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So I would like to know if filters usually used for pulse shaping (like SRRC) have a linear phase response, and if not, what this response looks like and does it matters in any circumstances?

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FIR filters that have coefficients symmetric about their center coefficient(s) are linear phase. Digital pulse filters commonly in use, including a Root-Raised-Cosine filter, are FIR and have this property, so they will have will have a linear phase response.

For all digital modulations, I would avoid non-linear phase response filters.

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