Questions: Z-Transform: Fundamentals for Discrete-Time Signals

Question 1

Two Z-transforms both simplify to the same algebraic expression X(z) = 1/(1 − az⁻¹). One has ROC |z| > |a| and the other has ROC |z| < |a|. What does this tell you?

• A. The two ROCs correspond to the same time-domain sequence — the ROC is just a mathematical detail
• B. The two ROCs correspond to different time-domain sequences: one causal, one anti-causal
• C. The ROC |z| < |a| is always invalid because the Z-transform only converges outside a circle
• D. Having two different ROCs means there is an error in the calculation

Answer: (B)

This is the central reason the ROC must always be specified. The algebraic expression alone does not uniquely determine the time-domain signal — the same X(z) can correspond to a causal right-sided sequence (ROC: |z| > |a|) or an anti-causal left-sided sequence (ROC: |z| < |a|). Without the ROC, the inverse Z-transform is ambiguous. Carrying the ROC is not a formality; it encodes causality and stability information that the fraction itself cannot.

Question 2

A digital filter has transfer function H(z) with poles at z = 0.9 and z = −0.5. Assuming the system is causal, is it stable?

• A. No — the pole at z = 0.9 is too close to the unit circle and will cause oscillations
• B. No — stability requires all poles to lie outside the unit circle for causal systems
• C. Yes — for a causal system, stability requires all poles to lie strictly inside the unit circle, and both poles satisfy |z| < 1
• D. Cannot be determined without knowing the zeros of H(z)

Answer: (C)

For a causal LTI discrete-time system, BIBO stability requires all poles to lie strictly inside the unit circle (|z| < 1). Both poles here satisfy this: |0.9| < 1 and |−0.5| < 1. This is the discrete-time analog of the Laplace stability condition (all poles with negative real part). The zeros of H(z) do not affect stability — only poles do. Closeness to the unit circle affects the filter's frequency response shape but does not by itself cause instability as long as the pole is strictly inside.

Question 3

A causal digital system is stable if and mainly if most its poles lie exactly on the unit circle (|z| = 1).

• T. True
• F. False

Answer: False

Poles on the unit circle (|z| = 1) make a causal system marginally stable, not stable. A pole at z = 1 corresponds to an accumulator — a bounded nonzero input produces an output that grows without bound. Stability requires all poles to lie *strictly inside* the unit circle (|z| < 1). The distinction between 'on' and 'inside' is critical: a pole at z = e^(jω) on the unit circle produces an undamped oscillation in the impulse response, which grows unboundedly for certain inputs.

Question 4

Evaluating the Z-transform X(z) on the unit circle (z = e^(jω)) yields the discrete-time Fourier transform (DTFT) of the sequence x[n].

• T. True
• F. False

Answer: True

This is the precise relationship between the Z-transform and the DTFT. Substituting z = e^(jω) into X(z) = Σ x[n]z^(−n) gives Σ x[n]e^(−jωn), which is exactly the DTFT definition. This mirrors the Laplace–Fourier relationship: just as the Fourier transform is the Laplace transform evaluated on the imaginary axis (s = jω), the DTFT is the Z-transform evaluated on the unit circle. For this evaluation to be valid, the unit circle must lie within the ROC of X(z).

Question 5

Why must the region of convergence (ROC) always be specified alongside the algebraic expression for a Z-transform? What goes wrong if it is omitted?

Model answer: The ROC must be specified because the same algebraic expression X(z) can correspond to multiple different time-domain sequences — differing in causality and stability — depending on which region of the z-plane is used. Without the ROC, the inverse Z-transform is ambiguous. Additionally, the ROC directly determines whether a system is stable: for causal systems, stability requires all poles to be strictly inside the unit circle, which means the ROC must include the unit circle. Omitting the ROC strips away the information needed to determine causality and stability.

The Z-transform is defined as a power series, and like all series it converges only in certain regions. The algebraic form that results from simplification looks identical regardless of which ROC applies, but the time-domain signals they represent can be completely different (e.g., causal vs. anti-causal versions of the same exponential). The ROC is not a byproduct — it is part of the answer.