Z-Transform and Discrete-Time System Analysis

Introduction and Context

Discrete-time signals arise whenever a physical quantity is sampled, digitally recorded, or generated algorithmically. Audio processors, digital controllers, biomedical monitors, communication receivers, radar systems, and embedded processors all manipulate sequences rather than continuous-time waveforms. The central analytical difficulty is that difference equations, convolution sums, and frequency responses can become cumbersome when treated directly. The Z-transform provides a unified framework for representing sequences, solving linear constant-coefficient difference equations, characterizing systems, and connecting transient behavior with steady-state frequency response.

For a sequence x[n], the Z-transform is the complex-variable transform

where n is an integer time index and z is a complex variable. The definition is meaningful only for those values of z for which the infinite series converges. This set is called the region of convergence (ROC). Unlike the discrete-time Fourier transform, the Z-transform explicitly represents growth or decay through the magnitude of z and therefore supports a broader class of sequences and systems.

Learning Objectives

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

1. Define the bilateral and unilateral Z-transforms and explain their regions of convergence.
2. Derive and apply major transform properties.
3. Determine inverse Z-transforms using algebraic expansion, partial fractions, and contour concepts.
4. Analyze poles, zeros, stability, causality, and system behavior.
5. Solve linear constant-coefficient difference equations with initial conditions.
6. Derive transfer functions and frequency responses of discrete-time systems.
7. Apply the framework to digital filters and embedded control systems.
8. Evaluate numerical, implementation, and modeling limitations.

1. The Z-Transform and Its Region of Convergence

1.1 Bilateral Z-transform

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

The ROC is the set of z for which the sum converges absolutely:

Writing z = re^{jω}, where r = |z| and ω is the angular coordinate, gives

Thus r controls exponential weighting, while ω controls oscillation. The DTFT is obtained, when it exists, by evaluating X(z) on the unit circle z = e^{jω}.

1.2 Unilateral Z-transform

The unilateral, or one-sided, transform is

It is especially useful for causal systems with nonzero initial conditions because time-shift operations naturally expose initial samples. The bilateral transform is generally preferred for classifying two-sided sequences, causality, and ROC-based system properties.

1.3 Standard transform pairs

Here δ[n] is the unit impulse, u[n] is the unit step, and a may be real or complex. The two sequences aⁿu[n] and −aⁿu[−n−1] have the same algebraic expression but different ROCs and therefore represent different sequences.

For a right-sided sequence, the ROC lies outside the outermost pole. For a left-sided sequence, it lies inside the innermost pole. For a two-sided rational sequence, the ROC is an annulus between poles. The ROC never contains a pole.

2. Transform Properties

Let x[n] ↔ X(z) and y[n] ↔ Y(z), with ROCs sufficiently overlapping for the operation to be valid.

Linearity

where a and b are constants.

Time shifting

For the bilateral transform,

where n₀ is an integer. A delay by n₀ samples therefore introduces a factor z⁻ⁿ⁰. A time advance may alter the ROC at z = 0 or z = ∞, so ROC statements must be checked rather than inferred mechanically.

Exponential scaling

provided a ≠ 0 and the transformed ROC is appropriately scaled.

Time reversal

The ROC is geometrically inverted: an exterior region becomes an interior region.

Convolution

For discrete convolution,

and therefore

The ROC includes the intersection of the individual ROCs, except where pole-zero cancellations and convergence conditions require additional care.

Difference and accumulation

The first difference x[n] − x[n−1] transforms, under suitable bilateral ROC conditions, as

The causal accumulator y[n] = ∑_{k=−∞}^{n}x[k] has transfer function 1/(1−z⁻¹), with a pole at z = 1.

Initial- and final-value relations

For a causal sequence,

when the limit exists. A final-value relation is

provided all poles of (1−z⁻¹)X(z), except possibly a simple pole at z = 1 before multiplication, lie strictly inside the unit circle. This condition is essential; applying the formula to an oscillatory or unstable sequence can produce an invalid conclusion.

3. Inverse Z-transform

The inverse transform is formally given by a contour integral around a closed contour C in the ROC:

where j = √−1 and C is counterclockwise. In engineering analysis, algebraic methods are usually more efficient.

3.1 Power-series expansion

If X(z) = 1/(1−az⁻¹) and the ROC is |z|>|a|, expand geometrically:

so x[n] = aⁿu[n]. If the ROC is |z|<|a|, rewrite in positive powers of z:

which corresponds to x[n] = −aⁿu[−n−1]. The ROC is therefore part of the transform specification.

3.2 Partial fractions

Consider

Assume the causal ROC |z|>0.5. Write

Equating coefficients gives A = 2 and B = −1. Hence

The same rational expression with ROC |z|<0.25 would yield a different left-sided sequence. With ROC 0.25<|z|<0.5, one term is left-sided and the other right-sided. This example demonstrates why poles and algebra alone do not uniquely specify a sequence.

4. Poles, Zeros, Causality, and Stability

A rational transform is commonly written as

where zᵢ are zeros and pᵢ are poles. A zero makes H(z) equal to zero; a pole makes it unbounded unless cancellation or another singularity changes the local behavior.

For a rational causal system, the ROC is outside the outermost pole. The system is BIBO stable if and only if its impulse response is absolutely summable:

For a rational causal discrete-time LTI system, this is equivalent to every pole lying strictly inside the unit circle:

The unit circle must lie in the ROC for the DTFT to exist in the usual stable-system sense. A pole on the unit circle generally indicates marginal or unstable behavior, although special pole-zero cancellations must be treated at the transfer-function and realization levels.

The frequency response is

where ω is normalized angular frequency in radians per sample. Because frequency is periodic, H(e^{j(ω+2π)}) = H(e^{jω}).

5. Difference Equations and Transfer Functions

A linear time-invariant discrete-time system can be described by

with a₀ commonly normalized to 1. Under initial rest, taking the Z-transform gives

The phrase “initial rest” means that stored energy or past-state contributions are zero before the applied input. Without initial rest, the output includes a zero-input response in addition to the zero-state response.

Worked example: first-order system

Let

with initial rest. The transfer function is

The pole is at z = 0.8. For the causal realization, the ROC is |z|>0.8, which includes the unit circle; therefore the system is causal and BIBO stable. Its impulse response is h[n] = (0.8)ⁿu[n]. A unit-step input has X(z)=1/(1−z⁻¹), so

and the step response is

The limiting value is 5, consistent with the DC gain H(1)=1/(1−0.8)=5.

6. Unilateral Transform and Initial Conditions

For causal sequences, the unilateral shift formula is particularly useful. For example,

and

The extra terms encode pre-initial samples. Consider

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

Therefore

and y[n] = 2(0.5)ⁿu[n] for n ≥ 0. The initial condition contributes directly to the numerator. This is why unilateral methods are common in switching circuits and digital control initialization.

7. Frequency-Domain Interpretation and Filter Design

The Z-plane provides a geometric interpretation of filtering. A frequency corresponds to a point on the unit circle. A zero near a point on that circle attenuates frequencies near the corresponding angle; a pole near the circle amplifies them and creates a narrow resonance, provided stability is preserved.

For a moving-average filter,

so

Using 1 + q + q² = (1−q³)/(1−q), its zeros occur at z = e^{j2π/3} and z = e^{j4π/3}. Thus the filter has nulls at normalized frequencies ±2π/3 and passes DC with gain H(1)=1. It is a finite impulse response (FIR) filter because its denominator is 1.

An infinite impulse response (IIR) filter contains feedback. For example,

has a pole at 0.9 and a zero at 0.5. The causal system is stable because 0.9 lies inside the unit circle, but its response decays more slowly than that of a system with a pole at 0.4. Pole radius controls decay rate; pole angle controls oscillation frequency.

8. State-Space Representation

A discrete-time state-space model is

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

where I is the identity matrix. The eigenvalues of A are the system poles, subject to cancellations and realization details. State-space models are advantageous for high-order systems, multivariable systems, numerical implementation, and control design. They also clarify that a transfer function describes input-output behavior but may conceal internal states that are poorly conditioned or numerically sensitive.

9. Detailed Applied Case: Digital Removal of Power-Line Interference

Background

Consider an illustrative biomedical acquisition unit sampling an electrocardiogram at 250 samples/s. The desired cardiac waveform is contaminated by approximately 50-Hz power-line interference. The normalized interference frequency is

Problem

The filter should strongly attenuate 50 Hz while preserving the slowly varying cardiac morphology. A narrow notch is preferable to a broad low-pass filter because broad attenuation could distort clinically important components.

Analysis and approach

A real-coefficient second-order notch filter can be constructed as

where zeros are placed exactly on the unit circle at e^{±jω₀}, poles are placed at radius r<1 at the same angles, and K is a gain normalization constant. Setting r close to 1 produces a narrow notch; reducing r broadens the notch and shortens ringing. For unity DC gain, choose K so H(1)=1:

This expression assumes the denominator is nonzero and ω₀ is not zero modulo 2π. With r=0.98, the filter is stable because its poles have magnitude 0.98. The zeros provide exact attenuation at 50 Hz in ideal arithmetic.

Implementation considerations

The difference equation is

Finite word length changes the actual pole and zero locations. Direct-form implementations can be sensitive when poles are close to the unit circle, so a second-order-section realization, careful coefficient quantization, saturation protection, and verification using impulse and frequency-response tests are recommended.

Outcome and learning points

The notch removes the targeted interference with limited impact on nearby frequencies. However, adaptive interference, sampling-frequency drift, electrode motion, and harmonics may require a bank of notches or adaptive filtering. The case illustrates the full workflow: identify a spectral disturbance, translate its frequency to the Z-plane, place zeros for rejection, place stable poles to control selectivity, normalize gain, and validate the finite-precision implementation.

10. Numerical and Practical Limitations

Theoretical pole-zero cancellation may fail in finite precision because numerator and denominator coefficients are rounded differently. A nominally stable pole can move outside the unit circle. High-order polynomials are also poorly conditioned when evaluated directly. Practical best practices include using cascaded first- or second-order sections, scaling internal signals, examining pole locations after quantization, and testing worst-case input amplitudes.

The Z-transform assumes a meaningful discrete-time model. Sampling below the required rate causes aliasing that no subsequent digital filter can perfectly undo. Quantization introduces noise and nonlinear overflow. Latency, memory, processor load, and coefficient-update timing may be as important as nominal frequency response.

For control systems, stability margins should not be inferred only from an ideal transfer function. Sensor delay, actuator saturation, computation delay, and unmodeled dynamics can alter the closed-loop poles. For biomedical, industrial, or safety-critical applications, traceable verification, calibration, access control, and documented test procedures are required. AI-based coefficient tuning may assist design exploration, but it does not replace mathematical stability analysis, reproducibility, or engineering approval.

Contemporary Developments and Best Practices

Modern digital signal processors increasingly use SIMD arithmetic, FPGA pipelines, and model-based code generation. Frequency-domain identification, sparse adaptive filters, multirate systems, and data-driven system identification extend classical Z-transform analysis. Nevertheless, the foundational checks remain unchanged:

• specify the sampling frequency and signal conventions;
• state the ROC or causal realization;
• locate poles and zeros;
• verify unit-circle inclusion for stable frequency response;
• distinguish zero-state and zero-input responses;
• test theoretical coefficients after quantization;
• document assumptions, numerical precision, and saturation behavior.

Chapter Summary

The Z-transform represents a discrete-time sequence as a complex function together with a region of convergence. The ROC resolves the ambiguity that occurs when different right-sided, left-sided, or two-sided sequences share the same rational expression. Transform properties convert shifting, convolution, scaling, and difference operations into algebraic manipulation. Inverse transforms are obtained through series expansion, partial fractions, or contour integration.

For LTI systems, poles determine natural modes and stability, while zeros determine attenuation and spectral shaping. A causal rational system is BIBO stable when all poles lie strictly inside the unit circle. Difference equations become transfer functions under initial rest, whereas the unilateral transform incorporates initial conditions directly. Z-plane geometry supports FIR and IIR filter design, and state-space models generalize the analysis to high-order and multivariable systems. Practical engineering additionally requires attention to sampling, finite precision, implementation structure, validation, and safety.

Key Terms

Z-transform: Complex-variable transform 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.

Pole: Value of z at which a rational transform becomes unbounded.

Zero: Value of z at which a transform equals zero.

Unit circle: Locus |z|=1 used to evaluate the DTFT.

Causality: Property in which output depends only on present and past inputs.

BIBO stability: Bounded-input, bounded-output behavior.

Impulse response: Output of an LTI system to δ[n].

Transfer function: Ratio Y(z)/X(z) under initial rest.

Difference equation: Discrete-time input-output relation involving shifts.

FIR filter: Filter with finite-duration impulse response.

IIR filter: Filter with potentially infinite-duration impulse response.

Convolution: Operation expressing an LTI output as weighted shifted impulses.

Frequency response: System response evaluated on the unit circle.

Pole radius: Magnitude of a pole, governing decay or growth rate.

State-space model: Matrix representation using internal state variables.

Initial rest: Assumption of zero stored-state contribution before excitation.

Quantization: Finite-precision representation of samples or coefficients.

Review Questions

Short Answer Questions

1. What is the region of convergence, and why is it needed in addition to an algebraic transform expression?
2. State the convolution property of the bilateral Z-transform.
3. What condition on the poles of a causal rational system guarantees BIBO stability?
4. Distinguish the bilateral and unilateral Z-transforms.
5. What is the significance of the unit circle in the Z-plane?

Long Answer Questions

1. Derive the Z-transform and ROC of aⁿu[n] and −aⁿu[−n−1], explaining why their algebraic expressions are identical.
2. Explain how poles and zeros influence the magnitude response and transient behavior of a discrete-time LTI system.
3. Derive the transfer function of a general linear constant-coefficient difference equation under initial rest.
4. Compare FIR and IIR filters in terms of structure, stability, computational cost, and design flexibility.
5. Explain how the unilateral Z-transform incorporates nonzero initial conditions, using a second-order example.

Analytical / Application Questions

1. For H(z)=1/[(1−0.6z⁻¹)(1−0.2z⁻¹)], determine the causal ROC, stability, and impulse response.
2. Design a second-order notch filter for a 60-Hz disturbance sampled at 300 samples/s. State the notch frequency, pole-radius choice, and stability condition.
3. A measured filter has a pole at 1.002 after coefficient quantization although its design pole was 0.998. Analyze the likely consequence and propose implementation safeguards.

Multiple Choice Questions with Answers

Question 1

For the causal sequence x[n] = aⁿu[n], which ROC is correct?

A. |z| < |a|

B. |z| > |a|

C. |z| = |a| only

D. The entire z-plane including z = a

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

Question 2

For an LTI system with impulse response h[n], BIBO stability requires which condition?

A. ∑|h[n]| is finite

B. ∑h[n] is zero

C. All zeros lie inside the unit circle

D. The ROC excludes the unit circle

Correct Answer: A. ∑|h[n]| is finite

Question 3

The bilateral Z-transform of x[n−n₀] is:

A. X(z+n₀)

B. z^{n₀}X(z)

C. z^{-n₀}X(z)

D. X(z^{-n₀})

Correct Answer: C. z^{-n₀}X(z)

Question 4

For a causal rational system, which pole location indicates instability?

A. p = 0.5

B. p = −0.8

C. p = 0.2 + j0.3

D. p = 1.02

Correct Answer: D. p = 1.02

Question 5

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

A. Setting z=0

B. Setting z=1 only

C. Setting z=e^{jω}

D. Replacing z with ω

Correct Answer: C. Setting z=e^{jω}

Question 6

The transfer function of y[n]−0.7y[n−1]=x[n], under initial rest, is:

A. 1−0.7z⁻¹

B. 1/(1−0.7z⁻¹)

C. 1/(1+0.7z⁻¹)

D. 0.7/(1−z⁻¹)

Correct Answer: B. 1/(1−0.7z⁻¹)

Question 7

A zero placed exactly on the unit circle at angle ω₀ produces, ideally, what effect?

A. Maximum gain at ω₀

B. A spectral null at ω₀

C. Instability at every frequency

D. A time advance only

Correct Answer: B. A spectral null at ω₀

Question 8

For a causal stable rational system, the unit circle must:

A. Contain at least one pole

B. Be outside the ROC

C. Lie within the ROC

D. Coincide with every zero

Correct Answer: C. Lie within the ROC

Question 9

For a moving-average filter y[n]=(x[n]+x[n−1]+x[n−2])/3, the DC gain is:

A. 0

B. 1/3

C. 1

D. 3

Correct Answer: C. 1

Question 10

In the state-space transfer matrix H(z)=C(zI−A)⁻¹B+D, the eigenvalues of A generally represent:

A. Input amplitudes

B. System poles

C. Sampling frequencies

D. Filter zeros only

Correct Answer: B. System poles