156. The transfer function of a system is given as
     

     Given =	2		z +	1
     				4
     		z -	1			z -	1
     			2				3

     
     Consider the two statements
     Statement (A) : System is causal and stable
     Statement (B) : Inverse system is causal and stable
     The correct option is
     

1. 1. (A) is true

   
   
   

   2. (B) is true

   
   
   

   3. Both are true

   
   
   

   4. Both are false

   
   
   

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1. View Hint View Answer Discuss in Forum

   Pole of system at: z = –	1	,	1
   	2		3

   
   

   Pole of inverse system at: z = –	1
   	2

   
   For this system and inverse system all poles are inside | z| = 1. So, both system are both causal and stable.

   Correct Option: C

   Pole of system at: z = –	1	,	1
   	2		3

   
   

   Pole of inverse system at: z = –	1
   	2

   
   For this system and inverse system all poles are inside | z| = 1. So, both system are both causal and stable.

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157. The transfer function of a system is given by
     

     H(z) =	z(3z - 2)
     	z² - z - 1/4

     
     The system is

1. 1. causal and stable

   
   
   

   2. causal, stable and minimum phase

   
   
   

   3. minimum phase

   
   
   

   4. none of these

   
   
   

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1. View Hint View Answer Discuss in Forum

   Zero at: z = 0,	2	, poles at z =	1 ±√2
   	3		2

   
   (i) Not all poles are inside | z| = 1, the system isnot causal and stable
   (ii) Not are poles and zero are inside | z| = 1, the system is not minimum phase

   Correct Option: D

   Zero at: z = 0,	2	, poles at z =	1 ±√2
   	3		2

   
   (i) Not all poles are inside | z| = 1, the system isnot causal and stable
   (ii) Not are poles and zero are inside | z| = 1, the system is not minimum phase

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158. A system is described by the difference equation
     y [n] = x[n] – x[n – 2] + x[n – 4] – x[n – 6]
     The impulse response of system is

1. 1. δ[n] – 2δ[n + 2] + 4δ[n + 4] – 6δ[n + 6]

   
   
   

   2. δ[n] + 2δ[n – 2] – 4δ[n – 4] + 6δ[n – 6]

   
   
   

   3. δ[n] – δ[n – 2] + δ[n – 4] – δ[n – 6]

   
   
   

   4. δ[n] – δ[n + 2] + δ[n + 4] – δ[n + 6]

   
   
   

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1. View Hint View Answer Discuss in Forum

   H(z) = Y X z z b g b g = (1 – z – 2 + z – 4 – z – 6)
   

   H(z) =	Y(z)	= (1 - 2z-1 + z-4 - z-6
   	X(z)

   
   ⇒ h[n] = δ[n] – δ[n – 2] + δ[n – 4] – δ[n – 6]

   Correct Option: C

   H(z) = Y X z z b g b g = (1 – z – 2 + z – 4 – z – 6)
   

   H(z) =	Y(z)	= (1 - 2z-1 + z-4 - z-6
   	X(z)

   
   ⇒ h[n] = δ[n] – δ[n – 2] + δ[n – 4] – δ[n – 6]

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159. The Z-transform function of a stable system is given as The impuse response h[n] is

1. 1. 2n u[-n + 1]-		1		n	u[n]
      		2

   
   
   

   2. 2n u[-n - 1] +		- 1		n	u[n]
      		2

   
   
   

   3. 2n u[-n - 1] +		- 1		n	u[n]
      		2

   
   
   

   4. 2n u[n] +		- 1		n	u[n]
      		2

   
   
   

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1. View Hint View Answer Discuss in Forum

   X(z) =	1			+		1				h[n] is stable
   	1 - 2z-1					1 +	1	z-1
   							2

   
   So ROC includes | z| = 1
   ROC: 1/2 < | z| < 2
   

   2n u[-n - 1] +		- 1		n	u[n]
   		2

   Correct Option: C

   X(z) =	1			+		1				h[n] is stable
   	1 - 2z-1					1 +	1	z-1
   							2

   
   So ROC includes | z| = 1
   ROC: 1/2 < | z| < 2
   

   2n u[-n - 1] +		- 1		n	u[n]
   		2

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160. Given =	1 +	1		z-1
     		4
     		1 -	1	z-1			1 +	1	z-1
     			2					3

     
     For three different ROC, consider three different solution of signal x[n].
     

     1. |z| >	1	, x[n]		1	-		-1		n		u[n]
     	2			2n-1			3

     
     

     2. |z| <	1	, x[n]		1	-		-1		n		u[-n+1]
     	3			2n-1			3

     
     

     3.	1	< |z| <	1	, x[n] =	1	u[- n - 1] -		-1		n	u[n]
     	3		2		2n-1			3

     
     Of these the correct solutions are

1. 1. 1 and 2

   
   
   

   2. 1 and 3

   
   
   

   3. 2 and 3

   
   
   

   4. 1, 2 and 3

   
   
   

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1. View Hint View Answer Discuss in Forum

   X(z) =	1				+			- 1				|z| > 1/2 (Right-Sidesd)
   	1 -	1	z-1					1 +	1	z-1
   		2							z

   
   

   ⇒ x[n] =	2	u[n] -		-1		n	u[n] |z| < 1/3 (Left-Sided)
   	2n			3

   
   

   ⇒ x[n] =		- 2	+		-1		n		u[-n-1] 1/3 < |Z| < 1/2(Two-Sided)
   		2n			3

   
   

   x[n] = -	2	u[-n-1] -		-1		n	u[-n+1]
   	2n			3

   
   

   Correct Option: B

   X(z) =	1				+			- 1				|z| > 1/2 (Right-Sidesd)
   	1 -	1	z-1					1 +	1	z-1
   		2							z

   
   

   ⇒ x[n] =	2	u[n] -		-1		n	u[n] |z| < 1/3 (Left-Sided)
   	2n			3

   
   

   ⇒ x[n] =		- 2	+		-1		n		u[-n-1] 1/3 < |Z| < 1/2(Two-Sided)
   		2n			3

   
   

   x[n] = -	2	u[-n-1] -		-1		n	u[-n+1]
   	2n			3

   
   

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