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 ω 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 ω of a Type II FIR filter always has a zero at ω : 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 ω seen earlier. Type automatic zeros I — II ω III ω 0 IV ω 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.