---
title: "Sub-sampling theory and aperture effect"
subject: "Communication Systems"
chapter: 4
key: "semester_6/communication_systems:subsampling-aperture-effect"
source: https://www.bctengineer.com/q/semester_6/communication_systems/subsampling-aperture-effect
site: BCT Engineering
---

# Sub-sampling theory and aperture effect

Communication Systems · Chapter 4: Pulse Modulation

- Asked in: 2074 Chaitra (CS II), 2+2 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/subsampling-aperture-effect
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-4

## Question

Briefly explain the terms "sub-sampling theory" and "aperture effect".

## Short answer

Sub-sampling (band-pass sampling) allows sampling bandpass signals (fL to fH, bandwidth B) at rates near 2B instead of 2fH, using deliberate aliasing to shift band to baseband. Allowed rates: 2fH/k ≤ fs ≤ 2fL/(k-1) for integer k ≤ fH/B. Aperture effect: in flat-top sampling, finite pulse width τ causes spectrum multiplication by sinc(fτ), attenuating high frequencies. Reduced by τ ≪ Ts and corrected by equalizer with inverse sinc response.

## Answer

#### Sub-sampling (band-pass sampling) theory
For a **band-pass signal** occupying $f_L$ to $f_H$ with bandwidth $B = f_H - f_L$, it is not necessary to sample at $2f_H$. The signal can be recovered from samples taken at a rate as low as about $2B$, i.e. **below the Nyquist rate of its highest frequency**. This is called sub-sampling (or under-sampling). The allowed rates are
$$
\frac{2f_H}{k} \le f_s \le \frac{2f_L}{k - 1}, \qquad k = 1, 2, \ldots, \left\lfloor \frac{f_H}{B} \right\rfloor
$$
Aliasing is used deliberately to shift the band down to baseband without overlap. Example: an 80–115 kHz signal can be sampled at 76.7–80 kHz instead of 230 kHz. Sub-sampling is used in digital receivers to sample IF signals directly.

#### Aperture effect
In flat-top sampling the sample is held for a finite width $\tau$, so the spectrum is multiplied by $H(f) = \tau\,\text{sinc}(f\tau)$. This attenuates the higher frequencies of the message, an amplitude distortion called the **aperture effect**. It is reduced by making $\tau \ll T_s$ and corrected by an equalizer with response $1/H(f)$ after the reconstruction filter.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Sub-sampling (band-pass sampling) allows a band-pass signal from fL to fH to be sampled at about:
   A. 2 fH always
   B. 2B, where B = fH - fL, below 2fH
   C. fL only
   D. B/2
   Answer: B. Because the band is narrow, a rate near 2B can work without overlap. (Key point: Sub-sampling idea)

2. In sub-sampling, aliasing is:
   A. Always harmful
   B. Used deliberately to shift the band down without overlap
   C. Impossible
   D. Removed by an equalizer
   Answer: B. The band folds to baseband without overlapping itself if the rate is chosen correctly. (Key point: Role of aliasing)

3. The allowed sub-sampling rates satisfy:
   A. 2fH/k <= fs <= 2fL/(k-1)
   B. fs >= 2fH only
   C. fs = fH - fL exactly
   D. fs <= fL/2
   Answer: A. For each integer k up to floor(fH/B), this range avoids overlap. (Key point: Allowed rates)

4. The aperture effect is:
   A. Attenuation of high message frequencies due to the sinc factor in flat-top sampling
   B. Overlap of spectra
   C. Slope overload
   D. Quantization noise
   Answer: A. H(f) = tau sinc(f tau) multiplies the spectrum; an equalizer 1/H(f) corrects it. (Key point: Aperture effect)

## Related in Communication Systems

- [Sampling theorem; aliasing and aperture effect remedies](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asampling-theorem-aliasing-aperture-remedy) — Communication Systems, chapter 4
- [Prove sampling theorem; aliasing and aperture effect](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asampling-prove-aliasing-aperture) — Communication Systems, chapter 4
- [Aperture effect correction; bandpass sampling 80–115 kHz](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aaperture-correction-bandpass-80-115) — Communication Systems, chapter 4
- [Nyquist-Kotelnikov sampling theorem; aliasing and aperture](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anyquist-kotelnikov-aliasing-aperture) — Communication Systems, chapter 4
- [Aperture effect, DPCM and DM slope overload numerical](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aaperture-effect-dpcm-dm-slope) — Communication Systems, chapter 4
- [Aliasing and aperture effects; types of sampling](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aaliasing-aperture-sampling-types) — Communication Systems, chapter 4

## Related in other subjects

- [Prove sampling theorem; aliasing effect](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aprove-sampling-theorem-aliasing) — Signals and Systems, chapter 4
- [Numerical aperture and dispersion](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Ana-and-dispersion) — Propagation and Antenna, chapter 6
- [Aperture antennas and rectangular horn radiation](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Aaperture-antenna-rectangular-horn) — Propagation and Antenna, chapter 3