---
title: "Sampling theorem via impulse train sampling"
subject: "Signals and Systems"
chapter: 4
key: "semester_4/signals_and_systems:nyquist-impulse-train-frequency"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/nyquist-impulse-train-frequency
site: BCT Engineering
---

# Sampling theorem via impulse train sampling

Signals and Systems · Chapter 4: Sampling

- Asked in: 2077 Chaitra, 6 marks
- Video: [Sampling Theorem for Bandlimited Signals](https://www.youtube.com/watch?v=yriemaFCQtc) — John Buck
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/nyquist-impulse-train-frequency
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-4

## Question

Define Nyquist sampling theorem. Explain using frequency domain analysis of impulse train sampling.

## Short answer

Impulse-train sampling multiplies x(t) by a periodic impulse train, making the spectrum X(jω) repeat at multiples of ω_s with scaling 1/T. The copies do not overlap only if ω_s > 2ω_M, so an ideal low-pass filter of gain T can extract the original baseband spectrum without aliasing.

## Answer

**Nyquist sampling theorem:** a signal band-limited to $\omega_M$ is completely determined by its samples $x(nT)$ if $\omega_s = 2\pi/T > 2\omega_M$. The minimum rate $2\omega_M$ is the Nyquist rate.

#### Frequency-domain analysis of impulse-train sampling
Let $x(t)$ be band-limited to $\omega_M$. Impulse-train sampling multiplies it by a periodic impulse train of period $T$:
$$
p(t) = \sum_{n=-\infty}^{\infty} \delta(t - nT), \qquad x_p(t) = x(t)p(t) = \sum_{n=-\infty}^{\infty} x(nT)\,\delta(t-nT)
$$
The Fourier transform of the impulse train is also an impulse train, with spacing $\omega_s = 2\pi/T$:
$$
P(j\omega) = \frac{2\pi}{T}\sum_{k=-\infty}^{\infty}\delta(\omega - k\omega_s)
$$
Multiplication in time is convolution in frequency:
$$
\begin{aligned}
X_p(j\omega) &= \frac{1}{2\pi}\, X(j\omega) * P(j\omega) \\
&= \frac{1}{T}\sum_{k=-\infty}^{\infty} X\big(j(\omega - k\omega_s)\big)
\end{aligned}
$$
So the spectrum of the sampled signal is the original spectrum, scaled by $1/T$, repeated at every multiple of $\omega_s$.

```text
X(jw):            /\
                 /  \
           -----/----\------> w
              -wM    wM

Xp(jw), ws > 2wM (no overlap):
    /\        /\        /\
   /  \      /  \      /  \
--/----\----/----\----/----\--> w
    -ws        0        ws
       gap between copies = ws - 2wM
```

#### Conclusion from the spectrum
- The copy centred at $k=0$ occupies $-\omega_M$ to $\omega_M$; the next copy starts at $\omega_s - \omega_M$.
- The copies do not overlap only if $\omega_s - \omega_M > \omega_M$, i.e. **$\omega_s > 2\omega_M$**. This is the Nyquist condition.
- When it holds, an ideal low-pass filter with gain $T$ and cutoff $\omega_c$ such that $\omega_M < \omega_c < \omega_s - \omega_M$ passes only the $k=0$ copy, giving back $X(j\omega)$ exactly. Hence $x(t)$ is fully recovered.
- If $\omega_s < 2\omega_M$, the copies overlap (aliasing) and $x(t)$ cannot be recovered.

## Check yourself

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

1. FT of p(t) = sum δ(t - nT) is
   A. (1/T) sum δ(ω - kωs)
   B. T sum δ(ω - kωs)
   C. (2π/T) sum δ(ω - kωs)
   D. A single impulse at ωs
   Answer: C. An impulse train in frequency with spacing ωs. (Key point: FT of impulse train)

2. X_p(jω) is
   A. X(jω) scaled by T
   B. X(jω) shifted once by ωs
   C. X(jω) multiplied by P(jω)
   D. X(jω) repeated at multiples of ωs, scaled by 1/T
   Answer: D. Convolution with impulses creates copies. (Key point: Sampled spectrum)

3. Copies do not overlap if
   A. ωs - ωM > ωM
   B. ωs < ωM
   C. ωs - ωM < ωM
   D. ωs = 0
   Answer: A. Gives ωs > 2ωM. (Key point: Condition)

4. The recovery LPF cutoff ωc must satisfy
   A. ωc > ωs
   B. ωM < ωc < ωs - ωM
   C. ωc < ωM
   D. ωc = 2ωs
   Answer: B. It must keep the baseband copy and reject the next one. (Key point: Cutoff)

## Related in Signals and Systems

- [Aliasing via impulse-train sampling](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aaliasing-impulse-train-frequency) — Signals and Systems, chapter 4
- [Impulse train sampling and reconstruction](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aimpulse-train-sampling-reconstruction) — Signals and Systems, chapter 4
- [Impulse train sampling, aliasing, reconstruction](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aimpulse-train-aliasing-reconstruction) — Signals and Systems, chapter 4
- [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

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- [State and prove Nyquist sampling theorem](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anyquist-sampling-prove) — Communication Systems, chapter 4
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