Why is the Region of Convergence Important?

In summary, the conversation discusses the Laplace transform of a function and the conditions for convergence. One person asks about the role of the imaginary part in the denominator and the other person explains that it is bad to have a denominator of zero and that the condition for convergence is Re[s] > 7. They also discuss different scenarios and their corresponding regions of convergence.
  • #1
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0
One more:
after doing Laplace transform for this:

[tex] f(t) = e^{(7+5j)t}u(t-1) [/tex]

where u(t) = 1 for t >= 0 and 0 otherwise;
so here's what I have:

[tex]L[f(t)] = \frac {e^{-(s-7-5j)}}{s-7-5j} [/tex]

so, my reasoning was that it would converge if Re > 7 because that's the value for which exponential would converge. But why exactly do we not care about Im? I know that by Euler's formula, it [tex]e^{jw}[/tex] would just be oscillating but don't I need a condition for denominator of L[f(t)]?
Thanks for your time and explanation.
 
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  • #2
It is bad to have denominator of zero.
Looks like, if Re > 7 strictly (not >=)
the denominator can't be zero.
 
  • #3
Thanks for reply.

lightgrav said:
It is bad to have denominator of zero. the denominator can't be zero.
yeah, i know, that's what I was asking about: do I need to state a condition for the denominator to exclude a case where it is = 0?

edit: what are the cases when Re >= or <= some value?
let's say I have [tex] L[f(t)] = \frac{1}{s-a}[/tex]
so, region of convergence would be Re >= a or Re > a?

thanks again.
 
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1. What is the Region of Convergence (ROC)?

The Region of Convergence (ROC) is a concept in signal processing and control theory that refers to the set of values for which a given mathematical series or transform is convergent. In other words, it is the range of input values for which a particular function or system will produce a valid output.

2. How is the ROC determined?

The ROC is typically determined by analyzing the properties of the given function or system. In the case of a power series, for example, the ROC can be determined by using the ratio test or the root test. In other cases, the ROC may be defined by certain constraints or conditions that the function or system must satisfy for convergence to occur.

3. Why is the ROC important?

The ROC is important because it provides information about the behavior and stability of a function or system. It helps to determine the range of inputs for which the function or system will produce a valid output, and can also provide insights into the behavior of the output for values outside of the ROC. In control systems, the ROC is used to ensure stability and proper functioning of the system.

4. What happens if a value lies outside of the ROC?

If a value lies outside of the ROC, the given function or system will not produce a valid output. This indicates that the function or system is not convergent for that particular value, and may lead to instability or other undesirable behaviors. It is important to stay within the ROC to ensure proper functioning of the system.

5. Can the ROC change for a given function or system?

Yes, the ROC can change for a given function or system. This can occur due to changes in the properties or parameters of the function or system, or due to changes in the input values. It is important to analyze the ROC for different scenarios to fully understand the behavior of the function or system.

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