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.
Best Answer
(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 \$:
\$ V_{out}=V_{max} \times sin(wt+\phi)=V_{max} \times sin[w(t+\$ \$\phi\over w\$ \$)] = V_{max} \times sin[w(t-t_p)] \$
Here, the expression \$t_p=-\$ \$ \phi \over w\$ is a delay time (phase delay) between input and output.
(2) For communication purposes of arbitrary 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|=t_p\times w\$ results in a LINEAR rising function between \$\phi\$ and \$w\$ (for \$t_p\$=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
\$t_g=-\$ \$d\phi\over dw\$ \$=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.