Filters: When, Why, and How (Not) to Use Them

Filters are commonly used to reduce noise and improve data quality. Filter theory is part of a scientist’s training, yet the impact of filters on interpreting data is not always fully appreciated. This paper reviews the issue and explains what a filter is, what problems are to be expected when using them, how to choose the right filter, and how to avoid filtering by using alternative tools. Time-frequency analysis shares some of the same problems that filters have, particularly in the case of wavelet transforms. We recommend reporting filter characteristics with sufficient details, including a plot of the impulse or step response as an inset. Filters are commonly used to reduce noise and improve data quality. Filter theory is part of a scientist’s training, yet the impact of filters on interpreting data is not always fully appreciated. This paper reviews the issue and explains what a filter is, what problems are to be expected when using them, how to choose the right filter, and how to avoid filtering by using alternative tools. Time-frequency analysis shares some of the same problems that filters have, particularly in the case of wavelet transforms. We recommend reporting filter characteristics with sufficient details, including a plot of the impulse or step response as an inset. One of the major challenges of brain science is that measurements are contaminated by noise and artifacts. These may include environmental noise, instrumental noise, or signal sources within the body that are not of interest in the context of the experiment (“physiological noise”). The presence of noise can mask the target signal, or interfere with its analysis. However, if signal and interference occupy different spectral regions, it may be possible to improve the signal-to-noise ratio (SNR) by applying a filter to the data. For example, a direct current (DC) component or slow fluctuation may be removed with a high-pass filter, power line components may be attenuated by a notch filter at 50 or 60 Hz, and unwanted high-frequency components may be removed by “smoothing” the data with a low-pass filter. Filtering takes advantage of the difference between spectra of noise and target to improve SNR, attenuating the data more in the spectral regions dominated by noise, and less in those dominated by the target. Filters are found at many stages along the measurement-to-publication pipeline (Figure 1). The measuring rig or amplifier may include a high-pass filter and possibly a notch filter, the analog-to-digital (AD) converter is preceded by a low-pass antialiasing filter, preprocessing may rely on some combination of high-pass, low-pass, and notch filters, data analysis may include band-pass or time-frequency (TF) analysis, and so on. Filters are ubiquitous in brain data measurement and analysis. The improvement in SNR offered by the filter is welcome, but filtering affects also the target signal in ways that are sometimes surprising. Obviously, any components of the target signal that fall within the stop band of the filter are lost. For example, applying a 50 Hz notch filter to remove power line artifact might also remove brain activity within the 50 Hz region. The experimenter who blindly relies on the filtered signal is blind to features suppressed by the filter. Harder to appreciate are the distortions undergone by the target. Such distortions depend on the frequency characteristics of the filter, including both amplitude and phase characteristics (which are often not reported). The output of a filter is obtained by convolution of its input with the impulse response of the filter, which is a fancy way of saying that each sample of the output is a weighted sum of several samples of the input. Each sample therefore depends on a whole segment of the input, spread over time. Temporal features of the input are smeared in the output, and conversely “features” may appear in the output that were not present in the input to the filter. We first explain what a filter is in detail, and how filters are involved in data analysis. Then, we review the main issues that can arise and make suggestions on how to fix them. Importantly, similar issues occur also in TF analyses, such as spectrograms and wavelet transforms, which are based on a collection of filters (a filterbank). Finally, we list a number of recommendations that may help investigators identify and minimize issues related to the use of filters, and we suggest ways to report them so that readers can make the best use of the information that they read. In this paper, “filter” refers to the familiar one-dimensional convolutional filter (e.g., high-pass or band-pass) applicable to a single-channel waveform, as opposed to “spatial filters” applicable to multichannel data. For many of us, a filter is “a thing that modifies the spectral content of a signal.” For the purposes of this paper, however, we need something more precise. A filter is an operation that produces each sample of the output waveform y as a weighted sum of several samples of the input waveform x. For a digital filter:y(t)=∑n=0Nh(n)x(t−n)(Equation 1) where t is the analysis point in time, and h(n),n=0,…,N is the impulse response. This operation is called convolution. We expect readers to fall into one of three categories: (1) those who understand and feel comfortable with this definition, (2) those who mentally transpose it to the frequency domain where they feel more comfortable, and (3) those who remain mind-boggled. Categories 2 and 3 both need assistance, and that is what this section is about. Category 2 needs assistance because a frequency-domain account is incomplete unless phase is taken into account, but doing so is mentally hard and often not so illuminating. It is often easier to reason in the time domain. For the mind-boggled, the one important idea to retain is that every sample of the output depends on multiple samples of the input, as illustrated in Figure 2 (top). Conversely, each sample x(t) of the input impacts several samples y(t+n) of the output (Figure 2, bottom). As a result, the signal that is being filtered is smeared along the temporal axis, and temporal relations between filtered and original waveforms are blurred. For example, the latency between a sensory stimulus and a brain response, a straightforward notion, becomes less well defined when that brain response is filtered. The exact way in which the output of a filter differs from its input depends upon the filter, i.e., the values h(n) of the impulse response. Some filters may smooth the input waveform; others may enhance fast variations. There is a considerable body of theory, methods, and lore on how best to design and implement a filter for the needs of an application. Expert readers will add that a filter is a linear system, that h(n) is not expected to change over time (linear time-invariant system), that, in addition to causal filters described by Equation 1, there are acausal filters for which the series h(n) includes also negative indices (gray lines in Figure 2), and that N may be finite (finite impulse response, FIR) or infinite (infinite impulse response, IIR). IIR filters are often derived from standard analog filter designs (e.g., Butterworth or elliptic). Essentially everything we discuss below is true for these more general notions of filtering. Expert readers will also recognize that Equation 1 can be substituted by the simpler equation Y(ω)=H(ω)X(ω) involving the Fourier transforms of x(t), y(t), and h(n), that neatly describes the effects of filtering in the frequency domain as a product of two complex functions, the transfer function of the filter H(ω), and the Fourier transform of the input, X(ω). The magnitude transfer function |H(ω)| quantifies the amount of attenuation at each frequency ω. A special mention should be made of acausal filters. These are filters for which each sample of the output depends also on future samples of the input, i.e., we must modify Equation 1 to include negative indices n=−N′,…,−1. All physical systems must be causal (the future cannot influence the past) so this filter cannot represent a physical system, nor could it be implemented in a real-time processing device. However, for offline data analysis we can take samples from anywhere in the dataset, so in that context acausal filters are realizable. In particular, it is common to use zero-phase filters, for which the impulse response is symmetrical relative to zero. The MATLAB function filtfilt applies the same filter to the data twice, forward and backward, effectively implementing a zero-phase filter. While acausal filters are easy to apply, interpreting their output requires special care. An important goal of neuroscience is to determine causal relations, for example, between a stimulus and brain activity, or between one brain event and another, and we must take care that these relations are not confused by an acausal stage in the data analysis. For an IIR filter, the output depends on all samples from the start of the data, previous samples being treated as 0. If the IIR filter is acausal it can also depend on all samples until the end of the data, samples beyond the end being treated as 0. This is also the case when filters are implemented in the Fourier domain: each output sample y(t) depends potentially on all input samples x(t) that are used to compute the Fourier transform, i.e., every sample within the analysis window. Figure 3 illustrates four common types of filter: low-pass, high-pass, band-pass, and notch (or band-reject). The upper plots show the magnitude transfer function (on a log-log scale), and the bottom plots show the impulse response of each filter. For high-pass and notch filters, the impulse response includes a one-sample impulse (“Dirac”) of amplitude much greater than the rest (plotted here using a split ordinate). For each filter two versions are shown, one with with shallow (blue) and the other with steep (red) frequency transitions. Note that a filter with a steep transition in the frequency domain tends to have an impulse response that is extended in the time domain. Also important to note is that different impulse responses can yield the same magnitude transfer function. Figure 4 (left) shows four impulse responses that all share the same magnitude frequency characteristic (low-pass, similar to that shown in Figure 3) but differ in their phase characteristics (plotted on the right). Magnitude and phase together fully specify a filter (as does the impulse response). Among all the filters that yield the same magnitude frequency response, one is remarkable in that it is causal and has minimum phase over all frequency (thick blue). Another is remarkable in that it has zero phase over all frequency (thick green). It is acausal. Ubiquitous, if rarely noticed, is the hardware “antialiasing” filter that precedes analog-to-digital conversion within the measuring apparatus. Data processing nowadays is almost invariably done in the digital domain, and this requires signals to be sampled at discrete points in time so as to be converted to a digital representation. Only values at the sampling points are retained by the sampling process, and thus the digital representation is ambiguous: the same set of numbers might conceivably reflect a different raw signal. The ambiguity vanishes if the raw signal obeys certain conditions, the best known of which is given by the sampling theorem: if the original signal’s spectrum contains no power beyond the Nyquist frequency (one half the sampling rate) then it can be perfectly reconstructed from the samples. The antialiasing filter aims to enforce this condition (“Nyquist condition”). A hardware antialiasing filter is usually applied before sampling, and a software antialiasing filter may later be applied if the sampled data are further downsampled or resampled. Phenomena of interest often obey slow dynamics. In that case, high-frequency variance can safely be attributed to irrelevant noise fluctuations and attenuated by low-pass filtering. Smoothing is also often used to make data plots visually more palatable, or to give more emphasis on longer-term trends than on fine details. Some recording modalities such as electroencephalography (EEG) or magnetoencephalography (MEG) are susceptible to DC shifts and slow drift potentials or fields, upon which ride the faster signals of interest (Huigen et al., 2002Huigen E. Peper A. Grimbergen C.A. Investigation into the origin of the noise of surface electrodes.Med. Biol. Eng. Comput. 2002; 40: 332-338Crossref PubMed Scopus (267) Google Scholar, Kappenman and Luck, 2010Kappenman E.S. Luck S.J. The effects of electrode impedance on data quality and statistical significance in ERP recordings.Psychophysiology. 2010; 47: 888-904PubMed Google Scholar, Vanhatalo et al., 2005Vanhatalo S. Voipio J. Kaila K. Full-band EEG (FbEEG): an emerging standard in electroencephalography.Clin. Neurophysiol. 2005; 116: 1-8Abstract Full Text Full Text PDF PubMed Scopus (127) Google Scholar). Likewise, in extracellular recordings, spikes of single neurons ride on slower events, such as negative deflections of the local field potential (LFP) that often precede spikes, or the larger and slower drifts due to the development of junction potentials between the electrode tip and the brain tissue. High-pass filters are the standard tool to remove such slow components prior to data analysis. A hardware high-pass filter might also be included in the measurement apparatus to remove DC components prior to conversion so as to make best use of the limited range of the digital representation. This is the meaning of “AC coupling” on an oscilloscope: it consists of the application of a high-pass filter—often implemented as a mere capacitor—to the signal. Amplifiers for recording extracellular brain activity are usually AC coupled. Electrophysiological signals are often plagued with power line noise (50 or 60 Hz and harmonics) coupled electrically or magnetically with the recording circuits. While such noise is best eliminated at its source by careful equipment design and shielding, this is not always successful, nor is it applicable to data already gathered. Notch filtering is often used to mitigate such power line noise. Additional notches may be placed at harmonics if needed. It has become traditional to interpret brain activity as coming from frequency bands with names such as alpha, beta, theta, etc., and data analysis often involves applying one or more band-pass filters to isolate particular bands, although the consensus is incomplete as to the boundary frequencies or the type of filter to apply. One prominent application of filtering is TF analysis. A TF representation can be viewed as the time-varying magnitude of the data at the outputs of a filterbank. A filterbank is an array of filters that differ over a range of parameter values (e.g., center frequencies and/or bandwidths). The indices of the filters constitute the frequency axis, while the time series of their output magnitude unfolds along the time axis of the TF representation. The time-varying magnitude is obtained by applying a non-linear transform to the filter output, such as half-wave rectification or squaring, possibly followed by a power or logarithmic transform. The time-varying phase in each channel may also be represented. The answer to this question depends on the data and on the filter. In this section, we review a number of archetypical “events” that might occur within a time series of brain activity, and look at how they are affected by commonly used filters. Brain events that are temporally localized, for example, a neuronal “spike” can be modeled as one or a few impulses. It is obvious from Equation 1 that such events must be less localized once filtered, as summarized schematically in Figure 5. The response is spread over time, implying that the temporal location of the event is less well defined. It is delayed if the filter is causal. The delay may be avoided by choosing a zero-phase filter (green), but the response is then acausal. If the impulse response has multiple modes, these may appear misleadingly as multiple spurious events, confusing the analysis. The nature and extent of these effects depends on the filter and can be judged by looking at its impulse response. Figure 6 shows impulse responses of a selection of commonly used filters (others were shown in Figure 3). The left-hand plot shows the time course of the impulse response, and the right-hand plot displays the logarithm of its absolute value using a color scale, to better reveal the low-amplitude tail. The first three examples (A–C) correspond to low-pass filters with the same nominal cutoff (10 Hz). The next two (D and E) are low-pass filters with nominal cutoff 20 Hz. The following three (F–H) are band-pass filters. Two of the filters are zero-phase (B and E, in green), and the others are causal (blue). The response of the first two filters is relatively short and unimodal; that of the others is more extended and includes excursions of both signs. The temporal span is greater for filters of high order (compare F and G) and for lower frequency parameters (compare C and D). Band-pass filters have relatively extended impulse responses, particularly if the band is narrow or the slopes of the transfer function steep. The oscillatory response of a filter to an impulse-like input is informally called “ringing” and may occur in all filter types (low-pass, band-pass, high-pass, and so on). Of course, real brain events differ from an infinitely narrow unipolar impulse, for example, they have finite width, and the response to such events will thus differ somewhat from the ideal impulse response. As a rule of thumb, features of the impulse response that are wider than the event are recognizable in the response of the filter to the event. Features that are narrower (for example, the one-sample impulse at the beginning of the impulse response of the high-pass and notch filters in Figure 3) may appear smoothed. Certain brain events can be modeled as a step function, for example, the steady-state pedestal that may follow the onset of a stimulus (Picton et al., 1978Picton T.W. Woods D.L. Proulx G.B. Human auditory sustained potentials. I. The nature of the response.Electroencephalogr. Clin. Neurophysiol. 1978; 45: 186-197Abstract Full Text PDF PubMed Scopus (168) Google Scholar, Lammertmann and Lütkenhöner, 2001Lammertmann C. Lütkenhöner B. Near-DC magnetic fields following a periodic presentation of long-duration tonebursts.Clin. Neurophysiol. 2001; 112: 499-513Abstract Full Text Full Text PDF PubMed Scopus (21) Google Scholar, Southwell et al., 2017Southwell R. Baumann A. Gal C. Barascud N. Friston K. Chait M. Is predictability salient? A study of attentional capture by auditory patterns.Philos. Trans. R. Soc. Lond. B Biol. Sci. 2017; (Published online February 19, 2017)https://doi.org/10.1098/rstb.2016.0105Crossref PubMed Scopus (66) Google Scholar). Figure 7 illustrates the various ways a step can be affected by filtering: the step may be smoothed and spread over time, implying that its temporal location is less well defined, and it may be delayed if the filter is causal. Multiple spurious events may appear, some of which may occur before the event if the filter is acausal. The nature of these effects depends on the filter and can be inferred from its step response (integral over time of the impulse response). Step responses of typical filters are shown in Figure 8. The sharp transition within the waveform is smoothed by a low-pass filter (A and B) and delayed relative to the event if the filter is causal (A), or else it starts before the event if the filter is acausal (B). The steady-state pedestal is lost for a high-pass (C–E) or band-pass (F–H) filter. The response may include spurious excursions, some of which precede the event if the filter is acausal. The response may be markedly oscillatory (ringing) (F–H), and it may extend over a remarkably long duration if the filter has a narrow transfer function. Of course, actual step-like brain events differ from an ideal step. As a rule of thumb, features of the step response that are wider than the event onset will be recognizable in the output, whereas features that are narrower will appear smoothed. Note that a response of opposite polarity would be triggered by the offset of a pedestal. Some activity within the brain is clearly oscillatory (Buzsáki, 2006Buzsáki G. Rhythms Of The Brain. Oxford University Press, 2006Crossref Scopus (3343) Google Scholar, Lopes da Silva, 2013Lopes da Silva F. EEG and MEG: relevance to neuroscience.Neuron. 2013; 80: 1112-1128Abstract Full Text Full Text PDF PubMed Scopus (447) Google Scholar). A burst of oscillatory activity can be modeled as a sinusoidal pulse. As Figure 9 shows, the time course of such a pulse is affected by filtering: it is always smoothed and spread over time, it may be delayed if the filter is causal, or else start earlier than the event if the filter is acausal. These effects are all the more pronounced as the filter has a narrow passband (as one might want to use to increase the SNR of such oscillatory activity). For a notch filter tuned to reject the pulse frequency, ringing artifacts occur at both onset and offset. If the filter is acausal, these artifacts may both precede and follow onset and offset events. For a notch filter tuned to reject power line components (50 or 60 Hz), such effects might also be triggered by fluctuations in amplitude or phase. They might also conceivably affect the shape of a short narrow-band gamma brain response in that frequency region (Fries et al., 2008Fries P. Scheeringa R. Oostenveld R. Finding gamma.Neuron. 2008; 58: 303-305Abstract Full Text Full Text PDF PubMed Scopus (97) Google Scholar, Saleem et al., 2017Saleem A.B. Lien A.D. Krumin M. Haider B. Rosón M.R. Ayaz A. Reinhold K. Busse L. Carandini M. Harris K.D. Carandini M. Subcortical Source and Modulation of the Narrowband Gamma Oscillation in Mouse Visual Cortex.Neuron. 2017; 93: 315-322Abstract Full Text Full Text PDF PubMed Scopus (81) Google Scholar). The use of filters raises many concerns, some serious, others merely inconvenient. It is important to understand them, and to report enough details that the reader too fully understands them. An obvious concern is loss of useful information suppressed together with the noise. Slightly less obvious is the distortion of the temporal features of the target: peaks or transitions may be smoothed, steps may turn into pulses, and artifactual features may appear. Most insidious, however, is the blurring of temporal or causal relations between features within the signal, or between the signal and external events such as stimuli. This section reviews a gallery of situations in which filtering may give rise to annoying or surprising results. This is an obvious gripe: information in frequency ranges rejected by the filter is lost. High-pass filtering may mask slow fluctuations of brain potential, whether spontaneous or stimulus evoked (Picton et al., 1978Picton T.W. Woods D.L. Proulx G.B. Human auditory sustained potentials. I. The nature of the response.Electroencephalogr. Clin. Neurophysiol. 1978; 45: 186-197Abstract Full Text PDF PubMed Scopus (168) Google Scholar, Lammertmann and Lütkenhöner, 2001Lammertmann C. Lütkenhöner B. Near-DC magnetic fields following a periodic presentation of long-duration tonebursts.Clin. Neurophysiol. 2001; 112: 499-513Abstract Full Text Full Text PDF PubMed Scopus (21) Google Scholar, Vanhatalo et al., 2005Vanhatalo S. Voipio J. Kaila K. Full-band EEG (FbEEG): an emerging standard in electroencephalography.Clin. Neurophysiol. 2005; 116: 1-8Abstract Full Text Full Text PDF PubMed Scopus (127) Google Scholar, Southwell et al., 2017Southwell R. Baumann A. Gal C. Barascud N. Friston K. Chait M. Is predictability salient? A study of attentional capture by auditory patterns.Philos. Trans. R. Soc. Lond. B Biol. Sci. 2017; (Published online February 19, 2017)https://doi.org/10.1098/rstb.2016.0105Crossref PubMed Scopus (66) Google Scholar). Low-pass filtering may mask high-frequency activity (e.g., gamma or high-gamma bands) or useful information about the shape of certain responses (Cole and Voytek, 2017Cole S.R. Voytek B. Brain Oscillations and the Importance of Waveform Shape.Trends Cogn. Sci. 2017; 21: 137-149Abstract Full Text Full Text PDF PubMed Scopus (237) Google Scholar, Lozano-Soldevilla, 2018Lozano-Soldevilla D. Nonsinusoidal neuronal oscillations: bug or feature?.J. Neurophysiol. 2018; 119: 1595-1598Crossref PubMed Scopus (5) Google Scholar). A notch filter might interfere with narrowband gamma activity that happens to coincide with the notch frequency (Fries et al., 2008Fries P. Scheeringa R. Oostenveld R. Finding gamma.Neuron. 2008; 58: 303-305Abstract Full Text Full Text PDF PubMed Scopus (97) Google Scholar, Saleem et al., 2017Saleem A.B. Lien A.D. Krumin M. Haider B. Rosón M.R. Ayaz A. Reinhold K. Busse L. Carandini M. Harris K.D. Carandini M. Subcortical Source and Modulation of the Narrowband Gamma Oscillation in Mouse Visual Cortex.Neuron. 2017; 93: 315-322Abstract Full Text Full Text PDF PubMed Scopus (81) Google Scholar). A band-pass filter may reduce the distinction between shapes of spikes emitted by different neurons and picked up by an extracellular microelectrode, degrading the quality of spike sorting. Slightly less obvious is the distortion of the temporal features of the target: peaks or transitions may be smoothed, steps may turn into pulses, and so on. Artifactual features may emerge, such as response peaks, or oscillations (“ringing”) created de novo by the filter in response to some feature of the target or noise signal. Figure 10 shows the response to a step of a high-pass filter (Butterworth order 8) of various cutoff frequencies. The response includes multiple excursions of both polarities (“positivities” and “negativities”) that may have no obvious counterpart in the brain signal. Disturbingly, the latencies of some fall in the range of standard event-related potential (ERP) response features (schematized as lines in Figure 10). The morphology of these artifacts depends on both the filter and the brain activity, as further illustrated in Figure 11. An investigator or a reader might wrongly be tempted to assign to the multipolar deflections of the filter response a sequence of distinct physiological processes. Similar issues have been pointed out with respect to spike waveform morphology from extracellular recordings (Quian Quiroga, 2009Quian Quiroga R. What is the real shape of extracellular spikes?.J. Neurosci. Methods. 2009; 177: 194-198Crossref PubMed Scopus (59) Google Scholar, Molden et al., 2013Molden S. Moldestad O. Storm J.F. Estimating extracellular spike waveforms from CA1 pyramidal cells with multichannel electrodes.PLoS ONE. 2013; 8: e82141Crossref PubMed Scopus (1) Google Scholar). Oscillatory phenomena play an important role in the brain (Buzsáki, 2006Buzsáki G. Rhythms Of The Brain. Oxford University Press, 2006Crossref Scopus (3343) Google Scholar, Lopes da Silva, 2013Lopes da Silva F. EEG and MEG: relevance to neuroscience.Neuron. 2013; 80: 1112-1128Abstract Full Text Full Text PDF PubMed Scopus (447) Google Scholar), and many response patterns are interpreted as reflecting oscillatory activity (Zoefel and VanRullen, 2017Zoefel B. VanRullen R. Oscillatory mechanisms of stimulus processing and selection in the visual and auditory systems: State-of-the-art, speculations and suggestions.Front. Neurosci. 2017; 11: 296Crossref PubMed Scopus (38) Google Scholar, Meyer, 2018Meyer L. 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When brain rhythms aren’t “rhythmic”: implication for their mechanisms and meaning.Curr. Opin. Neurobiol. 2016; 40: 72-80Crossref PubMed Scopus (123) Google Scholar, van Ede et al., 2018van Ede F. Quinn A.J. Woolrich M.W. Nobre A.C. Neural Oscillations: Sustained Rhythms or Transient Burst-Events?.Trends Neurosci. 2018; 41: 415-417Abstract Full Text Full Text PDF PubMed Scopus (74) Google Scholar). Non-oscillatory inputs (e.g., an impulse or step) can trigger a filter response with distinctly oscillatory features. Figure 12 shows the response of an 8–11 Hz band-pass filter (such as might be used to enhance alpha activity relative to background noise) to several inputs, including a 10 Hz sinusoidal pulse (top) and two configurations of impulses. Visually, the responses to the non-oscillatory impulse pairs are, if anything, more convincingly oscillatory than the response to the oscillatory input! Oscillations tend to occur with a frequency close to a filter cutoff and to be more salient for filters with a high order. They can occur for any filter with a sharp cutoff in the frequency domain and are particularly salient for band-pass filters, as high-pass and low-pass cutoffs are close and may interact. Furthermore, if the pass band is narrow, the investigator might be tempted to choose a filter with steep cutoffs, resulting in a long impulse response with prolonged ringing.