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I'm quite confused as to why we call a system with transfer function $$\frac{1}{s+\alpha}$$ a delayed integrator.

It does not seem to be an integrator in the first place since it convolves with an exponential unless alpha is 0.

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    \$\begingroup\$ Probably because of the pole introduced by \$\alpha\$, which causes the slope to come later, than starting with DC. Personally, I haven't heard this term, but don't count on my technical English. \$\endgroup\$ Commented Apr 4, 2022 at 13:59
  • \$\begingroup\$ This table of Laplace transforms classifies it as an exponential decay (assuming \$a > 0\$), though I suppose it could become exponential growth if \$a < 0\$. Other tables I referenced list something similar, sometimes tabulating for \$e^{a t}\$ instead of \$e^{-a t}\$ \$\endgroup\$ Commented Apr 4, 2022 at 14:55

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