Periodic Signals

Periodic signals are signals that repeat themselves after a certain period of time. In other words, if a signal x(t) is periodic with a period T, then x(t) will have the same values for all time instants t + mT, where m is an integer.

Understanding the Given Options

Let's analyze each option given in the question to determine if it satisfies the condition for a periodic signal.

a) x(-t): This option represents a time-reversed version of the signal x(t). Time-reversal does not guarantee periodicity, as the signal may have a different shape and characteristics when reversed.

b) x(mT): This option represents a signal that only takes on values at integer multiples of T. While this may result in a periodic signal if the values repeat, it does not encompass all time instants and may not capture the entire behavior of the original signal.

c) x(mt): This option represents a signal that is time-scaled by a factor of m. Time-scaling does not guarantee periodicity, as the signal may change its shape and characteristics when stretched or compressed.

Correct Answer: x(t)

The correct answer is option 'D', which states that x(t) should be equal to x(t) for a continuous-time signal x(t) to be periodic with a period T. This means that the signal remains unchanged for all time instants t.

Explanation

If a signal x(t) is periodic with a period T, it means that the values of the signal repeat after every T seconds. In other words, x(t) should have the same values for all time instants t + mT, where m is an integer.

By choosing option 'D' (x(t)), we ensure that the signal x(t) remains unchanged for all time instants. This guarantees that the signal will repeat itself after every T seconds, satisfying the condition for periodicity.

Conclusion

To summarize, for a continuous-time signal x(t) to be periodic with a period T, the signal x(t) should remain unchanged for all time instants. Choosing option 'D' (x(t)) ensures that the signal satisfies the condition for periodicity, as it guarantees that the signal will repeat itself after every T seconds.