= Time above threshold for an RC-filtered rectangular pulse
{title2=$W=t_\downarrow-t_\uparrow$}
The <time above threshold for an RC-filtered rectangular pulse> is the separation of its rising and falling threshold crossings, rather than the applied pulse duration. In the relaxed first-order model, put $a=V_{\rm th}/V_0$ with $0<a<1$ and $T> T_{\min}=-\tau\log(1-a)$. The parent crossing calculation gives
$$
W=T+\tau\log\frac{(1-e^{-T/\tau})(1-a)}{a}.
$$
This excludes comparator propagation dynamics and uses the same threshold on both edges. A <Schmitt trigger> with unequal switching thresholds needs a different formula.
The positive-width relation can be inverted exactly:
$$
\frac W\tau=\log(e^{T/\tau}-1)+\log\frac{1-a}{a},\qquad
T=\tau\log\left(1+\frac{a}{1-a}e^{W/\tau}\right).
$$
Its derivative is $dW/dT=1/(1-e^{-T/\tau})>1$. Thus a measured positive $W$ determines a unique input duration under the stated model. A zero-width observation does not: all rejected durations $T\le T_{\min}$ have that same outcome.
For an original example, take $a=0.25$, $\tau=1\,\mathrm{\mu s}$ and $T=2\,\mathrm{\mu s}$. The output remains above threshold for about $2.9532\,\mathrm{\mu s}$, longer than the input pulse because of its decaying tail. Substitution into the inverse relation recovers $2\,\mathrm{\mu s}$.
For long pulses, $W-T$ tends to $\tau\log[(1-a)/a]$. It is positive below a half-height threshold and negative above one. At exactly half height, $W<T$ for every finite accepted pulse, with the difference tending to zero for long pulses. Consequently a measured above-threshold width cannot be equated to the input width without the threshold and filter model.
A pulse-generator/<oscilloscope> comparison can record both widths while varying threshold. The measured RC decay should be checked independently, using the natural-response method in the primary MIT laboratory.<Laboratory 11: Sinusoidal Steady State Response of RC and RL Circuits by MIT OpenCourseWare>[Printed pp. 1–2]{ref} Initial residual charge, hysteresis, and different rising/falling comparator delays alter the observation; the algebra above only inverts the ideal capacitor waveform.
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