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Group delay versus pulse-front arrival for an RC load distinguishes a narrow-band modulation displacement from the earliest response to a newly applied input. For the relative amplifier-load model in group delay introduced by a capacitive amplifier load, the causal impulse response is $h(t)=e^{-t/\tau }/\tau$ for $t\ge 0$ and zero for $t<0$. Its convolution with an input that vanishes before zero also vanishes before zero. Nevertheless its group delay is $\tau /[1+(\omega \tau {)}^2]$, which is positive and frequency dependent. The phase-derivative definition concerns spectral components and their envelopes.^[ref]

An original finite-pulse calculation shows why a threshold measurement gives another time. For an initially relaxed load, apply a rectangular pulse of height $V_0$ and duration $T>0$. Convolution givesThe first mathematical response occurs immediately after zero. A rising threshold equal to fraction $a$ of the input pulse height is reached at $t_a=-\tau \log (1-a)$ only if the pulse lasts at least that long. This is the threshold-crossing time of a finite rectangular pulse through an RC load.

With $\tau =1\mu \mathrm{s}$ and $T=0.1\mu \mathrm{s}$, the largest output is only $(1-e^{-0.1})V_0\approx 0.0952V_0$: a $0.5V_0$ threshold is never reached. If $T=3\mu \mathrm{s}$ instead, that threshold is crossed at $0.693\mu \mathrm{s}$. Neither value is the $0.5\mu \mathrm{s}$ group delay evaluated at $\omega \tau =1$.

A pulse generator and oscilloscope can compare the threshold observation with a slow modulation on a carrier; the chosen threshold and modulation bandwidth must be recorded. MIT's laboratory likewise separates square-wave decay measurements from sinusoidal amplitude/phase measurements.^[ref] If wiring adds a transport delay $d$, shift the impulse response to $h(t-d)$: the front moves to $d$, while the modulation delay becomes $d+{\tau }_g$. The immediate onset of the unshifted lumped model is not a model of propagation along a finite-length cable.

RC pulse-width discrimination converts an input-duration test into a threshold observation on an RC circuit. For a relaxed first-order load with time constant $\tau >0$, a positive rectangular input of height $V_0$ and duration $T$ reaches peak output $V_0(1-e^{-T/\tau })$. An ideal electronic comparator has a nonzero interval above fixed threshold $0<V_{\mathrm{th}}<V_0$ precisely whenEquality only touches the threshold at the peak. The parent threshold-crossing time of a finite rectangular pulse through an RC load derives the full response and the rejected-pulse boundary.

This is a duration discriminator only while the input height is controlled. At a fixed duration, the same condition isA larger input can pass the threshold even when a smaller pulse of exactly the same duration fails. The minimum accepted duration decreases with height, with derivative $-\tau V_{\mathrm{th}}/[V_0(V_0-V_{\mathrm{th}})]$.

For an original example, choose $\tau =2\mu \mathrm{s}$, $V_{\mathrm{th}}=0.40\mathrm{V}$, and $T=1\mu \mathrm{s}$. The minimum height for a nonzero comparator interval is about $1.0166\mathrm{V}$. A $0.8\mathrm{V}$ pulse peaks at about $0.3148\mathrm{V}$ and is rejected; a $1.2\mathrm{V}$ pulse peaks at about $0.4722\mathrm{V}$ and is accepted. Treating this observation as a width-only decision would conflate two different inputs.

A pulse-generator experiment therefore sweeps width at a recorded height and threshold, measures the actual capacitor waveform on an oscilloscope, and allows the initial state to relax. A residual baseline in RC pulse-width discrimination changes the next pulse's peak and invalidates the relaxed-state formula. Probe and comparator input capacitance also belong in the effective time constant. The natural-decay apparatus in MIT's RC laboratory supplies a direct component-level check of that time constant.^[ref] Comparator propagation delay and hysteresis are additional dynamics, not part of the ideal RC acceptance inequality.

An electronic comparator registers a pulse only if its input reaches the chosen threshold. For an initially uncharged capacitor driven through an electrical resistance $R>0$, a short rectangular pulse may end before that happens. The resulting threshold-crossing time of a finite rectangular pulse through an RC load depends on threshold and pulse duration, as well as the time constant $\tau =RC$; it is not determined by group delay alone.

Take a positive input electric potential difference $V_0$ for $0\le t<T$, zero input outside that interval, and output $v(t)$ across a capacitance $C>0$. Include the source's effective series electrical resistance in $R$. The initial condition is $v(0)=0$. Kirchhoff's voltage law and the capacitor current relation give the first-order linear differential equationThe output is a continuous function and reaches its maximum at the pulse's falling edge:The input jumps there, but an ideal capacitor's stored electric charge cannot change discontinuously under the finite currents of this model.

Choose a fixed threshold $V_{\mathrm{th}}=a{V}_0$ with $0<a<1$. Solving the rising exponential function using a natural logarithm givesIf $T<T_{\min }$, the output never reaches the threshold. At $T=T_{\min }$, it touches the threshold only at its peak; the ideal time spent strictly above threshold is zero. A nonzero high interval in an ideal non-inverting electronic comparator requires $T>T_{\min }$. This is RC pulse-width discrimination: different input durations can give a missing pulse or a finite output pulse even when their input heights are equal.

For $T>T_{\min }$, solving the falling exponential decay givesHere $W$ is the time above threshold for an RC-filtered rectangular pulse, before any electronic comparator switching dynamics. It is not generally the input duration $T$. Just above $T_{\min }$, differentiation gives $dW/dT=1/(1-e^{-T/\tau })$ and hence $W\approx (T-T_{\min })/a$ to first order; a half-height threshold gives slope two at the detection boundary.

For an original numerical example, take $R=1\mathrm{k\Omega }$, $C=1\mathrm{nF}$, $V_0=1\mathrm{V}$, and $V_{\mathrm{th}}=0.50\mathrm{V}$, so $\tau =1\mu \mathrm{s}$. A $0.40\mu \mathrm{s}$ input peaks at only $0.3297\mathrm{V}$ and is rejected. A $2.00\mu \mathrm{s}$ input peaks at $0.8647\mathrm{V}$ and givesThese are model calculations, not recorded instrument traces.

A threshold near the asymptotic input height is also especially sensitive to noise and threshold measurement uncertainty. Holding the absolute input height $V_0$ and $\tau$ fixed, the threshold timing sensitivity of an RC step response isFor the same $1\mathrm{V}$, $1\mu \mathrm{s}$ model, a $1\mathrm{mV}$ effective threshold perturbation gives approximately $2\mathrm{ns}$ of timing change at $0.50\mathrm{V}$, but $10\mathrm{ns}$ at $0.90\mathrm{V}$, by first-order uncertainty propagation. The rectangular pulse must still last long enough to reach the selected threshold; changing $a$ together with $V_0$ is a different sensitivity calculation.

A bench test uses a pulse generator and an oscilloscope to record both the applied electric potential difference and the capacitor output. Sweep $T$ while keeping $V_0$ and the actual threshold fixed, and allow enough time between pulses for the measured initial output to be negligible at the desired precision. Include probe and comparator-input capacitance in $C$. MIT's laboratory measures RC decay with a $47\mathrm{nF}$ capacitor, a $1.5\mathrm{k\Omega }$ resistor and square-wave excitation; that gives a directly measurable $70.5\mu \mathrm{s}$ component-value time constant.^[ref] The pulse-duration sweep proposed here extends that decay measurement to a threshold observation. A real electronic comparator's input offset voltage, hysteresis and finite switching time must be separated from the ideal RC crossing times; a Schmitt trigger uses different rising and falling thresholds.

Threshold timing sensitivity of an RC step response measures how a crossing time changes when the absolute threshold or signal height changes. For a relaxed step response $v(t)=V_0(1-e^{-t/\tau })$, with $0<V_{\mathrm{th}}<V_0$, the rising crossing isHolding the other inputs fixed givesA threshold approaching the step height encounters a flatter part of the exponential response, making the first sensitivity large.

For an original timing estimate, choose $V_0=2\mathrm{V}$, $\tau =0.5\mu \mathrm{s}$, and effective small threshold-noise standard deviation $4\mathrm{mV}$. At $V_{\mathrm{th}}=0.4\mathrm{V}$, local linear propagation gives timing standard deviation about $1.25\mathrm{ns}$. At $1.9\mathrm{V}$, it gives about $20\mathrm{ns}$. The latter threshold also needs a longer input pulse to be reached at all, as shown in threshold-crossing time of a finite rectangular pulse through an RC load.

The two voltage sensitivities can cancel under a different measurement calibration model. If the threshold is set to a fixed fraction $a$ of the same varying height, $d{V}_{\mathrm{th}}=ad{V}_0$ and $V_{\mathrm{th}}=a{V}_0$, their contributions sum to zero. The crossing becomes $-\tau \log (1-a)$, independent of amplitude in this ideal model. Such correlated threshold and amplitude errors in RC timing must retain covariance rather than add two independent voltage uncertainties.^[ref]

The local noise rule also follows from dividing a small voltage perturbation by the crossing slope $(V_0-V_{\mathrm{th}})/\tau$. It assumes a well-defined nearby crossing and sufficiently small perturbations; it does not describe multiple noise crossings, comparator switching dynamics, or a pulse ending before the threshold. A bench test can compare controlled threshold shifts with oscilloscope crossing times while holding the actual step height and initial state fixed.