256. Double integration of unit step function would lead to—

1. 1. an impulse

   
   
   

   2. a parabola

   
   
   

   3. a ramp

   
   
   

   4. a doublet

   
   
   

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1. View Hint View Answer Discuss in Forum

   First integration of
   u(t) = tu(t).
   Second integration of
   

   u(t) =	t2	u(t)
   	2

   
   which represent parabola form.
   Hence, alternative (B) is correct answer.

   Correct Option: B

   First integration of
   u(t) = tu(t).
   Second integration of
   

   u(t) =	t2	u(t)
   	2

   
   which represent parabola form.
   Hence, alternative (B) is correct answer.

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257. 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. 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

   
   
   

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1. View Hint View Answer Discuss in Forum

   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.

   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.

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258. Fourier terms form F(jω) of an arbitrary signal has the property—

1. 1. F(jω) = F(– jω)

   
   
   

   2. F(jω) = – F(– jω)

   
   
   

   3. F(jω) = F(jω)

   
   
   

   4. F(jω) = – F(jω)

   
   
   

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1. View Hint View Answer Discuss in Forum

   NA

   Correct Option: B

   NA

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259. The inverse Fourier transform of the function
     

     F(ω) =	1	+ πδ(ω) is—
     	jω

1. 1. sin ωt

   
   
   

   2. cos ωt

   
   
   

   3. sgn (t)

   
   
   

   4. u(t)

   
   
   

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1. View Hint View Answer Discuss in Forum

   Since, Fourier transform of unit step function is
   

   	1	+ πs(ω) so, inverse Fourier transform will be u(t).
   	jω

   
   Hence, alternative (D) is the correct choice.

   Correct Option: D

   Since, Fourier transform of unit step function is
   

   	1	+ πs(ω) so, inverse Fourier transform will be u(t).
   	jω

   
   Hence, alternative (D) is the correct choice.

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260. The amplitude spectrum of a Gaussian pulse is—

1. 1. uniform

   
   
   

   2. a sine function

   
   
   

   3. Gaussian

   
   
   

   4. an impulse function

   
   
   

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1. View Hint View Answer Discuss in Forum

   NA

   Correct Option: C

   NA

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