A Matched filter has a frequency response is given by H(f)

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A Matched filter has a frequency response is given by
H(f) = 1 – e^{– j2πfT} ,
j2πf

the impulse response h(t) of the filter will be—

1. sgn t – sgn (t – T)
  2 2
2. 2 sgn t – 2 sgn (t – T)
3. sgn t – sgn (t – T)
4. None of these

Correct Option: A

Given that

H(f) =	1 – e^-j2πft
	j2πf

or H(f) =	1	–	e^-j2πfT
	j2πf		j2πf

or H(f) =	1	–	e^-jωT
	jω		jω

We know that
sgn t ←f(jω)→ 2 jω
sgn (t – T) ←f(jω)→2 jω⎯⎯ e^{– jωT}
Therefore, the impulse response can be obtained by taking inverse Laplace transform i.e.,

h(t)=L^{– 1}			1	–	1	· e^{– jωT}
			jω		jω

or h(t) =	sgn t	–	sgn (t – T)
	2		2

Hence, alternative (A) is the correct choice.