Question

Among the following, which signal is aperiodic?

• A. $\cos(10t)$
• B. $e^{j2\pi t}$
• C. $\sin(4t + \pi)$
• D. $e^{-2t}u(t)$

Hint:

Pure continuous sine, cosine, and imaginary exponential functions are always periodic over the interval $(-\infty, \infty)$. Real exponential signals that decay or are cut off by a step function $u(t)$ are always aperiodic.

Answer 1

The Correct Option is D

Concept: A continuous-time signal $x(t)$ is classified as periodic if there exists a positive, non-zero real constant $T$ such that $x(t + T) = x(t)$ for all $-\infty < t < \infty$. The smallest value of $T$ that satisfies this condition is called the fundamental period. If no such constant $T$ can satisfy this identity across the entire time axis, the signal is classified as aperiodic (or non-periodic).

Step 1: Testing the periodic options.
Let's analyze the properties of the first three sinusoidal and complex exponential configurations:
• Option (A) $\cos(10t)$: This is a standard continuous-time cosine wave. Its fundamental angular frequency is $\omega_0 = 10\text{ rad/s}$. The fundamental period is $T = \frac{2\pi}{\omega_0} = \frac{\pi}{5}\text{ s}$. Because $T$ is a well-defined real value, the signal is periodic.
• Option (B) $e^{j2\pi t$:} This represents a complex exponential signal on the unit circle. Its fundamental angular frequency is $\omega_0 = 2\pi\text{ rad/s}$. The fundamental period is $T = \frac{2\pi}{2\pi} = 1\text{ s}$. It is perfectly periodic.
• Option (C) $\sin(4t + \pi)$: This is a phased sine wave with $\omega_0 = 4\text{ rad/s}$. Its period is $T = \frac{2\pi}{4} = \frac{\pi}{2}\text{ s}$. It is periodic.

Step 2: Evaluating the aperiodic option.

• Option (D) $e^{-2tu(t)$:} This signal describes a causally gated, decaying real exponential function. Due to the presence of the Heaviside step function $u(t)$, the signal's value is exactly zero for all negative time ($t < 0$), jumps to $1$ at $t = 0$, and decays asymptotically toward zero as $t \to \infty$. Because the wave shape changes continuously and never repeats its past values, it cannot satisfy the condition $x(t+T) = x(t)$. This makes it an aperiodic signal, validating Option (D).