Z-Transform and Discrete-Time System Analysis

Introduction

The z-transform is one of the central analytical tools for discrete-time signals and systems. It generalizes the discrete-time Fourier transform (DTFT), converts convolution into multiplication, provides a systematic treatment of initial conditions, and exposes the pole–zero structure that determines stability, transient response, and frequency behavior. In digital signal processing, control, communications, biomedical instrumentation, and embedded systems, z-transform methods connect mathematical models with implementable algorithms.

A discrete-time signal is a sequence indexed by an integer n, written x[n]. Unlike a continuous-time signal, it is defined only at discrete instants. A discrete-time linear time-invariant (LTI) system may be described by convolution, a difference equation, an impulse response, or a transfer function. The z-transform provides a common language for all four descriptions.

Learning Objectives

After completing this chapter, the reader should be able to:

1. Define the bilateral and unilateral z-transforms and explain their regions of convergence.
2. Use transform properties to analyze discrete-time signals and systems.
3. Derive transfer functions from linear constant-coefficient difference equations.
4. Interpret poles, zeros, causality, stability, and frequency response.
5. Distinguish zero-state, zero-input, and complete responses.
6. Analyze systems using block diagrams, convolution, and pole–zero methods.
7. Apply z-transform techniques to a practical digital filtering problem.
8. Recognize numerical, modeling, and implementation limitations.

1. The z-Transform and Its Region of Convergence

For a sequence x[n], the bilateral z-transform is defined by

where z is a complex variable, usually expressed in polar form as z = re^jω, r = |z| is the radial coordinate, and ω is angular frequency in radians per sample. The transform exists only for those values of z for which the infinite series converges. This set is called the region of convergence (ROC).

The ROC is part of the transform description. The same algebraic expression can represent different sequences when different ROCs are associated with it. For example,

represents a right-sided sequence when |z| > |a| and a left-sided sequence when |z| < |a|.

The inverse z-transform is formally obtained from a contour integral,

where the contour is a counterclockwise closed path lying entirely inside the ROC and enclosing the origin. In engineering practice, inverse transforms are more commonly obtained using power-series expansion, partial fractions, or standard transform pairs.

1.1 Bilateral and unilateral forms

The bilateral transform includes all integer indices. The unilateral, or one-sided, transform is

The bilateral transform is natural for system classification and convolution. The unilateral transform is especially useful for causal systems with nonzero initial conditions because it incorporates starting values directly into transformed difference equations.

1.2 Important transform pairs

Let u[n] denote the unit-step sequence: u[n] = 1 for n ≥ 0 and u[n] = 0 otherwise. Important pairs include:

For x[n] = aⁿ u[n]], the geometric-series derivation is

provided |az⁻¹| < 1, or equivalently |z| > |a|.

2. Properties of the z-Transform

The transform properties follow directly from the defining summation and are valid only when the relevant ROCs are respected.

Linearity

If x[n] ↔ X(z) and y[n] ↔ Y(z), then

where α and β are constants.

Time shifting

For a bilateral transform,

where n₀ is an integer. A delay of n₀ samples multiplies the transform by z^-n₀. A left shift uses the same relation with negative n₀, subject to ROC considerations.

Time reversal

The ROC is geometrically inverted: if the original ROC is r₁ < |z| < r₂, the reversed ROC is 1/r₂ < |z| < 1/r₁.

Convolution

For discrete-time convolution,

and, under convergence conditions,

This property is fundamental: a system that performs convolution in the time domain becomes multiplication in the z-domain.

Multiplication by n

This property is useful for transforms of polynomially weighted exponentials.

Difference and accumulation

For bilateral transforms, the first difference y[n] = x[n] − x[n−1] has transform

An accumulator satisfying y[n] = y[n−1] + x[n] has transfer function

under zero initial conditions.

3. Inverse z-Transform Methods

Power-series expansion

The ROC determines the expansion. For

an expansion in negative powers of z, valid outside the pole, gives aⁿu[n]. An expansion in positive powers, valid inside the pole, gives −aⁿu[−n−1]. Thus the ROC identifies time direction.

Partial fractions

Suppose

Assume the ROC is |z| > 0.8. Write

Equating coefficients gives A = −0.5 and B = 1.5. Therefore,

The sequence is causal because the ROC lies outside the outermost pole.

Residue method

For isolated poles, the inverse transform can also be computed using residues. This approach is especially useful when poles are repeated or when symbolic software is employed. However, the ROC must still be specified; residues alone do not determine sidedness.

4. Discrete-Time LTI Systems

An LTI system is characterized by its impulse response h[n]. For an input x[n], the output is

Taking z-transforms gives

and therefore the system function is

provided the input–output relationship is considered under zero initial conditions and the relevant transforms converge.

Causality

An LTI system is causal if its output at n depends only on present and past input values. Equivalently,

For a rational causal system, the ROC is outside the outermost pole. A finite-duration impulse response is causal if it begins at n = 0 and has only finitely many nonzero samples.

BIBO stability

A discrete-time LTI system is bounded-input bounded-output stable if and only if

For a rational system, stability requires that the ROC include the unit circle |z| = 1. For a causal rational system, all poles must lie strictly inside the unit circle. A pole on the unit circle generally produces a non-decaying response and violates BIBO stability.

Frequency response

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

This substitution is valid when the unit circle lies in the ROC. The magnitude |H(e^jω)| describes gain, while the phase ∠H(e^jω) describes phase shift.

5. Difference Equations and Transfer Functions

A broad class of digital systems is represented by the linear constant-coefficient difference equation

where a_k and b_k are constant coefficients. Under zero initial conditions, the z-transform gives

Thus,

The roots of the numerator are zeros; the roots of the denominator are poles. A pole is a value of z at which H(z) becomes unbounded unless cancelled by a zero. Exact pole–zero cancellation may conceal internal modes, so cancellation must be treated cautiously in finite-precision implementations.

Worked Example 1: Transfer function and stability

Consider

with zero initial conditions. Taking the z-transform,

The denominator factors as

Hence the poles are z = 0.8 and z = 0.4, and the zero is z = −0.5. Assuming the causal ROC |z| > 0.8, the system is causal. Since both poles lie inside the unit circle, it is also BIBO stable.

The impulse response can be found from partial fractions. Let

Matching the numerator gives A + B = 1 and 0.4A + 0.8B = 0.5. Solving yields A = 0.75 and B = 0.25. Therefore,

which decays to zero, confirming stability.

6. Unilateral z-Transform and Initial Conditions

For causal sequences, the unilateral transform is particularly convenient. Its time-shift property is not identical to the bilateral property. For example,

and

The additional terms represent initial values. Consequently, a difference equation can be transformed without artificially setting stored energy to zero.

The complete response is commonly decomposed as

where y_zi[n] is the zero-input response caused by initial conditions with x[n] = 0, and y_zs[n] is the zero-state response caused by the input with all initial conditions set to zero. The natural response is associated with system poles; the forced response is associated with the input and system dynamics.

Worked Example 2: Initial-condition effect

Consider the first-order accumulator

with y[−1] = 2. For x[n] = δ[n], the unilateral transform gives

Therefore,

and

The output is y[n] = 2(0.5)ⁿ u[n]. The initial stored value and the impulse both contribute to the amplitude. If the initial condition were incorrectly ignored, the result would be only (0.5)ⁿ u[n].

7. Pole–Zero Geometry and System Behavior

The pole–zero plot is a geometric representation in the z-plane. Poles are conventionally marked by × and zeros by ○. Important interpretations include:

• Poles near the unit circle produce slowly decaying transients and sharp spectral features.
• A real pole near +1 emphasizes low frequencies and creates long memory.
• A real pole near −1 emphasizes behavior near ω = π and can produce alternating transients.
• Complex-conjugate poles produce damped oscillations.
• Zeros on the unit circle create complete attenuation at their corresponding angles.

For a causal second-order system with poles p = re^±jθ, the natural response has an approximate envelope proportional to rⁿ and oscillates at θ radians per sample. Stability requires r < 1. The closer r is to one, the longer the transient and the greater the sensitivity to coefficient quantization.

8. State-Space Representation

A difference equation can also be represented in state-space form. A discrete-time state model is

where x[n] is the state vector, u[n] is the input, y[n] is the output, and A, B, C, and D are matrices of compatible dimensions. The transfer matrix is

where I is the identity matrix. The eigenvalues of A are the system poles, assuming no pole-zero cancellation removes a mode. State-space forms are valuable for multivariable systems, numerical implementation, observer design, and control-oriented analysis.

9. Practical Case: Digital Removal of Power-Line Interference

Background

Consider an illustrative biomedical acquisition system sampling an electrocardiogram at 250 samples per second. The desired cardiac waveform is contaminated by interference at 50 Hz. A digital notch filter is required to suppress the interference while preserving nearby physiological components.

Problem analysis

The digital angular frequency corresponding to 50 Hz is

A pair of zeros at z = e^±jω₀ produces zero response at ±ω₀. To obtain a narrow notch rather than excessive attenuation around the target frequency, poles are placed at the same angle with radius r slightly less than one, for example r = 0.95.

Filter model

A normalized second-order notch filter is

The numerator zeros lie exactly on the unit circle at the interference angle. The denominator poles lie at radius r, so the filter is causal and stable when 0 < r < 1. A common normalization chooses K so that DC gain is unity:

For ω₀ = 0.4π, cos(ω₀) ≈ 0.3090 and r = 0.95. Thus the numerator at z = 1 is approximately 2 − 2(0.3090) = 1.3820, while the denominator at z = 1 is approximately 1 − 2(0.95)(0.3090) + 0.95² = 0.4750. Consequently, K ≈ 0.3437.

Implementation and outcome

The corresponding difference equation is

Substitution gives approximate feedback coefficients 0.5871 and 0.9025, and feedforward coefficients 0.3437, −0.2125, and 0.3437. The filter strongly attenuates 50-Hz interference while retaining nearby frequencies better than a broad low-pass filter would.

Learning points

1. Zero locations determine frequencies of complete cancellation.
2. Pole radius controls notch bandwidth and transient duration.
3. Stability requires the poles to remain inside the unit circle.
4. Coefficient quantization can move poles and reduce rejection accuracy.
5. A practical design must evaluate signal distortion, arithmetic precision, overflow, and latency—not only an ideal transfer function.

10. Numerical and Implementation Considerations

Direct forms and numerical sensitivity

Direct-form implementations are simple but can be sensitive to round-off and coefficient quantization, especially for high-order filters. Cascading first- and second-order sections, often called biquad sections, generally improves numerical robustness.

Finite word length

Real digital hardware represents values with finite precision. Quantization introduces noise, while overflow can cause severe nonlinear behavior. Designers must select scaling, word length, saturation policy, and arithmetic format deliberately.

Pole-zero cancellation

A pole and zero that are theoretically identical may not cancel exactly after quantization. An apparently stable mathematical model can therefore exhibit a residual unstable or slowly decaying mode. Internal states should be examined, not only the simplified transfer function.

Aliasing and sampling assumptions

The z-transform analyzes a sampled sequence; it does not remove aliasing introduced during sampling. Analog anti-alias filtering, an adequate sampling rate, and correct interpretation of digital frequency are prerequisites for meaningful system analysis.

Software verification

A professional workflow should compare: (1) difference-equation simulation, (2) convolution with the computed impulse response, (3) frequency-response evaluation, and (4) pole–zero analysis. Agreement among these methods helps identify indexing, sign, scaling, and initial-condition errors.

11. Advanced Perspective and Contemporary Relevance

The z-transform remains central even when modern tools use numerical optimization or machine learning. Digital control systems use pole placement and discrete-time state models; communications receivers use equalizers and matched filters; audio systems use cascaded IIR and FIR structures; and sensor systems use recursive estimators. In data-driven systems, learned filters can still be assessed through frequency response, impulse response, stability, and robustness.

A significant contemporary concern is certifiable stability. In safety-critical systems, a design should not rely solely on simulated inputs. Pole locations, coefficient ranges, worst-case arithmetic, startup behavior, and fault responses should be bounded analytically. Explainability is also improved when a learned or optimized filter is translated into interpretable poles, zeros, and state variables.

Challenges and Limitations

The z-transform is powerful but not universal. Linear time invariance is assumed in the simplest transfer-function framework. Time-varying, nonlinear, and periodically sampled systems require extensions or alternative tools. A transfer function does not by itself reveal internal state overflow, hidden cancellations, or implementation latency. The ROC may be ambiguous if only an algebraic expression is supplied. Finally, frequency response alone can conceal transient behavior and sensitivity to initial conditions.

Best-Practice Checklist

• State whether the transform is bilateral or unilateral.
• Specify the ROC whenever sequence sidedness matters.
• State initial conditions before forming a transfer function.
• Check causality independently from stability.
• Test whether the unit circle lies in the ROC before using frequency response.
• Factor the denominator and inspect pole radii and angles.
• Use second-order sections for high-order numerical implementations.
• Validate theoretical results with time-domain simulation.
• Assess quantization, overflow, startup transients, and sampling effects.

Chapter Summary

The z-transform maps discrete-time sequences into functions of a complex variable and converts convolution into multiplication. Its defining series is meaningful only within a region of convergence, and the ROC is essential for identifying whether a sequence is right-sided, left-sided, causal, or stable. Transform properties support delay, difference, accumulation, scaling, and inverse-transform calculations.

For LTI systems, the impulse response, convolution equation, difference equation, and transfer function are equivalent descriptions under appropriate assumptions. Causality corresponds to a right-sided impulse response, while BIBO stability requires absolute summability and, for rational systems, inclusion of the unit circle in the ROC. In causal rational systems, all poles must lie strictly inside the unit circle.

The unilateral transform incorporates initial conditions and separates zero-input from zero-state response. Pole–zero geometry explains transient decay, oscillation, resonance, and frequency cancellation. State-space representations extend the analysis to multivariable and control applications. Practical design additionally requires attention to finite precision, coefficient quantization, implementation structure, aliasing, and verification.

Key Terms

Z-transform: A complex-domain representation of a discrete-time sequence.

Bilateral z-transform: Transform summed over all integer time indices.

Unilateral z-transform: Transform summed from n = 0 to infinity.

Region of convergence: Set of z values for which the transform series converges.

Discrete-time signal: A sequence defined at discrete integer-indexed instants.

LTI system: A system possessing linearity and time invariance.

Convolution: Operation that gives the output of an LTI system from input and impulse response.

Impulse response: Output of a system when the input is δ[n].

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

Pole: A value of z at which a rational system function becomes unbounded.

Zero: A value of z at which a system function equals zero.

Causality: Property that output does not depend on future input values.

BIBO stability: Bounded-input bounded-output stability.

Frequency response: System response obtained on the unit circle.

Zero-state response: Response due only to the input with zero initial conditions.

Zero-input response: Response due only to initial stored energy.

Natural response: Homogeneous response determined by system poles.

Notch filter: Filter designed to strongly attenuate a selected frequency.

Finite word length: Limited numerical precision used in digital hardware.

State-space model: Matrix-based representation using state, input, and output equations.

Review Questions

Short Answer Questions

1. What is the region of convergence, and why is it necessary?
2. State the z-transform property of discrete-time convolution.
3. What condition on the poles guarantees stability for a causal rational system?
4. Distinguish zero-input and zero-state responses.
5. Why are zeros placed on the unit circle in a notch filter?

Long Answer Questions

1. Derive the z-transform of aⁿu[n] and explain how the ROC is obtained.
2. Explain causality and BIBO stability using impulse responses, pole locations, and ROCs.
3. Derive the transfer function of a general linear constant-coefficient difference equation.
4. Discuss bilateral and unilateral z-transforms, including the treatment of initial conditions.
5. Explain how pole radius and pole angle influence the transient and frequency response of a second-order system.

Analytical / Application Questions

1. For H(z) = (1 − z^-1)/(1 − 0.6z^-1), determine the pole, zero, causal ROC, stability, and impulse response.
2. Design a second-order notch filter for a 60-Hz interference component sampled at 300 Hz. Select a pole radius less than one and state how the choice affects bandwidth and transient duration.
3. A recursive filter is theoretically stable but becomes oscillatory after coefficient quantization. Analyze possible causes and propose at least three verification or mitigation methods.

Multiple Choice Questions with Answers

Question 1

For the sequence x[n] = aⁿ u[n], which ROC is correct?

A. |z| < |a|

B. |z| > |a|

C. |z| = |a| only

D. All z except z = a

Correct Answer: B. |z| > |a|

Question 2

The z-transform of discrete-time convolution x[n] * h[n] is:

A. X(z) + H(z)

B. X(z) − H(z)

C. X(z)H(z)

D. X(z)/H(z)

Correct Answer: C. X(z)H(z)

Question 3

For a causal rational LTI system to be BIBO stable, its poles must:

A. Lie strictly inside the unit circle

B. Lie strictly outside the unit circle

C. Lie on the unit circle

D. Have zero real parts

Correct Answer: A. Lie strictly inside the unit circle

Question 4

Which feature of a notch filter produces complete attenuation at the target frequency in the ideal case?

A. Poles at the origin

B. Zeros on the unit circle at the target angle

C. Poles outside the unit circle

D. A constant numerator

Correct Answer: B. Zeros on the unit circle at the target angle

Question 5

The frequency response H(e^{jω}) is obtained by evaluating H(z) at:

A. z = 0

B. z = 1 only

C. z = e^{jω}

D. z = ω²

Correct Answer: C. z = e^{jω}

Question 6

For H(z) = 1/(1 − 0.5z⁻¹) with a causal ROC, the impulse response is:

A. (−0.5)^n u[n]

B. 0.5^n u[n]

C. u[n−0.5]

D. δ[n] − 0.5δ[n−1]

Correct Answer: B. 0.5^n u[n]

Question 7

A unilateral z-transform is especially useful because it:

A. Eliminates all poles

B. Directly incorporates initial conditions in difference equations

C. Applies only to continuous-time signals

D. Makes every system stable

Correct Answer: B. Directly incorporates initial conditions in difference equations

Question 8

In a causal second-order system, moving a complex-conjugate pole pair closer to the unit circle generally causes:

A. Shorter transients and broader damping

B. Longer-lasting transients and sharper resonance

C. Immediate instability in every case

D. The zeros to disappear

Correct Answer: B. Longer-lasting transients and sharper resonance

Question 9

For a sampling rate of 250 Hz, the digital angular frequency of a 50-Hz sinusoid is:

A. 0.2π rad/sample

B. 0.4π rad/sample

C. 2.5π rad/sample

D. 50π rad/sample

Correct Answer: B. 0.4π rad/sample

Question 10

Which condition is the time-domain BIBO stability criterion for an LTI system?

A. ∑|h[n]| is finite

B. ∑h[n] is zero

C. h[n] is periodic

D. h[n] has at least one pole

Correct Answer: A. ∑|h[n]| is finite