 Time-dependent decay-rate envelope

A time-dependent decay-rate envelope bounds a nonnegative signal using a lower bound on its instantaneous decay rate. If $P$ is absolutely continuous on finite intervals, $P(b)\le A$ and $P^{\prime }\le -k(t)P$ almost everywhere for locally integrable $k(t)\ge 0$, thenThe differential inequality for an exponential tail envelope article proves this comparison using an integrating factor and gives a piecewise-constant example. A strictly positive constant lower bound on $k$ gives exponential decay, but a positive rate at every instant need not supply such a lower bound.

An original example takes $k(t)=\alpha /(t-b+t_0)$ with $t_0>0$ and $\alpha >0$. Its integrated rate is logarithmic, soThis power-law tail from a decaying rate envelope tends to zero for every positive $\alpha$. Its envelope energy is nevertheless finite only when $\alpha >1$, in which caseFor original chosen $A=0.20\mathrm{W}$, $t_0=5\mathrm{ns}$ and $\alpha =2$, the allowance is $1.0\mathrm{nJ}$. After a further $45\mathrm{ns}$ it is at most $0.10\mathrm{nJ}$. The remaining energy decreases algebraically rather than exponentially.

At $\alpha =1$ the integrated envelope diverges logarithmically despite its power tending to zero. This does not prove that every signal below it has infinite energy; it means these assumptions alone supply no finite all-time energy bound. Taking the envelope itself gives a mathematical counterexample to the claim that pointwise decay implies integrable energy.

An experimental rate law therefore needs both temporal validity and an integrability check before it bounds missed optical energy. A measured detector tail is not automatically the true power profile; NIST's primary calibration procedure measures the receiver response separately.^[ref] The power-law example is an independent comparison model, not a reported tail law of that apparatus.