# Why are linear phase filters called so, if they provide Constant delay instead of linear

Linear phase filters delay all frequencies by the same amount. Why aren't they called Constant phase filters instead of Linear phase?

As I understand, if there is an input signal with two components $f_1$ and $f_2$, and it is passed through a linear phase system which introduces a delay $t$, $f_1$ is delayed by $t$, and $f_2$ is also delayed by $t$.

To what property does the phase have a linear relationship with? Could someone provide some insight into this? Some Math would be helpful.

Phase is not the same as time delay – at $f_1$, time delay $t$ will lead to a phase of $\varphi_1=-\omega_1\cdot t$, whereas at $f_2$, the phase will be $\varphi_2=-\omega_2\cdot t$ ($\omega$ is $2\pi f$); as you can immediately see, phase is a linear function of frequency.
(It's always good to remind oneself that frequency is just the derivative of phase over time, $\omega=\frac{d\varphi}{dt}$, and thus phase is just an integral of frequency over time.)
• It can be added that the group delay is minus the derivative of phase with respect to frequency : $\tau = -\frac{d\phi(f)}{df}$, so a linear phase response (i.e. $\phi(f) = \alpha f$) will result in a constant group delay, as said in the title of the question. – Albits Dec 5 '17 at 11:29
• Just a minor correction: the phase corresponding to a delay of $\tau$ is given by $\phi(\omega)=-\omega\tau=-2\pi f\tau$. So there should be a negative sign and a factor of $2\pi$. – Matt L. Dec 5 '17 at 13:12
• @MattL. true; I probably wanted to go all $\omega$ on that notation, but frankly, it was written left-handedly on my bike ride to the office… – Marcus Müller Dec 5 '17 at 19:43