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A linear time invariant system is characterized by the system function H(z)=1/(1-0.5z-1)+2/(1-3z-1 ).What is the ROC of H(z) if the system is causal?
- a)|z|<3
- b)|z|>3
- c)|z|<0.5
- d)|z|>0.5
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
A linear time invariant system is characterized by the system function...
Explanation: The system has poles at z=0.5 and at z=3.
Since the system is causal, its ROC is |z|>0.5 and |z|>3. The common region is |z|>3. So, ROC of given H(z) is |z|>3.
Since the system is causal, its ROC is |z|>0.5 and |z|>3. The common region is |z|>3. So, ROC of given H(z) is |z|>3.
Most Upvoted Answer
A linear time invariant system is characterized by the system function...
> |0.5|< />< />
To determine the ROC (region of convergence) of H(z), we need to find the values of z for which the series representation of H(z) converges. One approach is to use partial fraction decomposition to express H(z) as a sum of simpler terms with known ROCs.
H(z) can be written as:
H(z) = A/(1-0.5z-1) + B/(1-3z-1)
where A=1/2 and B=1. We can then use the formula for the ROC of a geometric series:
|z|>|a|
where a is the coefficient of the z term in the denominator, to find the ROC for each term.
For the first term, we have a=0.5, so the ROC is:
|z|>|0.5|
which is equivalent to:
|z|<|0.5| or="" |z|="">|0.5|
But since the system is causal, we know that the ROC cannot include any poles (zeros of the denominator). The poles of H(z) are at z=2 and z=1/3, so the ROC cannot include these values.
The second term has a=3, so the ROC is:
|z|>|3|
which is equivalent to:
|z|<|3| or="" |z|="">|3|
Again, since the system is causal, the ROC cannot include any poles.
Therefore, the only valid ROC for H(z) that satisfies the causal condition is:
|0.5|< /><|3|>
To determine the ROC (region of convergence) of H(z), we need to find the values of z for which the series representation of H(z) converges. One approach is to use partial fraction decomposition to express H(z) as a sum of simpler terms with known ROCs.
H(z) can be written as:
H(z) = A/(1-0.5z-1) + B/(1-3z-1)
where A=1/2 and B=1. We can then use the formula for the ROC of a geometric series:
|z|>|a|
where a is the coefficient of the z term in the denominator, to find the ROC for each term.
For the first term, we have a=0.5, so the ROC is:
|z|>|0.5|
which is equivalent to:
|z|<|0.5| or="" |z|="">|0.5|
But since the system is causal, we know that the ROC cannot include any poles (zeros of the denominator). The poles of H(z) are at z=2 and z=1/3, so the ROC cannot include these values.
The second term has a=3, so the ROC is:
|z|>|3|
which is equivalent to:
|z|<|3| or="" |z|="">|3|
Again, since the system is causal, the ROC cannot include any poles.
Therefore, the only valid ROC for H(z) that satisfies the causal condition is:
|0.5|< /><|3|>
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A linear time invariant system is characterized by the system function H(z)=1/(1-0.5z-1)+2/(1-3z-1).What is the ROC of H(z) if the system is causal?a)|z|<3b)|z|>3c)|z|<0.5d)|z|>0.5Correct answer is option 'B'. 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 A linear time invariant system is characterized by the system function H(z)=1/(1-0.5z-1)+2/(1-3z-1).What is the ROC of H(z) if the system is causal?a)|z|<3b)|z|>3c)|z|<0.5d)|z|>0.5Correct answer is option 'B'. 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 A linear time invariant system is characterized by the system function H(z)=1/(1-0.5z-1)+2/(1-3z-1).What is the ROC of H(z) if the system is causal?a)|z|<3b)|z|>3c)|z|<0.5d)|z|>0.5Correct answer is option 'B'. Can you explain this answer?.
A linear time invariant system is characterized by the system function H(z)=1/(1-0.5z-1)+2/(1-3z-1).What is the ROC of H(z) if the system is causal?a)|z|<3b)|z|>3c)|z|<0.5d)|z|>0.5Correct answer is option 'B'. 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 A linear time invariant system is characterized by the system function H(z)=1/(1-0.5z-1)+2/(1-3z-1).What is the ROC of H(z) if the system is causal?a)|z|<3b)|z|>3c)|z|<0.5d)|z|>0.5Correct answer is option 'B'. 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 A linear time invariant system is characterized by the system function H(z)=1/(1-0.5z-1)+2/(1-3z-1).What is the ROC of H(z) if the system is causal?a)|z|<3b)|z|>3c)|z|<0.5d)|z|>0.5Correct answer is option 'B'. Can you explain this answer?.
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