---
title: "Reconstruction methods; Nyquist of product"
subject: "Signals and Systems"
chapter: 4
key: "semester_4/signals_and_systems:reconstruction-methods-nyquist"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/reconstruction-methods-nyquist
site: BCT Engineering
---

# Reconstruction methods; Nyquist of product

Signals and Systems · Chapter 4: Sampling

- Asked in: 2071 Bhadra, 4+4 marks
- Video: [Reconstruction of Signals](https://www.youtube.com/watch?v=rmDg3eVWT8E) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/reconstruction-methods-nyquist
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-4

## Question

Discuss different methods to reconstruct a signal. Determine the Nyquist rate for a continuous time signal x(t) = (1/2π) cos(200πt) cos(300πt).

## Short answer

Reconstruction methods include ideal sinc interpolation, zero-order hold, and first-order hold. For x(t) = (1/2π)cos(200πt)cos(300πt), using cosAcosB yields components at 50 Hz and 250 Hz; maximum frequency is 250 Hz, so Nyquist rate is 500 Hz.

## Answer

#### Methods of reconstruction
Reconstruction (interpolation) means rebuilding the continuous signal $x(t)$ from its samples $x(nT)$.

1. **Ideal (band-limited) interpolation:** pass the impulse-sampled signal through an ideal LPF with gain $T$ and cutoff $\omega_c = \omega_s/2$:
$$
x_r(t) = \sum_{n=-\infty}^{\infty} x(nT)\,\frac{\sin\!\big(\omega_c(t-nT)\big)}{\omega_c(t-nT)}\cdot\frac{\omega_c T}{\pi}
$$
Each sample is replaced by a sinc pulse; the result equals $x(t)$ exactly when $\omega_s > 2\omega_M$. It is not realizable because the ideal LPF is non-causal.
2. **Zero-order hold (ZOH):** each sample is held constant until the next sample, giving a staircase. Impulse response $h_0(t) = 1$ for $0 \le t < T$. Simple (used in DACs) but adds distortion, so a smoothing (reconstruction) filter follows it.
3. **First-order hold (linear interpolation):** adjacent samples are joined by straight lines. Impulse response is a triangle of width $2T$. Smoother than ZOH but still approximate.

```text
ZOH:   _    __          FOH:    /\
      | |__|  |__              /  \__/\
      staircase                straight lines
```

#### Nyquist rate of $x(t) = \frac{1}{2\pi}\cos(200\pi t)\cos(300\pi t)$
Use $\cos A\cos B = \tfrac12[\cos(A-B) + \cos(A+B)]$:
$$
\begin{aligned}
x(t) &= \frac{1}{2\pi}\cdot\frac12\big[\cos(100\pi t) + \cos(500\pi t)\big] \\
&= \frac{1}{4\pi}\cos(100\pi t) + \frac{1}{4\pi}\cos(500\pi t)
\end{aligned}
$$

| Term | $\omega$ (rad/s) | $f$ (Hz) |
|---|---|---|
| $\cos(100\pi t)$ | $100\pi$ | $50$ |
| $\cos(500\pi t)$ | $500\pi$ | $250$ |

$f_M = 250$ Hz.
$$
f_N = 2 f_M = 500\ \text{Hz}
$$
**Answer: Nyquist rate = 500 Hz (500 samples/s, $\omega_N = 1000\pi$ rad/s).**

## Check yourself

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

1. What does ideal band-limited reconstruction use, and why is it not realizable?
   A. An ideal LPF (sinc interpolation); it is non-causal
   B. A staircase hold; it needs future samples
   C. A triangular filter; it has infinite gain
   D. An ideal HPF; it is unstable
   Answer: A. Ideal interpolation replaces each sample by a sinc, whose impulse response extends to negative time, so it is non-causal. (Key point: Ideal interpolation)

2. What does a zero-order hold output?
   A. Straight lines joining samples
   B. Sinc pulses at each sample
   C. A staircase, each sample held until the next
   D. Impulses only
   Answer: C. ZOH has h0(t) = 1 for 0 <= t < T, giving a staircase. (Key point: Zero-order hold)

3. What is the impulse response shape of first-order hold?
   A. A rectangle of width T
   B. A triangle of width 2T
   C. A sinc
   D. An exponential decay
   Answer: B. FOH joins samples with straight lines, equivalent to a triangular impulse response. (Key point: First-order hold)

4. For x(t) = (1/2π)cos(200πt)cos(300πt), what is the Nyquist rate?
   A. 300 Hz
   B. 600 Hz
   C. 1000 Hz
   D. 500 Hz
   Answer: D. The product gives cos(100πt) and cos(500πt); fM = 250 Hz so the Nyquist rate is 500 Hz. (Key point: Nyquist rate of product)

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