Putting the 'Group' into 'Group Delay'.
Updated 1-June-2024.
A Wireless World article from May 1995 claimed that almost any amplifier low frequency cut-off was a bad idea, resulting in a low frequency time delay, with the resulting bass being out of time with the rest of the music. A calculation shows my MJR7 amplifier phase advance at 40Hz to be the same effect produced by a time shift of 694 usec, but there are no real time delay elements involved, apart from maybe a nanosec for the signal to travel the length of the circuit board. Also a phase advance looks more like a time advance than a delay. So where did this 'time delay' idea come from. What we find in the article are plots of 'group delay', showing a positive group delay at low frequencies. That is correct, but there is then an assumption that a positive group delay is also a time delay. I covered this in a limited way in my Group Delay, Time Delay, and Phase Shift page, where I demonstrated that a real time delay is not the same thing as a group delay, but they can have an almost identical effect over a limited frequency range. So ok, let's just look at the range from 0 to 100Hz. There the MJR7 amplifier phase starts at over 90 deg and falls in a negative direction as frequency increases. We can compare this to a real time delay with similar group delay, in the next plot green is the MJR7 phase, and red is a 2.5msec time delay. I used a linear scale for frequency, so the constant time delay is a straight line:
The 2.5msec has been chosen to give the same gradient at 20Hz and therefore the same group delays at that frequency for both plots. The group delay at a given frequency is just the negative of the gradient of the plot of phase versus frequency.
The group delay for both plots may be the same at 20Hz, but there is otherwise little similarity, one is a phase advance starting at over 90 deg and falling towards zero, while the other is a phase lag starting at zero and heading for minus infinity. So does a 'group delay' at 20Hz tell us anything useful if it could be the consequence of either of these options?
To see why this is still useful, the clue is in the word 'group', which refers to a group of frequencies rather than single sinewaves. For a simple example we can use just two sinewaves, 20Hz and 21Hz, with equal 1V peak amplitudes. As time progresses these will at some point have their positive peaks at the same time giving total 2V, but later their peaks will be in opposite directions and cancel. What we want to look at is the signal 'envelope'. We don't need to install any software to do this, Google will plot mathematical equations for us just by putting them in the search bar. Then any graphics program with a screen capture function can copy the result and edit it if needed. I use MGI Photosuite, an ancient Win95 program.For waves starting at +1 at time zero we just need cosines, cos (2pi.f)t, and for frequency f=20Hz this is cos 125.66t. For f=21Hz it is cos 131.95t. Our composite signal is then just these added, and then cos (125.66*t) + cos (131.95*t) looks like this if you copy and paste it into Google, then select zoom mode horizontal and click - a few times. (Those symbols in the top left of the image).
Now, a positive group delay just means the higher frequency, 21Hz is shifted in phase either more negative or less positive compared to 20Hz. I intended to start with the MJR7 phase response, but that has too little effect for clarity, so something more extreme is called for. So, here first is a phase advance of 1 radian at 20Hz and reduced phase advance 0.5 radians at 21Hz. So both have a phase advance as for the MJR7, with higher advance at the lower frequency. So, phase advance, together with positive group delay. The group delay for all the following waves has magnitude 80msec.
Then a phase delay 0.5 radians at 20Hz increasing to 1 radian at 21Hz. So again a positive group delay.
Looking at these last two results, they are not perfectly identical, but both wave envelopes are clearly displaced a similar distance along the time axis, both appear to be delayed in time compared to the original without phase shifts. So, it makes little difference whether we have phase advances or delays, what matters is that with increased frequency we get either less phase advance or more phase delay, so the group delay in both cases is positive. So, our comparison of the MJR7 with a 2.5msec real time delay looked like two very different responses, but the effects on the wave envelope of 20Hz plus 21Hz are determined by the identical group delays, and so will be more or less identical, though in that case very small effects.
So yes, a phase advance reducing with increasing frequency can have a similar effect on the wave envelope to a time delay over a limited frequency range. Also, if we reverse the direction of phase change to get a negative group delay we find as follows, the wave envelope shifts to the left, to earlier time. So do we have time travel? A phase delay reducing as frequency increases leads to a wave envelope shifted forward in time, so if we think we can hear positive group delay as a time delay should we also expect to hear bass notes sent back from the future by negative group delays?
Remember, there are always everywhere just two individual sinewaves extending from the infinite past to the infinite future, and neither is shifted in time, apart from the equivalent time shift resulting from the 1 or 0.5 radian phase shifts. I have added a black dot above the point on the original unshifted waveform where the 20Hz and 21Hz are in phase and add to give amplitude 2. In the other three waveforms the black dots show where these unshifted sinewaves have been shifted to, but not both by exactly the same phase shift, so they no longer add up to 2, but that is the only extent to which the sinewaves are actually moved along the time axis. The times at which they add in phase or cancel when out of phase have moved much further, by the 80msec group delay, but the individual sinewaves have moved on average only about the distance indicated by the dots, with the same effect as a time delay for a negative phase shift, and time advance for a positive phase shift.
The wave envelope is what we see, and in this case what we hear, it sounds like a modulated signal with a 1 second repetition rate (To check that I used a higher pair of frequencies, still differing by 1Hz, my speakers don't do much at 20Hz).However, for a real musical note from something like a bass guitar then there is a starting transient with a wide range of frequency components, so some components likely to be outside the limited frequency range of our group delays, and to learn more about how negative group delay then works take a look at a reference I mentioned in another article:
Causality and negative group delays in a simple bandpass amplifier by Mitchell and Chiao. That includes a good coverage of how transients are affected by group delay. To summarise, sharp starting or ending transients are not generally delayed or advanced, though the rest of the wave may appear to be shifted and modified to some degree. With some high level of group delay there certainly could be some audible effect on musical bass notes, but not just a simple time delay.What we really need to know is the minimum audible group delay at a given frequency. I have seen a few different figures claimed, but none suggesting 2.5 msec at 20Hz is a problem (one paper said 2.5msec at 100Hz is the limit of audibility, so nothing to worry about at 20Hz). So, no good reason to increase the MJR7 input capacitor, I continue to stick with 2u2 in my own amplifiers. By all means change to 4u7 and listen for a difference, but unless your speakers have similarly low group delay errors it's not going to lead to a very reliable conclusion.
I searched for an example to show how a wave envelope may be delayed but the starting transient is not, and found Audibility of group delay at low frequencies, which is mostly about misaligned subwoofers and room acoustics, and also considers only fairly high group delays, 20ms and 100ms at 50Hz, but the resulting waveforms shown at the top of that page confirm my assertion that the starting transients are not delayed, so in effect the notes start at the same time even with 100ms group delay and obvious 'delay' effects. The amplifier input filter effects I was considering with typically 2.5ms group delay at 20Hz will be something like 0.5ms at 50Hz, so far less than that 20ms at 50Hz example which the article author says is audible, claiming a 7/8 score on ABX.
