 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 $t\ge 0$ 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.