[SOLVED] Group delay confusion

[pavel_adameyko]

Hello, dear friends
We know that definition of group delay is τ=-Δφ/Δω (1). Suppose we have system with phase response φ=-αω +β. According to formula (1) GD=α is constant. But consider two sinusoids with frequencies ω1 and ω2. Time delay for them would be t1=-φ1/ω1 = α-β/ω1, for the second - α-β/ω2. They are not equal. So the output of such system wouldn't be a shifted replica of input - i.e. the signal would be distorted. But how does this correlate with constant GD ???
Thanks in advance.

 

[Seetha09]

Hi
I cant speak interms of the mathematical definition but my undestanding tells me that all signal components are delayed when passed through a device such an amplifier or a loudspeaker or as propagating though space or a medium, such as air. This delay can be different for different frequencies if the device is not LINEAR PHASE.A linear phase filter has constant group delay, all frequency components have equal delay times. That is, there is no distortion due to the time delay of frequencies relative to one another.
Thanks & Regards
Akella

 

[LvW]

Hi Pavel,
* at first, you shouldn't mix group delay and phase delay - but that's not the main point here.
* Your approach with a phase function that contains a fixed part β is not correct and is the cause of the misunderstanding. Thus, for w=0 there would be a fixed phase delay what is a contradiction to the assumption w=0 (which means dc).

 

[pavel_adameyko]

Hi, Lvw
as for your second remark - let's i change initial conditions - suppose phase function can be approximated by φ=-αω +β in some frequency range ω_min<ω<ω_max ( and near dc φ goes to zero as it should be). My calculations remain the same in this case - i get different time delay for sinusoids with ω_min<ω1,ω2<ω_max in spite of group delay is constant. So the output signal is distorted. Where is the mistake?

 

[LvW]

Pavel, as I have mentioned: Don't mix group delay with phase delay!
* Phase delay has a clear and general definition. For calculation you must use the exact function rather than to approximate only a small region of the whole function (that - perhaps - may be quasilinear).

* Group delay does not apply to a single frequency but to a group of frequencies that are relatively close together.
More correct and in detail: The definition of the group delay applies only for frequencies that fulfill the following requirements: The maximum difference of frequencies within the group must be small if compared with the medium value of all these frequencies. (Example: AM modulation)

 

[pavel_adameyko]

Thanks for your clarification)

 

[eagle5]

LvW, although you are right in everything you are saying, I think you are not answering Pavel question.

Pavel, you have a very reasonable doubt. The mistake in your reasoning is the following:
Constant GD doesn't mean that you receive a replica of the input.
Constant GD means that the envelope of the received signal is a shifted replica of the envelope of the input signal.