Question

The auto-correlation function of a rectangular pulse of duration \( T \) is

• A. A triangular pulse of duration \( 2T \)
• B. A rectangular pulse of duration \( 2T \)
• C. A triangular pulse of duration \( T \)
• D. A rectangular pulse of duration \( T \)

Hint:

When you compute the autocorrelation of a rectangular pulse, think of it as the convolution of the pulse with itself (flipped). Convolution of two rectangular pulses results in a triangular pulse, and the duration becomes the sum of the widths: \( T + T = 2T \).

Answer 1

The Correct Option is A

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 \( 2T \).