11. A rectangular pulse of duration T is applied to a filter matched to this input. The output of the filter is a—

1. 1. rectangular pulse of duration T

   
   
   

   2. rectangular pulse of duration 2T

   
   
   

   3. triangular pulse

   
   
   

   4. sine function

   
   
   

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1. View Hint View Answer Discuss in Forum

   NA

   Correct Option: B

   NA

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12. The Laplace transform of eat cos (αt) is equal to—

1. 1. (s – α)

      (s – α)2 + α2

   
   
   

   2. (s + α)

      (s + α)2 + α2

   
   
   

   3. 4

      (s – α)2

   
   
   

   4. None of these

   
   
   

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1. View Hint View Answer Discuss in Forum

   Explanation can seen in synopsis on signals and systems.

   Correct Option: A

   Explanation can seen in synopsis on signals and systems.

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13. The energy for the signal—
    x(n) = 3(0.5)²ⁿ n ≥ 0 will be—

1. 1. 8 joules

   
   
   

   2. 12 joules

   
   
   

   3. 24 joules

   
   
   

   4. None of these

   
   
   

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1. View Hint View Answer Discuss in Forum

   E =	N – 1	|x(n)|2
   	∑
   	n = 0

   
   

   =	∞	|3(0.5)2n|2
   	∑
   	n = 0

   
   

   = 9	∞	|3(.5)2|n
   	∑
   	n = 0

   
   

   = 9	∞	(.25)n
   	∑
   	n = 0

   
   = 9[1 + (.25) + (.25)² + (.25)³ + … ∞]
   

   = 9·	1
   	1 – .25

   
   

   = 9·	1	= 12
   	1 – .75

   
   Hence alternative (B) is the correct choice.

   Correct Option: B

   E =	N – 1	|x(n)|2
   	∑
   	n = 0

   
   

   =	∞	|3(0.5)2n|2
   	∑
   	n = 0

   
   

   = 9	∞	|3(.5)2|n
   	∑
   	n = 0

   
   

   = 9	∞	(.25)n
   	∑
   	n = 0

   
   = 9[1 + (.25) + (.25)² + (.25)³ + … ∞]
   

   = 9·	1
   	1 – .25

   
   

   = 9·	1	= 12
   	1 – .75

   
   Hence alternative (B) is the correct choice.

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14. A linear phase channel with phase delay τp and group delay τg must have—

1. 1. τ_p = τ_g = Constant

   
   
   

   2. τ_p ∝ f and τ_g ∝ f

   
   
   

   3. τ_p = constant and τ_g ∝ f

   
   
   

   4. τ_p ∝ f and τ_g = constant (f denotes frequency)

   
   
   

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1. View Hint View Answer Discuss in Forum

   For a linear phase channel phase delay should be directly proportional to the frequency and group delay should be constant.
   Here alternative (D) is the correct choice

   Correct Option: D

   For a linear phase channel phase delay should be directly proportional to the frequency and group delay should be constant.
   Here alternative (D) is the correct choice

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15. Which of the following cannot be the Fourier series expansion of a periodic signal?

1. 1. x(t) = 2 cos 6t + 3 cos 3t

   
   
   

   2. x(t) = 2 cos πt + 7 cos t

   
   
   

   3. x(t) = cos t + 0.5

   
   
   

   4. x(t) = 2 cos 1.5πt + sin 3.5πt

   
   
   

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1. View Hint View Answer Discuss in Forum

   To solve such type of problem solve each and every options one by one.
   (A) x(t) = 2 cos 6t + 3 cos 3t
   Fundamental period of
   

   x1(t)= T1 =	2π	=	π
   	6		3

   
   Fundamental period of
   

   x2(t)= T2 =	2π	=	2π
   	3		3

   
   For the signal x(t) to be periodic ratio
   

   	= rational number =	π	=
   T1		3		1
   T2		2π		2
   		3

   
   which is rational number
   i.e., Fourier series expansion
   (B) x(t) = 2 cos πt + 7 cos t
   

   T1 =	2π	=	π	= 2
   	ω		π

   
   

   T1 =	2π	=	π	= 2π
   	ω		1

   
   

   	T1	=	2	=	1	= irrational numbers
   	T2		2π		π

   
   i.e., Fourier series expansion is not possible.
   Hence, alternative (B) is the correct choice, and no need to solve further other option.

   Correct Option: B

   To solve such type of problem solve each and every options one by one.
   (A) x(t) = 2 cos 6t + 3 cos 3t
   Fundamental period of
   

   x1(t)= T1 =	2π	=	π
   	6		3

   
   Fundamental period of
   

   x2(t)= T2 =	2π	=	2π
   	3		3

   
   For the signal x(t) to be periodic ratio
   

   	= rational number =	π	=
   T1		3		1
   T2		2π		2
   		3

   
   which is rational number
   i.e., Fourier series expansion
   (B) x(t) = 2 cos πt + 7 cos t
   

   T1 =	2π	=	π	= 2
   	ω		π

   
   

   T1 =	2π	=	π	= 2π
   	ω		1

   
   

   	T1	=	2	=	1	= irrational numbers
   	T2		2π		π

   
   i.e., Fourier series expansion is not possible.
   Hence, alternative (B) is the correct choice, and no need to solve further other option.

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