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I have a signal $E(t)$ which start at $t=0$ and dies out overtime. What would be a proper notation to specify part of which is above a certain threshold (say for e.g. C)?. Is $E_{interested} = E(t)>C$ a proper notation?

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What you are looking for is called an Indicator Function which would assign a $1$ for all those elements of $E(t)$ that satisfy the given condition.

The same could also be expressed via the use of Set Builder Notation as $E_{interested} = \{1|e \in E \land e>C\}$ translating to "Elements of $E_{interested}$ take the value of 1 for all elements $e$ in $E$ satisfying the condition $e>C$". You might have to add $E_{interested} = \{0|e \in E\}$ as a pre-initialisation step to show the default value of $E_{interested}$.

The difference between the two is that the indicator function is defined and so you can simply reference it (e.g. "...where $\mathcal{\chi}(E, \Phi(E))$ denotes the Indicator Function using a predicate $\Phi$...", or even drop $\Phi$ if the predicate is the same throughout) while the set builder notation version could be used more as a "one-of" in-line way in writing down an algorithm.

To an extent, it is also a matter of notation already used and what you are trying to convey.

Hope this helps.

EDIT :

Just to clarify, if you are also interested in the actual values of $E(t)$ (interrupted by zeros wherever $E(t) \leq C$), then given $E_{interested}$ you can now $E_{interested} \cdot E$ to produce that sequence that contains the actual values of $E(t)$ satisfying the condition (rather than just the "pulse sequence").

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