--- title: "Continuous-time vs discrete-time signals" subject: "Signals and Systems" chapter: 1 key: "semester_4/signals_and_systems:ct-vs-dt-signals" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/ct-vs-dt-signals site: BCT Engineering --- # Continuous-time vs discrete-time signals Signals and Systems · Chapter 1: Signals - Asked in: 2078 Baisakh, 3 marks; 2075 Bhadra, 3 marks; 2083 Baisakh (new course), 4 marks (asked 3 times) - 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/ct-vs-dt-signals - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-1 ## Question Define signals both in continuous time and discrete time with examples. Give the difference between them. ## Short answer A continuous-time signal x(t) is defined for all t, e.g., 5sin(100πt); a discrete-time signal x[n] is defined only at integer indices, e.g., (0.5)^n u[n]. CT signals are processed by analog circuits, DT by digital systems; DT often obtained by sampling CT with period T_s. ## Answer A **signal** is a function of one or more independent variables (usually time) that carries information, e.g. speech, ECG, temperature readings. - **Continuous-time (CT) signal:** defined for every value of time $t$ in an interval. The independent variable is continuous and written in round brackets, $x(t)$. Examples: speech voltage from a microphone, $x(t) = 5\sin(100\pi t)$, room temperature varying with time. - **Discrete-time (DT) signal:** defined only at discrete instants of time, $t = nT_s$, where $n$ is an integer. It is a sequence of numbers written $x[n]$. Examples: daily closing share price, samples of speech stored in a computer, $x[n] = (0.5)^n u[n]$. ```text CT signal x(t) DT signal x[n] | .--. | o | / \ | | o | / \ . | o | | o |/ \ / | | | | | --+----------\--/---> t --+-+-+--+--+----> n '-- -1 0 1 2 ``` | Point | Continuous-time | Discrete-time | |---|---|---| | Independent variable | Continuous $t$ | Integer $n$ | | Notation | $x(t)$ | $x[n]$ | | Defined at | All instants | Only sampling instants | | Origin | Natural/physical (analog) | Sampling of CT or naturally discrete data | | Processing | Analog circuits (R, L, C, op-amps) | Digital hardware, computers, DSP | | Energy/power | Integral of $\lvert x(t)\rvert^2$ | Sum of $\lvert x[n]\rvert^2$ | | Periodicity | $e^{j\omega t}$ periodic for any $\omega$ | $e^{j\omega n}$ periodic only if $\omega/2\pi$ rational | | Frequency range | $-\infty$ to $\infty$ | Unique only over $2\pi$ | A DT signal is often obtained by **sampling** a CT signal: $x[n] = x(nT_s)$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. A discrete-time signal x[n] is defined: A. For every real value of t B. Only at integer-indexed sampling instants t = nTs C. Only for positive time D. Only for periodic inputs Answer: B. A DT signal is a sequence defined only at discrete instants t = nTs with n an integer. (Key point: DT signal definition) 2. Which pairing of notation and independent variable is correct? A. x(t) with integer n; x[n] with continuous t B. x(t) with continuous t; x[n] with integer n C. Both use continuous variables D. Both use integer variables Answer: B. CT signals use round brackets and continuous t, DT signals use square brackets and integer n. (Key point: Notation and variable) 3. Which statement about periodicity of complex exponentials is correct? A. e^(jwt) is periodic for any w; e^(jwn) only if w/2pi is rational B. Both are periodic only if w/2pi is rational C. e^(jwn) is periodic for any w; e^(jwt) only if w is rational D. Neither is ever periodic Answer: A. CT exponentials are periodic for every non-zero w, while DT ones need w/2 pi rational. (Key point: Periodicity difference) 4. How is a DT signal commonly obtained from a CT signal? A. x[n] = x(nTs) by sampling B. x[n] = dx/dt C. x[n] = integral of x(t) D. x[n] = x(-t) Answer: A. Sampling the CT signal at t = nTs gives the DT sequence x[n] = x(nTs). 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