Periodic Signals (Continuous & Discrete) – Concepts, Conditions & GATE Tricks

Periodic signals form the backbone of signal processing, communication, and electronics. Any signal that repeats itself after a fixed interval is called periodic. From AC waveforms to digital sequences, periodicity helps us simplify analysis using powerful tools like Fourier Series. If you truly understand this topic, many GATE questions become direct scoring.

Keywords: periodic signals, continuous-time periodic signals, discrete-time periodic signals, fundamental period of a signal, periodic vs aperiodic signals, sum of periodic signals, GATE ECE signals and systems, signal periodicity examples, LCM condition discrete signals

Continuous vs Discrete Periodic Signals (Core Differences)

Continuous-time signals repeat after time T
Discrete-time signals repeat after N samples
CT signals are always periodic for sinusoidal inputs
DT signals are periodic only if frequency is rational
Fundamental period must be smallest possible value

Feature	Continuous-Time	Discrete-Time
Definition	x(t) = x(t + T)	x[n] = x[n + N]
Period	Any real value (T)	Integer only (N)
Example	sin(2πt)	cos(πn/2)
Important Condition	Always periodic (for sinusoids)	ω / 2π must be rational

continuous vs discrete periodic signals comparison waveform

GATE-Level Insights & Common Mistakes

DT signal like cos(n) is NOT periodic ❌
Check ω / 2π carefully before assuming periodicity
Fundamental period is always the smallest valid value
Combination of periodic signals may NOT be periodic
LCM concept is used when multiple periods exist

Signal	Periodic?	Reason
cos(πn/2)	Yes	ω/2π = 1/4 (rational)
cos(n)	No	ω/2π is irrational
sin(2πt)	Yes	Always periodic in CT
sin(t) + cos(√2 t)	No	No common period

GATE exam periodic signal tricks and mistakes — ("n" is in radian)

Period of Sum of Two Signals (Continuous vs Discrete)

When two periodic signals are added, the result is not always periodic. The key idea is whether both signals repeat together after some common interval.

For Continuous-Time Signals:

If two signals have fundamental periods $T_1$ and $T_2$, then:

Resultant signal is periodic only if $$\frac{T_1}{T_2} \text{ is rational}$$

In that case, overall period: $$T = \mathrm{LCM}(T_1, T_2)$$

Example: $$\sin(2\pi t) + \cos(4\pi t)$$

Here, $$T_1 = 1, \quad T_2 = \frac{1}{2}$$

Since $\frac{T_1}{T_2} = 2$ (rational), signal is periodic.
Overall period: $$T = \mathrm{LCM}(1, \tfrac{1}{2}) = 1$$

For Discrete-Time Signals:

If signals have periods $N_1$ and $N_2$, then:

Both signals must be individually periodic.

Overall period: $$N = \mathrm{LCM}(N_1, N_2)$$

Example: $$\cos\left(\frac{\pi n}{2}\right) + \cos\left(\frac{\pi n}{3}\right)$$