The auto-correlation function of a rectangular pulse of duration \( T \) is
Show Hint
- A triangular pulse of duration \( 2T \)
- A rectangular pulse of duration \( 2T \)
- A triangular pulse of duration \( T \)
- A rectangular pulse of duration \( T \)
The Correct Option is A
Solution and Explanation
To solve this problem, we need to find the auto-correlation function of a rectangular pulse and analyze its characteristics.
1. Understanding the Concepts:
- Auto-correlation function: The auto-correlation function of a signal \( x(t) \), denoted as \( R_x(\tau) \), is defined as the convolution of the signal with its time-reversed version. Mathematically:
\[ R_x(\tau) = \int_{-\infty}^{\infty} x(t) x(t - \tau) dt \]
- Rectangular Pulse: A rectangular pulse of duration \( T \) is a signal that is 1 for \( t \) in the interval \( [0, T] \) and 0 otherwise. Its auto-correlation function can be calculated by overlapping the pulse with a shifted version of itself and measuring the overlap as a function of the shift.
2. Auto-correlation of a Rectangular Pulse:
The auto-correlation function of a rectangular pulse of duration \( T \) is known to produce a triangular pulse. This is because the overlap between the original pulse and the shifted version decreases linearly as the shift increases. Therefore, the auto-correlation function is a triangular pulse with a base duration of \( 2T \).
Final Answer:
The auto-correlation function of a rectangular pulse of duration \( T \) is A triangular pulse of duration \( 2T \).
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