Embedded Software and Motor Control Libraries for PXR40xx

Function GDFLIB_FilterIIR2_FLT
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
where: x[k] is the input signal, y[k] is the output signal, a i and b i are the filter
coefficients. 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 lower 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 4-6. Direct Form 1 second order IIR filter
The coefficients of the filter depicted in Figure 4-6 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.
Note
To enumerate the computation error in one calculation cycle,
the internal accumulator is not used for testing purposes and is
Embedded Software and Motor Control Libraries for PXR40xx, Rev. 1.0
194 Freescale Semiconductor, Inc.