But your comment that minimum phase filters lead to minimal group delay leaves me very bewildered.

I see it as opposite....min phase filters (like xovers and system high-pass as I mentioned), are the major source of phase rotation/slope...and hence the source of group delay.

Minimum phase high-Q PEQs are of course another source of phase warp / group delay..

So in my mind, minimum phase is all about being the source of group delay.

Whereas linear phase is all about avoiding group delay.

That's the misunderstanding. (And I don't want to sound like an ass, but most people get this wrong when it comes to linear phase filters.)

The slope of the phase tells you the group delay. A linear phase filter has a linear (duh) phase response, which means the slope is constant vs frequency but that does NOT mean that it's zero. A linear phase filter can have a very high slope (very high group delay), but the phase response is linear which also means that the group delay is

*constant*. So the group delay is a "flat line" vs frequency, for example a constant 8k samples of delay for a 16k taps linear phase brickwall filter.

To achieve the same magnitude response, a minimum phase filter will have a much lower phase shift overall, which also minimizes the group delay.

But since the slope will change vs frequency, so will the group delay.

Here's an example 100 Hz lowpass filter:

The linear phase response is linear vs the linear frequency axis and the phase shift is magnitudes greater than that of the min phase filter, naturally.

The min phase filter's phase response is compressed too much in the comparison, so let's look at it in isolation (now with logarithmic frequency axis which also shows the magnitude response in a more readable way):

And this results in the minimal albeit non-constant group delay:

The min phase filter only delayed by 4ms over a wide range and 6ms tops. (Btw, with a different filter choice you can also create such a min phase lowpass with a flattened group delay, i.e. no peak.)

While for the linear phase filter it's a constant ~46 ms.

This can also be proven mathematically.

Btw, any "deviation" from min phase necessitates "unnatural" pre-ringing in the impulse response.

Another beautiful attribute of min phase is that you can

* perfectly und*o any deviation just by filtering with another min phase filter with inverted magnitude response. This does not work for non-min phase systems, because of the all-pass (extra phase shifting, delaying) components.