---
title: "Short note: Discrete data function"
subject: "Signals and Systems"
chapter: 1
key: "semester_4/signals_and_systems:short-note-discrete-data-function"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/short-note-discrete-data-function
site: BCT Engineering
---

# Short note: Discrete data function

Signals and Systems · Chapter 1: Signals

- Asked in: 2075 Bhadra, 3 marks
- Video: [Continuous and Discrete Time Signals](https://www.youtube.com/watch?v=H4hk6N5vC1Q) — 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/short-note-discrete-data-function
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1

## Question

Write a short note on discrete data function.

## Short answer

A discrete-data function (discrete-time signal) x[n] is defined only at integer n, often obtained by sampling a continuous signal every T_s seconds. It is represented functionally, as a sequence, tabularly, or by a stem plot. Processing uses sums and differences; periodicity requires ω/(2π) rational and frequency range is limited to 2π.

## Answer

A **discrete data function** (discrete-time signal) is a function whose independent variable takes only discrete (integer) values. It is a sequence of numbers $x[n]$, where $n = \ldots, -2, -1, 0, 1, 2, \ldots$; it is **not defined** between integer values of $n$.

**How it arises**
- By **sampling** a continuous-time signal every $T_s$ seconds: $x[n] = x(nT_s)$, e.g. a digital audio signal with $f_s = 44.1$ kHz.
- Naturally discrete data: daily rainfall, monthly sales, number of students per year.

**Ways to represent it**
1. **Functional:** $x[n] = (0.5)^n$ for $n \ge 0$, 0 otherwise.
2. **Sequence:** $x[n] = \{1, 2, \underline{3}, 1\}$ (underline/arrow marks $n=0$).
3. **Tabular:** a table of $n$ and $x[n]$.
4. **Graphical:** stem (lollipop) plot.

```text
 x[n]       o
        o   |
        |   |   o
    o   |   |   |
 ---+---+---+---+---> n
   -1   0   1   2
```

**Features:** processed by digital computers and DSP chips; operations are sums and differences instead of integrals and derivatives; DT sinusoids are periodic only if $\omega/2\pi$ is rational; frequency is unique only over a $2\pi$ range. If the amplitude is also quantised, it becomes a **digital signal**.

## Check yourself

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

1. What is a discrete data function?
   A. A function defined for every real t
   B. A function whose amplitude only is quantised
   C. A function whose independent variable takes only integer values and is undefined between them
   D. A function that is always periodic
   Answer: C. A DT signal x[n] is a sequence defined only at integer n. (Key point: Definition)

2. How is a DT signal commonly obtained from a CT signal?
   A. By differentiating x(t)
   B. By integrating x(t)
   C. By multiplying x(t) by u(t)
   D. By sampling every Ts seconds: x[n] = x(nTs)
   Answer: D. Sampling picks the values at t = nTs to form the sequence. (Key point: How it arises)

3. When does a discrete-time signal become a digital signal?
   A. When its amplitude is also quantised
   B. When it is periodic
   C. When it is sampled twice
   D. When it is multiplied by a unit step
   Answer: A. Digital signals are discrete in both time and amplitude. (Key point: Digital vs discrete)

## Related in Signals and Systems

- [DT transfer function; drawing a Bode plot](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adt-transfer-function-bode) — Signals and Systems, chapter 6
- [Transfer function of DT low pass filter](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adt-lowpass-transfer-function) — Signals and Systems, chapter 6

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