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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 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 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 1 z 0 0 and H z 0 z N 1 H 1 z 0 0 or H 1 z 0 H 1 z 0 0

If z 0 is a zero of a (real-valued) linear-phase filter, then so are z 0 , 1 z 0 , and 1 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 0 ) exist in sets of 2. ( z 0 1 )
  • zeros on the real line ( z 0 a ) exist in sets of 2. ( z 0 1 )
  • zeros at 1 and -1 do not imply the existence of zeros at other specific points.

Examples of zero sets

Zero locations: automatic zeros

The frequency response H f of a Type II FIR filter always has a zero at : 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 H H -1 0
H f 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 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
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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Source:  OpenStax, Intro to digital signal processing. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10203/1.4
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