= Group delay versus pulse-front arrival for an RC load
<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 $t\ge0$ 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.<SciPy 1.16.2 documentation>[scipy.signal.group_delay, definition and frequency-unit convention]{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 gives
$$
v(t)=\begin{cases}
0,&t<0,\\
V_0(1-e^{-t/\tau}),&0\le t<T,\\
V_0(1-e^{-T/\tau})e^{-(t-T)/\tau},&t\ge T.
\end{cases}
$$
The first mathematical response occurs immediately after zero. A rising threshold equal to fraction $a$ of the \i[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 $\tau=1\,\mathrm{\mu s}$ and $T=0.1\,\mathrm{\mu s}$, the largest output is only $(1-e^{-0.1})V_0\approx0.0952V_0$: a $0.5V_0$ threshold is never reached. If $T=3\,\mathrm{\mu s}$ instead, that threshold is crossed at $0.693\,\mathrm{\mu s}$. Neither value is the $0.5\,\mathrm{\mu s}$ <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.<Laboratory 11: Sinusoidal Steady State Response of RC and RL Circuits by MIT OpenCourseWare>[Printed pp. 1–2]{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.
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