---
title: "Nyquist sampling theorem; natural sampling"
subject: "Communication Systems"
chapter: 4
key: "semester_6/communication_systems:nyquist-natural-sampling"
source: https://www.bctengineer.com/q/semester_6/communication_systems/nyquist-natural-sampling
site: BCT Engineering
---

# Nyquist sampling theorem; natural sampling

Communication Systems · Chapter 4: Pulse Modulation

- Asked in: 2076 Chaitra (CS II), 3+6 marks
- Video: [Sampling Theorem](https://www.youtube.com/watch?v=XyZerq7-IvA) — TutorialsPoint
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/nyquist-natural-sampling
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-4

## Question

State Nyquist sampling theorem. Explain natural sampling with its appropriate mathematical derivation.

## Short answer

Nyquist theorem: sample bandlimited signal at least twice the highest frequency. Natural sampling multiplies signal by pulse train, spectrum copies are exact shape scaled by sinc, no aperture distortion.

## Answer

#### Nyquist sampling theorem
A signal $g(t)$ band-limited to $W$ Hz can be exactly reconstructed from its uniformly spaced samples if the sampling rate satisfies $f_s \ge 2W$. The minimum rate $2W$ is the **Nyquist rate** and $1/(2W)$ the **Nyquist interval**. Below this rate the spectral copies overlap and aliasing occurs.

#### Natural sampling
In natural (chopper) sampling the signal is multiplied by a periodic train of rectangular pulses $c(t)$ of width $\tau$ and period $T_s$. During each pulse the sample **follows the shape of the signal**:
$$
s(t) = g(t)\,c(t)
$$

```text
g(t) ----->(x)-----> s(t)  (tops follow g(t))
            ^
  c(t): |‾|  |‾|  |‾|  width tau, period Ts
```

**Derivation**
1. Expand the pulse train in a Fourier series ($f_s = 1/T_s$):
$$
c(t) = \sum_{n=-\infty}^{\infty} C_n e^{j2\pi n f_s t}, \qquad C_n = \frac{\tau}{T_s}\,\text{sinc}(nf_s\tau)
$$
2. Multiply by $g(t)$:
$$
s(t) = \sum_{n} C_n\, g(t)\, e^{j2\pi n f_s t}
$$
3. By the frequency-shift property, $g(t)e^{j2\pi nf_st} \leftrightarrow G(f - nf_s)$, so
$$
S(f) = \frac{\tau}{T_s}\sum_{n=-\infty}^{\infty}\text{sinc}(nf_s\tau)\,G(f - nf_s)
$$

**Interpretation**
- The spectrum is the message spectrum $G(f)$ repeated at multiples of $f_s$.
- Each copy keeps the **exact shape** of $G(f)$; only its height is scaled by the constant $C_n$. Hence there is **no aperture distortion** (unlike flat-top sampling).
- The $n = 0$ copy is $(\tau/T_s)G(f)$, so a low-pass filter with cut-off between $W$ and $f_s - W$ recovers $g(t)$ (scaled by $\tau/T_s$), provided $f_s \ge 2W$.
- The envelope of the copy heights follows $\text{sinc}(f\tau)$, which falls as $n$ grows.

**Drawback:** the pulse tops vary, so quantizing them is difficult; therefore flat-top sampling is used in PCM.

## Check yourself

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

1. What happens if a band-limited signal is sampled below 2W?
   A. Spectral copies overlap (aliasing)
   B. The signal is amplified
   C. The spectrum is scaled by tau/Ts
   D. Nothing changes
   Answer: A. If fs < 2W the shifted copies overlap and the signal cannot be recovered. (Key point: Nyquist rate)

2. Natural sampling is modelled as:
   A. g(t) times a periodic train of rectangular pulses
   B. g(t) convolved with a rectangular pulse
   C. g(t) times an impulse train only
   D. g(t) passed through a quantizer
   Answer: A. s(t) = g(t)c(t), with c(t) a rectangular pulse train of width tau and period Ts. (Key point: Natural sampling model)

3. In natural sampling, S(f) is a sum of copies of G(f - n fs) weighted by:
   A. fs
   B. (tau/Ts) sinc(n fs tau)
   C. 1/tau
   D. sinc(f tau) applied across each copy
   Answer: B. The Fourier-series coefficient C_n = (tau/Ts) sinc(n fs tau) is a constant per copy. (Key point: Spectrum result)

4. Why does natural sampling have no aperture distortion?
   A. Each copy is only scaled by a constant, so its shape equals G(f)
   B. It uses a higher sampling rate
   C. It uses an equalizer
   D. The pulses are impulses
   Answer: A. The weights C_n are constants, not functions of f, so the shape of each copy is unchanged. (Key point: No aperture distortion)

## Related in Communication Systems

- [Nyquist sampling theorem; flat top sampling](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anyquist-sampling-flat-top) — Communication Systems, chapter 4
- [Flat top vs natural sampling; Nyquist rate of sin(800πt)/πt + 10cos2000t](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aflat-top-vs-natural-nyquist-rate) — Communication Systems, chapter 4
- [Natural and flat top sampling; Nyquist rate of sin²(400πt)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anatural-flat-top-sin2-400pit) — Communication Systems, chapter 4
- [State and prove Nyquist sampling theorem](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anyquist-sampling-prove) — 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
- [Practical considerations in sampling; natural sampling](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asampling-practical-natural) — Communication Systems, chapter 4

## Related in other subjects

- [Sampling theorem; Nyquist of 1 + cos + sin](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Asampling-theorem-nyquist-1-cos-sin) — Signals and Systems, chapter 4, asked 5×
- [Prove sampling theorem; Nyquist of product](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aprove-sampling-theorem-nyquist) — Signals and Systems, chapter 4, asked 2×
- [Sampling theorem; aliasing cause; Nyquist](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Asampling-aliasing-cause-nyquist) — Signals and Systems, chapter 4
- [Sampling theorem, Nyquist criterion and aliasing](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_i%3Asampling-nyquist-aliasing) — Instrumentation I, chapter 6