Bandpass filters
and
Hilbert Transform
Summary of Chapter 14 In
Analyzing Neural Time Series Data:
Theory and Practice
Lauritz W. Dieckman
A quick message from Mike X
I've created a googlegroups for the book. If you have other questions about the book, analyses, or code, feel free to post them there.
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nalyzingneuraltimeseriesdata
Filters ‐Everywhere
• Music
‐analog recordings
~44,100 Hz sampling
‐typically bandpass
filtered to range
of human hearing
‐Or, purposefully Used in artistic creation
‐One persons distortion is another persons art
Why? Filter‐Hilbert Method vs
Complex Wavelet Convolution
Similarities
• Used to create a complex (real and imaginary) time series (analytic signal) from real signal data
• Analytic signal used to determine phase and power ‐Methods described in Chapter 13
• The signal must be bandpass filtered before Why? Filter‐Hilbert Method vs
Complex Wavelet Convolution
Differences
• Signal must first be bandpass filtered
• Analytic signal acquired by creating the “phase quadrature component” (one quarter‐cycle) by rotating sections of a Fourier spectrum.
• PRO: Filter‐Hilbert provides superior control over frequency filtering ‐not limited in shape whereas bandpass filters can take many shapes
• CON: However, Filter‐Hilbert requires signal processing toolbox for kernel creation (unlike Morlet wavelets).
• CON: Bandpass filtering is slower than wavelets
Bandpass Filtering
• First step is Bandpass filtering ‐Don’t worry Hilbert fans, we’ll get back to Hilbert transform soon
• Highly recommended to separate frequencies before the transform ‐e.g. lower frequencies may dominate combined signal
Filters: To Infinity and… much shorter
• Finite Impulse Response (FIR) SOON I will reign –The response to an impulse ends supreme and at some point. infinite over your ‐Stability analytic signal
‐Computation time
‐ Risk of phase distortions
You will never triumph Infinite Impulse Response (IIR) Butterworth.
–The response to an impulse
It may take more than just my does not end.
analytic skill to protect the ‐Butterworth IIR
filter help me!
signal. Matlab
Filters: Types
• High‐pass filter – Allows frequencies higher than cutoff to remain in the signal (lower frequency signals are attenuated)
‐Simple illustration – Cutoff 5
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FL studio example
Filters: Types
• Low‐pass filter – Allows frequencies lower than cutoff to remain in the signal (higher frequency signals are attenuated)
‐Simple illustration – Cutoff 5
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Filters: Types
• Band‐Stop filter – Attenuates or removes frequencies within a specific range. ‐e.g. Notch filters at 60 Hz
‐Simple illustration – Low Cutoff 5 & High Cutoff 2
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Filters: Types
• Bandpass filter – Allows frequencies lower than lower bound cutoff to remain in the signal and frequencies higher than higher bound cut off to remain in the signal ‐Simple illustration – Low Cutoff 5 & High Cutoff 2
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‐Basically a high and low‐pass combo
LFO
Many Music FX use filters Filter
• Phaser‐
Separates signal 1. Unfiltered
2. Alters phase
‐When recombined the signals out of phase nullify
‐Frequently combined with low frequency ossiclator
How to: Filter Kernel Construction
• Like complex wavelet convolution – The analytic signal produced from filtering *weighted combination of kernel and signal
How to: Filter Kernel Construction
• Note difference shape Gaussian vs Plateau
• Plateau shapes allow for greater frequency specificity
Filter Functions
• Firls – 3 inputs
A. Order : Length of filter kernel (+1)
*Important for frequency precision
Larger Frequency precision + processing time
– For a particular frequency of interest: MUST be long enough for one cycle
*Recommended 2 – 5 time longer than frequency of interest
*Use sample points, not time (i.e. must know sample rate)
filter_order =round(3*(EEG.srate/lower_filter_bound));
Filter Functions
• Firls – 3 inputs
B. 6 numbers define shape of response
1. Zero frequency Filter Functions
• Firls – 3 inputs to define shape of response
B. 6 numbers define shape of response 1. Zero frequency
• ffrequencies = [ 0 (1‐transition_width) *lower_filter_bound lower_filter_bound
upper_filter_bound (1+transition_width)
*upper_filter_bound nyquist ]/nyquist;
Filter Functions
• Firls – 3 inputs
B. 6 numbers define shape of response
2. Frequency of start –lower transition zone
Filter Functions
• Firls – 3 inputs
B. 6 numbers define shape of response 2. Frequency of start –lower transition zone
transition_width = 0.2;
lower_filter_bound = 4; % Hz
ffrequencies = [ 0 (1‐transition_width) *lower_filter_bound lower_filter_bound
upper_filter_bound (1+transition_width)
*upper_filter_bound nyquist ]/nyquist;
Filter Functions
• Firls – 3 inputs
B. 6 numbers define shape of response
3. Lower bound of bandpass
Filter Functions
• Firls – 3 inputs
B. 6 numbers define shape of response 3. Lower bound of bandpass
ffrequencies = [ 0 (1‐transition_width) *lower_filter_bound lower_filter_bound
upper_filter_bound (1+transition_width)
*upper_filter_bound nyquist ]/nyquist;
Filter Functions
• Firls – 3 inputs
B. 6 numbers define shape of response
Side note: Transition zones
Transition zones
Filter Functions
• Firls – 3 inputs
B. 6 numbers define shape of response
4. Upper bound of bandpass
Filter Functions
• Firls – 3 inputs
B. 6 numbers define shape of response 4. Upper bound of bandpass
upper_filter_bound = 10; % Hz
ffrequencies = [ 0 (1‐transition_width) *lower_filter_bound lower_filter_bound
upper_filter_bound (1+transition_width)
*upper_filter_bound nyquist ]/nyquist;
Filter Functions
• Firls – 3 inputs
B. 6 numbers define shape of response
5. Frequency of the end of the upper transition
zone Filter Functions
• Firls – 3 inputs
B. 6 numbers define shape of response 5. Frequency of the end of the upper transition
zone ffrequencies = [ 0 (1‐transition_width) *lower_filter_bound lower_filter_bound
upper_filter_bound (1+transition_width)
*upper_filter_bound nyquist ]/nyquist;
Filter Functions
• Firls – 3 inputs
B. 6 numbers define shape of response
6. Nyquist frequency Filter Functions
• Firls – 3 inputs
B. 6 numbers define shape of response 6. Nyquist frequency nyquist = EEG.srate/2;
ffrequencies = [ 0 (1‐transition_width) *lower_filter_bound lower_filter_bound
upper_filter_bound (1+transition_width)
*upper_filter_bound nyquist ]/nyquist;
Filter Functions
• Firls – 3 inputs
B. 6 numbers define shape of response
‐ These numbers are scaled so that Nyquist
frequency is 1.
‐Once frequencies are set in Hz we divide by the nyquist frequency
ffrequencies = [ 0 (1‐transition_width) *lower_filter_bound lower_filter_bound
upper_filter_bound (1+transition_width)
*upper_filter_bound nyquist ]/nyquist;
Filter Functions
• Firls – 3 inputs
C. Ideal response amplitude
Length of ideal response should = 6 numbers define shape of response
0s = attenuate . 1s = keep. Can use range 0‐1
[0 0 1 1 0 0] Lower
Upper
Frequency bounds
Filter Functions
• Firls – 3 inputs
C. Ideal response amplitude
Length of ideal response should = 6 numbers define shape of response
[0 0 1 1 0 0] DC Nyquist
Lower
Upper
Frequency bounds of transition zones
Frequency Width
• Controls Time Frequency trade off
Narrow Width
Frequency Precision
Temporal Precision Wide Width
Temporal Precision Frequency Precision Time
Freq
Precision in Width–
Time Frequency Wrestling:
Fun with mixed metaphors • Unlike combat axiom‐ smaller does not mean quicker
• Narrow filter kernels are long
and thus influenced by more numbers. Like holding up sumo meat? • Wider width filters have smaller
kernels and are thus created
more quickly. Like sumo quickly
crushing a kid?
Precision in Width–
Time Frequency
Sharp Edges make Ripples in Time
<‐ Artifact with a sharp edge
Sharp filter edges in frequency increase response, but also increase risk of artifacts in time
Shoot for 10‐25% of upper and lower bond firls vs fir1
• Fir 1 has automatic transition zones then smooths
• Firls can produce a similar effect if kernel is smoothed with Hamming (or Hann) window
See the same… minus the ham
Prof. X’s recommendations
To make a narrow band filter, use fir1
If you are worried about ripple artifacts firls can allow for gentle transition zones
Gentle sloping plateaus… gentle filters… gentle
Patrick_Stewart = disp(‘To make a narrow band filter, use fir1. If you are worried about ripple artifacts, firls can allow for gentle transition zones… gentle.’)
Check.Your.Filters:
So Whatcha Want?
Goodness quantified!
Ideal forms:
Similarity between actual filter and the ideal filter
Sum of squared errors
Sum of squared errors (SSE)
center_freq = 60; % in Hz
filter_frequency_spread_wide = 10; % Hz +/‐ the center frequency
ffrequencies = [ 0 (1‐transition_width)*(center_freq‐
filter_frequency_spread_wide) (center_freq‐filter_frequency_spread_wide) (center_freq+filter_frequency_spread_wide) (1+transition_width)*(center_freq+filter_frequency_spread_wide) nyquist
]/nyquist;
idealresponse = [ 0 0 1 1 0 0 ];
filterweightsW = zscore(firls(200,ffrequencies,idealresponse));
num2str(sum( (idealresponse‐fft_filtkern(freqsidx)).^2 ))
SSE = num2str(sum( (idealresponse‐fft_filtkern(freqsidx)).^2 ))
Good Filter/Bad Filter: Filter Width
SSE should
be close
to 0
Do not use a
filter with
SSE > 1
‐Good at
10hz
‐Bad at
15hz
Application: filtfilt
• After creating the perfect kernel and testing it
• Filtfilt used to apply
data2filter = squeeze(double(EEG.data(47,:,1)));
filterweights = firls(200,ffrequencies,idealresponse); % recompute
without z‐scoring
filter_result = filtfilt(filterweights,1,data2filter);
Filter Kernel ‐‐ Weight coefficient ‐‐ Data
Filtfilt compared to convolution
Phase Delay and Time Travel
• Causal filter : Filter is based on information from the past (back in time)
• Filter function: Causes a phase delay
Phase Delay and Time Travel
• Luckily we can filter back in time to correct the errors of the past!
• Forward_to_the‐past = 1‐(Back_to_the_Future)
• Refiltering corrects the phase delay
I need a nuclear reaction to generate 1.21 JIGAwatts of electricity. Damn it DOC, my mat‐
telepathy demands GIGA! GIGAWATTS
You thought your puny mat‐telepathy powers could defeat me? With my Infinity Impulse Response Filtering death ray I will sow (relative) chaos, instability and non linear phase distortions across your signal! Bwahahaha
NOOOOO!
5th order Butterworth
Hope for analytic signal… fading…must use Hilbert powers to… transform Revenge of the Hilbert Transform
• Remember way back‐ before time traveling filters, sumo quick narrow filters, and (poor) comic book‐ish fight scenes?
Concatenation • Analytic signal produced by powers activate!
adding phase quadrature (1/4) Hilbert transform!
• Created by rotating aspects of complex
frequency spectrum (from Fourier)
How to filter‐Hilbert
Hilbert Transformer: With Autobot Matrix 1. Fourier transform from bandpass filtered signal and create an imaginary
copy (i.e. Fourier * i)
2. Find the positive (> 0 to Nyquist) and negative (> Nyquist frequencies). *Note: 0 & Nyquist are not used
3. Convert from complex cosine to complex sine by rotating positive 90 degree (‐pi/2) counter clockwise & negative frequencies 90 degrees clockwise
Original
Positive 2 x pos
freq coeff
Negative
Subtract from self = 0
How to Hilbert
• Final: Inverse Fourier
• These ingredients produce the analytic signal • Does not influence
real signal
Warning: Hilbert command will run FFT on the first
dimension in a data
matrix. Must be time in the first dimension Double check your signal
• Plot phase angles!
Come my allies, we have earned good
night’s sleep