Electronics > Beginners
Difference between phase response and group delay
fonograph:
This is something I am trying to understand for over a year.I watched bunch of videos,googled everything,read quora,wikipedia and bunch of articles yet I still have zero idea what group delay is.Every time I read about it,it sounds and looks exactly like phase response,I dont know what to do anymore,help me please :'(
Phase response also called phase shift or phase delay,thats easy for me to understand.If there is signal where two sinewaves of different frequency and they both start and stop at same time,after they go through filter with phase shift,one will start and stop later,it will be delayed,its nothing more than delay with different delay times at different frequencies.
Now what the hell is group delay?! It also shows delay vs frequency plot,but thats exactly what phase response looks like,it also sounds to me like same thing,delay vs frequency and delay vs frequency,how on earth can these be different?! It looks and sounds exactly same! What is this group stuff anyhow,any signal in existance can be deconstructed to sineways of different phase and amplitude,so what is this group thing then?
Wikipedia says something about amplitude envelope,but there is no such fundamental thing as amplitude envelope,every amplitude envelope is just result of sinewave of different frequencies,amplitudes and phases,maybe I am dumb but I dont see how amplitude envelope can be any different than as predicted by the frequency and phase response of filter,phase and amplite vs frequency graph is all thats needed to fully charcterize filter,or is it not?
bson:
It's \$d\phi/d\omega\$. The phase \$\phi\$ is in rad and the frequency \$\omega\$ is in rad/s. rad/(rad/s) = s. More specifically, it's the slope of the phase line in a Bode plot; if it's a straight line, then the group delay is constant (the slope is the same, regardless of frequency) and the phase response is said to be linear. If it's not linear, then a signal consisting of purely a voltage, like logic, will have some of its frequencies skewed - typically higher frequency voltage changes will lag more, but anything is possible. For audio it means a wide-spectrum waveform like from a percussion instrument will be spread out in time, making it more diffuse. In general, group delay is not considered distortion since there are no spurious frequency products, and requires a VNA to measure. A common way to look for group delay is to feed a square wave through a filter or circuit, and see if it still looks like a square wave coming out, but there are many other reasons it might not and the only way to know is to examine the phase response. The phase response is the curve \$\phi(\omega)\$; group delay is the slope of the phase curve.
orolo:
Note that the definition of group delay is a derivative with respect to frequency. This means that group delay is defined as a local propierty respect to frequency, or if you will, a propierty that makes sense for waves with a narrow spectrum.
Set a fixed angular frequency \$\omega_0\$. The fact that, at that frequency, group delay is \$k\$, means that \$\left.\frac{\mathrm{d}\phi}{\mathrm{d}\omega}\right\vert_{\omega_0} \ = \ k\$. By the definition of derivative, the phase shift near the central frequency is: \$\phi(\omega_0 + \Delta\omega) \ = \ \phi(\omega_0) + k\cdot \Delta\omega\$. Of course, this is very accurate as long as you don't go too far away from the central frequency.
Imagine a wave with a very narrow spectrum, between \$\omega_0 - \epsilon\$ and \$\omega_0 + \epsilon\$, for epsilon a relatively small number. The Fourier transform of the wave is:
\$\displaystyle \psi(t) \ = \ \int_{\omega_0-\epsilon}^{\omega_0+\epsilon}\, a(\omega) e^{-it\omega}\, d\omega\$
Let us change the variables to \$\Delta\omega = \omega-\omega_0\$
\$\psi(t) \ = \ \displaystyle \int_{-\epsilon}^{\epsilon}\, a(\Delta\omega) e^{-it\omega_0-it\Delta\omega}\, d\Delta\omega \quad = \quad e^{-it\omega_0}\, \int_{-\epsilon}^{\epsilon}\, a(\Delta\omega)\, e^{-it\Delta\omega} d\Delta\omega\$
As you can see, the wave is a small modification of a sinusoidal wave. Now, if you pass the wave through the filter, if the group delay equals \$k\$, we are phase shifting the wave by \$\phi(\Delta\omega) = \phi(0) + k\Delta\omega\$ (see the definition of derivative above).
So the Fourier component of the filtered wave, let us call it \$\tilde{\psi}\$, will be:
\$\displaystyle \tilde{\psi}(t) \ = \ e^{-it\omega_0}\, \int_{-\epsilon}^{\epsilon}\, a(\Delta\omega)\,e^{i\phi(0) + ik\Delta\omega}\, e^{-it\Delta\omega} d\Delta\omega \ = \ e^{-i\omega_0(t - \phi(0)/\omega_0)}\, \int_{-\epsilon}^{\epsilon}\, a(\Delta\omega) e^{-i(t-k)\Delta\omega} d\Delta\omega \$
Note the last integral:
1. The fundamental, \$\omega_0\$ component of the wave has been shifted by the phase delay \$\phi(0)/\omega_0\$.
2. The narrow spectral components have been shifted by \$k\$, the group delay.
I think here you can see the difference between phase and group delay. The phase delay affects the carrier wave, while the group delay affects the modulation of that wave, so to speak. It is an effect in the neighbourhood of the central frequency.
Now on to review all the LaTeX errors. Editions to follow :P.
fonograph:
bson orolo I dont understand equations,can you explain it in intuitive simple way?
Is group depay and phase response really a same thing,its just one is measured in degrees and other in time units?
Frequency specific group delay will always cause frequency specific delay and phase shift
Frequency specific phase shift will always cause frequency specific delay and phase shift
group delay = delay and phase shift
phase shift = delay and phase shift
For example if you have group delay 1 second at 100hz,its a same exact thing as phase shift of 36000 degrees at 100hz.
Carrier wave and modulation of that carrier wave are all just sinewaves! There arent any magic sinewaves that are intrinsicaly carriers and others who are intrinsicaly modulators,its just sines! Phase shift at specific frequency doesnt care if its carrier,sideband or 35th harmonic of mongol throat singer recorded through microphone,every signal is just bunch of sines with different frequency,amplitude and phase.
I cannot possibly imagine how filter looks at bunch of sines going through it and decide which are carrier so it will apply phase shift and which ones are sidebands so it applies group delay to the modulation.
orolo:
I'm not good at metaphors, and I believe that in science they make very poor substitutes for mathematical fact, but I'll try.
You might argue that velocity is a confusing concept. If you know the position of a body at every moment, why would you need velocity? Well, think about what the concept of velocity gives you: how the position of the body changes very soon after, and before, the time you measured that velocity. Relative to the position of the body at the time of measurement. Velocity gives you local (in terms of time and position) information about the movement of the body.
Now change: position -> phase, time -> frequency, velocity -> group delay, time of measurement -> carrier frequency, small change of time -> modulation / narrow frequency.
You might argue that group delay is a confusing concept. If you know the phase delay of the signal at every frequency, why would you need group delay? Well, think about what the concept of group delay gives you: how the phase of the signal changes very near the frequency you measured that group delay. Relative to the phase of the signal at the frequency of measurement. Group delay gives you local (in terms of frequency and phase) information about the phase change of the signal.
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