Spectral analysis and filtering

Spectral analysis and Filtering in
EEG : Ways to go wrong
Narayan Subramaniyam
Dept. Of Neuroscience and Biomedical Engineering
Aalto University

Motivation for filtering
●
Removing low voltage changes (e.g. skin potentials,
electrode impedance drifts, etc) and line noise.
●
Antialiasing – Attenuation of frequencies higher than
1 / 2 of the sampling rate.
●
Preprocessing : filter out slow frequencies (< 0.1 Hz )
and/or high frequencies (>40 Hz).
Figure source : Cheveigné and Nelken 2019

Types of filters - I

How filtering affects your data
●
Amplitude attenuation
Figure source : Luck 2005

How filtering affects your data ?
●
Shape distortion due to sharper cut-off
Low pass filtering
High pass filtering
Figure source : Luck 2005

●
Perils of high pass filtering
Figure source : Acunzo et al. 2012

Basic theory
●
A filter basically is a Linear Time Invariant (LTI)
system
●
An LTI obeys
– Superposition
– Time invariance

Basic theory
●
Knowing the impulse response is enough to know
everything about the LTI system
●
Impulse response (h) is the systems output when the
input is an impulse
●
By convolutuion of the input signal x to the impulse
response (filter coefficients) h, we get the filter output.
●
Convolution = series of multiplication followed by sum
of products

What is convolution ?
Figure source : Lyons 2004

Types of filters - II
Finite Impulse Response Infinite Impulse Response
Figure source : Lyons 2004

Types of filters - II
●
Finite impulse response
– Produces finite length output to an impulse
– Equal delays at all frequencies (linear phase)
●
Infinite impulse response
– Produces output for infinite duration to an impulse
– Achieved by feeding part of output to the input
– Unequal delays at different frequencies and unstable
– Typically not used in EEG pre-processing

Types of filters - III
●
Causal filters
– Output of filter depends on past and present inputs.
●
Non-causal filters
– Output depends on both past and future inputs.
– Achieved by filtering the data forward and backward.
– Offline filtering
– Can cause smearing effects back in time

●
Causal or acausal ?
Figure source : Acunzo et al. 2012

Know your defaults!
●
EEGLAB uses a zero phase acausal FIR filter
as a default (it uses the filtfilt() function in
MATLAB).
●
FieldTrip uses a Butterworth filter (IIR) as
default.
●
MNE python uses acausal FIR filter as default.
The same EEG data can be distorted in different
ways if you filter them across pacakages using
default setting!

Some recommendations
●
High-pass filtering more problematic than low-
pass filtering
●
Avoid (offline) high-pass filtering if possible.
●
If not, lower values in the range of 0.01-0.05 is
preferred. Higher cut-off may distort the data.
●
Prefer FIR (zero phase) acausal filter unless
interested in earliest moment of effect.

Conclusions
●
Filters are a form of controlled distortion. Use
sparingly
●
Good looking data is not good data.
●
When publishing, report your filter settings in
detail (causal or acausal, zero or minimum
phase etc)

Power spectral density
●
Power spectral density (PSD) estimation a
fundamental tool in EEG analysis that utilizes
Fourier transform (FT).
●
Gives information about the power in each
frequency component

How to compute PSD?
●
Estimating PSD with fast Fourier Transform (FFT)
algorithm (fft() in MATLAB )
●
FFT computes the discrete-time Fourier transform (DFT) of
a signal in an efficient way.
●
PSD is then estimated by squaring the magnitude of DFT
●
FFT requires the number of DFT points N, which
determines the frequency resolution.
●
In practice, N is set to the power of 2 that is next above the
length of the signal (ex: length of signall is 1000, then N is
1024 = 2^10).

Example
●
20 seconds (4097) EEG data from one channel.
eyes closed.
Figure source : https://sapienlabs.co/factors-that-impact-power-spectrum-density-estimation/

How to compute PSD ? - The Welch
method
●
Results from FFT computation are noisy, jagged
and hard to interpret.
●
Welch’s method is routinely used to compute
PSD from EEG data
●
The idea is to divide the data into (overlapping)
windows and compute FFT for each window.
●
PSD is the average of FFTs over these
windows.

Parameters for Welch’s method
●
Window length (w)
●
Number of FFT points (N)
●
Percentage of window overlap (noverlap)
●
Choice of windowing function (to get continuous
waveform without sharp transitions)

Effect of window size
●
0.25 sec window (~2 cycles of alpha) gives poor
freq. Resolution (wide lobe between 5 – 15 Hz)

Effect of % of overlap

Windowing function
●
Hamming window is commonly used as it has
lower spectral leakage
Figure source : https://en.wikipedia.org/wiki/Window_function

Frequency bands
●
PSD is analyzed by slicing it into a small
number of frequency “bands”.
Figure source : https://sapienlabs.co/eyes-open-eyes-closed-and-variability-in-the-eeg/

Our brain is not a sinusoidal
generator!
●
Basic assumption in applying FT : The EEG can
be broken down to bunch of sinusoids
●
Many EEG signals display nonsinusoidal
features.
●
Non sinusoidal signals contains multiple
frequencies and these end up showing as
harmonics when using Fourier transform

Examples of non-sinusoidal EEG
Figure source : Cole et al. 2017

Conclusions
●
By using Welch method one can obtain smoother PSD
●
Choosing very short windows to compute PSD can lead to poor results
(esp. for analyzing lower frequencies)
●
Longer windows can result in finer resolution of PSD, but can lead to
noisy PSD
●
The number of DFT points (N) should always be greater than or equal to
the length of your data or window
●
Highly overlapping windows does not guarantee smoother PSD. (due to
correlation between the windows).
●
Many EEG signals are non-sinusoidal and violate FT assumption.
●
Look at the whole PSD and not just the ‘band’ as band definitions are
arbitrary and somewhat inconsistent across studies!

Spectral analysis and filtering

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