---
title: "Reconstruction of signal from samples"
subject: "Digital Control System"
chapter: 1
key: "semester_6/digital_control_system:reconstruction-from-sampled-signal"
source: https://www.bctengineer.com/q/semester_6/digital_control_system/reconstruction-from-sampled-signal
site: BCT Engineering
---

# Reconstruction of signal from samples

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

- Asked in: 2071 Bhadra, 6 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_6/digital_control_system/reconstruction-from-sampled-signal
- Whole chapter: https://www.bctengineer.com/md/semester_6/digital_control_system/chapter-1

## Question

Explain about reconstruction of original signal from sampled signal with the help of figure and necessary.

## Short answer

Reconstruction recovers a continuous signal from samples. The sampled spectrum contains the primary component and copies at multiples of ω_s. If sampling rate exceeds twice the signal bandwidth, an ideal low-pass filter recovers the signal exactly via sinc interpolation. Practical systems use zero-order hold (ZOH) or first-order hold (FOH) with transfer functions (1-e^{-Ts})/s and (Ts+1)/T((1-e^{-Ts})/s)² respectively.

## Answer

A sampled signal carries values only at $t = kT$. **Reconstruction** produces a continuous signal from these values, so that the output of a digital controller can drive an analog plant.

#### Spectrum of the sampled signal

The impulse-sampled signal has spectrum

$$
X^*(j\omega) = \frac{1}{T}\sum_{n=-\infty}^{\infty} X(j(\omega + n\omega_s)), \quad \omega_s = \frac{2\pi}{T}
$$

i.e. the original spectrum (primary component, $n=0$) plus copies (complementary components) centred at $\pm\omega_s, \pm 2\omega_s, \dots$

```text
 |X*(jw)|
   _/\_      _/\_      _/\_
 --------+---------+---------> w
       -ws    0    ws
      copies  |  primary
         <-- ideal LPF -->
           -ws/2    ws/2
```

#### Condition for exact recovery

If $x(t)$ is band-limited to $\omega_m$ and $\omega_s > 2\omega_m$ (sampling theorem), the copies do not overlap. Then an **ideal low-pass filter** with gain $T$ and cut-off $\omega_s/2$ keeps only the primary component and gives back $x(t)$ exactly:

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

If $\omega_s < 2\omega_m$ the copies overlap (**aliasing**) and recovery is impossible.

#### Practical reconstruction

The ideal filter is non-causal, so practical systems use **hold circuits**, which build the signal from past samples only:

1. **Zero order hold (ZOH):** keeps $x(kT)$ constant until $(k+1)T$. Transfer function $G_{h0}(s) = \frac{1-e^{-Ts}}{s}$. Its magnitude $T\left|\frac{\sin(\omega T/2)}{\omega T/2}\right|$ is a rough low-pass filter; it passes some high-frequency components (ripple) and adds a lag of $\omega T/2$.
2. **First order hold (FOH):** extrapolates with the slope of the last two samples. $G_{h1}(s) = \frac{Ts+1}{T}\left(\frac{1-e^{-Ts}}{s}\right)^2$.

```text
 samples:   .   .   .
 ZOH:      |‾‾‾|‾‾‾|‾‾‾   staircase
 FOH:      /  /  /        ramps
```

Choosing a small $T$ (high sampling rate) and following the hold with a smoothing filter makes the reconstructed signal close to the original.

## Check yourself

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

1. The spectrum of an impulse-sampled signal consists of:
   A. The original spectrum plus copies centred at multiples of the sampling frequency
   B. Only the original spectrum
   C. Only high-frequency copies
   D. The original spectrum compressed by T
   Answer: A. X*(jw) = (1/T) sum X(j(w + n ws)). (Key point: Sampled spectrum)

2. Exact recovery by an ideal low-pass filter requires:
   A. ws < 2 wm and cut-off wm
   B. ws > 2 wm, filter gain T and cut-off ws/2
   C. ws = wm
   D. Any ws with gain 1
   Answer: B. Copies must not overlap; the filter keeps only the primary band. (Key point: Exact recovery)

3. If ws < 2 wm, what happens?
   A. The ZOH corrects it
   B. A higher-order hold recovers it
   C. The copies overlap (aliasing) and recovery is impossible
   D. Only the gain changes
   Answer: C. High frequencies masquerade as low ones and cannot be separated. (Key point: Aliasing)

4. The ZOH phase lag is:
   A. wT
   B. T/2
   C. Zero
   D. wT/2
   Answer: D. It delays by half a sampling period; its gain is T sin(wT/2)/(wT/2). (Key point: ZOH as filter)

## Related in Digital Control System

- [Signal reconstruction and reconstructors](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Asignal-reconstruction-mathematical) — Digital Control System, chapter 1

## Related in other subjects

- [Frequency of a signal sampled at 300 Hz](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Asampled-signal-frequency-300) — Signals and Systems, chapter 4
- [Reconstruction from samples taken at Nyquist rate](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Areconstruction-nyquist-samples) — Communication Systems, chapter 4
- [Spectrum of a signal and its samples](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Asampled-spectrum-relation) — Signals and Systems, chapter 4
- [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