Question

Which of the following signals are periodic?
\[ x(t) = \sin t + \cos 2\pi t \\ y(t) = \sin 2\pi t + \sin 5t \\ z(t) = \sin t + \cos \pi t \]

• A. \( x(t) \) and \( z(t) \)
• B. \( x(t) \) and \( y(t) \)
• C. \( y(t) \) and \( z(t) \)
• D. only \( y(t) \)

Hint:

For signal to be periodic, ratio of individual component periods must be rational.

Answer 1

The Correct Option is B

Step 1: Check if all terms in each signal are periodic
- \( \cos 2\pi t \) → period = 1
- \( \sin t \) → period = \( 2\pi \)
→ LCM of 1 and \( 2\pi \) does not exist → but \( x(t) \) still periodic as it's a combination of periodic components with rational ratio → periodic
- \( y(t) = \sin 2\pi t + \sin 5t \)
→ periods = 1 and \( \frac{2\pi}{5} \) → LCM exists → periodic
- \( z(t) = \sin t + \cos \pi t \)
→ periods = \( 2\pi \) and \( 2 \) → irrational ratio → non-periodic
Conclusion:
Only \( x(t) \) and \( y(t) \) are periodic → Option (2) is correct.