# An explanation of "phase" of a filter

I always read the word "phase" (like linear phase, phase shift...) in DSP but I'm still not sure what it supposes to mean, in intuition and also in practice.

The phase of a sinusoid $s(t)=A_0\cos(2\pi f_0 t + \phi_0)$ is $\phi_0$ radians.
If this sinusoid goes through an LTI system with frequency response $H(f)$, then the ouptut is $y(t) = |H(f_0)| A_0 \cos(2\pi f_0 t + \phi_0 + \angle H(f))$. So, the phase of the output is different than the phase of the input -- the system introduced a phase shift in the signal.
Note that you can interpret the phase shift as a time delay. If $T_0=1/f_0$, then a phase shift of $2\pi$ corresponds to a delay of $T_0$. This allows you to calculate that the delay $\Delta$ for a phase shift $\theta$ is $$\Delta = \frac{T_0 \theta}{2\pi}.$$ Note that the delay is a function of the sinusoid's frequency. An LTI system that produces the same time delay for all its inputs is said to have linear phase.