Embedded Software and Motor Control Libraries for PXR40xx

Function GDFLIB_FilterIIR1_F16
4.10.4 Description
This function calculates the first order infinite impulse (IIR) filter. The IIR filters are also
called recursive filters because both the input and the previously calculated output values
are used for calculation of the filter equation in each step. This form of feedback enables
transfer of the energy from the output to the input, which theoretically leads to an
infinitely long impulse response (IIR).
A general form of the IIR filter expressed as a transfer function in the Z-domain is
described as follows:
Equation GDFLIB_FilterIIR1_Eq1
where N denotes the filter order. The first order IIR filter in the Z-domain is therefore
given from equation GDFLIB_FilterIIR1_Eq1 as:
Equation GDFLIB_FilterIIR1_Eq2
In order to implement the first order IIR filter on a microcontroller, the discrete time
domain representation of the filter, described by equation GDFLIB_FilterIIR1_Eq2, must
be transformed into a time difference equation as follows:
Equation GDFLIB_FilterIIR1_Eq3
Equation GDFLIB_FilterIIR1_Eq3 represents a Direct Form I implementation of a first
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
Embedded Software and Motor Control Libraries for PXR40xx, Rev. 1.0
174 Freescale Semiconductor, Inc.