Which pairing of notation and independent variable is correct?

x(t) with continuous t; x[n] with integer n

x(t) with integer n; x[n] with continuous t

Which statement about periodicity of complex exponentials is correct?

e^(jwt) is periodic for any w; e^(jwn) only if w/2pi is rational

Both are periodic only if w/2pi is rational

e^(jwn) is periodic for any w; e^(jwt) only if w is rational

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Continuous-time vs discrete-time signals

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.

Define signals both in continuous time and discrete time with examples. Give the difference between them.

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, $x[n] = (0.5)^n u[n]$ .

 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: $x[n] = x(nT_s)$ .

✓ Answer checked by Claude Opus 5.5 (high effort)

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