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I’m having difficulty understanding the concept of group delay. The mathematical definition is not difficult, since it says it is the negative derivative of Bode plot’s phase curve wrt frequency. Most often qualitative definitions are easy but the math is difficult. In this case seems the opposite or it is due to my ignorance.

I have read some similar questions but still did not get the reason for such a concept. Is it possible to illustrate this concept with an example in elementary level. I know the meaning of Fourrier transform, frequency domain representation, and basic filter theory. Also a bit of modulation. What would the derivative of a phase frequency plot tell us regarding a low pass filter for instance? Im completely lost on the meaning of it so I cant even pose the question well.

(1) Let us start with the PHASE DELAY: The response of a linear two-port to a sinusoidal excitation is an output signal with the same frequency w but with a phase delay phi:

Vout=vmax * sin(wt+phi)=vmax * sin[w(t+phi/w)] = vmax * sin[w(t-tp)]

Here, the expression tp=-phi/w is a delay time (phase delay) between input and output.

(2) For communication purposes of arbritrari waveforms we need the superposition of several sinusoidal waves with different frequencies. Of course we do not want that the various sinusoidal waves suffer from DIFFERENT delay figures.

Hence, we want a constant delay time tp for all these frequencies and we require that the equation |phi|=tp*w results in a LINEAR rising function between phi and w (for tp=const).

From system theory we know that such a requirement (linearity between phi and w) can be realized within a relatively small frequency band only. Hence, we define this requirement to be valid only within a frequency band that is realtively small if compared with the mean value of these frequencies:

We express this linearity requirement in form of the slope of the function and arrive at the so-called group delay

tg=-d(phi)/d(w)=const .

In practice, this requirement can be fulfilled with some errors only. Therefore, the constancy of the value for the group delay tg is a good measure for the quality of a communication channel (low distortion).

For example, a constant group delay is very important for a "good" pulse transmission.

Group delay is the delay, in seconds, to a signal.

Imagine a cable that's (say) 1uS long, electrically. If you put a step into one end, the step will come out 1uS later. If you plot the phase response of the cable, the phase at DC will be zero, and at 1MHz will be 2pi. The slope of the phase, d(phase)/d(frequency) is therefore 1uS.

The same goes for a filter. The delay to a signal through the filter is dp/df. As this measure will be frequency dependant, it only applies to signals which are bandlimited to lie within the range for which dp/df is fairly constant.

As a step in voltage is wideband, if this is passed through a filter with group delay that varies with frequency, the different frequencies of the step will be passed with different delays, and the result is dispersion, or smearing out of the step at the filter output.

If the group delay for signals between $f_1$ and $f_2$ is $tau_1,2$, then a signal that's band-limited to lie between $f_1$ and $f_2$ will be delayed by $tau_1,2$.

If the group delay across the entire frequency band is the same number, then any signal will be delayed by that -- and it'll just be "delay", because the term "group delay" applies technically, but you don't need to get that specific.

If you have one signal (e.g. a pulse) that encounters a filter with a group delay that varies across the spectrum of your signal, then your signal will get spread out, because various components will arrive at different times. This is why a pulse that's run through a simple lumped-component low-pass filter will be smeared out, but the same pulse run through a constant group-delay FIR filter will just be rounded.

(Because you asked about phase delay in the comments): https://en.wikipedia.org/wiki/Group_delay_and_phase_delay