A real-valued system that doesn't distort the shape of the input signal must have the following input-output relation:
with arbitrary real-valued constants $A>0$ and $t_0$.
In the frequency domain, Eq. $(1)$ corresponds to
Consequently, the corresponding system is an LTI system ...
Real-valued and causal FIR filters have a (generalized) linear phase response if and only if their coefficients satisfy either
where $N$ is the filter length (number of taps).
There are four types of linear phase FIR filters, depending on the type of symmetry (even as in $(1)$ or odd as in $(2)$), ...