FILTERING

1
FILTERING

• In cognitive experiments
• Based on Luck 2005

2
Whats a filter

• Gain the ratio of output to input
• Amplifier system designed to boost signal
  magnitude
• Gain 1 ? output of amp. larger than input
• Filter system designed to reduce signal
  magnitude at particular frequencies
• Gain given frequencies

3
Why do filtering?

• The Nyquist Theorem
• An analog signal can be converted to digital as
  long as the rate of digitization is twice as high
  (at least) as the highest frequency in signal
  being digitized
• If the signal contains any frequencies higher
  than this limit (at least half or less than rate
  of digitization) they will show up as artifactual
  low frequencies (aliasing).
• This second part is why you have to do at least
  some (online) low-pass filtering

4
Why do filtering?

• EEG consists of a signal plus noise, and some of
  the noise is sufficiently different in frequency
  content from the signal that it can be suppressed
  simply by attenuating different frequencies, thus
  making the signal more visible
• Unfortunately, this benefit needs to be weighted
  against the cost of distortion in your signal,
  because ANY FILTER DISTORTS AT LEAST SOME PART OF
  THE SIGNAL.

5
More on distortion

• What kind of filter distortion are we talking
  about?
• Filters can change the relative amplitude of ERP
  components, the timing of ERP components, and can
  add peaks that werent even there before.
• So not just innocuous smoothing!
• But still useful in limited capacity.

6
How to describe filter properties

• Properties of a filter are defined by its
  TRANSFER FUNCTION.
• Transferring signal to filter output
• Transfer function has two components
• Frequency response function
• How filter changes amplitude of each frequency
• Phase response function
• How filter changes phase of each frequency

7
Frequency Response Function

• Filters in ERP research often described with one
  parameter only, the half-amplitude cutoff
• The frequency at which the amplitude is cut by
  half (50) of its original value
• Notice that in most cases this does not mean that
  no frequencies above this value pass, they just
  pass with amplitudes between 50-0 of their
  original value

8
How to describe filter properties

• High-pass filters are often described in terms of
  time constants instead of half-amplitude cutoffs.
• If input to high-pass filter in time domain is
  constant, will ultimately return toward zero
• Time constant is how long it takes for it to get
  down to 1/e of starting value

Input
Output
9
What kinds of filtering typically get done in ERP?

• MANDATORY
• Initial, online, low-pass filter built into the
  hardware to ensure the sampling rate is twice as
  high as highest signal frequency recorded (so you
  dont get aliasing)
• OPTIONAL
• High-pass filter to get rid of slow, non-neural
  potentials like skin potentials (
• Notch filters to eliminate electrical noise (60
  Hz)
• BAD LAST RESORT!
• Low or high-pass to clean up data

10
What kinds of filtering typically get done in ERP?

• Online filters can also be described as analog,
  as they are applied to a continuous voltage
  contrast
• Offline filters can also be described as digital,
  as they are applied to data that has already been
  converted from the continuous voltage to a
  digital representation on the computer

11
What kinds of filtering typically get done in ERP?

• Best policy is to do as little online/analog
  filtering as possiblejust the little you need to
  prevent aliasing and minimize the impact of very
  slow voltage shifts
• Digital filtering has important advantages
• Not permanentcan always return to unfiltered
  data
• Easier to construct phase-invariant filters
• Easier to adapt than a hardware filter

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Reminder DISTORTION

• Famous tenet of filterers
• Precision in the time domain is inversely
  related to precision in the frequency domain
• Kind of analogous to the Uncertainty Principle
• The more certainly you specify what frequencies
  are in your data, the less certain you can be
  about what happened in the time domain

Low-pass filter
High-pass filter
13
Lucks Recommendations

• Use a sampling rate of between 200 and 500 Hz,
  and make your online low-pass filter between 1/3
  and 1/4 of that
• Use online high-pass filter (AC recording) of .01
  Hz
• If averaged waveforms are fuzzy, can use a
  low-pass with a gentle, half-amplitude cutoff of
  between 20-40 Hz

14
Phase

• You really want the phase portion of the filters
  transfer function to be zero for all frequencies,
  so that the phase of the ERP waveform wont
  change (you dont want to move latencies)
• This is usually possible with offline digital
  filters
• For this to happen with your online filter, your
  amplifier needs to incorporate special filters
  like Bessel filters that shift all frequency
  phases equally so that the original phases are
  easily recoverable.

15
Why distortions?Look at 10 Hz notch filter on 10
Hz data.
16
Why distortions? Look at a notch filter.

• Time-domain waveforms in the Fourier analysis are
  represented as the sum of a set of
  infinite-duration sine waves
• PROBLEM transient ERP waveforms are finite in
  durationthey consist of finite-duration voltage
  deflections, not infinite-duration sine waves
• Thus, a notch filter of 10 Hz is equivalent to
  computing amplitude and phase in 10 Hz band of
  signal and then subtracting a sine wave with this
  amplitude from the waveform.
• This is going to create an inverse 10 Hz
  oscillation is the portions of the ERP waveform
  where you didnt have this frequencyin fact
  possibly increasing 10 Hz noise

17
Offline Filtering
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Filtering as a time-domain procedure a
common-sense view

• Whats the intuitive way we could imagine of
  getting rid of the jagged lines in our waveform?

19
Filtering as a time-domain procedure

• For each point, we could average it with the
  nearby points for a smoother waveform
• Point x ( x1x2x3xx-1x-2x-3)
  / 7
• More generally
• filtERP(t) weight x ERP(tj)

20
Filtering as a time-domain procedure

• If you just averaged together the /- 3
  surrounding points, all 6 contribute equally to
  xs new value (regardless of their distance from
  x) and this reduces temporal precision
• Instead you can weight them by their distance
  from x ( ERP(t))
• filterERP(t) Weight(j) ERP(tj)

21
Flip

• Last equation shows how the many values
  surrounding one point are incorporated into
  filter equation to give transformed value for the
  one point
• New question how does the original value of a
  given point influence all of the values in the
  waveform through the filter
• If we think of filtering as smearing individual
  point values over time, were asking, what is the
  extent and amount of the smear?

22
Impulse Response Function (IRF)

• The extent and amount of smear for a given point
  in a given filter is represented as a function
  over time, where t0 is when the given point
  occurs. Positive points represent smear forwards,
  negative points represent smear back (acausal
  smear)

23
Impulse Response Function (IRF)

• If you imagine plotting the IRF at every single
  individual point, the filter it corresponds to is
  just going to be the sum of all those IRFs scaled
  to each points original value.

24
CONVOLUTION

• Therefore, the filter equation in the time domain
  can be re-represented as
• filterERP(t) IRF(j) x ERP(t-j)
• Note that we have been treating the ERP waveform
  as a function over time. When you combine two
  functions in this way, you say that youre
  CONVOLVING them
• filterERP IRF ERP

25
Convolution

• For each point in the ERP waveERP(t)substitute
  the sum offor every scaled instantiation of the
  IRF along the timecourse of the ERP functionthe
  scaled IRFs value at that point in time t
• filterERP(t) IRF(j) x ERP(t-j)
• filterERP(100) IRF(1) x ERP(99) IRF(2) x
  ERP(98) IRF(3) x ERP(97)IRF(99) x ERP(1)
• the first term on the right gives you the effect
  of point t99 on point t100, the second gives
  you the effect of point t98 on point t100, etc.

26
Properties of Convolution

• Convolution is
• Commutative A B B A
• Associative A (B C) (A B) C
• Distributive A (B C) (A B) (A C)
• This helps answer a common question that you may
  find yourself asking What happens if you filter
  twice?
• (ERP IRF1) IRF2 ERP (IRF1 IRF2)
• E.g., the convolution of two gaussians ? a wider
  gaussian, SO, filtering twice with same gaussian
  is like filtering once with a wider gaussian

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Properties of Convolution

• Also, since convolution in time domain is equal
  to multiplication in frequency domain, frequency
  response function of double filtering equals the
  product of the two filters
• So if you high-pass and then low-pass, like
  multiplying the two functions, usually gives you
  a band pass of the middle frequencies
• But watch out b/c if the two freq resp functions
  have a gradual attenuation, you could get much
  MORE attenuation than you wanted of the middle
  frequencies

28
Time ?? Frequency

• What is the relationship between the IRF and the
  HP/LP filters we were talking about before?
• Multiplication in the frequency domain is
  equivalent to convolution in the time domain

29
Frequency MultiplicationEx low-pass filter

• x
  
• ? inv
  FFT?

30
Time ConvolutionEx 60 Hz Notch filter

• 
  

31
Time ?? Frequency
32
Time ?? Frequency

• Usually faster to filter using convolutions than
  Fourier Transforms, so most digital filters
  implemented this way
• Thus you create your desired transfer function
  for the filter and then transform into time
  domain to create IRF
• Ex you want to bandpass just 60 Hz
• ?IFFT?
  

33
Time ?? Frequency

• In fact, not perfect transfer
• IRF for 60 Hz should be just 60 Hz wave
• Problem is that sine-wave is finite, so you get
  power at other freqs too.
• Thats why tapered function better

34
Back to distortion

• Lets pretend our ERP waveform is just this
  portion of a perfect 10 Hz sine wave

35
Back to distortion

• In fact, even though we defined the waveform as
  containing only a 10 hz sine wave, the spectrum
  of this waveform contains power at other
  frequencies. Why?
• This is because the waveform is finite

36
Back to distortion

• Remember that when time range is limited,
  frequency range is broadened, and vice versa.
• That means that if you were to apply a filter to
  remove the 10 Hz component, you wont remove the
  entire waveform, even though we defined it as a
  purely 10 Hz sine wave

37
Back to distortion

• The reason you see artifactual peaks is that the
  original spectrum had a lot of power in
  frequencies near 10 Hz, such that they sum with
  the 10 Hz oscillations in the IRF

38
Back to distortion

• For the same reason, you dont see as much
  artifactual oscillation in 20 Hz or 60 Hz notch
  filters, b/c there isnt a lot of power in this
  or nearby frequencies in the original spectrum

39
Low-pass distortions

• Shape of window (freq response function) has a
  big impact on amount of distortion
• General rule of thumb unless you have a really
  good trick, the more ideal the filter in the
  frequency domain, the less ideal it is in the
  time domain
• Since we usually care most about time domain for
  interpreting data, and the filtering is just an
  optional way to improve the image, usually better
  to idealize the time domain

40
Low-pass distortions

• Shape of window has a big impact on amount of
  distortion (back to our 10 hz sine wave)

41
Low-pass distortions

• Running average filter (here, 12.5 Hz)
• Much less temporal spread than windowed ideal
• Problem substantial attenuation in 10-hz band
  where most of original power located, so
  amplitude reduction
• Problem side lobes in freq response f (due to
  sharp onset and offset of IRF) let substantial
  high-freq noise pass
• Can be especially good for dealing with precisely
  defined line freq noise

42
Low-pass distortions

• Gaussian filter (12.5 Hz)
• Still a little smear in onset/offset, but almost
  complete attenuation of freq 30 Hz
• Problem still some amplitude reduction in freq
  of interestmaybe should set cutoff higher
• Good compromise between time and freq domain
• Can be harder to implement than running avg.
• Easy to describe time-domain just full-width at
  half-maximum of gaussian (FWHM)

43
Low-pass distortions

• Causal filter (12.5 Hz)
• An event at particular times cant affect events
  at subsequent times
• One positive if the IRF has a fairly rapid,
  monotonic fall-off, they may produce relatively
  little distortion in onset latency, which may be
  key in some analyses.

44
A note about offline low-pass filtered data

• Reviewers often want you to do numbers on
  (offline) unfiltered data. Does this really make
  a difference?
• Probably doesnt have a huge effect if you use
  type of filter just recommended
• However, helps to keep things straight in your
  head, because taking mean amplitude in a
  particular latency window from filtered waveforms
  is equivalent to taking mean amplitude from a
  larger window in the original data
• Thus, if you use unfiltered data for this kind of
  average-over-time measurement, high frequency
  noise will average out anyway, and youll be more
  clear in your head about what latencies your
  measure is coming from
• Note that for non-averaged measures like peak
  amplitude, this is less of an option b/c the high
  frequency noise will corrupt these measures
  significantly

45
High-pass filtering in time domain

• A little more complicated
• We implement this as doing a low-pass and
  subtracting it from the original waveform,
  yielding the left-over high frequencies.
• ERPh ERP ERPl ERP (IRFl ERP)

46
High-pass filtering in time domain

• If we substitute IRFu ERP for ERP (where IRFu
  is the unity impulse response, 1 at 0 and 0
  everywhere else) and do some more algebra, we get
  the following
• IRFh IRFu - IRFl

47
High-pass distortions

• Application of same 2.5 Hz high-pass filter to
  three different waveforms

48
High-pass distortions

• Effect of high-pass filter can be very sensitive
  to shape of waveform, b/c of the negative shape
  of the IRF
• If waveform consists of roughly equal positive
  and negative subcomponents, will lead to
  opposite-polarity copies of the IRF which will
  cancel each other out.

49
High-pass distortions

• Effect of high-pass filter can be very sensitive
  to shape of waveform, b/c of the negative shape
  of the IRF
• If one polarity dominates waveform, the overshoot
  will be greater

50
High-pass distortions

• Can imagine this as reversed polarity of
  distortions low-pass produced
• In low-pass, you get spreading of peaks in same
  polarity as peaks, giving earlier onset/later
  offset
• High-pass filter inverts the gaussian of the
  low-pass response, so produces opposite-polarity
  spreading
• This can be worse b/c makes it look like they are
  artifactual peaks
• This is what makes high-pass filtering more
  dangerous than low-pass.

51
High-pass distortions

• Interesting difference between causal and
  non-causal high pass filters
• Noncausal hp filters take low-freq info away from
  time zone of waveform and push it forwards and
  backwards in time
• Causal hp filters only push it forward

52
Linearity

• Most (all?) of filters discussed here have been
  linear
• If your filter is linear, remember that you can
  combine it with other linear operations in any
  order (e.g. signal averaging or baselining) with
  no change in the result. (artifact rejection is
  nonlinear though)
• Keep this in mind when youre designing your
  analysis protocol, by switching things around you
  can save computing time and human time

53
Bottom line of Off-line Filtering

• Try not to have to filter very much (e.g., get
  good data)
• If you have to filter offline, try not to do any
  high-pass filtering (b/c of the danger of
  artifactual peaks)
• If you have to filter offline, try filtering on a
  simulated data set firstthen you can see exactly
  what distortions are possible.

54
Reporting filters

• Especially important if you report stats on data
  that has any offline filtering
• Often filters reported like A low-pass filter of
  30 Hz was applied to the averaged data
• 2 problems
• No description of shape
• No explicit description of what 30 Hz applies to

55
Reporting filter shape

• We saw that amount and type of distortion heavily
  depends on shape of frequency response
  functione.g. a sharp cutoff leads to more
  oscillatory artifacts than a gradual one
• Therefore important to report shape of filter you
  used
• Can be a classic type with well-known
  featuresHanning, Butterworth, etc.
• If gaussian, can specify FWHM

56
Specifying measure Amplitude vs Power

• Although directly related, voltage amplitude is
  not the same as power
• Voltage current x resistance
• Power voltage x current
• Power voltage2 / resistance

57
Specifying measure Amplitude vs Power

• Filters can reported in terms of either, but note
  that HALF-POWER CUTOFF DOESNT EQUAL
  HALF-AMPLITUDE CUTOFF
• .7072 (V) .5 (P)
• So frequency at which power is reduced by 50
  will be frequency at which voltage is reduced by
  29
• Bottom-line important to know (and report) which
  one youre using

58
Online Filtering
59
Back to the Nyquist Theorem

• The Nyquist Theorem
• An analog signal can be converted to digital as
  long as the rate of digitization is twice as high
  (at least) as the highest frequency in signal
  being digitized
• If the signal contains any frequencies higher
  than this limit (at least half or less than rate
  of digitization) they will show up as artifactual
  low frequencies (aliasing).

60
Nyquist Theorem

• Ex 100 Hz sampling rate x ½ 50 Hz Limit

10 cycles ? correct 20 cycles ? correct 40
cycles ? correct (but choppy) 40 cycles ?
incorrect (aliased to 40 Hz) 3 cycles ?
incorrect (aliased to 3 Hz)
10 Hz signal 20 Hz 40 Hz 60 Hz 97 Hz
1 sec
61
Nyquist rule

• Requires sampling at twice the fastest frequency
  present in original waveform to prevent aliasing
• Not just the fastest that you care about
• This is why an online filter is necessary, to
  make sure that certain frequencies dont even
  make it to the digitization stage

62
Nyquist rule

• Does NOT ensure that data will be smooth, only
  that the frequencies will be accurate
• The rule assumes that the signal is perfectly
  sinusoidal, not necessarily true in physiological
  systems
• For these reasons, you often want to limit the
  frequencies to even less than half the sampling
  rate1/3, 1/5, 1/10..
• (some refs Attinger, Anne McDonald 1966 in
  Coles, Donchin Porges 1986 book)

63
Nyquist rule

• If you dont have very much high frequency noise,
  you can sample less and ignore the small amount
  of aliased noise
• Also if all your noise is at a given frequency
  (e.g. 60 Hz), you can play some tricks with the
  sampling rate to limit the aliasing
• But these techniques probably demand some amount
  of sophistication

64
Other filtering references

• Glaser Ruchkin, 1976, Principles of
  neurobiological signal analysis
• Cook Miller, 1992, Psychophysiology
• Ruchkin, 1988, in Pictons Handbook of
  electroencephalography and clinical
  neurophysiology