---
title: "Data hold circuit and ZOH derivation"
subject: "Digital Control System"
chapter: 1
key: "semester_6/digital_control_system:data-hold-zoh-derivation"
source: https://www.bctengineer.com/q/semester_6/digital_control_system/data-hold-zoh-derivation
site: BCT Engineering
---

# Data hold circuit and ZOH derivation

Digital Control System · Chapter 1: Introduction to Discrete Time Control System

- Asked in: 2082 Chaitra (new course), 3 marks
- Video: [Zero Order Hold Sampling (Detailed explanation)](https://www.youtube.com/watch?v=NCLo4Kqm3P8) — vinay singh
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/digital_control_system/data-hold-zoh-derivation
- Whole chapter: https://www.bctengineer.com/md/semester_6/digital_control_system/chapter-1

## Question

What do you mean by data hold circuit? Derive the transfer function of Zero Order Hold (ZOH).

## Short answer

A data hold circuit reconstructs a continuous signal from samples. A zero-order hold keeps each sample constant for one period T, giving impulse response g(t)=u(t)-u(t-T). Its Laplace transform yields the transfer function G_h0(s) = (1-e^(-Ts))/s.

## Answer

A **data hold circuit** converts a sequence of samples into a continuous signal by holding or extrapolating the sample value(s) between sampling instants. The simplest is the **zero order hold (ZOH)**, which keeps each sample constant for one period: $h(kT+\tau) = x(kT)$, $0 \le \tau < T$, giving a staircase output.

#### Derivation of ZOH transfer function

Apply a unit impulse at $t=0$. The ZOH holds the value 1 for one sampling period and then drops to zero, so its impulse response is a rectangular pulse:

$$
g_{h0}(t) = u(t) - u(t-T)
$$

```text
 g(t)
  1 |------+
    |      |
  0 +------+-------> t
    0      T
```

Taking the Laplace transform and using the shift theorem $\mathcal{L}[u(t-T)] = e^{-Ts}/s$:

$$
\begin{aligned}
G_{h0}(s) &= \frac{1}{s} - \frac{e^{-Ts}}{s} \\
&= \frac{1-e^{-Ts}}{s}
\end{aligned}
$$

This is the transfer function of the zero order hold.

## Check yourself

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

1. The ZOH impulse response is:
   A. A unit step
   B. A rectangular pulse of height 1 and width T, u(t) - u(t-T)
   C. A ramp of slope 1
   D. An impulse
   Answer: B. Holds 1 for one period then drops to zero. (Key point: Impulse response)

2. The shift theorem used is:
   A. L[u(t-T)] = e^(Ts)/s
   B. L[u(t-T)] = 1/(s+T)
   C. L[u(t-T)] = e^(-Ts)/s
   D. L[u(t-T)] = T/s
   Answer: C. Gives G(s) = 1/s - e^(-Ts)/s. (Key point: Laplace step)

3. G_h0(s) equals:
   A. (1 + e^(-Ts))/s
   B. 1/(Ts)
   C. (1 - e^(-Ts))s
   D. (1 - e^(-Ts))/s
   Answer: D. Combine the two terms. (Key point: Result)

## Related in Digital Control System

- [Transfer function of zero order hold](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Azoh-transfer-function) — Digital Control System, chapter 1, asked 7×
- [ZOH and FOH defined; ZOH transfer function](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Azoh-foh-definition) — Digital Control System, chapter 1
- [Transfer function of first order hold](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Afoh-transfer-function) — Digital Control System, chapter 1, asked 7×
- [ZOH transfer function and pulse transfer function](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Azoh-pulse-transfer-function) — Digital Control System, chapter 1
- [Pulse transfer function with first order hold](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Afoh-system-pulse-tf) — Digital Control System, chapter 3
- [Short note: Data hold circuit](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Ashort-note-data-hold) — Digital Control System, chapter 1

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