Questions: Sampling Theorem and Nyquist Sampling Rate

Question 1

A signal contains frequency components up to 4 kHz and is sampled at 6 kHz. What happens?

• A. The signal is reconstructed perfectly — 6 kHz is above 4 kHz so the Nyquist condition is satisfied
• B. Aliasing occurs because the sampling rate (6 kHz) is below the required Nyquist rate (2 × 4 kHz = 8 kHz)
• C. The signal is slightly degraded but recoverable with a good lowpass filter applied after sampling
• D. There is no problem as long as a high-resolution analog-to-digital converter is used

Answer: (B)

The Nyquist theorem requires f_s ≥ 2 × f_max — the sampling rate must be at least twice the highest frequency in the signal. Here f_max = 4 kHz requires f_s ≥ 8 kHz; sampling at 6 kHz violates this. The spectral copies created by sampling will overlap (alias), corrupting the signal. The common mistake is comparing f_s to f_max directly (6 > 4 looks fine) rather than to 2 × f_max. Option C is also wrong — once aliasing has occurred, it cannot be undone by post-sampling filtering.

Question 2

A 3 kHz tone is sampled at 5 kHz, causing aliasing. After sampling, which of the following is true?

• A. A high-quality lowpass filter applied to the sampled data can recover the original 3 kHz tone
• B. The aliasing only affects frequencies above 2.5 kHz, so the lower portion of the signal is intact
• C. The aliased image at 5 − 3 = 2 kHz is indistinguishable from an original 2 kHz component, making perfect recovery impossible
• D. The original signal can be recovered if the sampling rate was only slightly below the Nyquist rate

Answer: (C)

Aliasing is irreversible. When f_s = 5 kHz and a 3 kHz tone is present, the spectral copy centered at 5 kHz places an image at 5 − 3 = 2 kHz — right inside the signal band. The sampled signal now contains energy at 2 kHz, but you cannot tell whether it originated from an actual 2 kHz component or from the aliased 3 kHz component. No post-processing can distinguish the two contributions. This is why anti-aliasing filters must be applied *before* sampling — once aliasing occurs, the information is permanently corrupted.

Question 3

If a bandlimited signal is sampled at exactly the Nyquist rate (f_s = 2 × f_max), the original continuous signal can theoretically be recovered exactly from its discrete samples using ideal sinc interpolation.

• T. True
• F. False

Answer: True

This is the positive result of the Nyquist-Shannon sampling theorem. When the sampling criterion is met, sampling is not an approximation — it is a lossless representation. The spectral copies created by sampling are spaced far enough apart that they don't overlap, and an ideal lowpass filter (or equivalently, sinc interpolation in the time domain) can perfectly reconstruct the original signal. This bridges discrete-time and continuous-time representations without any information loss.

Question 4

Sampling a continuous signal inevitably introduces some information loss, regardless of how high the sampling rate is, because continuous signals have infinite resolution.

• T. True
• F. False

Answer: False

This is a fundamental misconception. For bandlimited signals, the Nyquist theorem guarantees that sampling above the Nyquist rate is perfectly lossless — the original continuous signal can be exactly recovered. The key condition is bandlimiting: if all energy is confined below f_max and you sample at f_s ≥ 2 × f_max, no information is lost. Sampling is not an approximation; it is a lossless change of representation when the bandwidth condition is satisfied. (Truly infinite-bandwidth signals cannot be sampled without loss, but practical signals are always bandlimited.)

Question 5

Why must an anti-aliasing filter be applied to a signal *before* sampling rather than after, and what does this tell us about the reversibility of aliasing?

Model answer: An anti-aliasing filter removes frequency components above f_s/2 (the Nyquist frequency) before sampling, ensuring no energy exists that could alias. It must come before sampling because aliasing is irreversible: once spectral copies overlap in the sampled signal, the aliased components are mixed with original signal components and cannot be separated. Applying a filter after sampling cannot undo this — you would be filtering both the original signal and the aliased imposters together. The pre-sampling filter prevents the problem from occurring at all.

This reveals that aliasing is not noise that can be cleaned up later — it is a fundamental loss of information structure. The overlapping copies create ambiguity about which energy came from where, and ambiguity cannot be resolved by filtering alone. Anti-aliasing filters are therefore part of the correct system design, not an afterthought.