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

# Nyquist sampling theorem; flat top sampling

Communication Systems · Chapter 4: Pulse Modulation

- Asked in: 2080 Baisakh (CS II), 4+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-sampling-flat-top
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-4

## Question

State and explain Nyquist sampling theorem. Explain flat top sampling with appropriate derivation.

## Short answer

Nyquist: a signal bandlimited to W needs fs ≥ 2W for recovery. Flat-top sampling holds each sample for τ; its spectrum is fs Σ G(f–nfs) times H(f)=τ sinc(fτ)e^{-jπfτ}. The sinc causes aperture distortion, requiring an equalizer at the receiver.

## Answer

#### Nyquist sampling theorem
A signal band-limited to $W$ Hz (no components above $W$) is completely described by, and can be exactly recovered from, its samples taken uniformly at a rate
$$
f_s \ge 2W
$$
The minimum rate $f_s = 2W$ is the **Nyquist rate** and $T_s = 1/(2W)$ is the **Nyquist interval**.

**Explanation:** ideal sampling multiplies $g(t)$ by an impulse train $\delta_{T_s}(t)$. In the frequency domain this repeats the spectrum every $f_s$:
$$
G_\delta(f) = f_s\sum_{n=-\infty}^{\infty} G(f - nf_s)
$$
If $f_s \ge 2W$ the copies do not overlap, so a low-pass filter of cut-off $W \le f_c \le f_s - W$ recovers $G(f)$ exactly. If $f_s < 2W$ the copies overlap (**aliasing**) and recovery is impossible.

#### Flat-top sampling
In flat-top sampling each sample is held constant for a pulse width $\tau$ (sample-and-hold), giving a PAM wave with flat tops. It is modelled as ideal sampling followed by convolution with a rectangular pulse $h(t)$ of width $\tau$:
$$
s(t) = \sum_n g(nT_s)\,h(t - nT_s) = \big[g(t)\,\delta_{T_s}(t)\big] * h(t)
$$
Taking the Fourier transform (convolution becomes multiplication):
$$
\begin{aligned}
S(f) &= \Big[f_s\sum_n G(f - nf_s)\Big] H(f) \\
H(f) &= \tau\,\text{sinc}(f\tau)\,e^{-j\pi f\tau}
\end{aligned}
$$

```text
 g(t)--(x)-->ideal samples-->[h(t): hold tau]-->s(t)
        |
   impulse train
```

**Meaning of the result**
- The spectrum is still repeated every $f_s$, but each copy is multiplied by $H(f)$, a sinc shape.
- So the high frequencies of the message are attenuated: this amplitude distortion is the **aperture effect**.
- Distortion is small when $\tau \ll T_s$.
- At the receiver it is corrected with an **equalizer** of response $1/H(f)$ (in the band $|f| \le W$) after the reconstruction low-pass filter.

Flat-top sampling is the one used in practice (sample-and-hold before the ADC) because a constant amplitude during $\tau$ is easy to quantize and generate.

## Check yourself

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

1. A signal band-limited to W Hz can be exactly recovered if it is sampled at a rate:
   A. fs <= W
   B. fs >= W
   C. fs >= 2W
   D. fs = W/2
   Answer: C. The theorem requires fs >= 2W; 2W is the Nyquist rate. (Key point: Nyquist theorem)

2. What does ideal impulse sampling do to the spectrum of the signal?
   A. It shifts it once to fs
   B. It repeats the spectrum G(f) at every multiple of fs
   C. It compresses it to half the bandwidth
   D. It removes the high frequencies
   Answer: B. G_delta(f) = fs times the sum of G(f - n fs), i.e. copies at multiples of fs. (Key point: Spectrum of ideal sampling)

3. Flat-top sampling is modelled as:
   A. Ideal sampling followed by convolution with a rectangular pulse of width tau
   B. Multiplication of the signal by a sinc pulse train
   C. Ideal sampling followed by differentiation
   D. Quantizing the signal and then low-pass filtering it
   Answer: A. Ideal samples are convolved with h(t), so in frequency the repeated spectrum is multiplied by H(f). (Key point: Flat-top model)

4. Why are high message frequencies attenuated in flat-top sampling?
   A. Because the copies overlap (aliasing)
   B. Because H(f) = tau sinc(f tau) multiplies the spectrum and falls with frequency
   C. Because fs is too low
   D. Because quantization noise grows with frequency
   Answer: B. The sinc-shaped H(f) rolls off, causing the aperture effect, corrected by a 1/H(f) equalizer. (Key point: Aperture effect)

## Related in Communication Systems

- [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
- [Nyquist sampling theorem; natural sampling](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anyquist-natural-sampling) — Communication Systems, chapter 4
- [Merits and demerits of flat-top sampling](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aflat-top-sampling-merits) — 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

## 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