[Eeglablist] Units in output of timefreq - wavelet normalization

[Andreas Widmann]

Dear all,

for me the discussion has not yet come to a satisfying conclusion. I would like to argue that it is essential that we agree on a common terminology. I expect that some (normalization) issues will automagically disappear (but as always some new issues come up ;)

(1) There are two families of transformations involved and intermixed in the discussion: Wavelet vs. STFT

In wavelet transforms time-frequency decomposition is performed with template waveforms (the mother wavelet) which are scaled (dilated or compressed) and translated. Due to scaling the time and frequency resolution changes across the time-frequency-plane (TF-plane): high frequency and low time resolution at low frequencies and low frequency and high time resolution at high frequencies. The wavelet transform is a multiresolution analysis (MRA) and this is not a side effect but the core feature. There exist various wavelets with different properties; Morlet wavelets are sine based and thus closely related to STFT and FT. However, by scaling the number of cycles in a Morlet wavelet is never changed. I have never read a definition of (Morlet) wavelet transform in a textbook changing cycles instead of scaling a wavelet. If you find one, please, send me the reference.

In contrast to the wavelet transform the STFT family has a constant fixed time and frequency resolution across the whole TF plane. There are various versions, in particular application of different window types but also variations of other parameters. As in Morlet wavelets a sinusoid is multiplied by a window function. To achieve constant frequency resolution of the atoms the number of cycles has to change with frequency (not arbitrarily but in a fixed relation as already annotated by Norman) while window width is kept constant.

There are time-frequency transformations not clearly belonging to any of these two categories as "multiresolution STFT" and others. Neither wavelet transform nor STFT is to be preferred in general. They are equally valid but targeting different applications. MRA is preferred if frequency resolution is considered more important at low frequencies and time resolution at high frequencies as e.g. in audition. Googling „wavelet vs. STFT“ gives dozens of presentations and papers comparing both methods including their (dis-)advantages and applications.

(2) Normalization has different requirements for wavelet transforms vs. STFT

Scaling wavelets changes their energy. This is disadvantageous as now the energy of the output of the wavelet transform would change with scale. Therefore, most commonly sets of (scaled) wavelets are normalized to have all the same energy. It not necessarily matters whether this is unit energy or not. But personally I would prefer unit energy for two reasons. First, because it gives a straight-forward and intuitively accessible interpretation of the output amplitudes: the amplitude of a sample in the output reflects the integrated energy of the input in the surface area covered by the wavelet in the TF plane at the given scale and delay weighted by the window function (also see my previous message how to compute the wavelet transform amplitude of a sinusoid). Second, because data and amplitudes can be more easily compared across studies and methods (also see example code below).

The atoms of the STFT, however, are already energy normalized "by design" as their window width is constant. Thus, energy normalization is not a core issue in STFT. Unit energy normalization might be advantageous for the reasons stated above but it will work also without. If one agrees to the above differentiation I would consider the fourth version in the code contributed by Mike ("Define empirical FWHM of wavelet in the frequency domain“) rather as a member of the STFT family (time and frequency resolution is constant). I have added a single set of these (once with and once without unit energy normalization) to my previous code hopefully demonstrating the differences between transforms and normalizations.

(3) Suggestions for EEGLAB

I would suggest to modify dftfilt3 so that by defaults wavelets are scaled to unit energy (as indirectly suggested by the documentation) and adding an option for backward compatible amplitude scaling. I would not add Gabor transform equivalent normalization to the multiresolution Morlet wavelet code (in case this is really needed this is a one-liner on the command line). Rather in my opinion a nice option would be to separately implement a textbook Gabor transform STFT in a separate function (actually reflecting Mike’s suggestion with minor changes; see Norman’s remarks).

(4) General remark

Sorry, I cannot withhold a more general „philosophical“ remark. Imho, the highest priority is replicability. Whatever analysis one performs should be described most thoroughly and detailed allowing for perfect replication and comparison of results across datasets, studies, researchers, labs, etc. This is considerably easier using textbook methods and terminology. For new methods and own approaches it is essential to describe them in a sufficiently detailed and accessible way. In my view this is the biggest problem with past and present literature applying TF transforms to electrophysiological data.

Best,
Andreas

fs = 256;
T = 4;
pnts = T * fs;
t = ( 0:pnts - 1 ) / fs;
cycles = 7;
F = 4:2:70;

% White noise
rng( 0 )
signal = randn( fs * T, 1 );
% f = [ 10 50 ];
% signal = sin( 2 * pi * f( 1 ) * t )' + sin( 2 * pi * f( 2 ) * t )';

%% Morlet wavelet
freqs = F * ( pnts / fs );
sigma_f = repmat( freqs / cycles, [ pnts 1 ] );
f = repmat( 0:pnts - 1, [ size( freqs, 2 ) 1 ] )' - repmat( freqs, [ pnts 1 ] );
cmorlf = exp( -f .^ 2 ./ ( 2 * sigma_f .^ 2 ) );

figure

% Analog to Gabor transform
A = 2;
cmorlf_gabor = A .* cmorlf;

tf = ifft( repmat( fft( signal ), [ 1 length( F ) ] ) .* cmorlf_gabor );

h = subplot( 4, 3, 3, 'YDir', 'normal' );
imagesc( t, F, abs( tf' ) .^ 2 )
colorbar
set( h, 'YDir', 'normal' );
subplot( 4, 3, 2 );
plot( F, mean( abs( tf ) .^ 2 ) )
subplot( 4, 3, 1 )
plot( (0:pnts / 2) * fs / pnts, cmorlf_gabor( 1:pnts / 2 + 1, : ) )

% Unit energy
A = sqrt( pnts ./ ( sqrt( pi ) * sigma_f ) );
cmorlf_energy = A .* cmorlf;

tf = ifft( repmat( fft( signal ), [ 1 length( F ) ] ) .* cmorlf_energy );

h = subplot( 4, 3, 6, 'YDir', 'normal' );
imagesc( t, F, abs( tf' ) .^ 2 )
colorbar
set( h, 'YDir', 'normal' );
subplot( 4, 3, 5 );
plot( F, mean( abs( tf ) .^ 2 ) )
subplot( 4, 3, 4 )
plot( (0:pnts / 2) * fs / pnts, cmorlf_energy( 1:pnts / 2 + 1, : ) )

%% STFT
% adapted from https://sccn.ucsd.edu/pipermail/eeglablist/2016/011536.html
% by mikexcohen at gmail.com
hz = (0:pnts - 1) * fs / pnts;
fwhm = 10;
for iF = 1:length( F )
    s  = fwhm*(2*pi-1)/(4*pi);       % normalized width
    x  = hz-F( iF );                 % shifted frequencies
    fx = exp(-.5*(x/s).^2)';         % gaussian
    fxArray( :, iF )  = fx./max(fx); % gain-normalize
end

tf = sqrt( 2 ) * ifft( repmat( fft( signal ), [ 1 length( F ) ] ) .* fxArray );

h = subplot( 4, 3, 9, 'YDir', 'normal' );
imagesc( t, F, abs( tf' ) .^ 2 )
colorbar
set( h, 'YDir', 'normal' );
subplot( 4, 3, 8 );
plot( F, mean( abs( tf ) .^ 2 ) )
subplot( 4, 3, 7 )
plot( (0:pnts / 2) * fs / pnts, fxArray( 1:pnts / 2 + 1, : ) )

% Unit energy
fxArray_unit = fxArray * sqrt( pnts / max( sum( fxArray .^ 2 ) ) );

tf = ifft( repmat( fft( signal ), [ 1 length( F ) ] ) .* fxArray_unit );

h = subplot( 4, 3, 12, 'YDir', 'normal' );
imagesc( t, F, abs( tf' ) .^ 2 )
colorbar
set( h, 'YDir', 'normal' );
subplot( 4, 3, 11 );
plot( F, mean( abs( tf ) .^ 2 ) )
subplot( 4, 3, 10 )
plot( (0:pnts / 2) * fs / pnts, fxArray_unit( 1:pnts / 2 + 1, : ) )

> Am 26.08.2016 um 03:11 schrieb Makoto Miyakoshi <mmiyakoshi at ucsd.edu>:
> 
> Dear Norman,
> 
> > So what is it then? We haven't talked about a label for this type. Is there any? Maybe just wavelet analysis with variable cycles numbers? 
> 
> MJWAWVCN method? That's certainly too long and complicated. It describes what it does, but as a definition it's self-contradictory and bad. I wonder what Andreas would say on this.
> 
> > But I'm not sure if it is applicable for a wavelet analysis with variable cycles.
> 
> Normalization is a normalization, it does not matter whether the cycles are fixed or variable. Am I wrong?
> 
> Makoto
> 
> 
> On Wed, Aug 24, 2016 at 12:15 AM, Norman Forschack <forschack at cbs.mpg.de>wrote:
> Dear Makoto,
> 
> > But technically speaking, you vary the cycle > numbers and it's not wavelet anymore,
> > correct?
> Yes, I would say so. I just tried to outline that it's neither a Gabor transform/ STFT. For that, wavelets would need to have unit gain, fixed time and frequency resolution and require a different centering in time. So what is it then? We haven't talked about a label for this type. Is there any? Maybe just wavelet analysis with variable cycles numbers? 
> 
> About 3, I agree it would be nice to have multiple options available once raw output normalization is needed and I also cannot see why unit energy normalization should be wrong. It's been out for a long while. But I'm not sure if it is applicable for a wavelet analysis with variable cycles. So far, Mike and Andreas were refering to scaled wavelets, i.e. with fixed number of cycles. Maybe they could comment on that.
> 
> Norman
> 
> ---- Makoto Miyakoshi schrieb ----
> 
> 
> Dear Norman,
> 
> Thanks for your comment. I would like to confirm the following with you.
> 
> About (6),
> 
> > But in fact, there are a lot of possibilities how steps of these cycle numbers, and therefore the window widths, are defined across frequencies.
> 
> But technically speaking, you vary the cycle numbers and it's not wavelet anymore, correct? I understand there are several reasonable ways to adjust the cycles numbers, but that's not the main point here.
> 
> It does not matter even if newtimef() is actually not a wavelet transform, if we use is with varying number of cycles. I just want to give it a correct label and description that points us to exact methods with which engineers and specialists in other fields can understand. 
> 
> About (3), yes, if we see flat frequency responses it means underweighting the high-frequency power. That seems correct intuitively. However, raw output normalization is mainly for convenience anyway, and there could be more than one user need in doing this. It does not hurt to support multiple ways to normalize it, unless we all see that one method is apparently wrong. The energy normalization, as I understand, is not a 'wrong' method (or is it?) but one of reasonable and potentially useful ways to normalize the output--across frequency too.
> 
> Thank you again for your continued interest and contribution.
> If we can all agree with the summary, I'll feed it to EEGLAB developers. It was indeed one of the deepest discussion on the list in the past years!
> 
> Makoto
> 
> 
> 
> On Tue, Aug 23, 2016 at 3:19 PM, Norman Forschack <forschack at cbs.mpg.de>wrote:
> Dear Makoto and list,
> 
> I agree to almost all points with which Makoto summarized our discussion so well. Yet, after reading this excellent reference provided by Andreas, I like to further distinguish things within points 3 and 6. Please also correct me if I'm wrong.
> 
> Starting with 6, I wouldn't say that if the number of cycles in wavelet increases over frequency, this actually is a Gabor transform/STFT per se. According to Addison, there are at least two differences. For Gabor 'the internal frequency, f, is allowed to vary within the gaussian window of fixed width' (qoting Addison). In other words, time and frequency resolution stays the same across analysed frequencies once the window length is given. For wavelets it's possible to 'immitate' this by applying variable cycle numbers across frequencies (with which it's not a wavelet transform anymore as Andreas noted). But in fact, there are a lot of possibilities how steps of these cycle numbers, and therefore the window widths, are defined across frequencies. I think the most convenient control for resolution parameters can be gained by ''Define empirical FWHM of wavelet in the frequency domain' as Mike proposed' (point 4). This would, however, still require that wavelet amplitude is normalized to unit gain (i.e. maximum value normalization and not unit energy or, alternatively, unit energy plus 'normalized to the RMS amplitude of the sinusoid').
> Another difference is, 'that the wavelet complex sinusoid is centered at [the given time point] on the time axis whereas the complex sinusoid contained within the gabor atom is 'centered' at the origin (t=0). [...So for] STFT sinusoid remains fixed in relation to the origin [...] whereas for the wavelet [...] the sinusoid and window move together. This should get important if you want to extract phase related measures which, without a correction term, wouldn't be comparable between the two transforms.
> 
> For point 3, I think we haven't reached a consensus yet. I think Mike was referring to unit energy normalization when he stated: 'Trying to correct for this [maintain constant power across frequencies] while keeping the number of cycles constant is actually underestimating power at higher frequencies, because it involves trying to make the red bars more blue, even though the redness is accurate.' (if you're still listening Mike, is that right?) So it might be mentioned that applying unit energy normalization ensures that for an analysis with fixed cycle number across frequencies, the area covered by wavelet spectra (energy) is the same for all analysed center frequencies (but variable cycles is a different thing). (or maybe there's a more elegant way to put this).
> 
> I also want to thank all the folks contributing to this discussion, I learned a lot from you!
> Best
> Norman
> 
> 
> ----- Original Message -----
> From: "Makoto Miyakoshi" <mmiyakoshi at ucsd.edu>
> To: "Andreas Widmann" <widmann at uni-leipzig.de>, "Norman Forschack" <forschack at cbs.mpg.de>, "Niko Busch" <niko.busch at gmail.com>, "Mike X Cohen" <mikexcohen at gmail.com>
> Cc: "eeglablist" <eeglablist at sccn.ucsd.edu>, ramon at sccn.ucsd.edu
> Sent: Samstag, 20. August 2016 02:21:12
> Subject: Re: [Eeglablist] Units in output of timefreq - wavelet normalization
> 
> 
> Update--
> 
> 
> 
> Ok, let me confirm conclusions for the EEGLAB developers.
> 
> 
> 1) This is the bottom line: Let's not use the raw uV^2 outputs since these changes due to non-interesting factors as Mike mentioned. Instead, let's use dB changes against baseline etc.
> 
> 
> 2) If the raw uV^2 is absolutely necessary, there are probably two reasonable ways to normalize the raw values. One is, in Andreas's sample code, 'Analog to Gabor transform' in which A = 2. You'll find increasing power as frequency increases (which is the nature of the wavelet and not a problem.)
> 
> 
> 3) The other good one is 'Unit energy' in which A = sqrt(pnts./(sqrt(pi)*sigma_f). Probably, EEGLAB developers can implement both as options and leave the choice to the users. You'll find flatter response over the frequency range.
> 
> 4) Yet another good one is 'Define empirical FWHM of wavelet in the frequency domain' as Mike proposed. Personally, I found it most reasonable for intuitiveness. It's also normalized across frequencies.
> 
> 
> 5) EEGLAB's explanation about dftfilt3() needs to be at least more detailed, and potentially the default/optional normalization can be changes to one of the above two, as long as it does not conflict with backword compatibility.
> 
> 
> 6) Also, it's probably good to add the following description to EEGLAB functions that if the number of cycles in wavelet increases in the time domain as the frequency increases, technically it's not a wavelet transform but classified as a Gabor transform (special case of short-time FT).
> 
> 
> If I wrote something wrong, please correct me.
> 
> 
> Makoto
> 
> 
> 
> 
> 
> 
> On Fri, Aug 19, 2016 at 3:00 PM, Makoto Miyakoshi < mmiyakoshi at ucsd.edu > wrote:
> 
> 
> 
> Dear Andreas, Niko, Norman, Mike, and the list,
> 
> What a detailed discussion!
> Let me quickly confirm & summarize.
> 
> > The wavelets produced by dftfilt3 appear to always have an energy of srate (thus they are *not* unit energy normalized?!):
> [wavelet,cycles,freqresol,timeresol] = dftfilt3(F, ncycles, srate);
> E = sum( abs( wavelet{ 1 } ) .^ 2 )
> Consequently, to my understanding the correct „normfactor“ should be sqrt( E ) or better sqrt( srate ).
> 
> I confirmed with your code that it's E ~= srate which is not normalized. Yes, if it's all the same, it's better to be normalized using one of these reasonable ways...
> 
> 
> I discussed this normalization issue with Rey in Feb 2007 in SCCN as a visitor (also asked the same question to Arno in August 2006 in email). It was one of the most puzzling problems that I found that Herrmann's Morlet equation was different from that of Tallon-Baudry's and Gruber's in the normalization factor A; to make things worse, EEGLAB's timef()'s dftfilt() was further different. Rey agreed with me, saying that he was puzzled too. I was glad to know that I was not the one who was confused! Several months before I met him, he had already written dftfilt3() to support the standard Morlet wavelet in EEGLAB, for which I've been grateful.
> 
> 
> Ok, let me confirm conclusions for the EEGLAB developers.
> 
> 
> 1) This is the bottom line: Let's not use the raw uV^2 outputs since these changes due to non-interesting factors as Mike mentioned. Instead, let's use dB changes against baseline etc.
> 
> 
> 2) If the raw uV^2 is absolutely necessary, there are probably two reasonable ways to normalize the raw values. One is, in Andreas's sample code, 'Analog to Gabor transform' in which A = 2.
> 
> 
> 3) The other good one is 'Unit energy' in which A = sqrt(pnts./(sqrt(pi)*sigma_f). Probably, EEGLAB developers can implement both as options and leave the choice to the users.
> 
> 
> 4) EEGLAB's explanation about dftfilt3() needs to be at least more detailed, and potentially the default/optional normalization can be changes to one of the above two, as long as it does not conflict with backword compatibility.
> 
> 
> Let me know if I'm wrong and/or missing something important.
> 
> 
> As a list member, I'd like to express my deep gratitude for your depth of knowledge and kindness to put so much effort on this discussion. I learned a lot!
> 
> 
> Makoto
> 
> 
> 
> 
> On Thu, Aug 18, 2016 at 10:49 AM, Andreas Widmann < widmann at uni-leipzig.de > wrote:
> 
> 
> P.S.: I forgot that I wanted to add to the final paragraph how to estimate the resulting amplitude of a sinusoid in the unit energy normalized wavelet transform: the energy of the wavelet with the corresponding frequency peak amplitude normalized to the RMS amplitude of the sinusoid, e.g. (for Niko’s example):
> 
> mysig = 5 * sin(2*F*t*pi);
> [wavelet,cycles,freqresol,timeresol] = dftfilt3(F, ncycles, srate);
> temp = wavelet{1} / max( abs( wavelet{1} ) ) * sqrt( mean( abs( mysig ) .^ 2 ) );
> sum( abs( temp ) .^ 2 )
> sqrt( sum( abs( temp ) .^ 2 ) )
> 
> Best,
> Andreas
> 
> 
> 
> > Am 17.08.2016 um 21:07 schrieb Norman Forschack < forschack at cbs.mpg.de >:
> >
> > Dear Andreas,
> >
> > thanks a lot for your elaborate input! Below some further clarifications.
> >
> > First of all, I did not intend to further discuss question (1) as I found it quite convincing from your previous post that dftfilt3 is not computing results as we expect it to do.
> > But probably I should have put my initial question into other words (e.g. 'One question was' instead of 'The initial question was'), in order not to confuse things. So I am sorry for the mess!
> >
> > Coming to question (2) I do not fully agree here. Literally, Nico's questions were: 'what do values in tf represent and how should I "normalize" these values to match the amplitude of the input signal?' So, I was aiming to give some input on the second part. Below, using your code, Nico's signal is plotted together with both types of amplitude envelopes, i.e. gabor normalization and unit energy (see also picture attached). It's clear that gabor transform (i.e. maximal wavelet spectrum power) normalization closely matches the amplitude of the original signal, while the unit energy envelope shows much higher values. Unless there's a major bug in the code below (please check), I'd say gabor is more appropriate in this scenario and the normalization of a complex wavelet spectrum to maximal power is much more straight forward (see below).
> >
> > However, I totally agree that when it comes to EEG, things are different. As Makoto and Mike already pointed out before, we usually do not interpret raw EEG power, but rather refer to a baseline or another condition to compare with. In this case it shouldn't matter which wavelet normalization has been applied.
> >
> > The other points you mentioned, I found super super interesting and helpful and it would be great if you could comment on some emerging questions. Maybe it is also interesting for others to approach this intricate topic on a more detailed and basic level.
> >
> > You mentioned the gabor transform is rather suited for narrowband signals and EEG, however, is closer to white noise. But to my eyes EEG signals seem to be a lot more 'structured' e.g. when one thinks of the narrow spectral peaks of the rolandic rhythm in the alpha and beta band, or when a steady state evoked potential paradigm was applied.
> >
> > The gabor wavelet spectrum indeed seems to produce higher average amplitudes for higher frequencies, but the actual average values seem to vary much less than for the unit energy normalization, which then certainly does not produce 'flat' average amplitudes across frequencies. Do you think this is related to scaling issues?
> >
> > Unit energy normalization seems to increase frequency smearing for higher frequencies. Although one may apply a baseline normalization later, do you think this makes unit energy normalization less comparable to gabor normalization (i.e. contrary to what I've written above?)
> >
> > You said it doesn’t matter whether you normalize in the time or frequency domain. I'm confused now about this line of unit energy code:
> > A = sqrt( pnts ./ ( sqrt( pi ) * sigma_f ) );
> > Could you explain how sqrt(srate) normalization term transformed to the line above?
> > Also, when plotting the envelope based on this normalization, it yields an amplitude which is 25 times higher than in the original signal. When I normalized my complex wavelet with sqrt(srate)
> >> m = exp(-t.^2/(2*s^2)).*exp(1i*2*pi*F.*t) ./ sqrt(srate);
> >
> > as you did before for the dft3filt wavelet
> >> wavelet_norm{ 1 } = wavelet{ 1 } / sqrt( srate );
> >
> > I got an envelope which was just 2.5 times higher than the original signal. So I guess there's an error in my code. could you elaborate on that please?
> >
> > I am very much looking forward to your comments, thanks a lot Andreas!
> > Best
> > Norman
> >
> > figure links:
> > .fig: https://www.dropbox.com/s/ktoffojfcxxu6vd/wavelet_normalization.fig?dl=0
> > .png: https://www.dropbox.com/s/v5i4d95y53t2woi/wavelet_normalization.png?dl=0
> >
> > clear
> > D = 4; % total signal duration in seconds.
> > sigD = 1; % duration of the test oscillation within the signal.
> > F = 10; % frequency of the test oscillationin Hz.
> > P = .25; % Phase of the test oscillation. 2 pi radians = 360 degrees
> > srate = 256; % sampling rate, i.e. N points per sec used to represent sine wave.
> > dt = 1/srate; % sampling period, i.e. for this e.g. points at 1 ms intervals
> > time = dt:dt:D; % time vector.
> >
> > sigpoints = length(time)/2 - (sigD*srate)/2:(length(time)/2 + (sigD*srate)/2)-1;
> > mysig = zeros(1,D*srate);
> > mysig(sigpoints) = sin(2*F*time(sigpoints)*pi+ 2*pi*P);
> >
> > figure; hold all
> > plot(time,mysig);
> >
> > cycles = 7;
> > pnts = D * srate;
> >
> > freqs = F * ( pnts / srate );
> > sigma_f = repmat( freqs / cycles, [ pnts 1 ] );
> > f = repmat( 0:pnts - 1, [ size( freqs, 2 ) 1 ] )' - repmat( freqs, [ pnts 1 ] );
> > cmorlf = exp( -f .^ 2 ./ ( 2 * sigma_f .^ 2 ) );
> >
> > % Analog to Gabor transform
> > A = 2;
> > cmorlf_A = A .* cmorlf;
> >
> > tf = ifft( fft( mysig' ) .* cmorlf_A );
> >
> > plot( time, abs( tf ) .^ 2 , 'r-')
> >
> > % Unit energy
> >
> > A_unit = sqrt( pnts ./ ( sqrt( pi ) * sigma_f ) );
> > cmorlf_Au = A_unit .* cmorlf;
> >
> > tf_unit = ifft( fft( mysig' ) .* cmorlf_Au );
> >
> > plot( time, abs( tf_unit ) .^ 2, 'm:' )
> > legend({'Nico''s signal';'gabor';'unit energy'})
> >
> >
> >> ----- On Aug 17, 2016, at 3:05 PM, Andreas Widmann widmann at uni-leipzig.dewrote:
> >>
> >>> Dear Norman,
> >>>
> >>> sorry, I do not fully agree. First, I think it is important not to mix up the
> >>> two questions:
> >>>
> >>> (1) Does dftfilt3 provide the expected/documented results?
> >>> To my understanding this is not the case. I hope I do not miss anything obvious.
> >>>
> >>> vs. (2) What is the recommended/best/optimal wavelet normalization for EEG data
> >>> analysis?
> >>> I think there cannot be a generally valid recommendation and different solutions
> >>> are necessary for different applications. The normalization you suggested is
> >>> equivalent to the normalization of the Gabor transform (incorrectly abbreviated
> >>> to Gabor normalization in my previous post, sorry). This normalization is well
> >>> suited for narrowband signals and peaky spectra and the frequently used for
> >>> example in audio signal analysis. For broadband/noise signals, however, the
> >>> Gabor transform normalization overestimates high frequencies relative to low
> >>> frequencies.
> >>>
> >>> Please find below some code directly comparing both normalizations using a white
> >>> noise signal (btw. it doesn’t matter whether you normalize in the time or
> >>> frequency domain). In the global wavelet spectrum (middle column) power is
> >>> increasing with frequency for Gabor transform normalization (top row) but
> >>> „flat“ for unit energy normalization (bottom row).
> >>>
> >>> As EEG reflects a broadband and noisy signal I personally usually prefer to
> >>> apply energy normalization. As always, there are exceptions and other opinions
> >>> and special applications and ...
> >>>
> >>> Best,
> >>> Andreas
> >>>
> >>> fs = 256;
> >>> T = 4;
> >>> pnts = T * fs;
> >>> t = ( 0:pnts - 1 ) / fs;
> >>> cycles = 7;
> >>> F = 2:2:70;
> >>>
> >>> % White noise
> >>> rng( 0 )
> >>> signal = randn( fs * T, 1 );
> >>>
> >>> % Morlet wavelet
> >>> freqs = F * ( pnts / fs );
> >>> sigma_f = repmat( freqs / cycles, [ pnts 1 ] );
> >>> f = repmat( 0:pnts - 1, [ size( freqs, 2 ) 1 ] )' - repmat( freqs, [ pnts 1 ] );
> >>> cmorlf = exp( -f .^ 2 ./ ( 2 * sigma_f .^ 2 ) );
> >>>
> >>> figure
> >>>
> >>> % Analog to Gabor transform
> >>> A = 2;
> >>> cmorlf = A .* cmorlf;
> >>>
> >>> tf = ifft( repmat( fft( signal ), [ 1 length( F ) ] ) .* cmorlf );
> >>>
> >>> h = subplot( 2, 3, 3 );
> >>> imagesc( t, F, abs( tf' ) .^ 2 )
> >>> set( h, 'YDir', 'normal' );
> >>> subplot( 2, 3, 2 );
> >>> plot( F, mean( abs( tf ) .^ 2 ) )
> >>> subplot( 2, 3, 1 )
> >>> plot( (0:pnts / 2) * fs / pnts, cmorlf( 1:pnts / 2 + 1, : ) )
> >>>
> >>> % Unit energy
> >>> A = sqrt( pnts ./ ( sqrt( pi ) * sigma_f ) );
> >>> cmorlf = A .* cmorlf;
> >>>
> >>> tf = ifft( repmat( fft( signal ), [ 1 length( F ) ] ) .* cmorlf );
> >>>
> >>> h = subplot( 2, 3, 6 );
> >>> imagesc( t, F, abs( tf' ) .^ 2 )
> >>> set( h, 'YDir', 'normal' );
> >>> subplot( 2, 3, 5 );
> >>> plot( F, mean( abs( tf ) .^ 2 ) )
> >>> subplot( 2, 3, 4 )
> >>> plot( (0:pnts / 2) * fs / pnts, cmorlf( 1:pnts / 2 + 1, : ) )
> >>>
> >>>> Am 15.08.2016 um 18:19 schrieb Norman Forschack < forschack at cbs.mpg.de >:
> >>>>
> >>>> Dear all,
> >>>>
> >>>> I'd like to contribute from the perspective of a discussion on Mike Cohen's
> >>>> Blog.
> >>>>
> >>>> The initial question was, how to obtain an amplitude envelope of a given signal
> >>>> which has in fact the same amplitude as the given signal, right?
> >>>> So coming from Nicos signal:
> >>>>
> >>>> clear all
> >>>> D = 4; % total signal duration in seconds.
> >>>> sigD = 1; % duration of the test oscillation within the signal.
> >>>> F = 10; % frequency of the test oscillationin Hz.
> >>>> P = .25; % Phase of the test oscillation. 2 pi radians = 360 degrees
> >>>> srate = 256; % sampling rate, i.e. N points per sec used to represent sine wave.
> >>>> T = 1/srate; % sampling period, i.e. for this e.g. points at 1 ms intervals
> >>>> time = T:T:D; % time vector.
> >>>>
> >>>> sigpoints = length(time)/2 - (sigD*srate)/2:(length(time)/2 + (sigD*srate)/2)-1;
> >>>> mysig = zeros(1,D*srate);
> >>>> mysig(sigpoints) = sin(2*F*time(sigpoints)*pi+ 2*pi*P);
> >>>>
> >>>> one way to obtain equal amplitudes is to normalize the wavelet by its maximal
> >>>> value within the frequency domain:
> >>>>
> >>>> % some preparations
> >>>> mysig = mysig';
> >>>> ss = size(mysig);
> >>>> cycles = 4;
> >>>> dt = 1/srate;
> >>>> sf = F/cycles;
> >>>> s = 1./(2*pi*sf);
> >>>> t = (-4*s:dt:4*s)';
> >>>> nM = length(t);
> >>>> halfMsiz = (nM-1)/2;
> >>>> hz = linspace(0,srate/2,floor(nM/2)+1);
> >>>> Ly = ss(1)*ss(2)+nM-1;
> >>>> Ly2=pow2(nextpow2(ss(1)*ss(2)+nM-1));
> >>>>
> >>>> % fft of signal
> >>>> X=fft(reshape(mysig,ss(1)*ss(2),1), Ly2);
> >>>>
> >>>> % building morlet wavelet (without a normalization factor)
> >>>> m = exp(-t.^2/(2*s^2)).*exp(1i*2*pi*F.*t);
> >>>> H = fft(m,Ly2); % fft of wavelet
> >>>>
> >>>> % normalize wavelet spectrum
> >>>> H = H./max(H);
> >>>>
> >>>> y = ifft(X.*H,Ly2);
> >>>> y = y(floor(halfMsiz+1):Ly-ceil(halfMsiz));
> >>>> y_amp = 2* abs(y);
> >>>>
> >>>> figure; plot(time,mysig,'b',time,y_amp,'r')
> >>>>
> >>>> This seems to work for any combination of srate and cycles (except when number
> >>>> of cycles become large) because the signal spectrum is convolved by spectral
> >>>> wavelet values being maximally one.
> >>>>
> >>>> I have not fully worked my way through Andreas' example. It normalizes the
> >>>> wavelet in time, not in frequency domain as here. So it is probably not
> >>>> comparable.
> >>>> But doing time domain normalization within the lines above by just replacing the
> >>>> kernel formula:
> >>>>
> >>>> m = exp(-t.^2/(2*s^2)).*exp(1i*2*pi*F.*t) ./ sqrt(srate); % unit energy
> >>>>
> >>>> and commenting out the max(H) normalization, however, yields an amplitude
> >>>> envelope which is 2.5 times larger than the original signal amplitude and
> >>>> increases when the number of wavelet cycles is increased. But as Andreas
> >>>> suggestion referred to the dftfilt3 output, the matter becomes more complicated
> >>>> as this function uses it's own normalization factor:
> >>>> A = 1./sqrt(s*sqrt(pi));
> >>>> and there are problably some more relevant differences (not even going into the
> >>>> timefreq function).
> >>>>
> >>>> In sum, this post may have fostered the general confusion (or at least mine) but
> >>>> for a more puristic approach to the matter of wavelet normalization, the lines
> >>>> above might be of some value (kudos to mike x cohen, of course).
> >>>>
> >>>> All the best
> >>>> Norman
> >>>>
> >>>>
> >>>> ----- On Aug 12, 2016, at 6:06 PM, Andreas Widmann widmann at uni-leipzig.de wrote:
> >>>>
> >>>>> Dear Niko,
> >>>>>
> >>>>> I’m puzzled by this difference since a long time too (and as you have written a
> >>>>> book chapter on WT actually I would have hoped you could help resolving this
> >>>>> issue ;).
> >>>>>
> >>>>> (Morlet) wavelet normalization always appeared somewhat arbitrary to me (as
> >>>>> signal amplitude will never be directly reflected across the whole TF plane for
> >>>>> peaky spectra/time courses). To my understanding the most common normalization
> >>>>> for wavelets is unit energy (and Gabor). The help text for timefreq states that
> >>>>> dftfilt3 is "exact Tallon Baudry“. TB (1998, JNeurosci) states that "Wavelets
> >>>>> were normalized so that their total energy was 1,…“.
> >>>>>
> >>>>> The wavelets produced by dftfilt3 appear to always have an energy of srate (thus
> >>>>> they are *not* unit energy normalized?!):
> >>>>> [wavelet,cycles,freqresol,timeresol] = dftfilt3(F, ncycles, srate);
> >>>>> E = sum( abs( wavelet{ 1 } ) .^ 2 )
> >>>>> Consequently, to my understanding the correct „normfactor“ should be sqrt( E )
> >>>>> or better sqrt( srate ).
> >>>>>
> >>>>> You might want to confirm by looking at the TF transform of the (real part of
> >>>>> the) wavelet itself
> >>>>> wavelet{ 1 } = wavelet{ 1 } / sqrt( srate ); % Unit energy normalize
> >>>>> mysig = zeros(1,D*srate);
> >>>>> delay = ceil( ( length( mysig ) - length( wavelet{ 1 } ) ) / 2 );
> >>>>> mysig( delay:delay + length( wavelet{ 1 } ) - 1 ) = real( wavelet{ 1 } ) * 2;
> >>>>> % Discard imag part
> >>>>> normfactor = sqrt( srate );
> >>>>> which should now have a peak amplitude of 1 (independent of sampling rate and
> >>>>> signal duration).
> >>>>>
> >>>>> As the issue appears to be not only in EEGLAB but also other implementations, I
> >>>>> always assumed my reasoning to be incorrect. Is it? What am I missing?
> >>>>>
> >>>>> Best,
> >>>>> Andreas
> >>>>>
> >>>>>> Am 10.08.2016 um 11:58 schrieb Niko Busch < niko.busch at gmail.com >:
> >>>>>>
> >>>>>> Dear Makoto (and everyone who replied to me personally regarding this post),
> >>>>>>
> >>>>>> thank you for your reply! I see that the result of the wavelet transform inside
> >>>>>> the timefreq function is dependent on the length of the signal, which in turn
> >>>>>> is dependent on the number of cycles and sampling rate. However, simply
> >>>>>> dividing by the length of the wavelet does not seem to be the solution either.
> >>>>>> I modified the code below by including a "normalization factor", which
> >>>>>> currently is simply the length of the wavelet. Dividing the wavelet transformed
> >>>>>> amplitudes by this factor gives the right order of magnitude, but the results
> >>>>>> are still quite off. By increasing the sampling rate or number of cycles, the
> >>>>>> results are even more off. I believe we are on the right track, but something
> >>>>>> is still missing. Do you have any ideas?
> >>>>>>
> >>>>>> Cheers,
> >>>>>> Niko
> >>>>>>
> >>>>>> %% Create sine wave
> >>>>>> clear all
> >>>>>> D = 4; % total signal duration in seconds.
> >>>>>> sigD = 1; % duration of the test oscillation within the signal.
> >>>>>> F = 10; % frequency of the test oscillationin Hz.
> >>>>>> P = .25; % Phase of the test oscillation. 2 pi radians = 360 degrees
> >>>>>> srate = 256; % sampling rate, i.e. N points per sec used to represent sine wave.
> >>>>>> T = 1/srate; % sampling period, i.e. for this e.g. points at 1 ms intervals
> >>>>>> t = [T:T:D]; % time vector.
> >>>>>>
> >>>>>> sigpoints = length(t)/2 - (sigD*srate)/2:(length(t)/2 + (sigD*srate)/2)-1;
> >>>>>> mysig = zeros(1,D*srate);
> >>>>>> mysig(sigpoints) = sin(2*F*t(sigpoints)*pi+ 2*pi*P);
> >>>>>>
> >>>>>> %% TF computation
> >>>>>> ncycles = 4;
> >>>>>>
> >>>>>> [wavelet,cycles,freqresol,timeresol] = dftfilt3(F, ncycles, srate);
> >>>>>> normfactor = length(wavelet{1});
> >>>>>>
> >>>>>> [tf, outfreqs, outtimes] = timefreq(mysig', srate, ...
> >>>>>> 'cycles', ncycles, 'wletmethod', 'dftfilt3', 'freqscale', 'linear', ...
> >>>>>> 'freqs', F);
> >>>>>>
> >>>>>> %% Plot
> >>>>>> figure; hold all
> >>>>>> plot(t,mysig);
> >>>>>> plot(outtimes./1000,abs(tf)./normfactor)
> >>>>>> xlabel('Time (seconds)');
> >>>>>> ylabel('Amplitude');
> >>>>>> legend('input signal', 'wavelet result')
> >>>>>>
> >>>>>> _______________________________________________
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> >>>>>> For digest mode, send an email with the subject "set digest mime" to
> >>>>>> eeglablist-request at sccn.ucsd.edu
> >>>>>
> >>>>> _______________________________________________
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> >>>> _______________________________________________
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> > _______________________________________________
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> _______________________________________________
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> 
> 
> --
> 
> 
> Makoto Miyakoshi
> Swartz Center for Computational Neuroscience
> Institute for Neural Computation, University of California San Diego
> 
> 
> 
> 
> --
> 
> 
> Makoto Miyakoshi
> Swartz Center for Computational Neuroscience
> Institute for Neural Computation, University of California San Diego
> 
> 
> 
> -- 
> Makoto Miyakoshi
> Swartz Center for Computational Neuroscience
> Institute for Neural Computation, University of California San Diego
> 
> 
> 
> -- 
> Makoto Miyakoshi
> Swartz Center for Computational Neuroscience
> Institute for Neural Computation, University of California San Diego
> _______________________________________________
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