematical models. Reliable engineering practice combines transform analysis with simulation, real-time testing, safety mechanisms, and disciplined implementation.

Key Terms

Z-transform: Representation of a discrete-time sequence as a function of a complex variable.

Bilateral Z-transform: Transform whose summation extends over all integer sample indices.

Unilateral Z-transform: Transform whose summation begins at zero and is useful for initial-value problems.

Region of convergence: Set of complex values for which a Z-transform series converges.

Discrete-time system: System whose signals are defined at discrete sample indices.

Difference equation: Equation relating present and delayed input and output samples.

Impulse response: Output of an LTI system produced by a unit impulse.

Convolution: Sum that combines an input sequence with a shifted impulse response.

Transfer function: Ratio H(z)=Y(z)/X(z) under zero initial conditions.

Pole: Value of z that makes the denominator of a rational system function zero.

Zero: Value of z that makes the numerator of a rational system function zero.

Unit circle: Locus |z|=1 in the Z-plane.

BIBO stability: Property that bounded input produces bounded output.

Causality: Dependence only on present and past, not future, inputs.

FIR filter: Filter with a finite-duration impulse response.

IIR filter: Filter with an impulse response that may continue indefinitely due to feedback.

Sampling period: Time interval T_s between successive samples.

Aliasing: Irreversible frequency ambiguity caused by insufficient sampling.

Zero-order hold: Interface that maintains a digital command constant between updates.

Quantization: Mapping of a continuous-valued measurement to finite numerical levels.

Review Questions

Short Answer Questions

1. Define the bilateral Z-transform and explain the role of its region of convergence.

1. State the convolution property of the Z-transform.

1. What condition on the poles of a causal rational system guarantees BIBO stability?

1. Distinguish an FIR filter from an IIR filter.

1. Why can a computational delay affect digital-control stability?

Long Answer Questions

1. Derive the transfer function associated with a general linear constant-coefficient difference equation and explain the zero-initial-condition assumption.

1. Explain three methods for finding an inverse Z-transform, including the role of the ROC.

1. Discuss how pole radius and pole angle affect the transient response of a second-order discrete-time system.

1. Explain the principal stages of a sampled-data control loop and the engineering effects of sampling, holding, and computation delay.

1. Compare direct-form implementation, finite precision, quantization, and saturation as sources of practical deviation from an ideal Z-domain model.

Analytical / Application Questions

1. For H(z)=(1+0.4z^(-1))/[(1-0.7z^(-1))(1-0.3z^(-1))], identify the poles and determine whether the causal system is BIBO stable. Explain your reasoning.

1. A controller uses y[k]=0.5y[k-1]+0.25x[k]+0.25x[k-1]. Derive H(z) and calculate the first three output samples for a unit impulse input with zero initial conditions.

1. A sensor is sampled at 800 Hz. Determine its Nyquist frequency and explain two consequences of sampling a 500 Hz vibration without adequate anti-aliasing.

Multiple Choice Questions with Answers

Question 1

For a causal rational discrete-time system, which pole condition guarantees BIBO stability?

A. Every pole satisfies |p|<1.

B. Every pole satisfies |p|=1.

C. At least one pole satisfies |p|>1.

D. All zeros lie outside the unit circle.

Correct Answer: A. Every pole satisfies |p|<1.

Question 2

The Z-transform of a^k u[k] is:

A. 1/(1-a z^(-1)), with |z|>|a|

B. 1/(1+a z^(-1)), with |z|<|a|

C. a/(1-z^(-1)), with |z|>1

D. z/(z-a), with |z|<|a|

Correct Answer: A. 1/(1-a z^(-1)), with |z|>|a|

Question 3

Which operation in the time domain corresponds to multiplication of Z-transforms?

A. Addition

B. Time reversal only

C. Convolution

D. Differentiation with respect to time

Correct Answer: C. Convolution

Question 4

For H(z)=0.2/(1-0.8z^(-1)), the pole is located at:

A. z=0.2

B. z=0.8

C. z=-0.8

D. z=1.25

Correct Answer: B. z=0.8

Question 5

A sampling frequency of 1 kHz has a Nyquist frequency of:

A. 100 Hz

B. 250 Hz

C. 500 Hz

D. 1 kHz

Correct Answer: C. 500 Hz

Question 6

For y[k]-0.5y[k-1]=x[k], the zero-state system function is:

A. (1-0.5z^(-1))

B. 1/(1-0.5z^(-1))

C. 1-0.5z

D. z/(z+0.5)

Correct Answer: B. 1/(1-0.5z^(-1))

Question 7

What does the radius of a complex pole pair primarily control?

A. The decay or growth rate of the response

B. The ADC resolution

C. The number of input channels

D. The sampling-unit definition

Correct Answer: A. The decay or growth rate of the response

Question 8

An FIR filter is generally stable when:

A. Its finite coefficients are bounded

B. It contains a pole outside the unit circle

C. Its sampling frequency is zero

D. Its zeros are all at the origin

Correct Answer: A. Its finite coefficients are bounded

Question 9

For y[k]=0.6y[k-1]+x[k] with zero initial state and a unit impulse input, what is y[1] if the impulse occurs at k=0?

A. 0

B. 0.6

C. 1

D. 1.6

Correct Answer: B. 0.6

Question 10

Which practical effect is directly associated with finite-word-length arithmetic in an IIR filter?

A. Exact preservation of pole locations

B. Possible coefficient-rounding and limit-cycle errors

C. Elimination of all measurement noise

D. Removal of computational delay

Correct Answer: B. Possible coefficient-rounding and limit-cycle errors