What is the relationship between group delay and propagation delay for a filter? I was trying to design a filter for a system and I was able to obtain the group delay plot in matlab using grpdelay. The group delay output is in samples. However, I was not able to relate that data to the propagation delay of the signal through the filter. Is there a formula between these two quantities?
Also, if I were to change the magnitude of the input signal in a step (keeping the frequency constant), how long would this change take to propagate to the output? Would that depend on the group delay or phase delay or would it depend on the filter order?
For example, if the filter cutoff is 5000Hz and I have unity gain at 500Hz and Group delay of 15 samples and Phase delay of pi radians, then if I send a 500Hz signal through the filter, after how many samples would I get it at the output? Assuming sampling frequency is 10000Hz.
The quantity you might be looking for is signal-front delay, which is the delay of the beginning of a signal passing through a linear system. It is simply the largest value $\tau_{sf}$ (in samples) for which
$$h[n]=0,\qquad n&lt;\tau_{sf}\tag{1}$$
is satisfied, where $h[n]$ is the system's impulse response.
If an input, or change in the input, starts at $n_0$, then the system's response to it begins at $n_0+\tau_{sf}$.
If the impulse response has a long period of pre-ringing, the signal-front delay is generally much smaller than the actual propagation delay. In that case, definition $(1)$ can be modified as
$$\big|h[n]\big|&lt;\delta,\qquad n&lt;\tau_{sf}\tag{2}$$
where $\delta$ is a positive constant which is small compared to the maximum value of $h[n]$.
Depending on the nature of the impulse response, another meaningful definition of signal delay introduced by a linear system could be the center of gravity of the impulse response:
$$\tau_{gr}=\frac{\displaystyle\sum_n nh[n]}{\displaystyle\sum_n h[n]}=\frac{jH'(0)}{H(0)}\tag{3}$$
where $H(\omega)$ is the system's frequency response, and $H'(\omega)$ is its derivative w.r.t. to $\omega$. Note that $\tau_{gr}$ is simply the group delay evaluated at $\omega=0$.
Using the center of gravity as an estimate for signal delay is only useful for systems with a lowpass characteristic. For other systems (such as highpass or bandpass filters), the denominator of $(3)$ could be zero or very close to zero. For that reason, for general systems one would exchange $h[n]$ in $(3)$ for either its magnitude or its square.
Note that unlike phase delay and group delay, neither signal-front delay (including its modified form $(2)$) nor the center of gravity are functions of frequency.