The correct answer is A. Rectangular pulse of duration T.
A matched filter is a filter that is designed to maximize the output signal to noise ratio for a given input signal. In other words, it is a filter that is designed to remove all noise from the input signal and leave only the desired signal.
A rectangular pulse is a signal that is zero for all time except for a short period of time when it is equal to one. The duration of the rectangular pulse is denoted by T.
When a rectangular pulse is applied to a matched filter, the output of the filter is a rectangular pulse of the same duration, T. This is because the matched filter is designed to remove all noise from the input signal and leave only the desired signal, which is a rectangular pulse of duration T.
The other options are incorrect because they do not represent the output of a matched filter when a rectangular pulse is applied as input. Option B is incorrect because it is a rectangular pulse of duration 2T. Option C is incorrect because it is a triangular pulse. Option D is incorrect because it is a sine function.