166. What is the output of the system with
     

     h[n] =		1		2	u(n)
     		2

     
     in response to the input

     x[n] = 3 + cos		πn +	π		?
     			3

1. 1. y[n] = 3 +	1		cos		πn +	π		dt
      	3					3

   
   
   

   2. y[n] = 3 +	2		cos		πn +	π		dt
      	3					3

   
   
   

   3. y[n] = 1 +	2		sin		πn +	π		dt
      	3					3

   
   
   

   4. y[n] = 6 +	2		cos		πn +	π		dt
      	3					3

   
   
   

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1. View Hint View Answer Discuss in Forum

   NA

   Correct Option: B

   NA

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167. A discrete time system has impulse response h[n] = αⁿu(n + 2), |α| < 1. Which one of the following is correct?

1. 1. stable, causal and memory less

   
   
   

   2. unstable, causal and has memory

   
   
   

   3. stable, non-causal and has memory

   
   
   

   4. unstable, non-causal and has memory less

   
   
   

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1. View Hint View Answer Discuss in Forum

   Given
   h(n) = aⁿ u(n + 2), |a| < 1
   ● As given that |a| < 1, therefore, the system is obviously stable.
   ● Non-causal since output depends upon the future value for any value of n.
   ● Given system possess memory,
   Hence, alternative (C) is the correct choice.

   Correct Option: C

   Given
   h(n) = aⁿ u(n + 2), |a| < 1
   ● As given that |a| < 1, therefore, the system is obviously stable.
   ● Non-causal since output depends upon the future value for any value of n.
   ● Given system possess memory,
   Hence, alternative (C) is the correct choice.

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168. Consider the following statements about the z-transform, X(z) of the sequence
     x(n) = 0 for n < 0 = 2ⁿ for n ≥ 0
     (ROC denotes region of convergence in the z-plane)
     

     1. X(z) =	1	ROC: |z| > 2
     	1 – 2z– 1

     
     

     2. X(z) = 1 +	2z– 1	ROC: |z| > 2
     	1 – 2z– 1

     
     

     3. X(z) =	1	ROC: |z| > 2
     	1 – 2z– 1

     
     Which of these statements are correct?

1. 1. 1 and 3

   
   
   

   2. 1 and 2

   
   
   

   3. 2 and 3

   
   
   

   4. 1, 2 and 3

   
   
   

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1. View Hint View Answer Discuss in Forum

   Given,
   

   x(t) =		2n;	2n, for n ≥ 0
   		0,	for n < 0

   
   then
   

   X(z) =	z	=	1	|z| > 2
   	z - 2		1 - 2z-1

   
   

   (1) X(z) =	1	ROC, |z| > 2 → is true
   	1 - 2z-1

   
   

   (2) X(z) = 1 +	2z-1	ROC : |z| > 2
   	1 - 2z-1

   
   

   =	1 - 2z-1 + 2z-1	; ROC : |z| > 2
   	1 - 2z-1

   
   

   =	1	;ROC: |z| > 2 → Also true
   	1 - 2z-1

   
   (3) is not true since x(n) = 0 for n < 0.
   Hence, alternative (B) is the correct choice.

   Correct Option: B

   Given,
   

   x(t) =		2n;	2n, for n ≥ 0
   		0,	for n < 0

   
   then
   

   X(z) =	z	=	1	|z| > 2
   	z - 2		1 - 2z-1

   
   

   (1) X(z) =	1	ROC, |z| > 2 → is true
   	1 - 2z-1

   
   

   (2) X(z) = 1 +	2z-1	ROC : |z| > 2
   	1 - 2z-1

   
   

   =	1 - 2z-1 + 2z-1	; ROC : |z| > 2
   	1 - 2z-1

   
   

   =	1	;ROC: |z| > 2 → Also true
   	1 - 2z-1

   
   (3) is not true since x(n) = 0 for n < 0.
   Hence, alternative (B) is the correct choice.

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169. Which one of the following gives the linear convolution of two finite length sequences x(n) = {1, 3, 1, 2} and y(n) = {1, 2, 3}?

1. 1. {3, 10, 8, 14, 7, 5, 2}

   
   
   

   2. {2, 10, 7, 14, 6, 6, 3}

   
   
   

   3. {3, 9, 8, 14, 7, 5, 2}

   
   
   

   4. {1, 5, 8, 10, 14, 5, 6}

   
   
   

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1. View Hint View Answer Discuss in Forum

   The linear convolution of x(n) and y(n) can be given as
   

   	∞
   r(n) =		x(k) y(n – k)
   	n = – ∞

   
   
   r(n) = 1 (3 + 2) (1 + 6 + 1) (2 + 2 + 3 + 3) (4 + 1 + 9) (2 + 3) 6
   or r(n) = {1 5 8 10 14 5 6}

   Correct Option: D

   The linear convolution of x(n) and y(n) can be given as
   

   	∞
   r(n) =		x(k) y(n – k)
   	n = – ∞

   
   
   r(n) = 1 (3 + 2) (1 + 6 + 1) (2 + 2 + 3 + 3) (4 + 1 + 9) (2 + 3) 6
   or r(n) = {1 5 8 10 14 5 6}

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170. Z transform and associated ROC for the sequence x[n] = a^{– n} u[– n] is—

1. 1. (1 + az), |z| >	1
      	|a|

   
   
   

   2. 	1	, |z| <	1
      	1 + az		|a|

   
   
   

   3. (1 – az), |z| >	1
      	|a|

   
   
   

   4. 	1	, |z| <	1
      	(1 – az)		|a|

   
   
   

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1. View Hint View Answer Discuss in Forum

   As we know that
   

   x[n] = an u[n] ←z→	z	= x(z), |z| > |a|
   	z - a

   
   By using time reversal property
   

   x[- n] ←z→	z	= x(z), |z| < |a|
   	z - a

   
   

   So, a–n u[– n]←z→	z- 1	=|z| <	1
   	z- 1 - a		|a|

   
   or
   

   or, a–n u[– n]←→	1
   	1 - az

   
   Hence, alternative (D) is the correct choice.

   Correct Option: D

   As we know that
   

   x[n] = an u[n] ←z→	z	= x(z), |z| > |a|
   	z - a

   
   By using time reversal property
   

   x[- n] ←z→	z	= x(z), |z| < |a|
   	z - a

   
   

   So, a–n u[– n]←z→	z- 1	=|z| <	1
   	z- 1 - a		|a|

   
   or
   

   or, a–n u[– n]←→	1
   	1 - az

   
   Hence, alternative (D) is the correct choice.

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