---
title: "Reconstruction from samples taken at Nyquist rate"
subject: "Communication Systems"
chapter: 4
key: "semester_6/communication_systems:reconstruction-nyquist-samples"
source: https://www.bctengineer.com/q/semester_6/communication_systems/reconstruction-nyquist-samples
site: BCT Engineering
---

# Reconstruction from samples taken at Nyquist rate

Communication Systems · Chapter 4: Pulse Modulation

- Asked in: 2070 Asar (CS II), 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/reconstruction-nyquist-samples
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-4

## Question

With mathematical derivation show that original band limited signal can be reconstructed from its samples taken at Nyquist rate.

## Short answer

Using ideal sampling and LPF, the Fourier transform shows that G(f) is recovered from the samples. Inverse transform yields g(t) = sum of sinc pulses weighted by samples, perfectly reconstructing the band-limited signal.

## Answer

Let $g(t)$ be band-limited to $W$ Hz and sampled ideally at the Nyquist rate $f_s = 2W$ ($T_s = 1/2W$).

**1. Sampled signal and its spectrum**
$$
g_\delta(t) = \sum_n g(nT_s)\,\delta(t - nT_s), \qquad G_\delta(f) = f_s\sum_n G(f - nf_s)
$$
With $f_s = 2W$ the copies just touch without overlapping, so in $|f| < W$:
$$
G_\delta(f) = 2W\,G(f) \;\Rightarrow\; G(f) = \frac{1}{2W}G_\delta(f), \quad |f| < W
$$

**2. Express $G(f)$ through the samples.** Taking the Fourier transform of $g_\delta(t)$ directly:
$$
G_\delta(f) = \sum_n g\!\left(\frac{n}{2W}\right)e^{-j\pi nf/W}
$$
so
$$
G(f) = \frac{1}{2W}\sum_n g\!\left(\frac{n}{2W}\right)e^{-j\pi nf/W}, \quad |f| < W
$$
$G(f)$, and therefore $g(t)$, is fixed by the samples alone.

**3. Inverse transform (ideal LPF reconstruction)**
$$
\begin{aligned}
g(t) &= \int_{-W}^{W}G(f)e^{j2\pi ft}df \\
&= \sum_n g\!\left(\frac{n}{2W}\right)\frac{1}{2W}\int_{-W}^{W}e^{j2\pi f(t - n/2W)}df \\
&= \sum_n g\!\left(\frac{n}{2W}\right)\frac{\sin\big(2\pi W(t - n/2W)\big)}{2\pi W(t - n/2W)} \\
&= \sum_n g\!\left(\frac{n}{2W}\right)\text{sinc}(2\pi Wt - n\pi)
\end{aligned}
$$

This is the **interpolation formula**. Each sample is multiplied by a sinc pulse centred at its own instant; at $t = kT_s$ all other sincs are zero, so $g(kT_s)$ is reproduced exactly, and between samples the sum fills in $g(t)$. Physically this is the output of an ideal LPF (cut-off $W$, gain $T_s$) driven by the sample impulses. Hence a band-limited signal is fully recovered from samples at the Nyquist rate.

## Check yourself

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

1. At fs = 2W the sampled spectrum within |f| < W equals:
   A. G(f)/2W
   B. 2W G(f)
   C. W G(f)
   D. G(f) fs^2
   Answer: B. G_delta(f) = fs G(f) = 2W G(f), hence G(f) = G_delta(f)/2W. (Key point: Spectrum at Nyquist rate)

2. How is G(f) expressed through the samples?
   A. As a sum of g(n/2W) e^(-j pi n f / W) divided by 2W
   B. As the integral of g(t)
   C. As the product of samples
   D. By the sampling rate only
   Answer: A. Fourier transform of the impulse train gives G_delta(f) = sum g(n/2W) e^(-j pi n f/W). (Key point: Fixed by samples)

3. The inverse transform over -W to W gives g(t) as:
   A. A sum of samples times sinc pulses centred at the sample instants
   B. A sum of rect pulses
   C. A product of samples
   D. A Fourier series in t
   Answer: A. g(t) = sum g(n/2W) sinc(2 pi W t - n pi). (Key point: Interpolation formula)

4. Physically, the interpolation formula is the output of:
   A. A high-pass filter
   B. An ideal LPF with cut-off W and gain Ts driven by sample impulses
   C. A quantizer
   D. An equalizer
   Answer: B. Each impulse produces a sinc impulse response, and the outputs sum. (Key point: Physical meaning)

## Related in Communication Systems

- [Ideal sampling and reconstruction; Nyquist rate of ½π cos(400πt)cos(1000πt)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aideal-sampling-reconstruction-nyquist) — Communication Systems, chapter 4
- [Information rate of a quantized 100 Hz signal](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ainformation-rate-100hz-four-symbols) — Communication Systems, chapter 6
- [Band pass and band limited signals; LTI properties](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Abandpass-bandlimited-lti-properties) — Communication Systems, chapter 1
- [USB-SSB expression for a random band limited message](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Ausb-ssb-random-message) — Communication Systems, chapter 2
- [Entropy and information rate of 6-level PCM signal](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aentropy-info-rate-5khz-six-levels) — Communication Systems, chapter 6
- [Aperture and aliasing; sampling 10cos(20πt)cos(200πt) at 250 samples/s](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aaperture-aliasing-10cos20pit-250) — Communication Systems, chapter 4

## Related in other subjects

- [Band-limited signals; aliasing; Nyquist rate](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Abandlimited-aliasing-nyquist) — Signals and Systems, chapter 4
- [Reconstruction of signal from samples](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Areconstruction-from-sampled-signal) — Digital Control System, chapter 1
- [Aliasing in a band-limited signal](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aaliasing-bandlimited-w) — Signals and Systems, chapter 4
- [Reconstruction methods; Nyquist of product](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Areconstruction-methods-nyquist) — Signals and Systems, chapter 4