4.11: Background- How Should We Define the Optimal Filter?

Before I explain our procedure for determining optimal filter settings, we need to discuss the question of what it means for a filter to be optimal for ERP research.

First, a filter needs to be able to reduce noise, as illustrated in Figure 4.7. Panel A shows a 30-second period of EEG from Subject 6 in the ERP Core N400 experiment without any filtering. You can see that it contains both high-frequency noise (very rapid up-and-down shifts that make the waveforms look “fuzzy”) and low-frequency noise (gradual drifts over time, such as the pink waveform from the P03 electrode site that falls completely out of view just before 24 seconds). Panel B shows the same data after application of a low-pass filter with a half-amplitude cutoff at 25 Hz and a 12 dB/octave roll-off. This filter reduces the high-frequency “fuzz”. Panel B shows what happens when we add a high-pass filter with a half-amplitude cutoff at 0.2 Hz and a 12 dB/octave roll-off. This filter eliminates the gradual drifts over time, keeping the waveforms largely parallel to the horizontal axis.

The point of Figure 4.7 is that part of the definition of an optimal filter is that it should do a good job of reducing the noise in the data.

The logic behind using these filters is that the EEG recordings contain a lot of noise in really low frequencies and in really high frequencies, whereas the ERP signals of interest are mainly in intermediate frequencies. Consequently, if we filter out the really low and really high frequencies, we will be getting rid of a lot of noise without having much impact on the signals of interest. However, if we don’t set our cutoff frequencies appropriately, some of the signal may be lost along with the noise.

This “loss of signal” problem is illustrated in Figure 4.8, which shows what happens when we use a low-pass cutoff that is too low and a high-pass cutoff that is too high. Panel A illustrates this by showing the impact of a 5 Hz low-pass filter on a simulated N170 component. The N170 is very narrow and sharp, meaning that it has a lot of energy above 5 Hz. This energy is eliminated by the filter, greatly reducing the amplitude of the N170. Panel B shows a simulated P3b component; it has much less high-frequency energy, so its amplitude isn’t reduced very much by the 5 Hz low-pass filter. This brings up an important point: Whether or not a filter reduces the size of the signal depends on the frequency content of the signal, which will be different for different ERP components.

Panels C and D show what happens when we instead use a high-pass filter. This filter has only a minimal impact on the N170, but it dramatically reduces the amplitude of the P3b. This is because the N170 has very little energy below 2 Hz, whereas the P3b has a lot. Once again, the impact of a filter depends on the waveshape (and therefore the frequency content) of the ERP component. Low-pass filters tend to reduce the amplitude of sharp, narrow components (which are typically the initial sensory responses), whereas high-pass filters tend to reduce the amplitude of slow, broad components (which are typically the later cognitive responses).

When we combine what we’ve learned from Figures 4.7 and 4.8, we see that we want a filter that will decrease the noise as much as possible without decreasing the signal by very much. We can make this more formal by considering the signal-to-noise ratio, which is simply the size of the signal divided by the size of the noise. A filter is typically useful only if it decreases the noise more than it decreases the signal, resulting in an increased signal-to-noise ratio. And the best filter is the one that gives us the best signal-to-noise ratio, subject to two caveats.

The first caveat is that filters may also distort the waveform of the ERP. As you saw in the preceding exercises and can see in Figure 4.8A, low-pass filters tend to cause the onset of a waveform to be shifted earlier and the offset to be shifted later. And as you have also learned, high-pass filters tend to produce artifactual peaks in the waveform such as those in Figures 4.9C and 4.9D. Thus, you will want to choose a filter that gives you the best signal-to-noise ratio without producing unacceptable levels of waveform distortion.

The second caveat is that we don’t care much about whether the filter reduces the amplitude of an ERP component when our goal is to assess the latency rather than the amplitude of the component. Of course, the latency scores might be affected if the amplitude gets so small that it’s hard to determine the latency of the component. But outside of that situation, we don’t really care about the amplitude reduction shown in Figure 4.8 when we’re scoring latencies instead of amplitudes.

We’re now ready for our formal definition of what constitutes the optimal filter for a given ERP study: the optimal filter is the one that maximizes the data quality without producing unacceptable levels of waveform distortion. We define data quality as the signal-to-noise ratio when we’re scoring amplitudes, and we define data quality simply in terms of the noise when we’re scoring latencies.

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