Question

Determine the fundamental period of the signal \( \cos\left(\frac{\pi}{4}t\right) + \sin\left(\frac{\pi}{3}t\right) \).

• A. 12
• B. 13
• C. 24, periodic
• D. 24\( t \), non-periodic

Hint:

To find period of sum: Get individual periods, take LCM. Both must be periodic!

Answer 1

The Correct Option is C

Step 1: Period of Individual Signals
For \( \cos\left(\frac{\pi}{4}t\right) \), the period \( T_1 = \frac{2\pi}{\pi/4} = 8 \)
For \( \sin\left(\frac{\pi}{3}t\right) \), the period \( T_2 = \frac{2\pi}{\pi/3} = 6 \)
Step 2: Fundamental Period of Sum
To find total signal period: \[ T = LCM(6, 8) = 24 \] Step 3: Periodicity Condition
Both component signals are periodic → sum is also periodic with LCM as fundamental period.
Conclusion:
Option (3) is correct — the signal is periodic with period 24.