---
title: "Signal reconstruction and reconstructors"
subject: "Digital Control System"
chapter: 1
key: "semester_6/digital_control_system:signal-reconstruction-mathematical"
source: https://www.bctengineer.com/q/semester_6/digital_control_system/signal-reconstruction-mathematical
site: BCT Engineering
---

# Signal reconstruction and reconstructors

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

- Asked in: 2079 Chaitra, 4 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/digital_control_system/signal-reconstruction-mathematical
- Whole chapter: https://www.bctengineer.com/md/semester_6/digital_control_system/chapter-1

## Question

Explain signal reconstruction in digital control system. How do you represent mathematically signal reconstructors?

## Short answer

Signal reconstruction recovers a continuous signal from samples using hold circuits. Ideal reconstruction uses sinc interpolation but is non-causal. Practical zero-order hold (ZOH) gives staircase output with transfer function (1-e^{-Ts})/s, while first-order hold (FOH) uses linear extrapolation. ZOH is most common due to simplicity and small phase lag.

## Answer

**Signal reconstruction** is the process of recovering a continuous-time signal from its samples $x(kT)$, so that it can drive a continuous plant. In a digital control system it is done after the D/A converter by a **hold circuit (signal reconstructor)**.

#### Ideal reconstruction

If the signal is band-limited and $\omega_s > 2\omega_m$, an ideal low-pass filter with cut-off $\omega_s/2$ recovers $x(t)$ exactly:

$$
x(t) = \sum_{k=-\infty}^{\infty} x(kT)\,\frac{\sin[\omega_s(t-kT)/2]}{\omega_s(t-kT)/2}
$$

This filter is non-causal (it needs future samples), so it cannot be built.

#### Practical reconstructors (hold circuits)

A hold circuit extrapolates the signal between $kT$ and $(k+1)T$ from the present and past samples using a polynomial:

$$
h(kT+\tau) = a_n\tau^n + a_{n-1}\tau^{n-1} + \cdots + a_1\tau + a_0, \quad 0 \le \tau < T
$$

with $h(kT) = x(kT)$, so $a_0 = x(kT)$. An $n$th-order hold uses the last $n+1$ samples.

- **Zero order hold ($n=0$):** $h(kT+\tau) = x(kT)$; staircase output.
  $G_{h0}(s) = \dfrac{1-e^{-Ts}}{s}$
- **First order hold ($n=1$):** straight-line extrapolation from two samples:
  $h(kT+\tau) = x(kT) + \dfrac{x(kT)-x((k-1)T)}{T}\,\tau$
  $G_{h1}(s) = \dfrac{Ts+1}{T}\left(\dfrac{1-e^{-Ts}}{s}\right)^2$

```text
 ZOH:  _|‾‾|__|‾‾‾|_   (steps)
 FOH:  /  /  /         (sloped segments)
```

The ZOH is the most widely used because it is simple and cheap, and its phase lag ($\omega T/2$) is small when $T$ is small. Higher-order holds track smooth signals better, but they need more memory and hardware and give more phase lag at high frequencies.

## Check yourself

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

1. In a digital control system, reconstruction after the D/A is performed by:
   A. A hold circuit (signal reconstructor)
   B. An A/D converter
   C. A sampler
   D. A multiplexer
   Answer: A. The hold recovers a continuous signal to drive the plant. (Key point: Reconstruction device)

2. The ideal reconstructor cannot be built because it is:
   A. Unstable
   B. Non-causal (needs future samples)
   C. Too slow
   D. Nonlinear
   Answer: B. The sinc interpolation needs samples from the future. (Key point: Ideal reconstruction)

3. The transfer function of a zero order hold is:
   A. 1/(1 + Ts)
   B. (1 - e^(-Ts))/s^2
   C. (1 - e^(-Ts))/s
   D. e^(-Ts)/s
   Answer: C. Impulse response is a rectangular pulse of width T. (Key point: ZOH transfer function)

4. An nth-order hold uses:
   A. Only the present sample
   B. The next n samples
   C. n samples regardless of order
   D. The last n+1 samples
   Answer: D. ZOH uses 1 sample, FOH uses 2. (Key point: Order of hold)

## Related in Digital Control System

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