Question

A signal is defined as \(x(t)=\sin(10\pi t)+\sin(15\pi t)\). What is its fundamental period (in seconds)?

• A. \(0.1\)
• B. \(0.2\)
• C. \(0.4\)
• D. \(0.8\)

Hint:

For a periodic sum of sinusoids, \[ \boxed{ T_0=\operatorname{LCM}(T_1,T_2,\ldots) } \] provided the frequency ratios are rational.

Answer 1

The Correct Option is C

Step 1: Find the individual periods. For \[ x(t)=\sin(10\pi t)+\sin(15\pi t), \] the angular frequencies are \[ \omega_1=10\pi,\qquad \omega_2=15\pi. \] Hence, \[ T_1=\frac{2\pi}{10\pi}=0.2\ \text{s}, \] and \[ T_2=\frac{2\pi}{15\pi}=\frac{2}{15}\ \text{s}. \]

Step 2: Determine the fundamental period. The fundamental period is the least common multiple of the individual periods. \[ \boxed{T_0=0.4\ \text{s}} \] Therefore, \[ \boxed{(C)} \] is the correct answer.