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What is the one sided z-transform X+(z) of the signal x(n)={1,2,5,7,0,1}?
- a)z2+2z+5+7z-1+z-3
- b)5+7z+z3
- c)z-2+2z-1+5+7z+z3
- d)5+7z-1+z-3
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
What is the one sided z-transform X+(z) of the signal x(n)={1,2,5,7,0,...
Explanation: Since the one sided z-transform is valid only for n>=0, the z-transform of the given signal will be X+(z)= 5+7z-1+z-3.
Most Upvoted Answer
What is the one sided z-transform X+(z) of the signal x(n)={1,2,5,7,0,...
The one-sided z-transform of a discrete-time signal x(n) is defined as X(z) = ∑[x(n) * z^(-n)], where n goes from 0 to infinity.
Given signal x(n) = {1, 2, 5, 7, 0, 1}, we can find its one-sided z-transform.
Let's calculate the z-transform step by step:
1. For n = 0:
X(z) += x(0) * z^(-0) = 1 * 1 = 1
2. For n = 1:
X(z) += x(1) * z^(-1) = 2 * z^(-1) = 2z^(-1)
3. For n = 2:
X(z) += x(2) * z^(-2) = 5 * z^(-2) = 5z^(-2)
4. For n = 3:
X(z) += x(3) * z^(-3) = 7 * z^(-3) = 7z^(-3)
5. For n = 4:
X(z) += x(4) * z^(-4) = 0 * z^(-4) = 0
6. For n = 5:
X(z) += x(5) * z^(-5) = 1 * z^(-5) = z^(-5)
Putting it all together, we have:
X(z) = 1 + 2z^(-1) + 5z^(-2) + 7z^(-3) + z^(-5)
Comparing this with the options provided, we can see that option 'D' matches the calculated z-transform:
X(z) = 5 + 7z^(-1) + z^(-3)
Therefore, the correct answer is option 'D': 5 + 7z^(-1) + z^(-3).
This z-transform represents the mathematical relationship between the given discrete-time signal x(n) and its complex frequency representation X(z) in the z-domain. It allows us to analyze and manipulate the signal using z-domain techniques such as pole-zero analysis, frequency response calculation, and system design.
Given signal x(n) = {1, 2, 5, 7, 0, 1}, we can find its one-sided z-transform.
Let's calculate the z-transform step by step:
1. For n = 0:
X(z) += x(0) * z^(-0) = 1 * 1 = 1
2. For n = 1:
X(z) += x(1) * z^(-1) = 2 * z^(-1) = 2z^(-1)
3. For n = 2:
X(z) += x(2) * z^(-2) = 5 * z^(-2) = 5z^(-2)
4. For n = 3:
X(z) += x(3) * z^(-3) = 7 * z^(-3) = 7z^(-3)
5. For n = 4:
X(z) += x(4) * z^(-4) = 0 * z^(-4) = 0
6. For n = 5:
X(z) += x(5) * z^(-5) = 1 * z^(-5) = z^(-5)
Putting it all together, we have:
X(z) = 1 + 2z^(-1) + 5z^(-2) + 7z^(-3) + z^(-5)
Comparing this with the options provided, we can see that option 'D' matches the calculated z-transform:
X(z) = 5 + 7z^(-1) + z^(-3)
Therefore, the correct answer is option 'D': 5 + 7z^(-1) + z^(-3).
This z-transform represents the mathematical relationship between the given discrete-time signal x(n) and its complex frequency representation X(z) in the z-domain. It allows us to analyze and manipulate the signal using z-domain techniques such as pole-zero analysis, frequency response calculation, and system design.
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