Title: FILTERING
1FILTERING
- In cognitive experiments
- Based on Luck 2005
2Whats 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
3Why 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
4Why 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.
5More 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.
6How 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
7Frequency 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
8How 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
9What 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
10What 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
11What 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
12Reminder 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
13Lucks 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
14Phase
- 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.
15Why distortions?Look at 10 Hz notch filter on 10
Hz data.
16Why 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
17Offline Filtering
18Filtering 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?
19Filtering 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)
20Filtering 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)
21Flip
- 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?
22Impulse 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)
23Impulse 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.
24CONVOLUTION
- 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
25Convolution
- 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.
26Properties 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
27Properties 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
28Time ?? 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
29Frequency MultiplicationEx low-pass filter
30Time ConvolutionEx 60 Hz Notch filter
31Time ?? Frequency
32Time ?? 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?
33Time ?? 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
34Back to distortion
- Lets pretend our ERP waveform is just this
portion of a perfect 10 Hz sine wave
35Back 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
36Back 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
37Back 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
38Back 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
39Low-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
40Low-pass distortions
- Shape of window has a big impact on amount of
distortion (back to our 10 hz sine wave)
41Low-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
42Low-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)
43Low-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.
44A 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
45High-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)
46High-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
47High-pass distortions
- Application of same 2.5 Hz high-pass filter to
three different waveforms
48High-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.
49High-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
50High-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.
51High-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
52Linearity
- 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
53Bottom 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.
54Reporting 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
55Reporting 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
56Specifying 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
57Specifying 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
58Online Filtering
59Back 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).
60Nyquist 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
61Nyquist 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
62Nyquist 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)
63Nyquist 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
64Other filtering references
- Glaser Ruchkin, 1976, Principles of
neurobiological signal analysis - Cook Miller, 1992, Psychophysiology
- Ruchkin, 1988, in Pictons Handbook of
electroencephalography and clinical
neurophysiology