Continuous & Discrete-Time Signals – Concepts, Examples & GATE Tricks

Continuous-Time Signals (CT Signals)

A continuous-time signal is defined for all values of time t. Common examples include analog signals like ECG, audio, and temperature variations.

• Unit step signal: $u(t)$
• Unit impulse (Dirac) signal: $\delta(t)$
• Ramp signal: $r(t)$
• Sinusoidal signal: $x(t) = A \cos(\omega t + \phi)$
• Exponential signal: $x(t) = A e^{st}$

Fundamental period: For a sinusoidal CT signal $x(t) = A \cos(\omega t + \phi)$, period $T$ is $$T = \frac{2\pi}{\omega}$$

⚠️ GATE Tip: Only signals where $\frac{\omega_1}{\omega_2}$ is rational are periodic when summed.

Discrete-Time Signals (DT Signals)

A discrete-time signal is defined only at integer values of n. These signals are obtained by sampling continuous-time signals or exist naturally in digital systems.

• Unit step: $u[n]$
• Unit impulse: $\delta[n]$
• Sinusoidal: $x[n] = A \cos(\omega_0 n + \phi)$
• Exponential: $x[n] = A \alpha^n$

Fundamental period: For a DT sinusoid $x[n] = A \cos(\omega_0 n + \phi)$, it is periodic if $\omega_0 / (2\pi) = \text{rational} \Rightarrow x[n+N] = x[n]$ and the period $N$ is the smallest integer satisfying this.

⚠️ GATE Tip: Discrete-time sinusoids may appear non-periodic for some $\omega_0$ even if it looks sinusoidal. Check rational condition carefully.

Continuous vs Discrete Signals – Quick Comparison

• CT signals: defined for all real t; DT signals: defined for integer n
• CT sinusoid always periodic; DT sinusoid periodic only if $\omega_0/2\pi$ is rational
• CT sum of sinusoids: periodic if ratio of frequencies rational; DT sum: period = LCM(N1,N2)
• GATE Tip: Carefully check DT signal periods, many students make mistakes assuming all DT sinusoids are periodic

Example – Period of Sum (DT): $$x[n] = \cos(\pi n/2) + \cos(\pi n/3)$$ $$\Rightarrow N = LCM(4,6) = 12$$