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Zero Locations of Linear-Phase FIR Filters

Module by: Ivan Selesnick. E-mail the author

Summary: (Blank Abstract)

Zero Locations of Linear-Phase Filters

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

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

H⁢z=z−(N−1)⁢H⁢1z H z z N 1 H 1 z
(1)
Recall that we are assuming that h⁢n h n is real-valued. If z 0 z 0 is a zero of H⁢z H z , H⁢ z 0 =0 H z 0 0 then H⁢ z 0 ¯=0 H z 0 0 (Because the roots of a polynomial with real coefficients exist in complex-conjugate pairs.)

Using the symmetry condition, Equation 1, it follows that H⁢ z 0 =z−(N−1)⁢H⁢1 z 0 =0 H z 0 z N 1 H 1 z 0 0 and H⁢ z 0 ¯=z−(N−1)⁢H⁢1 z 0 ¯=0 H z 0 z N 1 H 1 z 0 0 or H⁢1 z 0 =H⁢1 z 0 ¯=0 H 1 z 0 H 1 z 0 0

note:

If z 0 z 0 is a zero of a (real-valued) linear-phase filter, then so are z 0 ¯ z 0 , 1 z 0 1 z 0 , and 1 z 0 ¯ 1 z 0 .

ZEROS LOCATIONS

It follows that

  1. generic zeros of a linear-phase filter exist in sets of 4.
  2. zeros on the unit circle ( z 0 =ei⁢ ω 0 z 0 ω 0 ) exist in sets of 2. ( z 0 ≠±⁢1 z 0 ± 1 )
  3. zeros on the real line ( z 0 =a z 0 a ) exist in sets of 2. ( z 0 ≠±⁢1 z 0 ± 1 )
  4. zeros at 1 and -1 do not imply the existence of zeros at other specific points.

Figure 1: Examples of zero sets
(a)
Figure 1(a) (zerosets.png)
(b)
Figure 1(b) (zerosets_a.png)

ZERO LOCATIONS: AUTOMATIC ZEROS

The frequency response H f ⁢ω H f ω of a Type II FIR filter always has a zero at ω=π ω : h⁢n= h 0 h 1 h 2 h 2 h 1 h 0 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 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 -1 h 0 h 1 h 2 h 2 h 1 h 0 0 H f ⁢π=H⁢ei⁢π=H⁢-1=0 H f H H -1 0

rule:

H f ⁢π=0 H f 0 always for Type II filters.
Similarly, we can derive the following rules for Type III and Type IV FIR filters.

rule:

H f ⁢0= H f ⁢π=0 H f 0 H f 0 always for Type III filters.

rule:

H f ⁢0=0 H f 0 0 always for Type IV filters.
The automatic zeros can also be derived using the characteristics of the amplitude response A⁢ω A ω seen earlier.

Table 1
Type automatic zeros
I —
II ω=π ω
III ω=0∨π ω 0
IV ω=0 ω 0

ZERO LOCATIONS: EXAMPLES

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

Figure 2
Figure 2 (fourzeros.png)

Note that the zero locations satisfy the properties noted previously.

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