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If $$z_{0}$$ is a zero of a (real-valued) linear-phase FIR filter then following is/are also zero/zeros of a (real-valued) linear-phase FIR filter,
$$z_{0}^{*}$$
$$\frac{1}{z_{0}}$$
$$\frac{1}{z_{0}}, z_{0}^{*}$$ and $$\frac{1}{ z_{0}^{*}}$$
$$\frac{1}{z_{0}}$$ and $$\frac{1}{z_{0}^{*}} $$
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