Simple IIR Filter Design

From electronics.stackexchange.com

As others have mentioned, you should consider a IIR (infinite impulse response) filter rather than the FIR (finite impulse response) filter you are using now. There is more to it, but at first glance FIR filters are implemented as explicit convolutions and IIR filters with equations.

The particular IIR filter I use a lot in microcontrollers is a single pole low pass filter. This is the digital equivalent of a simple R-C analog filter. For most applications, these will have better characteristics than the box filter that you are using. Most uses of a box filter that I have encountered are a result of someone not paying attention in digital signal processing class, not as a result of needing their particular characteristics. If you just want to attenuate high frequencies that you know are noise, a single pole low pass filter is better. The best way to implement one digitally in a microcontroller is usually:

FILT <– FILT + FF ( NEW - FILT )

FILT is a piece of persistant state. This is the only persistant variable you need to compute this filter. NEW is the new value that the filter is being updated with this iteration. FF is the filter fraction, which adjusts the “heaviness” of the filter. Look at this algorithm and see that for FF = 0 the filter is infinitely heavy since the output never changes. For FF = 1, it’s really no filter at all since the output just follows the input. Useful values are in between. On small systems you pick FF to be 1/2^N so that the multiply by FF can be accomplished as a right shift by N bits. For example, FF might be 1/16 and the multiply by FF therefore a right shift of 4 bits. Otherwise this filter needs only one subtract and one add, although the numbers usually need to be wider than the input value.

I usually take A/D readings significantly faster than they are needed and apply two of these filters cascaded. This is the digital equivalent of two R-C filters in series, and attenuates by 12 dB/octave above the rolloff frequency. However, for A/D readings it’s usually more relevant to look at the filter in the time domain by considering its step response. This tells you how fast your system will see a change when the thing you are measuring changes.

To facilitate designing these filters (which only means picking FF and deciding how many of them to cascade), I use my program FILTBITS. You specify the number of shift bits for each FF in the cascaded series of filters and it computes the step response and other values. Actually I usually run this via my wrapper script PLOTFILT. This run FILTBITS, which makes a CSV file, then plots the CSV file. For example, here is the result of “PLOTFILT 4 4”:

The two parameters to PLOTFILT mean there will be two filters cascaded of the type described above. The values of 4 indicate the number of shift bits to realize the multiply by FF. The two FF values are therefore 1/16 in this case. The red trace is the unit impulse response, and is the main thing to look at. For example, this tells you that if the input changes instantaneously, the output of the combined filter will settle to 90% of the new value in 60 iterations. If you care about 95% settling time then you have to wait about 73 iterations, and for 50% settling time only 26 iterations. The green trace shows you the output from a single full amplitude spike. This gives you some idea of the random noise suppression. It looks like no single sample will cause more than a 2.5% change in the output. The blue trace is to give a subjective feeling of what this filter does with white noise. This is not a rigorous test since there is no guarantee what exactly the content was of the random numbers picked as the white noise input for this run of PLOTFILT. It’s only to give you a rough feeling of how much it will be squashed and how smooth it is.

PLOTFILT, maybe FILTBITS, and lots of other useful stuff, especially for PIC firmware development is available in the PIC Development Tools software release at my Software downloads page.

The only thing I’d add, is that moving average filters do have their place when performed synchronously to some task (like producing a drive waveform to drive an ultrasound generator) so that they filter out harmonics of 1/T where T is the moving average time.

(self-extract archives coped to the wiki for posterity. plotfilt/filtbits are in the install_picdev.exe file, and the csvplot is in the install_wav.exe file)

plotfilt_output.gif (11.1 KB) Andrew Kohlsmith, 04/20/2012 09:09 AM

install_picdev.exe (3.83 MB) Andrew Kohlsmith, 04/20/2012 09:10 AM

install_wav.exe (970 KB) Andrew Kohlsmith, 04/20/2012 09:10 AM

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