Transfer Function

The transfer function H(z) is the Z-domain representation of a linear time-invariant (LTI) digital system. It encodes the complete input-output behavior of the system in a compact rational polynomial form. Understanding how poles, zeros, and the ROC of H(z) determine stability, causality, and frequency response is central to digital filter design and is a consistently high-weightage topic in GATE.

Core Concept Explanation

The transfer function H(z) of an LTI digital system is defined as the ratio of the Z-transform of the output to the Z-transform of the input under zero initial conditions: H(z) = Y(z)/X(z). This definition holds when both Y(z) and X(z) exist and the system is linear and time-invariant. In the time domain, H(z) corresponds to the Z-transform of the impulse response h[n], meaning H(z) = Z{h[n]}.

The general form of H(z) for a practical digital filter is a rational function: a ratio of two polynomials in z⁻¹. The roots of the numerator polynomial are the zeros of H(z), and the roots of the denominator polynomial are the poles of H(z). Poles and zeros completely characterize H(z) up to a gain constant. Plotting them on the complex z-plane gives the pole-zero plot, which visually communicates stability, phase behavior, and frequency selectivity in a single diagram.

A system is BIBO stable if and only if the ROC of H(z) includes the unit circle. For a causal system whose ROC is |z| > max pole magnitude, stability requires that all poles lie strictly inside the unit circle (|pₖ| < 1 for all poles pₖ). If any pole lies exactly on or outside the unit circle, the causal system is unstable. Zeros have no direct effect on stability — they can lie anywhere in the z-plane.

The frequency response H(e^jω) is obtained by evaluating H(z) on the unit circle: substitute z = e^jω. The magnitude |H(e^jω)| gives the amplitude response (gain at each frequency), and the angle H(e^jω) gives the phase response. Poles close to the unit circle create peaks in the magnitude response (resonances), while zeros on or near the unit circle create notches (nulls at specific frequencies).

Mathematical Expression

For an Nth-order digital filter described by the difference equation y[n] + a₁y[n-1] + ... + aₙy[n-N] = b₀x[n] + b₁x[n-1] + ... + bₘx[n-M], taking the Z-transform gives Y(z)(1 + a₁z⁻¹ + ... + aₙz⁻ᴺ) = X(z)(b₀ + b₁z⁻¹ + ... + bₘz⁻ᴹ). Therefore H(z) = (b₀ + b₁z⁻¹ + ... + bₘz⁻ᴹ) / (1 + a₁z⁻¹ + ... + aₙz⁻ᴺ).

In factored form, H(z) = K · ∏(z - zₖ) / ∏(z - pₖ), where zₖ are zeros and pₖ are poles. The gain constant K sets the overall scale. Cascading two systems H₁(z) and H₂(z) gives an overall transfer function H(z) = H₁(z)·H₂(z), and the poles and zeros of the cascade are the union of poles and zeros of both subsystems. This factored form is used directly in direct-form II and cascade filter implementations.

Practical Understanding

In digital signal processing tools, filters are specified by their numerator and denominator coefficient vectors, which directly encode the transfer function. Given H(z), stability is verified by checking that all denominator roots (poles) have magnitude less than 1. MATLAB's roots() or Python's numpy.roots() computes these poles numerically. A single pole at |p| ≥ 1 in a causal filter will cause the output to grow without bound for any bounded input.

Poles near the unit circle create sharp peaks in the frequency response, useful for designing narrowband filters. By precisely placing a zero at z = e^(jω₀), a notch filter can completely eliminate a specific frequency ω₀. This frequency-selective placement of poles and zeros on the z-plane is the core of digital filter design methodology.

Given:
H(z) = (z² + z) / (z² - 0.9z + 0.2)
Find poles, zeros, and determine stability (causal system).

Why this formula applies:
Numerator roots give zeros, denominator roots give poles.
Causal stability requires all |pₖ| < 1.

Formula:
Zeros: z² + z = z(z + 1) = 0  →  z = 0, z = -1
Poles: z² - 0.9z + 0.2 = 0

Substitution:
Using quadratic formula: z = [0.9 ± √(0.81 - 0.8)] / 2
                        z = [0.9 ± √0.01] / 2
                        z = [0.9 ± 0.1] / 2

Calculation:
p₁ = (0.9 + 0.1)/2 = 1.0/2 = 0.5
p₂ = (0.9 - 0.1)/2 = 0.8/2 = 0.4

Stability check:
|p₁| = 0.5 < 1 ✓
|p₂| = 0.4 < 1 ✓
Both poles inside unit circle.

Final Answer:
Zeros at z = 0 and z = -1.
Poles at z = 0.5 and z = 0.4.
System is causal and stable (ROC: |z| > 0.5 includes unit circle).

Exam Tip: To check stability of a causal digital system quickly, compute the poles from the denominator of H(z) and verify all magnitudes are less than 1. If the denominator is a second-order polynomial az² + bz + c, the system is stable when all Jury stability criterion conditions are satisfied, or simply check that |roots| < 1 numerically.

• H(z) = Y(z)/X(z) = Z{h[n]}; completely characterizes a causal LTI system.
• Numerator roots: zeros. Denominator roots: poles.
• Causal stable: all poles have |pₖ| < 1 (strictly inside unit circle).
• Pole near unit circle → resonance (sharp peak in frequency response).
• Zero on unit circle → notch (complete null at that frequency).

Quick Revision

• H(z) = B(z)/A(z); zeros from numerator roots, poles from denominator roots.
• Causal stability: all |poles| < 1; ROC extends to |z| = ∞.
• Frequency response: H(e^jω) = H(z) evaluated on unit circle z = e^jω.
• Cascade: H(z) = H₁(z)·H₂(z); poles and zeros add up.
• Parallel: H(z) = H₁(z) + H₂(z); combine over common denominator.
• Trap: Zeros do not affect stability — only pole magnitudes matter for BIBO stability.
• Trap: A pole at z = 1 means the system has a DC resonance and is marginally stable (not stable for all bounded inputs).