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function [sts] = ft_spiketriggeredspectrum_convol(cfg, data, spike)
% FT_SPIKETRIGGEREDSPECTRUM_CONVOL computes the Fourier spectrum (amplitude and
% phase) of the LFP around the spikes using convolution of the complete LFP traces. A
% phase of zero corresponds to the spike being on the peak of the LFP oscillation. A
% phase of 180 degree corresponds to the spike being in the through of the oscillation.
% A phase of 45 degrees corresponds to the spike being just after the peak in the LFP.
%
% The difference to FT_SPIKETRIGGEREDSPECTRUM_FFT is that this function allows for
% multiple frequencies to be processed with different time-windows per frequency, and
% that FT_SPIKETRIGGEREDSPECTRUM_FFT is based on taking the FFT of a limited LFP
% segment around each spike.
%
% Use as
% [sts] = ft_spiketriggeredspectrum_convol(cfg,data,spike)
% or
% [sts] = ft_spiketriggeredspectrum_convol(cfg,data)
% The spike data can either be contained in the data input or in the spike
% input.
%
% The input DATA should be organised as the raw datatype, obtained from
% FT_PREPROCESSING or FT_APPENDSPIKE.
%
% The input SPIKE should be organised as the spike or the raw datatype, obtained from
% FT_SPIKE_MAKETRIALS or FT_PREPROCESSING, in which case the conversion is done
% within this function.
%
% Important is that data.time and spike.trialtime should be referenced
% relative to the same trial trigger times!
%
% Configurations (following largely FT_FREQNALYSIS with method mtmconvol)
% cfg.tapsmofrq = vector 1 x numfoi, the amount of spectral smoothing through
% multi-tapering. Note that 4 Hz smoothing means
% plus-minus 4 Hz, i.e. a 8 Hz smoothing box.
% cfg.foi = vector 1 x numfoi, frequencies of interest
% cfg.taper = 'dpss', 'hanning' or many others, see WINDOW (default = 'hanning')
% cfg.t_ftimwin = vector 1 x numfoi, length of time window (in
% seconds)
% cfg.taperopt = parameter that goes in WINDOW function (only
% applies to windows like KAISER).
% cfg.spikechannel = cell-array with selection of channels (default = 'all')
% see FT_CHANNELSELECTION for details
% cfg.channel = Nx1 cell-array with selection of channels (default = 'all'),
% see FT_CHANNELSELECTION for details
% cfg.borderspikes = 'yes' (default) or 'no'. If 'yes', we process the spikes
% falling at the border using an LFP that is not centered
% on the spike. If 'no', we output NaNs for spikes
% around which we could not center an LFP segment.
% cfg.rejectsaturation= 'yes' (default) or 'no'. If 'yes', we set
% EEG segments where the maximum or minimum
% voltage range is reached
% with zero derivative (i.e., saturated signal) to
% NaN, effectively setting all spikes phases that
% use these parts of the EEG to NaN. An EEG that
% saturates always returns the same phase at all
% frequencies and should be ignored.
%
% Note: some adjustment of the frequencies can occur as the chosen time-window may not
% be appropriate for the chosen frequency.
% For example, suppose that cfg.foi = 80, data.fsample = 1000, and
% cfg.t_ftimwin = 0.0625. The DFT frequencies in that case are
% linspace(0,1000,63) such that cfg.foi --> 80.645. In practice, this error
% can only become large if the number of cycles per frequency is very
% small and the frequency is high. For example, suppose that cfg.foi = 80
% and cfg.t_ftimwin = 0.0125. In that case cfg.foi-->83.33.
% The error is smaller as data.fsample is larger.
%
% Outputs:
% sts is a spike structure, containing new fields:
% sts.fourierspctrm = 1 x nUnits cell-array with dimord spike_lfplabel_freq
% sts.lfplabel = 1 x nChan cell-array with EEG labels
% sts.freq = 1 x nFreq frequencies. Note that per default, not
% all frequencies can be used as we compute the DFT
% around the spike based on an uneven number of
% samples. This introduces a slight adjustment of the
% selected frequencies.
%
% Note: sts.fourierspctrm can contain NaNs, for example if
% cfg.borderspikes = 'no', or if cfg.rejectsaturation = 'yes', or if the
% trial length was too short for the window desired.
%
% WHen using multitapering, the phase distortion is corrected for.
%
% The output STS data structure can be input to FT_SPIKETRIGGEREDSPECTRUM_STAT
% Copyright (C) 2008-2012, Martin Vinck
%
% This file is part of FieldTrip, see http://www.fieldtriptoolbox.org
% for the documentation and details.
%
% FieldTrip is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% FieldTrip is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with FieldTrip. If not, see <http://www.gnu.org/licenses/>.
%
% $Id$
% these are used by the ft_preamble/ft_postamble function and scripts
ft_revision = '$Id$';
ft_nargin = nargin;
ft_nargout = nargout;
% do the general setup of the function
ft_defaults
ft_preamble init
ft_preamble provenance data spike
% check if the input data is valid for this function
data = ft_checkdata(data, 'datatype', {'raw'}, 'feedback', 'yes');
if nargin==3
spike = ft_checkdata(spike, 'datatype', {'spike'}, 'feedback', 'yes');
end
% ensure that the required options are present
cfg = ft_checkconfig(cfg, 'required', {'foi','t_ftimwin'});
% get the options
cfg.borderspikes = ft_getopt(cfg, 'borderspikes','yes');
cfg.taper = ft_getopt(cfg, 'taper','hanning');
cfg.taperopt = ft_getopt(cfg, 'taperopt',[]);
cfg.spikechannel = ft_getopt(cfg,'spikechannel', 'all');
cfg.channel = ft_getopt(cfg,'channel', 'all');
cfg.rejectsaturation = ft_getopt(cfg,'rejectsaturation', 'yes');
% ensure that the options are valid
cfg = ft_checkopt(cfg, 'taper',{'char', 'function_handle'});
cfg = ft_checkopt(cfg, 'borderspikes','char',{'yes', 'no'});
cfg = ft_checkopt(cfg, 't_ftimwin',{'doublevector', 'doublescalar'});
cfg = ft_checkopt(cfg, 'foi',{'doublevector', 'doublescalar'});
cfg = ft_checkopt(cfg, 'spikechannel',{'cell', 'char', 'double'});
cfg = ft_checkopt(cfg, 'channel', {'cell', 'char', 'double'});
cfg = ft_checkopt(cfg, 'taperopt', {'double','empty'});
cfg = ft_checkopt(cfg, 'rejectsaturation','char', {'yes', 'no'});
if isequal(cfg.taper, 'dpss')
cfg = ft_checkconfig(cfg, 'required', {'tapsmofrq'});
cfg = ft_checkopt(cfg,'tapsmofrq',{'doublevector', 'doublescalar'});
end
cfg = ft_checkconfig(cfg, 'allowed', {'taper', 'borderspikes', 't_ftimwin', 'foi', 'spikechannel', 'channel', 'taperopt', 'rejectsaturation','tapsmofrq'});
% length of tapsmofrq, foi and t_ftimwin should all be matched
if isfield(cfg,'tapsmofrq')
if length(cfg.tapsmofrq) ~= length(cfg.foi) || length(cfg.foi)~=length(cfg.t_ftimwin)
error('lengths of cfg.tapsmofrq, cfg.foi and cfg_t_ftimwin should be equal and 1 x nFreqs')
end
end
% get the spikechannels
if nargin==2
% autodetect the spikechannels and EEG channels
[spikechannel, eegchannel] = detectspikechan(data);
if strcmp(cfg.spikechannel, 'all'),
cfg.spikechannel = spikechannel;
else
cfg.spikechannel = ft_channelselection(cfg.spikechannel, data.label);
if ~all(ismember(cfg.spikechannel,spikechannel)), error('some selected spike channels appear eeg channels'); end
end
if strcmp(cfg.channel,'all')
cfg.channel = eegchannel;
else
cfg.channel = ft_channelselection(cfg.channel, data.label);
if ~all(ismember(cfg.channel,eegchannel)), warning('some of the selected eeg channels appear spike channels'); end
end
tmpcfg = [];
tmpcfg.channel = cfg.spikechannel;
data_spk = ft_selectdata(tmpcfg, data);
tmpcfg.channel = cfg.channel;
data = ft_selectdata(tmpcfg, data); % leave only LFP
spike = ft_checkdata(data_spk,'datatype', 'spike');
clear data_spk % remove the continuous data
else
cfg.spikechannel = ft_channelselection(cfg.spikechannel, spike.label);
cfg.channel = ft_channelselection(cfg.channel, data.label);
end
% determine the channel indices and number of chans
chansel = match_str(data.label, cfg.channel); % selected channels
nchansel = length(cfg.channel); % number of channels
spikesel = match_str(spike.label, cfg.spikechannel);
nspikesel = length(spikesel); % number of spike channels
if nspikesel==0, error('no units were selected'); end
if nchansel==0, error('no channels were selected'); end
% preallocate
nTrials = length(data.trial); % number of trials
[spectrum,spiketime, spiketrial] = deal(cell(nspikesel,nTrials)); % preallocate the outputs
[unitsmp,unittime,unitshift] = deal(cell(1,nspikesel));
nSpikes = zeros(1,nspikesel);
% construct the frequency axis and restrict to unique frequencies
if ~isfield(data, 'fsample'), data.fsample = 1/mean(diff(data.time{1})); end
if any(cfg.foi > (data.fsample/2))
error('frequencies in cfg.foi are above Nyquist frequency')
end
numsmp = round(cfg.t_ftimwin .* data.fsample);
numsmp(~mod(numsmp,2)) = numsmp(~mod(numsmp,2))+1; % make sure we always have uneven samples, since we want the spike in the middle
foi = zeros(1,length(cfg.foi));
for iSmp = 1:length(numsmp)
faxis = linspace(0,data.fsample,numsmp(iSmp));
findx = nearest(faxis,cfg.foi(iSmp));
[foi(iSmp)] = deal(faxis(findx)); % this is the actual frequency used, from the DFT formula
end
%
[cfg.foi,B,C] = unique(foi); % take the unique frequencies from this
cfg.t_ftimwin = cfg.t_ftimwin(B);
% compute the minima and maxima of the data, this is done to remove EEG portions where there are potential saturation effects
if strcmp(cfg.rejectsaturation,'yes')
[minChan,maxChan] = deal([]);
for iChan = 1:nchansel
mx = -inf;
mn = +inf;
for iTrial = 1:nTrials
minTrial = nanmin(data.trial{iTrial}(iChan,:));
maxTrial = nanmax(data.trial{iTrial}(iChan,:));
if maxTrial>mx, mx = maxTrial; end
if minTrial<mn, mn = minTrial; end
end
minChan(iChan) = mn;
maxChan(iChan) = mx;
end
end
% compute the spectra
ft_progress('init', 'text', 'Please wait...');
for iTrial = 1:nTrials
% select the spike times for a given trial and restrict to those overlapping with the EEG
x = data.time{iTrial};
timeBins = [x x(end)+1/data.fsample] - (0.5/data.fsample);
for iUnit = 1:nspikesel
unitindx = spikesel(iUnit);
hasTrial = spike.trial{unitindx} == iTrial; % find the spikes that are in the trial
ts = spike.time{unitindx}(hasTrial); % get the spike times for these spikes
vld = ts>=timeBins(1) & ts<=timeBins(end); % only select those spikes that fall in the trial window
ts = ts(vld); % timestamps for these spikes
[ignore,I] = histc(ts,timeBins);
if ~isempty(ts)
ts(I==0 | I==length(timeBins)) = [];
end
I(I==0 | I==length(timeBins)) = [];
unitsmp{iUnit} = I;
unittime{iUnit} = ts(:); % this is for storage in the output structure
smptime = data.time{iTrial}(unitsmp{iUnit}); % times corresponding to samples
unitshift{iUnit} = ts(:) - smptime(:); % shift induced by shifting to sample times, important for high-frequency oscillations
nSpikes(iUnit) = length(unitsmp{iUnit});
end
if ~any(nSpikes),continue,end % continue to the next trial if this one does not contain valid spikes
% set the saturated parts of the data to NaNs
if strcmp(cfg.rejectsaturation,'yes')
for iChan = 1:nchansel
isSaturated = find(diff(data.trial{iTrial}(chansel(iChan),:))==0)+1;
remove = data.trial{iTrial}(chansel(iChan),isSaturated)==maxChan(iChan) | data.trial{iTrial}(chansel(iChan),isSaturated)==minChan(iChan);
remove = isSaturated(remove);
data.trial{iTrial}(chansel(iChan),remove) = NaN;
if ~isempty(remove)
fprintf('setting %d points from channel %s in trial %d to NaN\n', length(remove), cfg.channel{iChan}, iTrial);
end
end
end
% preallocate
nFreqs = length(cfg.foi);
for iUnit = 1:nspikesel
if nSpikes(iUnit)>0, spectrum{iUnit,iTrial} = zeros(nSpikes(iUnit),nchansel,nFreqs); end
end
% process the phases for every chan-freq combination separately to save memory
for iFreq = 1:nFreqs
% construct the input options for the sub-function phase_est
tmpcfg = cfg;
tmpcfg.foi = cfg.foi(iFreq);
try tmpcfg.tapsmofrq = cfg.tapsmofrq(iFreq);end
tmpcfg.t_ftimwin = cfg.t_ftimwin(iFreq);
if tmpcfg.t_ftimwin>=(data.time{iTrial}(end)-data.time{iTrial}(1))
warning('time window for frequency %.2f Hz too large for trial %d', tmpcfg.foi, iTrial);
spectrum{iUnit,iTrial}(:,:,iFreq) = NaN;
continue;
end
% just some message
if nTrials==1
ft_progress(iFreq/nFreqs, 'Processing frequency %d from %d', iFreq, nFreqs);
else
ft_progress(iTrial/nTrials, 'Processing trial %d from %d', iTrial, nTrials);
end
% compute the LFP phase at every time-point
spec = zeros(length(data.time{iTrial}),nchansel);
for iChan = 1:nchansel
[spec(:,iChan),foi, numsmp] = phase_est(tmpcfg,data.trial{iTrial}(chansel(iChan),:),data.time{iTrial}, data.fsample);
end
% select now the proper phases for every unit
for iUnit = 1:nspikesel
if nSpikes(iUnit)==0, continue,end
% gather the phases at the samples where we had the spikes
spectrum{iUnit,iTrial}(:,:,iFreq) = spec(unitsmp{iUnit},:);
% preallocate rephasing vector
rephase = ones(nSpikes(iUnit),1);
% do the rephasing and use proper phases for spikes at borders
if strcmp(cfg.borderspikes,'yes')
% use an LFP window not centered around the spike, to use spikes around the trial borders as well
beg = 1+(numsmp-1)/2; % find the borders empirically
ed = length(data.time{iTrial}) - beg + 1;
vldSpikes = unitsmp{iUnit}>=beg & unitsmp{iUnit}<=ed; % determine the valid spikes
earlySpikes = unitsmp{iUnit}<beg;
lateSpikes = unitsmp{iUnit}>ed;
if any(earlySpikes)
rephase(earlySpikes) = exp(1i*2*pi*foi* (unittime{iUnit}(earlySpikes) - data.time{iTrial}(beg))' );
for iChan = 1:nchansel
spectrum{iUnit,iTrial}(earlySpikes,iChan,iFreq) = spec(beg,iChan);
end
end
if any(lateSpikes)
rephase(lateSpikes) = exp(1i*2*pi*foi * (unittime{iUnit}(lateSpikes) - data.time{iTrial}(ed)) );
for iChan = 1:nchansel
spectrum{iUnit,iTrial}(lateSpikes,iChan,iFreq) = spec(ed,iChan);
end
end
if any(vldSpikes)
rephase(vldSpikes) = exp(1i*2*pi*foi*unitshift{iUnit}(vldSpikes));
end
else
% in this case the spikes that fall around borders are set to NaN
if ~isempty(unitshift{iUnit})
rephase(1:end) = exp(1i*2*pi*foi*unitshift{iUnit});
end
end
% Rephase the spectrum, this serves two purposes:
% 1) correct for potential misallignment of spikes to LFP sampling axis
% 2) for spikes falling at borders, we need to account for the fact that we
% used the instantaneous phase at a different moment in time
for iChan = 1:nchansel
spectrum{iUnit,iTrial}(:,iChan,iFreq) = spectrum{iUnit,iTrial}(:,iChan,iFreq).*rephase;
end
% Store the spiketimes and spiketrials, constructing a type of SPIKE format
spiketime{iUnit,iTrial} = unittime{iUnit};
spiketrial{iUnit,iTrial} = iTrial*ones(1,nSpikes(iUnit));
end
end
end
ft_progress('close');
% collect the results
sts.lfplabel = data.label(chansel);
sts.freq = cfg.foi;
sts.label = spike.label(spikesel);
for iUnit = 1:nspikesel
sts.fourierspctrm{iUnit} = cat(1, spectrum{iUnit,:});
spectrum(iUnit,:) = {[]}; % free from the memory
sts.time{iUnit} = cat(1, spiketime{iUnit,:});
sts.trial{iUnit} = cat(2, spiketrial{iUnit,:})';
end
sts.dimord = '{chan}_spike_lfpchan_freq';
sts.trialtime = spike.trialtime;
% do the general cleanup and bookkeeping at the end of the function
ft_postamble previous data spike
ft_postamble provenance sts
ft_postamble history sts
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% SUBFUNCTION
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [spctrm,foi, numsmp] = phase_est(cfg,dat,time,fsample)
% Phase estimation function
% Determine fsample and set total time-length of data
if nargin<4
fsample = 1./mean(time(2:end)-time(1:end-1)); % round off errors!
end
numsmp = round(cfg.t_ftimwin .* fsample);
numsmp(~mod(numsmp,2)) = numsmp(~mod(numsmp,2))+1; % make sure we always have uneven samples, since we want the spike in the middle
faxis = linspace(0,fsample,numsmp);
findx = nearest(faxis,cfg.foi);
[cfg.foi,foi] = deal(faxis(findx)); % this is the actual frequency used, from the DFT formula
timwinSamples = numsmp;
% Compute tapers per frequency, multiply with wavelets and compute their fft
switch cfg.taper
case 'dpss'
% create a sequence of DPSS tapers
taper = double_dpss(timwinSamples, timwinSamples .* (cfg.tapsmofrq ./ fsample))';
taper = taper(1:(end-1), :); % removing the last taper
% give error/warning about number of tapers
if isempty(taper)
error('%.3f Hz: datalength to short for specified smoothing\ndatalength: %.3f s, smoothing: %.3f Hz, minimum smoothing: %.3f Hz',...
cfg.foi, timwinSamples/fsample,cfg.tapsmofrq,fsample/timwinSamples);
elseif size(taper,1) == 1
warning('using only one taper for specified smoothing for %.2f Hz', cfg.foi)
end
case 'sine'
taper = sine_taper(timwinSamples, timwinSamples .* (cfg.tapsmofrq ./ fsample))';
taper = taper(1:(end-1), :);
otherwise
% create a single taper according to the standard window specification
if isempty(cfg.taperopt)
taper = window(cfg.taper, timwinSamples)';
else
try
taper = window(cfg.taper, timwinSamples,cfg.taperopt)';
catch
error('taper option was not appropriate for taper');
end
end
end
% do some check on the taper size
if size(taper,1)==numsmp, taper = taper'; end
% normalize taper if there's 1 and all are positive
if size(taper,1)==1 && all(taper>=0)
taper = numsmp*taper./sum(taper); % magnitude of cosine is now returned
end
%%%% fit linear regression for every datapoint: to remove mean and ramp of signal
sumKern = ones(1,timwinSamples);
avgKern = sumKern./timwinSamples;
xKern = timwinSamples:-1:1; % because of definition conv2 function
meanX = mean(xKern);
sumX = sum(xKern);
% beta1 = (sum(x.*y) - sum(x)*sum(y)/n) ./ (sum((x-meanx).^2): standard linear regr formula
beta1 = (conv2(dat(:),xKern(:),'same') - sumX.*conv2(dat(:),sumKern(:),'same')/timwinSamples ) ./ sum((xKern-meanX).^2);
beta0 = conv2(dat(:),avgKern(:),'same') - beta1.*meanX; % beta0 = mean(dat) - beta1*mean(x)
% DFT formula: basefunctions: cos(2*pi*k*[0:numsmp-1]/numsmp) and sin(2*pi*k*[0:numsmp-1]/numsmp)
% center the base functions such that the peak of the cos function is at the center. See:
% f = findx-1; ax = -(numsmp-1)/2:(numsmp-1)/2; y = cos(2*pi.*ax.*f/numsmp);figure, plot(ax,y,'sr');
% y2 = cos(2*pi.*linspace(-(numsmp-1)/2,(numsmp-1)/2,1000).*f/numsmp);
% hold on, plot(linspace(-(numsmp-1)/2,(numsmp-1)/2,1000),y2,'r-')
% correcting for phase rotation (as with multitapering)
nTapers = size(taper,1);
indN = -(numsmp-1)/2:(numsmp-1)/2;
spctrmRot = complex(zeros(1,1));
for iTaper = 1:nTapers
coswav = taper(iTaper,:).*cos(2*pi*(findx-1)*indN/numsmp);
sinwav = taper(iTaper,:).*sin(2*pi*(findx-1)*indN/numsmp);
wavelet = complex(coswav(:), sinwav(:));
cosSignal = cos(2*pi*(findx-1)*indN/numsmp);
spctrmRot = spctrmRot + sum(wavelet.*cosSignal(:));
end
phaseCor = angle(spctrmRot);
%%%% compute the spectra
nTapers = size(taper,1);
indN = -(numsmp-1)/2:(numsmp-1)/2;
spctrm = complex(zeros(length(dat),1));
for iTaper = 1:nTapers
coswav = taper(iTaper,:).*cos(2*pi*(findx-1)*indN/numsmp);
sinwav = taper(iTaper,:).*sin(2*pi*(findx-1)*indN/numsmp);
wavelet = complex(coswav(:), sinwav(:));
fftRamp = sum(xKern.*coswav) + 1i*sum(xKern.*sinwav); % fft of ramp with dx/ds = 1 * taper
fftDC = sum(ones(1,timwinSamples).*coswav) + 1i*sum(ones(1,timwinSamples).*sinwav); % fft of unit direct current * taper
spctrm = spctrm + (conv_fftbased(dat(:),wavelet) - (beta0*fftDC + beta1.*fftRamp))/(numsmp/2);
% fft % mean %linear ramp % make magnitude invariant to window length
end
spctrm = spctrm./nTapers; % normalize by number of tapers
spctrm = spctrm.*exp(-1i*phaseCor);
% set the part of the spectrum without a valid phase to NaNs
n = (timwinSamples-1)/2;
spctrm(1:n) = NaN;
spctrm(end-n+1:end) = NaN;
function [taper] = double_dpss(a, b, varargin)
taper = dpss(double(a), double(b), varargin{:});
function [spikelabel, eeglabel] = detectspikechan(data)
maxRate = 1000; % default on what we still consider a neuronal signal
% autodetect the spike channels
ntrial = length(data.trial);
nchans = length(data.label);
spikechan = zeros(nchans,1);
for i=1:ntrial
for j=1:nchans
hasAllInts = all(isnan(data.trial{i}(j,:)) | data.trial{i}(j,:) == round(data.trial{i}(j,:)));
hasAllPosInts = all(isnan(data.trial{i}(j,:)) | data.trial{i}(j,:)>=0);
fr = nansum(data.trial{i}(j,:),2) ./ (data.time{i}(end)-data.time{i}(1));
spikechan(j) = spikechan(j) + double(hasAllInts & hasAllPosInts & fr<=maxRate);
end
end
spikechan = (spikechan==ntrial);
spikelabel = data.label(spikechan);
eeglabel = data.label(~spikechan);
% CONVOLUTION: FFT BASED IMPLEMENTATION
function c = conv_fftbased(a, b)
P = numel(a);
Q = numel(b);
L = P + Q - 1;
K = 2^nextpow2(L);
c = ifft(fft(a, K) .* fft(b, K));
c = c(1:L);
toRm1 = [1:(Q-1)/2];
toRm2 = [(1 + length(c) - (Q-1)/2) : length(c)];
toRm = [toRm1(:); toRm2(:)];
c(toRm) = [];