Implementing IIR/FIR Filters

2.10 Summary of Digital
Coefficients
Figure 2-22 gives a summary of the coefficient values
for the four basic filter types. Note that the
coefficient, β has the same form for all four filter
types and that it can only assume values between
0 and 1/2 for practical filters. β is bounded by 1/2
because Q (or d) and θ 0 are not independent. For
Q >> 1, β→1/2; whereas, for θ 0 = f s /4 and Q = 1/2,
β→0. These properties are independent of the form
of implementation; they are only dependent on the
form of the transfer function. Alternate implementations
(difference equations) will be described in the
following sections.
Note that the Q described in Figure 2-22 meets the
traditional requirements (i.e., Q is the ratio of the
bandwidth at the -3 dB points divided by the center
frequency). The formula for β can be modified in
the case of the bandpass or bandstop filter by replacing
the damping coefficient, d, with the formula
for Q. When the coefficients are described in this
manner, a constant Q filter results. When the
bandwidth is any function of center frequency, this
relationship between d and Q makes it impossible
to implement a bandpass or bandstop filter by replacing
Q with the desired function of bandwidth
and center frequency.
Figure 2-23 shows the relationship between the
pole of the second-order section and the center frequency.
Note that the pole is on the real axis for
d>2, where d is also constrained by d < 2/sin θ 0 .
MOTOROLA 2-23