Digital Filters
39
bandwidth
(a)
(b)
0
1
2
3
0
0.2
0.4
0.6
0.8
frequency [rad/sample]
magnitude
0
1
2
3
-2
-1
0
frequency [rad/sample]
phase [rad]
Figure 16: Frequency response (magnitude (a) and phase (b)) of a two-pole IIR
filter
-1
0
1
-1
-0.5
0
0.5
1
Figure 17: Couple of poles on the complex plane
is clear that, in the neighborhood of a pole close to the unit circumference, the
vector that comes from that pole is dominant over the others. This means that,
accepting some approximation, we can neglect the longer vectors and consider
only the shortest vector while evaluating the frequency response in that region.
This approximation is useful to calculate the bandwidth of the resonant fil-
ter, which is defined as the difference between the two frequencies corresponding
to a magnitude attenuation by 3dB, i.e., a ratio 1/
2. Under the simplifying
assumption that only the local pole is exerting some influence in the neighbor-
ing area, we can use the geometric construction of fig. 18 in order to find an
expression for the bandwidth [67]. The segment P
0
A is
2 times larger than
the segment P
0
P . Therefore, the triangle formed by the points P
0
AP has two,
orthogonal, equal edges and AB = 2P
0
P = 2(1 - r). If AB is small enough, its
length can be approximated with that of the arc subtended by it, which is the
bandwidth that we are looking for. Summarizing, for poles that are close to the
unit circumference, the bandwidth is given by
= 2(1 - r) .
(44)
The formula (44) can be used during a filter design stage in order to guide the
pole placement on the complex plane.