A rectangular pulse of duration T is applied to a matched filter, output obtained out of this system would be most likelya)Sine functionb)Rectangular pulse of duration 2Tc)Triangular pulse of duration Td)Triangular pulse of duration 2TCorrect answer is option 'D'. Can you explain this answer?

Question

Answer

The output obtained from a matched filter when a rectangular pulse of duration T is applied is most likely a triangular pulse of duration 2T.

Definition of a Matched Filter:
A matched filter is a filter that maximizes the signal-to-noise ratio (SNR) at the output when a known signal is present at the input. It is designed to have an impulse response that is the time-reversed and conjugate of the input signal. In other words, the matched filter is designed to maximize the correlation between the received signal and the known signal.

Working Principle of a Matched Filter:
When a signal is applied to a matched filter, the filter convolves the input signal with its impulse response. The output of the matched filter is the correlation between the input signal and the impulse response of the filter. The matched filter is designed in such a way that it maximizes this correlation.

Response of a Matched Filter to a Rectangular Pulse:
When a rectangular pulse of duration T is applied to a matched filter, the impulse response of the filter is a time-reversed and conjugate version of the rectangular pulse. The matched filter convolves this impulse response with the rectangular pulse.

Convolution of a Rectangular Pulse with Itself:
The convolution of a rectangular pulse with itself results in a triangular pulse. This is because the rectangular pulse has a flat top and sharp edges, and when it convolves with itself, the edges of the rectangular pulse add up constructively, resulting in a triangular shape.

Doubling of the Pulse Duration:
When the rectangular pulse is convolved with itself, the resulting triangular pulse has a duration of 2T. This is because the convolution operation extends the duration of the pulse.

Therefore, the output obtained from the matched filter when a rectangular pulse of duration T is applied is most likely a triangular pulse of duration 2T.

Answer

We know that matched filter for a signal s(t) is given as

h(t) = s(T - t)

Thus, we would obtain convolution of rect(t), input, and rect(T-t), response of filter, at the output. This implies output would be convolution of two rectangular pulses of duration T, this would result into triangular pulse of duration 2T.

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