 Power-law tail from a decaying rate envelope

A power-law tail from a decaying rate envelope arises when a positive instantaneous decay rate weakens with time. The time-dependent decay-rate envelope with $k(t)=\alpha /(t-b+t_0)$ givesfor $A\ge 0$, $t_0>0$ and $\alpha >0$. The parent article obtains this envelope by integrating the rate. Its use for recording design additionally requires integrating over the entire unrecorded tail.

For $\alpha >1$, integration after an extension $L\ge 0$ givesHence the acquisition duration from a power-law tail-energy tolerance $\epsilon >0$ is zero when $A{t}_0/(\alpha -1)\le \epsilon$; otherwise it is sufficient to choose

In an original example, $A=0.10\mathrm{W}$, $t_0=10\mathrm{ns}$ and $\alpha =2$ give an initial energy allowance $1.0\mathrm{nJ}$. Reducing the remaining allowance to $0.010\mathrm{nJ}$ requires $L\ge 990\mathrm{ns}$. An exponential envelope with the same initial allowance and a $10\mathrm{ns}$ decay time would require only $10\log 100\approx 46.05\mathrm{ns}$. Equal initial tail-energy allowances therefore do not imply equal acquisition durations.

For $0<\alpha \le 1$, the envelope tends to zero but its integral after every finite extension diverges. These assumptions then supply no finite missed-energy guarantee. They do not imply that every signal lying below the envelope has infinite energy.

This is an original comparison model, not a measured decay law attributed to a particular detector. In optical pulse metrology, NIST's response measurement separates receiver dynamics from the incident pulse before applying bandwidth corrections.^[ref] The same physical distinction is necessary before interpreting any measured slow tail as a global bound on optical power.