3.8 Zero locations of linear-phase fir filters

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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

$H(z)=z^{-(N-1)}H(\frac{1}{z})$
Recall that we are assuming that $h(n)$ is real-valued. If ${z}_{0}$ is a zero of $H(z)$ , $H({z}_{0})=0$ then $H(\overline{{z}_{0}})=0$ (Because the roots of a polynomial with real coefficients exist in complex-conjugate pairs.)

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 $H(\frac{1}{{z}_{0}})=H(\frac{1}{\overline{{z}_{0}}})=0$

If ${z}_{0}$ is a zero of a (real-valued) linear-phase filter, then so are $\overline{{z}_{0}}$ , $\frac{1}{{z}_{0}}$ , and $\frac{1}{\overline{{z}_{0}}}$ .

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
$H(z)={h}_{0}+{h}_{1}z^{-1}+{h}_{2}z^{-2}+{h}_{2}z^{-3}+{h}_{1}z^{-4}+{h}_{0}z^{-5}$ $H(-1)={h}_{0}-{h}_{1}+{h}_{2}-{h}_{2}+{h}_{1}-{h}_{0}=0$ ${H}^{f}(\pi )=H(e^{i\pi })=H(-1)=0$
${H}^{f}(\pi )=0$ always for Type II filters.
Similarly, we can derive the following rules for Type III and Type IV FIR filters.
${H}^{f}(0)={H}^{f}(\pi )=0$ always for Type III filters.
${H}^{f}(0)=0$ always for Type IV filters.
The automatic zeros can also be derived using the characteristics of the amplitude response $A()$ seen earlier.

Type automatic zeros
I
II $=\pi$
III $=0\lor \pi$
IV $=0$

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.

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