Background information on filtering

Here we give some background information on filtering in general, and how it is done in MNE-Python in particular. Recommended reading for practical applications of digital filter design can be found in Parks & Burrus [1] and Ifeachor and Jervis [2], and for filtering in an M/EEG context we recommend reading Widmann et al. 2015 [7]_. To see how to use the default filters in MNE-Python on actual data, see the tut_artifacts_filter tutorial.

Problem statement

The practical issues with filtering electrophysiological data are covered well by Widmann et al. in [7]_, in a follow-up to an article where they conclude with this statement:

Filtering can result in considerable distortions of the time course
(and amplitude) of a signal as demonstrated by VanRullen (2011) [[3]_].
Thus, filtering should not be used lightly. However, if effects of
filtering are cautiously considered and filter artifacts are minimized,
a valid interpretation of the temporal dynamics of filtered
electrophysiological data is possible and signals missed otherwise
can be detected with filtering.

In other words, filtering can increase SNR, but if it is not used carefully, it can distort data. Here we hope to cover some filtering basics so users can better understand filtering tradeoffs, and why MNE-Python has chosen particular defaults.

Filtering basics

Let's get some of the basic math down. In the frequency domain, digital filters have a transfer function that is given by:

\begin{align}H(z) &= \frac{b_0 + b_1 z^{-1} + b_2 z^{-2} + ... + b_M z^{-M}} {1 + a_1 z^{-1} + a_2 z^{-2} + ... + a_N z^{-M}} \\ &= \frac{\sum_0^Mb_kz^{-k}}{\sum_1^Na_kz^{-k}}\end{align}

In the time domain, the numerator coefficients $b_k$ and denominator coefficients $a_k$ can be used to obtain our output data $y(n)$ in terms of our input data $x(n)$ as:

\begin{align}:label: summations y(n) &= b_0 x(n) + b_1 x(n-1) + ... + b_M x(n-M) - a_1 y(n-1) - a_2 y(n - 2) - ... - a_N y(n - N)\\ &= \sum_0^M b_k x(n-k) - \sum_1^N a_k y(n-k)\end{align}

In other words, the output at time $n$ is determined by a sum over:

1. The numerator coefficients $b_k$, which get multiplied by
   the previous input $x(n-k)$ values, and
2. The denominator coefficients $a_k$, which get multiplied by
   the previous output $y(n-k)$ values.

Note that these summations in :eq:summations correspond nicely to (1) a weighted moving average and (2) an autoregression.

Filters are broken into two classes: FIR (finite impulse response) and IIR (infinite impulse response) based on these coefficients. FIR filters use a finite number of numerator coefficients $b_k$ ($\forall k, a_k=0$), and thus each output value of $y(n)$ depends only on the $M$ previous input values. IIR filters depend on the previous input and output values, and thus can have effectively infinite impulse responses.

As outlined in [1]_, FIR and IIR have different tradeoffs:

* A causal FIR filter can be linear-phase -- i.e., the same time delay
  across all frequencies -- whereas a causal IIR filter cannot. The phase
  and group delay characteristics are also usually better for FIR filters.
* IIR filters can generally have a steeper cutoff than an FIR filter of
  equivalent order.
* IIR filters are generally less numerically stable, in part due to
  accumulating error (due to its recursive calculations).

In MNE-Python we default to using FIR filtering. As noted in Widmann et al. 2015 [7]_:

Despite IIR filters often being considered as computationally more
efficient, they are recommended only when high throughput and sharp
cutoffs are required (Ifeachor and Jervis, 2002 [2]_, p. 321),
...FIR filters are easier to control, are always stable, have a
well-defined passband, can be corrected to zero-phase without
additional computations, and can be converted to minimum-phase.
We therefore recommend FIR filters for most purposes in
electrophysiological data analysis.

When designing a filter (FIR or IIR), there are always tradeoffs that need to be considered, including but not limited to:

1. Ripple in the pass-band
2. Attenuation of the stop-band
3. Steepness of roll-off
4. Filter order (i.e., length for FIR filters)
5. Time-domain ringing

In general, the sharper something is in frequency, the broader it is in time, and vice-versa. This is a fundamental time-frequency tradeoff, and it will show up below.

FIR Filters

First we will focus first on FIR filters, which are the default filters used by MNE-Python.

Designing FIR filters

Here we'll try designing a low-pass filter, and look at trade-offs in terms of time- and frequency-domain filter characteristics. Later, in tut_effect_on_signals, we'll look at how such filters can affect signals when they are used.

First let's import some useful tools for filtering, and set some default values for our data that are reasonable for M/EEG data.

Take for example an ideal low-pass filter, which would give a value of 1 in the pass-band (up to frequency $f_p$) and a value of 0 in the stop-band (down to frequency $f_s$) such that $f_p=f_s=40$ Hz here (shown to a lower limit of -60 dB for simplicity):

This filter hypothetically achieves zero ripple in the frequency domain, perfect attenuation, and perfect steepness. However, due to the discontunity in the frequency response, the filter would require infinite ringing in the time domain (i.e., infinite order) to be realized. Another way to think of this is that a rectangular window in frequency is actually sinc_ function in time, which requires an infinite number of samples, and thus infinite time, to represent. So although this filter has ideal frequency suppression, it has poor time-domain characteristics.

Let's try to naïvely make a brick-wall filter of length 0.1 sec, and look at the filter itself in the time domain and the frequency domain:

This is not so good! Making the filter 10 times longer (1 sec) gets us a bit better stop-band suppression, but still has a lot of ringing in the time domain. Note the x-axis is an order of magnitude longer here, and the filter has a correspondingly much longer group delay (again equal to half the filter length, or 0.5 seconds):

Let's make the stop-band tighter still with a longer filter (10 sec), with a resulting larger x-axis:

Now we have very sharp frequency suppression, but our filter rings for the entire second. So this naïve method is probably not a good way to build our low-pass filter.

Fortunately, there are multiple established methods to design FIR filters based on desired response characteristics. These include:

1. The Remez_ algorithm (:func:`scipy.signal.remez`, `MATLAB firpm`_)
2. Windowed FIR design (:func:`scipy.signal.firwin2`, `MATLAB fir2`_)
3. Least squares designs (:func:`scipy.signal.firls`, `MATLAB firls`_)
4. Frequency-domain design (construct filter in Fourier
   domain and use an :func:`IFFT <scipy.fftpack.ifft>` to invert it)

If we relax our frequency-domain filter requirements a little bit, we can use these functions to construct a lowpass filter that instead has a transition band, or a region between the pass frequency $f_p$ and stop frequency $f_s$, e.g.:

Accepting a shallower roll-off of the filter in the frequency domain makes our time-domain response potentially much better. We end up with a smoother slope through the transition region, but a much cleaner time domain signal. Here again for the 1 sec filter:

Since our lowpass is around 40 Hz with a 10 Hz transition, we can actually use a shorter filter (5 cycles at 10 Hz = 0.5 sec) and still get okay stop-band attenuation:

But then if we shorten the filter too much (2 cycles of 10 Hz = 0.2 sec), our effective stop frequency gets pushed out past 60 Hz:

If we want a filter that is only 0.1 seconds long, we should probably use something more like a 25 Hz transition band (0.2 sec = 5 cycles @ 25 Hz):

So far we have only discussed acausal filtering, which means that each sample at each time point $t$ is filtered using samples that come after ($t + \Delta t$) and before ($t - \Delta t$) $t$. In this sense, each sample is influenced by samples that come both before and after it. This is useful in many cases, espcially because it does not delay the timing of events.

However, sometimes it can be beneficial to use causal filtering, whereby each sample $t$ is filtered only using time points that came after it.

Note that the delay is variable (whereas for linear/zero-phase filters it is constant) but small in the pass-band. Unlike zero-phase filters, which require time-shifting backward the output of a linear-phase filtering stage (and thus becoming acausal), minimum-phase filters do not require any compensation to achieve small delays in the passband. Note that as an artifact of the minimum phase filter construction step, the filter does not end up being as steep as the linear/zero-phase version.

We can construct a minimum-phase filter from our existing linear-phase filter with the minimum_phase function (that will be in SciPy 0.19's :mod:scipy.signal), and note that the falloff is not as steep:

Applying FIR filters

Now lets look at some practical effects of these filters by applying them to some data.

Let's construct a Gaussian-windowed sinusoid (i.e., Morlet imaginary part) plus noise (random + line). Note that the original, clean signal contains frequency content in both the pass band and transition bands of our low-pass filter.

Filter it with a shallow cutoff, linear-phase FIR (which allows us to compensate for the constant filter delay):

This is actually set to become the default type of filter used in MNE-Python in 0.14 (see tut_filtering_in_python).

Let's also filter with the MNE-Python 0.13 default, which is a long-duration, steep cutoff FIR that gets applied twice:

Let's also filter it with the MNE-C default, which is a long-duration steep-slope FIR filter designed using frequency-domain techniques:

And now an example of a minimum-phase filter:

Both the MNE-Python 0.13 and MNE-C filhters have excellent frequency attenuation, but it comes at a cost of potential ringing (long-lasting ripples) in the time domain. Ringing can occur with steep filters, especially on signals with frequency content around the transition band. Our Morlet wavelet signal has power in our transition band, and the time-domain ringing is thus more pronounced for the steep-slope, long-duration filter than the shorter, shallower-slope filter:

IIR filters

MNE-Python also offers IIR filtering functionality that is based on the methods from :mod:scipy.signal. Specifically, we use the general-purpose functions :func:scipy.signal.iirfilter and :func:scipy.signal.iirdesign, which provide unified interfaces to IIR filter design.

Designing IIR filters

Let's continue with our design of a 40 Hz low-pass filter, and look at some trade-offs of different IIR filters.

Often the default IIR filter is a Butterworth filter_, which is designed to have a maximally flat pass-band. Let's look at a few orders of filter, i.e., a few different number of coefficients used and therefore steepness of the filter:

The falloff of this filter is not very steep.

Let's increase the order, and note that now we have better attenuation, with a longer impulse response:

There are other types of IIR filters that we can use. For a complete list, check out the documentation for :func:scipy.signal.iirdesign. Let's try a Chebychev (type I) filter, which trades off ripple in the pass-band to get better attenuation in the stop-band:

And if we can live with even more ripple, we can get it slightly steeper, but the impulse response begins to ring substantially longer (note the different x-axis scale):

Applying IIR filters

Now let's look at how our shallow and steep Butterworth IIR filters perform on our Morlet signal from before:

Some pitfalls of filtering

Multiple recent papers have noted potential risks of drawing errant inferences due to misapplication of filters.

Low-pass problems

Filters in general, especially those that are acausal (zero-phase), can make activity appear to occur earlier or later than it truly did. As mentioned in VanRullen 2011 [3], investigations of commonly (at the time) used low-pass filters created artifacts when they were applied to smulated data. However, such deleterious effects were minimal in many real-world examples in Rousselet 2012 [5].

Perhaps more revealing, it was noted in Widmann & Schröger 2012 [6] that the problematic low-pass filters from VanRullen 2011 [3]:

1. Used a least-squares design (like :func:scipy.signal.firls) that included "do-not-care" transition regions, which can lead to uncontrolled behavior.
2. Had a filter length that was independent of the transition bandwidth, which can cause excessive ringing and signal distortion.

High-pass problems

When it comes to high-pass filtering, using corner frequencies above 0.1 Hz were found in Acunzo et al. 2012 [4]_ to:

"...generate a systematic bias easily leading to misinterpretations of neural activity.”

In a related paper, Widmann et al. 2015 [7] also came to suggest a 0.1 Hz highpass. And more evidence followed in Tanner et al. 2015 [8] of such distortions. Using data from language ERP studies of semantic and syntactic processing (i.e., N400 and P600), using a high-pass above 0.3 Hz caused significant effects to be introduced implausibly early when compared to the unfiltered data. From this, the authors suggested the optimal high-pass value for language processing to be 0.1 Hz.

We can recreate a problematic simulation from Tanner et al. 2015 [8]_:

"The simulated component is a single-cycle cosine wave with an amplitude of 5µV, onset of 500 ms poststimulus, and duration of 800 ms. The simulated component was embedded in 20 s of zero values to avoid filtering edge effects... Distortions [were] caused by 2 Hz low-pass and high-pass filters... No visible distortion to the original waveform [occurred] with 30 Hz low-pass and 0.01 Hz high-pass filters... Filter frequencies correspond to the half-amplitude (-6 dB) cutoff (12 dB/octave roll-off)."

Similarly, in a P300 paradigm reported by Kappenman & Luck 2010 [12]_, they found that applying a 1 Hz high-pass decreased the probaility of finding a significant difference in the N100 response, likely because the P300 response was smeared (and inverted) in time by the high-pass filter such that it tended to cancel out the increased N100. However, they nonetheless note that some high-passing can still be useful to deal with drifts in the data.

Even though these papers generally advise a 0.1 HZ or lower frequency for a high-pass, it is important to keep in mind (as most authors note) that filtering choices should depend on the frequency content of both the signal(s) of interest and the noise to be suppressed. For example, in some of the MNE-Python examples involving ch_sample_data, high-pass values of around 1 Hz are used when looking at auditory or visual N100 responses, because we analyze standard (not deviant) trials and thus expect that contamination by later or slower components will be limited.

Baseline problems (or solutions?)

In an evolving discussion, Tanner et al. 2015 [8] suggest using baseline correction to remove slow drifts in data. However, Maess et al. 2016 [9] suggest that baseline correction, which is a form of high-passing, does not offer substantial advantages over standard high-pass filtering. Tanner et al. [10]_ rebutted that baseline correction can correct for problems with filtering.

To see what they mean, consider again our old simulated signal x from before:

In respose, Maess et al. 2016 [11]_ note that these simulations do not address cases of pre-stimulus activity that is shared across conditions, as applying baseline correction will effectively copy the topology outside the baseline period. We can see this if we give our signal x with some consistent pre-stimulus activity, which makes everything look bad.

Putting some activity in the baseline period:

Both groups seem to acknowledge that the choices of filtering cutoffs, and perhaps even the application of baseline correction, depend on the characteristics of the data being investigated, especially when it comes to:

1. The frequency content of the underlying evoked activity relative to the filtering parameters.
2. The validity of the assumption of no consistent evoked activity in the baseline period.

We thus recommend carefully applying baseline correction and/or high-pass values based on the characteristics of the data to be analyzed.

Filtering defaults

Defaults in MNE-Python

Most often, filtering in MNE-Python is done at the :class:mne.io.Raw level, and thus :func:mne.io.Raw.filter is used. This function under the hood (among other things) calls :func:mne.filter.filter_data to actually filter the data, which by default applies a zero-phase FIR filter designed using :func:scipy.signal.firwin2. In Widmann et al. 2015 [7]_, they suggest a specific set of parameters to use for high-pass filtering, including:

"... providing a transition bandwidth of 25% of the lower passband
edge but, where possible, not lower than 2 Hz and otherwise the
distance from the passband edge to the critical frequency.”

In practice, this means that for each high-pass value l_freq or low-pass value h_freq below, you would get this corresponding l_trans_bandwidth or h_trans_bandwidth, respectively, if the sample rate were 100 Hz (i.e., Nyquist frequency of 50 Hz):

+------------------+-------------------+-------------------+ | l_freq or h_freq | l_trans_bandwidth | h_trans_bandwidth | +==================+===================+===================+ | 0.01 | 0.01 | 2.0 | +------------------+-------------------+-------------------+ | 0.1 | 0.1 | 2.0 | +------------------+-------------------+-------------------+ | 1.0 | 1.0 | 2.0 | +------------------+-------------------+-------------------+ | 2.0 | 2.0 | 2.0 | +------------------+-------------------+-------------------+ | 4.0 | 2.0 | 2.0 | +------------------+-------------------+-------------------+ | 8.0 | 2.0 | 2.0 | +------------------+-------------------+-------------------+ | 10.0 | 2.5 | 2.5 | +------------------+-------------------+-------------------+ | 20.0 | 5.0 | 5.0 | +------------------+-------------------+-------------------+ | 40.0 | 10.0 | 10.0 | +------------------+-------------------+-------------------+ | 45.0 | 11.25 | 5.0 | +------------------+-------------------+-------------------+ | 48.0 | 12.0 | 2.0 | +------------------+-------------------+-------------------+

MNE-Python has adopted this definition for its high-pass (and low-pass) transition bandwidth choices when using l_trans_bandwidth='auto' and h_trans_bandwidth='auto'.

To choose the filter length automatically with filter_length='auto', the reciprocal of the shortest transition bandwidth is used to ensure decent attenuation at the stop frequency. Specifically, the reciprocal (in samples) is multiplied by 6.2, 6.6, or 11.0 for the Hann, Hamming, or Blackman windows, respectively as selected by the fir_window argument.

In 0.14, we default to using a Hamming window in filter design, as it provides up to 53 dB of stop-band attenuation with small pass-band ripple.

For more information on how to use the MNE-Python filtering functions with real data, consult the preprocessing tutorial on tut_artifacts_filter.

Defaults in MNE-C

MNE-C by default uses:

1. 5 Hz transition band for low-pass filters.
2. 3-sample transition band for high-pass filters.
3. Filter length of 8197 samples.

The filter is designed in the frequency domain, creating a linear-phase filter such that the delay is compensated for as is done with the MNE-Python phase='zero' filtering option.

Squared-cosine ramps are used in the transition regions. Because these are used in place of more gradual (e.g., linear) transitions, a given transition width will result in more temporal ringing but also more rapid attenuation than the same transition width in windowed FIR designs.

The default filter length will generally have excellent attenuation but long ringing for the sample rates typically encountered in M-EEG data (e.g. 500-2000 Hz).

Defaults in other software

A good but possibly outdated comparison of filtering in various software packages is available in [7]_. Briefly:

• EEGLAB MNE-Python in 0.14 defaults to behavior very similar to that of EEGLAB, see the EEGLAB filtering FAQ_ for more information.
• Fieldrip By default FieldTrip applies a forward-backward Butterworth IIR filter of order 4 (band-pass and band-stop filters) or 2 (for low-pass and high-pass filters). Similar filters can be achieved in MNE-Python when filtering with :meth:raw.filter(..., method='iir') <mne.io.Raw.filter> (see also :func:mne.filter.construct_iir_filter for options). For more inforamtion, see e.g. FieldTrip band-pass documentation_.

Summary

When filtering, there are always tradeoffs that should be considered. One important tradeoff is between time-domain characteristics (like ringing) and frequency-domain attenuation characteristics (like effective transition bandwidth). Filters with sharp frequency cutoffs can produce outputs that ring for a long time when they operate on signals with frequency content in the transition band. In general, therefore, the wider a transition band that can be tolerated, the better behaved the filter will be in the time domain.

References

.. [1] Parks TW, Burrus CS (1987). Digital Filter Design. New York: Wiley-Interscience. .. [2] Ifeachor, E. C., & Jervis, B. W. (2002). Digital Signal Processing: A Practical Approach. Prentice Hall. .. [3] Vanrullen, R. (2011). Four common conceptual fallacies in mapping the time course of recognition. Perception Science, 2, 365. .. [4] Acunzo, D. J., MacKenzie, G., & van Rossum, M. C. W. (2012). Systematic biases in early ERP and ERF components as a result of high-pass filtering. Journal of Neuroscience Methods, 209(1), 212–218. http://doi.org/10.1016/j.jneumeth.2012.06.011 .. [5] Rousselet, G. A. (2012). Does filtering preclude us from studying ERP time-courses? Frontiers in Psychology, 3(131) .. [6] Widmann, A., & Schröger, E. (2012). Filter effects and filter artifacts in the analysis of electrophysiological data. Perception Science, 233. .. [7] Widmann, A., Schröger, E., & Maess, B. (2015). Digital filter design for electrophysiological data – a practical approach. Journal of Neuroscience Methods, 250, 34–46. .. [8] Tanner, D., Morgan-Short, K., & Luck, S. J. (2015). How inappropriate high-pass filters can produce artifactual effects and incorrect conclusions in ERP studies of language and cognition. Psychophysiology, 52(8), 997–1009. http://doi.org/10.1111/psyp.12437 .. [9] Maess, B., Schröger, E., & Widmann, A. (2016). High-pass filters and baseline correction in M/EEG analysis. Commentary on: “How inappropriate high-pass filters can produce artefacts and incorrect conclusions in ERP studies of language and cognition.” Journal of Neuroscience Methods, 266, 164–165. .. [10] Tanner, D., Norton, J. J. S., Morgan-Short, K., & Luck, S. J. (2016). On high-pass filter artifacts (they’re real) and baseline correction (it’s a good idea) in ERP/ERMF analysis. .. [11] Maess, B., Schröger, E., & Widmann, A. (2016). High-pass filters and baseline correction in M/EEG analysis-continued discussion. Journal of Neuroscience Methods, 266, 171–172. Journal of Neuroscience Methods, 266, 166–170. .. [12] Kappenman E. & Luck, S. (2010). The effects of impedance on data quality and statistical significance in ERP recordings. Psychophysiology, 47, 888-904.