251. If g(t) ↔ G(f) represents a Fourier transform pair, then according to the duality property of Fourier transforms—

1. 1. G(t) = g(f)

   
   
   

   2. G(t) = g*(f)

   
   
   

   3. G(t) = g(– f)

   
   
   

   4. G(t) = g*(– f)

   
   
   

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   NA

   Correct Option: C

   NA

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252. If δ(t) denotes a unit impulse, then the Laplace transform of
     

     	d2δ(t)	will be—
     	dt2

1. 1. 1

   
   
   

   2. s²

   
   
   

   3. s

   
   
   

   4. s^{– 2}

   
   
   

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1. View Hint View Answer Discuss in Forum

   If L{s(t)} = 1
   
   Alternative (B) is the correct choice.

   Correct Option: B

   If L{s(t)} = 1
   
   Alternative (B) is the correct choice.

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253. The unit step response of a system is given by (1 – e^ατ) u(t), the impulse response is given by—

1. 1. e^{– ατ} u(t)

   
   
   

   2. αe^{– ατ} u(t)

   
   
   

   3. 	1	e– ατ u(t)
      	α

   
   
   

   4. – αe^{– ατ} u(t)

   
   
   

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1. View Hint View Answer Discuss in Forum

   Given, that unit step response of system
   y(t) = (1 – e^{– αt}) u(t).
   Let, H(s) is the transfer function of the system.
   when x(t)=1
   

   X(s) =	1	, [∵ Laplace transform of x(t)]
   	s

   
   

   and Y(s) =	1	–	1	=	α
   	s		s + α		s(s + α)

   
   or Y(s) = H(s). X(s).
   

   	α	= H(s) ·	1
   	s(s + α)		s

   
   

   or H(s) =	α
   	s + α

   
   Now, according to question, we have to calculate y(t) for input x(t) = δ(t)
   then X(s)= 1
   

   H(s) =	α
   	s + α

   
   X(s)=?
   Now, Y(s) = X(s).H(s)
   

   or y(s) = 1·	α
   	s + α

   
   Taking inverse Laplace transform to calculate y(t)
   y(t) = α·e^{– αt} u(t).
   Hence, alternative (B) is the correct answer.

   Correct Option: B

   Given, that unit step response of system
   y(t) = (1 – e^{– αt}) u(t).
   Let, H(s) is the transfer function of the system.
   when x(t)=1
   

   X(s) =	1	, [∵ Laplace transform of x(t)]
   	s

   
   

   and Y(s) =	1	–	1	=	α
   	s		s + α		s(s + α)

   
   or Y(s) = H(s). X(s).
   

   	α	= H(s) ·	1
   	s(s + α)		s

   
   

   or H(s) =	α
   	s + α

   
   Now, according to question, we have to calculate y(t) for input x(t) = δ(t)
   then X(s)= 1
   

   H(s) =	α
   	s + α

   
   X(s)=?
   Now, Y(s) = X(s).H(s)
   

   or y(s) = 1·	α
   	s + α

   
   Taking inverse Laplace transform to calculate y(t)
   y(t) = α·e^{– αt} u(t).
   Hence, alternative (B) is the correct answer.

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254. Which one of the following represents the impulse response of a system defined by H(z) = z^{– m}?

1. 1. u[n – m] (x)

   
   
   

   2. δ[n – m]

   
   
   

   3. δ[m](x)

   
   
   

   4. δ[m – n]

   
   
   

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1. View Hint View Answer Discuss in Forum

   Since, if δ(n) ⎯⎯⎯⎯ z-transform → 1
   Then, δ(n – m) ⎯⎯⎯⎯ z-transform → z^{– m}.1
   (by using shifting property)
   Hence, alternative (B) is correct answer.

   Correct Option: B

   Since, if δ(n) ⎯⎯⎯⎯ z-transform → 1
   Then, δ(n – m) ⎯⎯⎯⎯ z-transform → z^{– m}.1
   (by using shifting property)
   Hence, alternative (B) is correct answer.

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255. The auto-correlation function R_x(τ) satisfies which one of the following properties?

1. 1. R_x(τ) = – R_x(– τ)

   
   
   

   2. R_x(τ) = R_x(– τ)

   
   
   

   3. R_x(τ) ≥ R_x(0)

   
   
   

   4. R_x(τ) ≥ 1

   
   
   

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1. View Hint View Answer Discuss in Forum

   NA

   Correct Option: B

   NA

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