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What is the partial fraction expansion of X(z)=1/((1+z-1 )(1-z-1)2)?
- a)z/(4(z+1)) + 3z/(4(z-1)) + z/(2〖(z+1)〗2 )
- b)z/(4(z+1)) + 3z/(4(z-1)) – z/(2〖(z+1)〗2 )
- c)z/(4(z+1)) + 3z/(4(z-1)) + z/(2〖(z-1)〗2 )
- d)z/(4(z+1)) + z/(4(z-1)) + z/(2〖(z+1)〗2 )
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
What is the partial fraction expansion of X(z)=1/((1+z-1)(1-z-1)2)?a)z...
Explanation: First we express X(z) in terms of positive powers of z, in the form X(z)=z3/((z+1)〖(z-1)〗2 )
X(z) has a simple pole at z=-1 and a double pole at z=1. In such a case the approximate partial fraction expansion is
(X(z))/z = z2/((z+1)〖(z-1)〗2 ) =A/(z+1) + B/(z-1) + C/〖(z-1)〗2
On simplifying, we get the values of A, B and C as 1/4, 3/4 and 1/2 respectively.
Therefore, we get X(z)= z/(4(z+1)) + 3z/(4(z-1)) + z/(2〖(z-1)〗2 ) .
X(z) has a simple pole at z=-1 and a double pole at z=1. In such a case the approximate partial fraction expansion is
(X(z))/z = z2/((z+1)〖(z-1)〗2 ) =A/(z+1) + B/(z-1) + C/〖(z-1)〗2
On simplifying, we get the values of A, B and C as 1/4, 3/4 and 1/2 respectively.
Therefore, we get X(z)= z/(4(z+1)) + 3z/(4(z-1)) + z/(2〖(z-1)〗2 ) .
Most Upvoted Answer
What is the partial fraction expansion of X(z)=1/((1+z-1)(1-z-1)2)?a)z...
Partial Fraction Expansion Explanation:
Given Function:
X(z) = 1 / ((1 + z - 1)(1 - z - 1)^2)
Partial Fraction Expansion Steps:
1. Factorize the denominator:
(1 + z - 1)(1 - z - 1)^2 = z(1 - z - 1)^2
= z(1 - z + 1)^2
= z(2 - z)^2
2. Write X(z) in partial fraction form:
X(z) = A / (z+1) + B / (z-1) + C / (2 - z)
3. Find the values of A, B, and C:
1 = A(2 - z)^2 + B(z+1)(2 - z) + C(z+1)^2
1 = A(4 - 4z + z^2) + B(2z - z^2 + 2 - z) + C(z^2 + 2z + 1)
4. Solve for A, B, and C:
By comparing the coefficients, we get:
A = 1/4, B = 3/4, C = 1/2
Partial Fraction Expansion:
X(z) = z / (4(z+1)) + 3z / (4(z-1)) + z / (2(2-z))
Therefore, the correct partial fraction expansion of X(z) is option 'C'.
Given Function:
X(z) = 1 / ((1 + z - 1)(1 - z - 1)^2)
Partial Fraction Expansion Steps:
1. Factorize the denominator:
(1 + z - 1)(1 - z - 1)^2 = z(1 - z - 1)^2
= z(1 - z + 1)^2
= z(2 - z)^2
2. Write X(z) in partial fraction form:
X(z) = A / (z+1) + B / (z-1) + C / (2 - z)
3. Find the values of A, B, and C:
1 = A(2 - z)^2 + B(z+1)(2 - z) + C(z+1)^2
1 = A(4 - 4z + z^2) + B(2z - z^2 + 2 - z) + C(z^2 + 2z + 1)
4. Solve for A, B, and C:
By comparing the coefficients, we get:
A = 1/4, B = 3/4, C = 1/2
Partial Fraction Expansion:
X(z) = z / (4(z+1)) + 3z / (4(z-1)) + z / (2(2-z))
Therefore, the correct partial fraction expansion of X(z) is option 'C'.
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What is the partial fraction expansion of X(z)=1/((1+z-1)(1-z-1)2)?a)z/(4(z+1)) + 3z/(4(z-1)) + z/(2〖(z+1)〗2)b)z/(4(z+1)) + 3z/(4(z-1)) – z/(2〖(z+1)〗2)c)z/(4(z+1)) + 3z/(4(z-1)) + z/(2〖(z-1)〗2)d)z/(4(z+1)) + z/(4(z-1)) + z/(2〖(z+1)〗2)Correct answer is option 'C'. Can you explain this answer? for Electrical Engineering (EE) 2025 is part of Electrical Engineering (EE) preparation. The Question and answers have been prepared according to the Electrical Engineering (EE) exam syllabus. Information about What is the partial fraction expansion of X(z)=1/((1+z-1)(1-z-1)2)?a)z/(4(z+1)) + 3z/(4(z-1)) + z/(2〖(z+1)〗2)b)z/(4(z+1)) + 3z/(4(z-1)) – z/(2〖(z+1)〗2)c)z/(4(z+1)) + 3z/(4(z-1)) + z/(2〖(z-1)〗2)d)z/(4(z+1)) + z/(4(z-1)) + z/(2〖(z+1)〗2)Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Electrical Engineering (EE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What is the partial fraction expansion of X(z)=1/((1+z-1)(1-z-1)2)?a)z/(4(z+1)) + 3z/(4(z-1)) + z/(2〖(z+1)〗2)b)z/(4(z+1)) + 3z/(4(z-1)) – z/(2〖(z+1)〗2)c)z/(4(z+1)) + 3z/(4(z-1)) + z/(2〖(z-1)〗2)d)z/(4(z+1)) + z/(4(z-1)) + z/(2〖(z+1)〗2)Correct answer is option 'C'. Can you explain this answer?.
What is the partial fraction expansion of X(z)=1/((1+z-1)(1-z-1)2)?a)z/(4(z+1)) + 3z/(4(z-1)) + z/(2〖(z+1)〗2)b)z/(4(z+1)) + 3z/(4(z-1)) – z/(2〖(z+1)〗2)c)z/(4(z+1)) + 3z/(4(z-1)) + z/(2〖(z-1)〗2)d)z/(4(z+1)) + z/(4(z-1)) + z/(2〖(z+1)〗2)Correct answer is option 'C'. Can you explain this answer? for Electrical Engineering (EE) 2025 is part of Electrical Engineering (EE) preparation. The Question and answers have been prepared according to the Electrical Engineering (EE) exam syllabus. Information about What is the partial fraction expansion of X(z)=1/((1+z-1)(1-z-1)2)?a)z/(4(z+1)) + 3z/(4(z-1)) + z/(2〖(z+1)〗2)b)z/(4(z+1)) + 3z/(4(z-1)) – z/(2〖(z+1)〗2)c)z/(4(z+1)) + 3z/(4(z-1)) + z/(2〖(z-1)〗2)d)z/(4(z+1)) + z/(4(z-1)) + z/(2〖(z+1)〗2)Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Electrical Engineering (EE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What is the partial fraction expansion of X(z)=1/((1+z-1)(1-z-1)2)?a)z/(4(z+1)) + 3z/(4(z-1)) + z/(2〖(z+1)〗2)b)z/(4(z+1)) + 3z/(4(z-1)) – z/(2〖(z+1)〗2)c)z/(4(z+1)) + 3z/(4(z-1)) + z/(2〖(z-1)〗2)d)z/(4(z+1)) + z/(4(z-1)) + z/(2〖(z+1)〗2)Correct answer is option 'C'. Can you explain this answer?.
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