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This may help: the result is the cascade of an FIR filter with coefficients $h(n)$ with the reverse conjugate filter with coefficients $h^*(N-n)$. If we can limit $H(z)$ to be the unit circle (for computing the frequency response), then $$H(z^*)|_{z=e^{j\omega}} = H(z^{-1})|_{z=e^{j\omega}}$$ since for $z = e^{j\omega}$, $z^* = e^{-j\omega}$ And therefore ...