# Physical significance of positive group delay with negative phase delay

This question is an addendum to: Is group delay the same as the delay of a certain frequency?

Lately, I've been going back to basics to better understand the concepts of group and phase delay, so I did some simulations with the following simple LPF and HPF:

simulate this circuit – Schematic created using CircuitLab

For the LPF the transfer function is:

$$H(jw)=\frac{\tau}{j\omega + \tau}$$

and for the HPF:

$$H(jw)=\frac{j\omega}{j\omega+\tau}$$

where $\tau=\frac{1}{RC}$

In case of the LPF its group and phase delays are:

$$\tau_g(\omega)=-\frac{d\phi}{d\omega}=\frac{\tau}{\omega^2+\tau^2}$$

and

$$\tau_\phi=-\frac{\phi}{\omega}=\frac{\text{arctan}\left(\frac{\omega}{\tau}\right)}{\omega}$$

To get a better feel, I simulated in Multisim with the resistor and capacitor values of 1K and 1uF respectively, I exported the phase angle VS frequency values to an Excel spreadsheet to numerically check the results, I entered the formula for the phase delay and then I differentiated numerically to get the group delay, these are the graphs:

Both graphs seem consistent with the analytical formulas.

Now for the HPF the group and phase delay are:

$$\tau_g(\omega)=-\frac{d\phi}{d\omega}=\frac{\tau}{\omega^2+\tau^2}$$

and

$$\tau_\phi=-\frac{\phi}{\omega}=\frac{\text{arctan}\left(\frac{\omega}{\tau}\right)-\frac{\pi}{2}}{\omega}$$

As you can see, the group delay for the HPF is the same as the group delay of the LPF, however the phase delay is different. Here are the simulated graphs

So here is the question: What does it mean (in the time domain) to have an increasing negative phase delay while having a decreasing positive group delay?

The way I see it, is that the envelope of the output waveform is still lagging but the phase seems to be leading the input waveform?