3.8 Zero locations of linear-phase fir filters

Zero locations of linear-phase filters

The zeros of the transfer function $H(z)$ of a linear-phase filter lie in specific configurations.

We can write the symmetry condition $$h(n)=h(N-1-n)$$ in the $Z$ domain. Taking the $Z$ -transform of both sides gives

Using the symmetry condition, , it follows that $$H(z_0)=z^{-(N-1)}H(\frac{1}{z_0})=0$$ and $$H(\overline{z_0})=z^{-(N-1)}H(\frac{1}{\overline{z_0}})=0$$ or $$H(\frac{1}{z_0})=H(\frac{1}{\overline{z_0}})=0$$

Zeros locations

It follows that

• generic zeros of a linear-phase filter exist in sets of 4.
• zeros on the unit circle ( $z_0=e^{i{}_0}$ ) exist in sets of 2. ( $z_0\neq (1)$ )
• zeros on the real line ( $z_0=a$ ) exist in sets of 2. ( $z_0\neq (1)$ )
• zeros at 1 and -1 do not imply the existence of zeros at other specific points.

Zero locations: automatic zeros

The frequency response $H^f()$ of a Type II FIR filter always has a zero at $=\pi$ : $$h(n)$$

h 0 h 1 h 2 h 2 h 1 h 0

Zero locations: examples

The Matlab command zplane can be used to plot the zero locations of FIR filters.

Note that the zero locations satisfy the properties noted previously.