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When a system is such that the square sum of its impulse response tends to infinity when summed over all real time space,
- a)System is marginally stable
- b)System is unstable
- c)System is stable
- d)None of the mentioned
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
When a system is such that the square sum of its impulse response tend...
The system turns out to be unstable. Only if it is zero/finite it is stable.
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When a system is such that the square sum of its impulse response tend...
Explanation:
When the square sum of the impulse response of a system tends to infinity when summed over all real time space, it indicates that the system is unstable. Let's understand this in detail:
Impulse Response:
The impulse response of a system is the output of the system when an impulse signal is given as input. It provides information about how the system responds to a delta function input.
Square Sum of Impulse Response:
The square sum of the impulse response is obtained by squaring each value of the impulse response and summing them over all real time space. Mathematically, it can be represented as:
Sum of (Impulse Response^2) = ∫(Impulse Response(t))^2 dt, where t ranges from -∞ to ∞.
If this sum tends to infinity, it means that the energy of the impulse response is unbounded and keeps increasing indefinitely.
System Stability:
A system is said to be stable if its output remains bounded for any bounded input. Stability is an important characteristic of a system as it determines whether the system will produce a meaningful and reliable output.
Unstable System:
If the square sum of the impulse response tends to infinity, it implies that the energy of the impulse response is unbounded. This indicates that even for a bounded input, the output of the system will become unbounded, which is an undesirable behavior. Hence, the system is considered to be unstable.
Conclusion:
In summary, when the square sum of the impulse response of a system tends to infinity when summed over all real time space, it signifies that the system is unstable.
When the square sum of the impulse response of a system tends to infinity when summed over all real time space, it indicates that the system is unstable. Let's understand this in detail:
Impulse Response:
The impulse response of a system is the output of the system when an impulse signal is given as input. It provides information about how the system responds to a delta function input.
Square Sum of Impulse Response:
The square sum of the impulse response is obtained by squaring each value of the impulse response and summing them over all real time space. Mathematically, it can be represented as:
Sum of (Impulse Response^2) = ∫(Impulse Response(t))^2 dt, where t ranges from -∞ to ∞.
If this sum tends to infinity, it means that the energy of the impulse response is unbounded and keeps increasing indefinitely.
System Stability:
A system is said to be stable if its output remains bounded for any bounded input. Stability is an important characteristic of a system as it determines whether the system will produce a meaningful and reliable output.
Unstable System:
If the square sum of the impulse response tends to infinity, it implies that the energy of the impulse response is unbounded. This indicates that even for a bounded input, the output of the system will become unbounded, which is an undesirable behavior. Hence, the system is considered to be unstable.
Conclusion:
In summary, when the square sum of the impulse response of a system tends to infinity when summed over all real time space, it signifies that the system is unstable.
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When a system is such that the square sum of its impulse response tends to infinity when summed over all real time space,a)System is marginally stableb)System is unstablec)System is stabled)None of the mentionedCorrect answer is option 'B'. Can you explain this answer?
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