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A real-valued system that doesn't distort the shape of the input signal must have the following input-output relation: $$y(t)=Ax(t-t_0)\tag{1}$$ with arbitrary real-valued constants $A>0$ and $t_0$. In the frequency domain, Eq. $(1)$ corresponds to $$Y(\omega)=Ae^{-j\omega t_0}X(\omega)\tag{2}$$ 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 $$h[n]=h[N-1-n]\tag{1}$$ or $$h[n]=-h[N-1-n]\tag{2}$$ 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 $(2)$), ...