Embedded Software and Motor Control Libraries for PXR40xx

Function GDFLIB_FilterIIR2_F32
In order to implement the second order IIR filter on a microcontroller, the discrete time
domain representation of the filter, described by eq. GDFLIB_FilterIIR2_Eq2, must be
transformed into a time difference equation as follows:
Equation GDFLIB_FilterIIR2_Eq3
Equation GDFLIB_FilterIIR2_Eq3 represents a Direct Form I implementation of a
second order IIR filter. It is well known that Direct Form I (DF-I) and Direct Form II
(DF-II) implementations of an IIR filter are generally sensitive to parameter quantization
if a finite precision arithmetic is considered. This, however, can be neglected when the
filter transfer function is broken down into low order sections, i.e. first or second order.
The main difference between DF-I and DF-II implementations of an IIR filter is in the
number of delay buffers and in the number of guard bits required to handle the potential
overflow. The DF-II implementation requires less delay buffers than DF-I, hence less
data memory is utilized. On the other hand, since the poles come first in the DF-II
realization, the signal entering the state delay-line typically requires a larger dynamic
range than the output signal y(k). Therefore, overflow can occur at the delay-line input of
the DF-II implementation, unlike in the DF-I implementation.
Figure 4-4. Direct Form 1 second order IIR filter
The coefficients of the filter depicted in Figure 4-4 can be designed to meet the
requirements for the second order Band Pass (BPF) or Band Stop (BSF) filters. Filter
coefficients can be calculated using various tools, for example the Matlab butter function.
In order to avoid overflow during the calculation of the GDFLIB_FilterIIR2_F32
function, filter coefficients must be divided by eight. The coefficient quantization error
due to finite precision arithmetic can be neglected in the case of a second order filter.
Therefore, calculation of coefficients can be done using Matlab as follows:
Embedded Software and Motor Control Libraries for PXR40xx, Rev. 1.0
184 Freescale Semiconductor, Inc.