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As pointed out in Peter K.'s answer, it is not necessary that a discrete-time system with linear phase has any symmetry. CAJ's argument goes in the right direction, but note that $e^{-jk\Omega}$ is only a simple discrete sample delay if $k \in \mathbb{Z}$. More importantly, $e^{-jk\Omega}$ doesn't always preserve the symmetry of the input signal. One way to ...

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