About Filters – Speed vs Accuracy

If you look at the filter graphs in your training software, compare an IIR filter, like an Elliptic, and an FIR filter.  You’ll see that the IIR filter tends to focus on producing a stable response across the whole frequency band of the filter, but the time response changes throughout the filter band.

FIR Magnitude & Delay

Look at an FIR filter, and you’ll see that the delay graph is a straight line at a specific number of milliseconds and it is the same all the way across the band being filtered. On the other hand, the frequency response tends to be quite center-loaded: very accurate in the middle of the band but falling off fairly sharply toward the edges.

In other words, with an Elliptic IIR filter, amplitude activity will be reported for a 4-8 Hz filter almost 100% from every frequency from 4 to 8 Hz, and activity outside that band will be reported at a much lower percent or zero.  But at 4Hz, let’s say, there might be a delay of 130 milliseconds; at 4.5 Hz a delay of 150 mS; at 5 Hz a delay of 180 mS, etc.  With the FIR filter the delay would be exactly the same at 4 Hz, 4.5, 5, etc.  But only at 6 Hz, the middle of the band, would the amount of amplitude be 100% accurately reported.  And by the time you got to 4 Hz or 8 Hz, the edges of the band, only 60% of it might be reported

The Butterworth filter is a center-loaded filter which loses accuracy at the edges. If you look at the Magnitude graph to the right of your Filters Properties box, you’ll see a graph of the filter’s response curve. On, for example, a 12-16 Hz filter, the 3rd order Butterworth will have a nice smooth top running directly on the 1.0 line, but it won’t go all the way from 12 to 16 Hz (the numbers along the bottom of the graph.) In fact it starts to “roll off” at around 13Hz and 15Hz. The higher the filter order, the more complete coverage the filter will have, but there is a trade-off: Check the Delay tab and you’ll see that as you increase the number in the Order box, the delay added by the filter goes up pretty sharply.

The Chebyshev and Elliptic are almost exactly the same. The FIR provides some different characteristics, as does the Butterworth.  The more you get into signal processing and using it in training, the more you recognize that each has its benefits.  I would only use FIR for coherence, phase or synchrony, and I’ve learned that many clients prefer the “softer” Butterworth for rewards, while the Elliptic works well for inhibits.  Also with some bandwidths, a Butterworth will provide a faster response than an Elliptic.  For example, those who want to try to train down in the 0-0.10 Hz band more recently popularized by the Othmers, a Butterworth is really the only option.  Its delay, even as a first-order filter, is over a second, but it does have a nice little peak right in the center of the band, and the Elliptic just can’t do it at any speed.

The Chebyshev and Elliptic filters tend to be better at covering the whole band, though they have a ripple along the top, and they are often faster than the Butterworths. If you recall from the BioExplorer workshop, we went through different options of filter type and order and looked at the curve and delay in each. It’s a painful experience of trade-off. But here’s the real clinker: clients sometimes experience the sharper filters as being a bit “harsh”, while the more center-loaded filters are experienced as being “gentler”.

The FIR is not a frequency-based filter as much as a time-based. Look at the difference in the graphs, and you’ll see the delay is exactly the same all the way across the frequency band on an FIR, where it varies with the IIR (Infinite Impulse Response) filters. They aren’t used much in training in anything I’ve seen to date.

I usually use Elliptic filters for inhibits (e.g. 2-9.5 Hz), as high an order as I can get without passing 200 mS delay, and Butterworth filters (e.g. 10-14 Hz) with the order set to keep the delay about the same as that for the inhibit filters.  With the Butterworths, because they are so center-loaded, I often change the frequency band to get the center/peak around 10 Hz or a little above.

The FIR filter should always be used for coherence or synchrony training.  The timing issues make this critical.  If you look at the Delay tab on the graphs to the right of the Bandpass Filter properties window, you’ll see, as you switch through the filter types, that delay is variable for different frequencies for Butterworth, Chebyshev and Elliptic filters.  The FIR filter shows a straight line across the graph, registering exactly the same delay for each frequency.  Since coherence is based on phase, and phase is based on timing relationships, trying to train coherence (or synchrony, which is coherence in phase) with a filter that introduces timing differences at different frequencies is a problem.  I don’t use them much for amplitude training, because I find that the IIR’s are faster and more accurate.

Training to increase a frequency is usually done in brain-trainer Designs using Butterworth filters, while those in the assessment generally use Elliptic.  The Butterworth “rolls off” within the training band.  If you look at the graph of the filter in the Properties window of the Filter object, you’ll notice that a filter picking up 12-15 Hz doesn’t stay at 100% (1.0) all the way from 12-15.  The edges of the filter “roll off” before the defined edges.  The elliptic filter rolls off OUTSIDE the band defined or, as in the assessment, we use a high enough filter order that the edges drop straight down, so they pick up 100% of the defined band and 0% of the rest.

Assessment Filters vs Training Filters

The filters are different between the assessment and training files, because in the assessment there is no feedback, so delay is not an issue.  We use the most accurate filters to give the clearest possible view of the frequency bands.

There is no way I can see to set frequency bands in BioExplorer below 0.0 to 0.1 (which means a central frequency of 0.05 or one pulse every 20 seconds!)  The software won’t accept a filter that filters 0.0 to 0.01 (central frequency of 0.005 or 1/200th second).

At 0-0.1 Hz as a reward band (which BE does just fine), you have some problems caused by the physics of digital signal processing:

1. An elliptic filter, 1st order, which only produces a delay of 150 mS, isn’t very accurate.  In fact, it passes 200% of the signal around the area they want to measure but 100% of all activity up as high as nearly 2 Hz.  Not exactly training 0-0.1 Hz.  Raising the filter order to 2 provides a little more specific and accurate filter–but the time delay goes up around 1.5 seconds and above.

2. The Butterworth filter the Othmers usually use, in its fastest form (1st order) does provide a fairly accurate reading of the band they want to train, but the delay ranges from about 1/2 second to about 2.25 seconds.  In other words, in order to actually SEE the frequency they say they are training, they have to set up the filters so they literally give the feedback between one and a half and two and a half seconds AFTER the even happens in the brain.  That’s the equivalent of trying to train a puppy by giving him a treat about 10 minutes after he does something you tell him to do.

I leave it to you to imagine what kind of accuracy and delay are provided by asking a filter to measure a band that is 2/100 of a Hz (the above are 50/100 Hz wide).