Biomedical Signal Processing - Computational Intelligence in Biomedical Engineering - page 60

either of the following factors to determine the location of the zeroes:

z −m )
(1 + z −m )

(1 −

It can be seen that m is equal to the number of zeroes placed at a particular
location, beginning with θ =0 ◦ and displaced evenly by 360 ◦ /m . The next
step is to ensure that the poles are positioned on the unit circle using com-
plex conjugate factors. This can be seen by noting the Euler relationship for
complex numbers, that is,

e jθ = cos θ + j sin θ (2.162)

and observing that the denominator of Equation 2.161 may be written as

( z − 1 e jθ )( z − 1 e −jθ )

− − (2.163)

This can be seen by using the fact that

cos θ = e jθ +e −jθ
2

and multiplying Equation 2.163 out to obtain

2 cos θz − 1 + z − 2

1 −

The possible positions of the poles are given by the values of θ when 2 cos θ
is an integer. This occurs only for θ =0 ◦ , 0 ◦ , 60 ◦ , 90 ◦ , 120 ◦ , and 180 ◦ ± ± ± ± ±

as seen in the plot on Figure 2.19.

90 °

120 °

60 °

180 ° 0 ° /360 °

300 °

240 °

270 °

FIGURE 2.19
Possible pole zero placements on the unit circle in the z -plane for integer filter
design.