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What is the inverse z-transform of X(z)= 1/(1-1.5z-1+0.5z-2 ) if ROC is |z|<0.5?
- a)[-2-0.5n]u(n)
- b)[-2+0.5n]u(n)
- c)[-2+0.5n]u(-n-1)
- d)[-2-0.5n]u(-n-1)
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
What is the inverse z-transform of X(z)= 1/(1-1.5z-1+0.5z-2) if ROC is...
Explanation: The partial fraction expansion for the given X(z) is
X(z)= 2z/(z-1)-z/(z-0.5)
In case when ROC is |z|<0.5,the signal is anti causal. Thus both the terms in the above equation are anti causal terms. So, if we apply inverse z-transform to the above equation we get x(n)= [-2+0.5n]u(-n-1).
X(z)= 2z/(z-1)-z/(z-0.5)
In case when ROC is |z|<0.5,the signal is anti causal. Thus both the terms in the above equation are anti causal terms. So, if we apply inverse z-transform to the above equation we get x(n)= [-2+0.5n]u(-n-1).
Most Upvoted Answer
What is the inverse z-transform of X(z)= 1/(1-1.5z-1+0.5z-2) if ROC is...
> First, we need to factor the denominator of X(z):
>
> X(z) = 1 / (1 - 1.5z^-1 + 0.5z^-2)
>
> X(z) = 1 / [(1 - z^-1)(1 - 0.5z^-1)]
>
> Now, we can use the partial fraction expansion to express X(z) as a sum of simpler terms:
>
> X(z) = A/(1 - z^-1) + B/(1 - 0.5z^-1)
>
> where A and B are constants to be determined. To find A and B, we can multiply both sides of this equation by the denominators of the two terms on the right-hand side:
>
> X(z)(1 - z^-1)(1 - 0.5z^-1) = A(1 - 0.5z^-1) + B(1 - z^-1)
>
> Next, we can substitute z = 0.5 and z = 1 in this equation to get two equations in A and B:
>
> X(0.5)(1 - 0.5^-1)(1 - 0.5^-1) = A(1 - 0.5^-1) + B(1 - 0.5)
>
> X(1)(1 - 1^-1)(1 - 0.5^-1) = A(1 - 0.5) + B(1 - 1^-1)
>
> Simplifying these equations, we get:
>
> A + B = X(0.5)(1 - 0.5^-1)(1 - 0.5^-1) = 4/3
>
> 0.5A + B = X(1)(1 - 0.5) + B = 2/3
>
> Solving for A and B, we get:
>
> A = 2/3
>
> B = 2/3 - X(1)(1 - 0.5) = -2/3
>
> So, we have:
>
> X(z) = 2/3/(1 - z^-1) - 2/3/(1 - 0.5z^-1)
>
> Now, we can use the formula for the inverse z-transform of a geometric series to find the inverse z-transform of each term:
>
> x_1[n] = (2/3)(z^n)
>
> x_2[n] = (-2/3)(0.5^n)
>
> Therefore, the overall inverse z-transform of X(z) is the sum of these two terms:
>
> x[n] = x_1[n] + x_2[n] = (2/3)(z^n) - (2/3)(0.5^n)
>
> The ROC of this sequence is |z| > 0.5, since that is the intersection of the ROCs of the two individual terms.
>
> X(z) = 1 / (1 - 1.5z^-1 + 0.5z^-2)
>
> X(z) = 1 / [(1 - z^-1)(1 - 0.5z^-1)]
>
> Now, we can use the partial fraction expansion to express X(z) as a sum of simpler terms:
>
> X(z) = A/(1 - z^-1) + B/(1 - 0.5z^-1)
>
> where A and B are constants to be determined. To find A and B, we can multiply both sides of this equation by the denominators of the two terms on the right-hand side:
>
> X(z)(1 - z^-1)(1 - 0.5z^-1) = A(1 - 0.5z^-1) + B(1 - z^-1)
>
> Next, we can substitute z = 0.5 and z = 1 in this equation to get two equations in A and B:
>
> X(0.5)(1 - 0.5^-1)(1 - 0.5^-1) = A(1 - 0.5^-1) + B(1 - 0.5)
>
> X(1)(1 - 1^-1)(1 - 0.5^-1) = A(1 - 0.5) + B(1 - 1^-1)
>
> Simplifying these equations, we get:
>
> A + B = X(0.5)(1 - 0.5^-1)(1 - 0.5^-1) = 4/3
>
> 0.5A + B = X(1)(1 - 0.5) + B = 2/3
>
> Solving for A and B, we get:
>
> A = 2/3
>
> B = 2/3 - X(1)(1 - 0.5) = -2/3
>
> So, we have:
>
> X(z) = 2/3/(1 - z^-1) - 2/3/(1 - 0.5z^-1)
>
> Now, we can use the formula for the inverse z-transform of a geometric series to find the inverse z-transform of each term:
>
> x_1[n] = (2/3)(z^n)
>
> x_2[n] = (-2/3)(0.5^n)
>
> Therefore, the overall inverse z-transform of X(z) is the sum of these two terms:
>
> x[n] = x_1[n] + x_2[n] = (2/3)(z^n) - (2/3)(0.5^n)
>
> The ROC of this sequence is |z| > 0.5, since that is the intersection of the ROCs of the two individual terms.
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What is the inverse z-transform of X(z)= 1/(1-1.5z-1+0.5z-2) if ROC is |z|<0.5? a)[-2-0.5n]u(n)b)[-2+0.5n]u(n)c)[-2+0.5n]u(-n-1)d)[-2-0.5n]u(-n-1)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 inverse z-transform of X(z)= 1/(1-1.5z-1+0.5z-2) if ROC is |z|<0.5? a)[-2-0.5n]u(n)b)[-2+0.5n]u(n)c)[-2+0.5n]u(-n-1)d)[-2-0.5n]u(-n-1)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 inverse z-transform of X(z)= 1/(1-1.5z-1+0.5z-2) if ROC is |z|<0.5? a)[-2-0.5n]u(n)b)[-2+0.5n]u(n)c)[-2+0.5n]u(-n-1)d)[-2-0.5n]u(-n-1)Correct answer is option 'C'. Can you explain this answer?.
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