---
title: "Impulse sampling and its equation"
subject: "Digital Control System"
chapter: 1
key: "semester_6/digital_control_system:impulse-sampling"
source: https://www.bctengineer.com/q/semester_6/digital_control_system/impulse-sampling
site: BCT Engineering
---

# Impulse sampling and its equation

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

- Asked in: 2080 Chaitra, 4 marks; 2071 Magh, 4 marks (asked 2 times)
- Video: [Sampling Signals](https://www.youtube.com/watch?v=AcuQnIXiZ2A) — Iain Explains Signals, Systems, and Digital Comms
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/digital_control_system/impulse-sampling
- Whole chapter: https://www.bctengineer.com/md/semester_6/digital_control_system/chapter-1

## Question

What is impulse sampling? Derive its equation with necessary diagrams.

## Short answer

Impulse sampling model the sampler output as a train of impulses of strength x(kT). The sampled signal is x*(t) = Σ x(kT) δ(t−kT). Its Laplace transform is X*(s) = Σ x(kT) e^(−kTs), which with z = e^(Ts) give the z-transform X(z) = Σ x(kT) z^(−k).

## Answer

**Impulse sampling** is the ideal mathematical model of sampling in which the sampler output is a train of impulses at $t = kT$, each with strength (area) equal to the signal value at that instant. Real samplers give narrow pulses, but when the pulse width is very small compared with $T$ (and a hold follows) they can be modelled this way.

#### Diagram

```text
 x(t) ---->o/o----> x*(t)
          T (sampler)

 x(t)            x*(t)
   /‾‾\            |  |
  /    \          ||  ||
 /      \__      |||  |||_
 ---------> t    -+-+-+-+-+--> t
                 0 T 2T 3T
```

#### Derivation

The ideal sampler multiplies the input by a unit impulse train:

$$
\delta_T(t) = \sum_{k=0}^{\infty} \delta(t-kT)
$$

$$
\begin{aligned}
x^*(t) &= x(t)\,\delta_T(t) = \sum_{k=0}^{\infty} x(t)\,\delta(t-kT) \\
&= \sum_{k=0}^{\infty} x(kT)\,\delta(t-kT)
\end{aligned}
$$

(using $x(t)\delta(t-kT) = x(kT)\delta(t-kT)$, with $x(t)=0$ for $t<0$).

Taking the Laplace transform, $\mathcal{L}[\delta(t-kT)] = e^{-kTs}$:

$$
X^*(s) = \sum_{k=0}^{\infty} x(kT)\,e^{-kTs}
$$

Putting $z = e^{Ts}$ gives the z-transform:

$$
X(z) = X^*(s)\Big|_{s=\frac1T\ln z} = \sum_{k=0}^{\infty} x(kT)\,z^{-k}
$$

#### Frequency-domain form

Expanding $\delta_T(t)$ as a Fourier series gives

$$
X^*(s) = \frac{1}{T}\sum_{n=-\infty}^{\infty} X(s + jn\omega_s), \qquad \omega_s = \frac{2\pi}{T}
$$

so the spectrum of the sampled signal is the original spectrum repeated every $\omega_s$. This leads to the sampling theorem $\omega_s > 2\omega_m$.

## Check yourself

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

1. In impulse sampling, what is the sampler output?
   A. A train of impulses at t = kT, each with area equal to the signal value at that instant
   B. A staircase signal
   C. Pulses of constant width following the signal shape
   D. A continuous signal
   Answer: A. Impulse sampling is the ideal model with impulse strengths x(kT). (Key point: Impulse sampling definition)

2. How is x*(t) derived in the ideal sampler model?
   A. x(t) is added to an impulse train
   B. x(t) is multiplied by the unit impulse train, and x(t)δ(t-kT) = x(kT)δ(t-kT)
   C. x(t) is integrated
   D. x(t) is convolved with a step
   Answer: B. x*(t) = x(t) Σ δ(t-kT) = Σ x(kT)δ(t-kT). (Key point: Derivation of x*(t))

3. What is X*(s) and how does it give the z-transform?
   A. Σ x(kT)e^(kTs) with z = e^(-Ts)
   B. Σ x(kT)e^(-kTs); putting z = e^(Ts) gives Σ x(kT)z^(-k)
   C. Σ x(kT)/s; z = s
   D. x(0)e^(-Ts)
   Answer: B. L[δ(t-kT)] = e^(-kTs), and z = e^(Ts) gives X(z). (Key point: Laplace transform of sampled signal)

4. What does X*(s) = (1/T)ΣX(s + jnωs) say about the spectrum?
   A. It is shifted once by ωs
   B. It is the original spectrum repeated every ωs, leading to the sampling theorem ωs > 2ωm
   C. It is unchanged by sampling
   D. It is limited to zero frequency
   Answer: B. Sampling repeats the spectrum at multiples of ωs, so no overlap needs ωs > 2ωm. (Key point: Spectrum of sampled signal)

## Related in Digital Control System

- [Standard and ladder programming block diagrams](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Afilter-standard-ladder-block) — Digital Control System, chapter 4

## Related in other subjects

- [Impulse train sampling and reconstruction](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aimpulse-train-sampling-reconstruction) — Signals and Systems, chapter 4
- [Sampling theorem via impulse train sampling](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Anyquist-impulse-train-frequency) — Signals and Systems, chapter 4
- [Impulse train sampling, aliasing, reconstruction](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aimpulse-train-aliasing-reconstruction) — Signals and Systems, chapter 4
- [Aliasing via impulse-train sampling](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aaliasing-impulse-train-frequency) — Signals and Systems, chapter 4