Questions: Discrete-Time Systems: Sampling and z-Domain Analysis

Question 1

A control engineer samples a sensor at 500 Hz. An electric motor vibrates at 400 Hz, producing noise. In the sampled data, this noise appears at 100 Hz. The engineer applies a digital filter to remove the 100 Hz component. Why does this fail to eliminate the motor vibration?

• A. Digital filters cannot attenuate frequencies above the Nyquist limit
• B. Aliasing has already folded the 400 Hz signal onto 100 Hz; the two are indistinguishable in the sampled data, so filtering 100 Hz also removes genuine low-frequency signals
• C. The z-transform maps 400 Hz to 100 Hz in the digital domain, and the filter must target 400 Hz directly
• D. Sampling at 500 Hz is too slow to represent 400 Hz signals at all, so the noise does not appear in the data

Answer: (B)

This is aliasing: a 400 Hz signal sampled at 500 Hz folds to 500 − 400 = 100 Hz. Once aliasing occurs, the 400 Hz component and any genuine 100 Hz content are mathematically identical in the sampled sequence — no downstream digital processing can separate them. The correct remedy is an anti-aliasing filter applied to the analog signal before the ADC, attenuating everything above the Nyquist frequency (250 Hz in this case) before sampling begins. Filtering after the fact cannot undo the information loss.

Question 2

In z-domain analysis, the stability condition for a discrete-time system requires that:

• A. All poles lie in the left half of the z-plane (Re(z) < 0)
• B. All poles lie inside the unit circle (|z| < 1)
• C. All poles lie on the imaginary axis of the z-plane
• D. All poles lie outside the unit circle (|z| > 1) to ensure sufficient gain

Answer: (B)

The mapping z = e^(sTs) transforms the stability boundary from the imaginary axis (Re(s) = 0) in the s-plane to the unit circle (|z| = 1) in the z-plane. The left half s-plane (stable region, Re(s) < 0) maps to the interior of the unit circle. So a discrete-time system is stable if and only if all its poles satisfy |z| < 1. This is the z-domain analog of the continuous-time rule — the geometry changes but the logic is identical.

Question 3

Anti-aliasing filters must be applied to the analog signal before the analog-to-digital converter, not applied digitally after sampling, because aliasing creates frequency content indistinguishable from genuine low-frequency signals.

• T. True
• F. False

Answer: True

Aliasing is a fundamental and irreversible consequence of sampling. Once a high-frequency signal has been sampled below the Nyquist rate, its alias overlaps with lower frequencies in the digital domain. There is no way to distinguish the aliased content from genuine content at that frequency — the information is permanently confounded. Anti-aliasing filters must act on the continuous-time signal before it is discretized.

Question 4

If a signal has been aliased during sampling, increasing the sampling rate of subsequent processing can recover the original high-frequency content.

• T. True
• F. False

Answer: False

Aliasing is irreversible. Once a signal is sampled below its Nyquist rate, high-frequency components are folded into lower frequencies and the distinction is permanently lost. Resampling the already-aliased digital sequence at a higher rate simply interpolates the corrupted data — it cannot reconstruct information that was never captured. Recovery requires going back to the original analog signal and resampling it at a sufficient rate with an anti-aliasing filter in place.

Question 5

Explain why aliasing is described as a fundamental consequence of sampling rather than a numerical error, and what this implies about when anti-aliasing filters must be applied.

Model answer: Aliasing arises from the mathematics of periodic sampling itself, not from any computational imprecision. When a continuous signal is sampled at rate fs, the spectrum of the sampled sequence is a sum of shifted copies of the original spectrum, repeated at every multiple of fs. If the original signal contains energy above fs/2, those spectral copies overlap and add together — frequencies above Nyquist are permanently confused with frequencies below it. No amount of computational care or post-processing can undo this because the information distinguishing the two was never captured. Anti-aliasing filters must therefore attenuate all signal content above the Nyquist frequency before the analog-to-digital conversion occurs.

This is why the Nyquist theorem is a hard constraint rather than a guideline. It is not that sampling above Nyquist is 'safer' — it is that sampling below Nyquist is mathematically guaranteed to produce aliasing regardless of how carefully the digital processing is done. Engineers place analog lowpass filters before ADCs for precisely this reason: the only point at which aliasing can be prevented is before it happens.