function [retval, s, errorb, tau] = allan_cov(data,tau,name,verbose) % ALLAN Compute the Allan deviation for a set of time-domain frequency data % [RETVAL, S, ERRORB, TAU] = ALLAN(DATA,TAU,NAME,VERBOSE) % % Inputs: % DATA should be a structure and have the following fields: % DATA.freq or DATA.phase % A vector of fractional frequency measurements (df/f) in % DATA.freq *or* phase offset data (seconds) in DATA.phase . % If frequency data is not present, it will be generated by % differentiating the phase data. % If both fields are present, then DATA.freq will be used. % Note: for general-purpose calculations of Allan deviation, % (i.e. a two-sample variance) use DATA.freq . % % DATA.rate or DATA.time % The sampling rate in Hertz (DATA.rate) or a vector of % timestamps for each measurement in seconds (DATA.time). % DATA.rate is used if both fields are present. % If DATA.rate == 0, then the timestamps are used. % % DATA.units (optional) % The units for the data. If present, the string DATA.units % is added to the plot y-axis label. % % TAU is an array of tau values for computing Allan deviation. % TAU values must be divisible by 1/DATA.rate (data points cannot be % grouped in fractional quantities!) and invalid values are ignored. % Leave empty to use default values. % NAME is an optional label that is added to the plot titles. % VERBOSE sets the level of status messages: % 0 = silent & no data plots; % 1 = status messages & minimum plots; % 2 = all messages and plots (default) % % Outputs: % RETVAL is the array of Allan deviation values at each TAU. % S is an optional output of other statistical measures of the data (mean, std, etc). % ERRORB is an optional output containing the error estimates for a 1-sigma % confidence interval. These values are shown on the figure for each point. % TAU is an optional output containing the array of tau values used in the % calculation (which may be a truncated subset of the input or default values). % % Example: % % To compute the Allan deviation for the data in the variable "lt": % >> lt % lt = % freq: [1x86400 double] % rate: 0.5 % % Use: % % >> ad = allan(lt,[2 10 100],'lt data',1); % % The Allan deviation will be computed and plotted at tau = 2,10,100 seconds. % 1-sigma confidence intervals will be indicated by vertical lines at each point. % You can also use the default settings, which are usually a good starting point: % % >> ad = allan(lt); % % % Notes: % This function calculates the standard Allan deviation (ADEV), *not* the % overlapping ADEV. Use "allan_overlap.m" for overlapping ADEV. % The calculation is performed using fractional frequency data. If only % phase data is provided, frequency data is generated by differentiating % the phase data. % No pre-processing of the data is performed, except to remove any % initial offset (i.e., starting gap) in the time record. % For rate-based data, ADEV is computed only for tau values greater than the % minimum time between samples and less than the half the total time. For % time-stamped data, only tau values greater than the maximum gap between % samples and less than half the total time are used. % The calculation for fixed sample rate data is *much* faster than for % time-stamp data. You may wish to run the rate-based calculation first, % then compare with time-stamp-based. Often the differences are insignificant. % To show the "tau bins" (y_k samples) on the data plot, set the variable % TAUBIN to 1 (search for "#TAUBIN"). % You can choose between loglog and semilog plotting of results by % commenting in/out the appropriate line. Search for "#PLOTLOG". % I recommend installing "dsplot.m", which improves the performance of % plotting large data sets. Download from File Exchange, File ID: #15850. % allan.m will use dsplot.m if it is present on your MATLAB path. % This function has been validated using the test data from NBS Monograph % 140, the 1000-point test data set given by Riley [1], and the example data % given in IEEE standard 1139-1999, Annex C. % The author welcomes other validation results, see contact info below. % % For more information, see: % [1] W. J. Riley, "The Calculation of Time Domain Frequency Stability," % Available on the web: % http://www.ieee-uffc.org/frequency_control/teaching.asp?name=paper1ht % % % M.A. Hopcroft % mhopeng at gmail dot com % % I welcome your comments and feedback! % % MH Mar2014 % v2.24 fix bug related to generating freq data from phase with timestamps % (thanks to S. David-Grignot for finding the bug) % MH Oct2010 % v2.22 tau truncation to integer groups; tau sort % plotting bugfix % v2.20 sychronize updates across allan, allan_overlap, allan_modified % v2.16 add TAU as output, fixed unusual error with dsplot v1.1 % v2.14 update plotting behaviour, default tau values % versionstr = 'allan v2.24'; % MH Jun2010 % v2.12 bugfix for rate data row/col orientation % add DATA.units for plotting % use dsplot.m for plotting % % MH MAR2010 % v2.1 minor interface and bugfixes % update data consistency check % % MH FEB2010 % v2.0 Consistent code behaviour for all "allan_x.m" functions: % accept phase data % verbose levels % % % MH JAN2010 % v1.84 code cleanup % v1.82 typos in comments and code cleanup % tau bin plotting changed for performance improvement % v1.8 Performance improvements: % vectorize code for rate data % logical indexing for irregular rate data % MH APR2008 % v1.62 loglog plot option % v1.61 improve error handling, plotting % fix bug in regular data calc for high-rate data % fix bug in timestamp data calc for large starting gap % (thanks to C. B. Ruiz for identifying these bugs) % uses timestamps for DATA.rate=0 % progress indicator for large timestamp data processing % MH JUN2007 % v1.54 Improve data plotting and optional bin plotting % MH FEB2007 % v1.5 use difference from median for plotting % added MAD calculation for outlier detection % MH JAN2007 % v1.48 plotting typos fixes % MH DEC2006 % v1.46 hack to plot error bars % v1.44 further validation (Riley 1000-pt) % plot mean and std % MH NOV2006 % v1.42 typo fix comments % v1.4 fix irregular rate algorithm % irregular algorithm rejects tau less than max gap in time data % validate both algorithms using test data from NBS Monograph 140 % v1.3 fix time calc if data.time not present % add error bars (not possible due to bug in MATLAB R14SP3) % remove offset calculation % v1.24 improve feedback % MH SEP2006 % v1.22 updated comments % v1.2 errors and warnings % v1.1 handle irregular interval data %#ok<*AGROW> % defaults if nargin < 4, verbose=2; end if nargin < 3, name=''; end if nargin < 2 || isempty(tau), tau=2.^(-10:10); end % plot "tau bins"? #TAUBIN TAUBIN = 0; % set 0 or 1 % WARNING: this has a significant impact on performance % Formatting for plots FontName = 'Arial'; FontSize = 14; plotlinewidth=2; if verbose >= 1, fprintf(1,'allan: %s\n\n',versionstr); end %% Data consistency checks if ~(isfield(data,'phase') || isfield(data,'freq')) error('Either ''phase'' or ''freq'' must be present in DATA. See help file for details. [con0]'); end if isfield(data,'time') if isfield(data,'phase') && (length(data.phase) ~= length(data.time)) if isfield(data,'freq') && (length(data.freq) ~= length(data.time)) error('The time and freq vectors are not the same length. See help for details. [con2]'); else error('The time and phase vectors are not the same length. See help for details. [con1]'); end end if isfield(data,'phase') && (any(isnan(data.phase)) || any(isinf(data.phase))) error('The phase vector contains invalid elements (NaN/Inf). [con3]'); end if isfield(data,'freq') && (any(isnan(data.freq)) || any(isinf(data.freq))) error('The freq vector contains invalid elements (NaN/Inf). [con4]'); end if isfield(data,'time') && (any(isnan(data.time)) || any(isinf(data.time))) error('The time vector contains invalid elements (NaN/Inf). [con5]'); end end % sort tau vector tau=sort(tau); %% Basic statistical tests on the data set if ~isfield(data,'freq') if isfield(data,'rate') && data.rate ~= 0 data.freq=diff(data.phase).*data.rate; elseif isfield(data,'time') data.freq=diff(data.phase)./diff(data.time); end if verbose >= 1, fprintf(1,'allan: Fractional frequency data generated from phase data (M=%g).\n',length(data.freq)); end data.time(1)=[]; % make time stamps correspond to freq data end if size(data.freq,2) > size(data.freq,1), data.freq=data.freq'; end % ensure columns s.numpoints=length(data.freq); s.max=max(data.freq); s.min=min(data.freq); s.mean=mean(data.freq); s.median=median(data.freq); if isfield(data,'time') if size(data.time,2) > size(data.time,1), data.time=data.time'; end % ensure columns s.linear=polyfit(data.time(1:length(data.freq)),data.freq,1); elseif isfield(data,'rate') && data.rate ~= 0; s.linear=polyfit((1/data.rate:1/data.rate:length(data.freq)/data.rate)',data.freq,1); else error('Either "time" or "rate" must be present in DATA. Type "help allan" for details. [err1]'); end s.std=std(data.freq); if verbose >= 2 fprintf(1,'allan: input data statistics:\n'); disp(s); end % center at median for plotting medianfreq=data.freq-s.median; sm=[]; sme=[]; % Screen for outliers using 5x Median Absolute Deviation (MAD) criteria s.MAD = median(abs(medianfreq)/0.6745); if verbose >= 2 fprintf(1, 'allan: 5x MAD value for outlier detection: %g\n',5*s.MAD); end if verbose >= 1 && any(abs(medianfreq) > 5*s.MAD) fprintf(1, 'allan: NOTE: There appear to be outliers in the frequency data. See plot.\n'); end %%%% % There are two cases, either using timestamps or fixed sample rate: %% Fixed Sample Rate Data % If there is a regular interval between measurements, calculation is much % easier/faster if isfield(data,'rate') && data.rate > 0 % if data rate was given if verbose >= 1, fprintf(1, 'allan: regular data (%g data points @ %g Hz)\n',length(data.freq),data.rate); end % string for plot title name=[name ' (' num2str(data.rate) ' Hz)']; % what is the time interval between data points? tmstep = 1/data.rate; % Is there time data? Just for curiosity/plotting, does not impact calculation if isfield(data,'time') % adjust time data to remove any starting gap; first time step % should not be zero for comparison with freq data dtime=data.time-data.time(1)+mean(diff(data.time)); if verbose >= 2 fprintf(1,'allan: End of timestamp data: %g sec.\n',dtime(end)); if (data.rate - 1/mean(diff(dtime))) > 1e-6 fprintf(1,'allan: NOTE: data.rate (%f Hz) does not match average timestamped sample rate (%f Hz)\n',data.rate,1/mean(diff(dtime))); end end else % create time axis data using rate (for plotting only) dtime=(tmstep:tmstep:length(data.freq)*tmstep)'; % column oriented end % check the range of tau values and truncate if necessary % find halfway point of time record halftime = round(tmstep*length(data.freq)/2); % truncate tau to appropriate values tau = tau(tau >= tmstep & tau <= halftime); if verbose >= 2, fprintf(1, 'allan: allowable tau range: %g to %g sec. (1/rate to total_time/2)\n',tmstep,halftime); end % save the freq data for the loop dfreq=data.freq; dfreq2=data.freq2; % find the number of data points in each tau group m = data.rate.*tau; % only integer values allowed (no fractional groups of points) %tau = tau(m-round(m)<1e-8); % numerical precision issues (v2.1) tau = tau(m==round(m)); % The round() test is only correct for values < 2^53 %m = m(m-round(m)<1e-8); % change to round(m) for integer test v2.22 m = m(m==round(m)); %m=round(m); if verbose >= 1, fprintf(1,'allan: calculating Allan deviation...\n '); end % calculate the Allan deviation for each value of tau k=0; tic; for i = tau if verbose >= 2, fprintf(1,'%g ',i); end k=k+1; % truncate frequency set to an even multiple of this tau value freq=dfreq(1:end-rem(length(dfreq),m(k))); freq2=dfreq2(1:end-rem(length(dfreq2),m(k))); % group the data into tau-length groups or bins f = reshape(freq,m(k),[]); % Vectorize! f2 = reshape(freq2,m(k),[]); % Vectorize! % find average in each "tau group", y_k (each colummn of f) fa=mean(f,1); fa2=mean(f2,1); % first finite difference fd=diff(fa); fd2=diff(fa2); % calculate two-sample variance for this tau M=length(fa); sm(k)=sqrt(0.5/(M-1)*abs(real(sum(fd.*fd2)))); % estimate error bars sme(k)=sm(k)/sqrt(M+1); if TAUBIN == 1 % save the binning points for plotting fs(k,1:length(freq)/m(k))=m(k):m(k):length(freq); fval{k}=mean(f,1); end end % repeat for each value of tau if verbose >= 2, fprintf(1,'\n'); end calctime=toc; if verbose >= 2, fprintf(1,'allan: Elapsed time for calculation: %e seconds\n',calctime); end %% Irregular data (timestamp) elseif isfield(data,'time') % the interval between measurements is irregular % so we must group the data by time if verbose >= 1, fprintf(1, 'allan: irregular rate data (no fixed sample rate)\n'); end % string for plot title name=[name ' (timestamp)']; % adjust time to remove any initial offset or zero dtime=data.time-data.time(1)+mean(diff(data.time)); %dtime=data.time; % where is the maximum gap in time record? gap_pos=find(diff(dtime)==max(diff(dtime))); % what is average data spacing? avg_gap = mean(diff(dtime)); if verbose >= 2 fprintf(1, 'allan: WARNING: irregular timestamp data (no fixed sample rate).\n'); fprintf(1, ' Calculation time may be long and the results subject to interpretation.\n'); fprintf(1, ' You are advised to estimate using an average sample rate (%g Hz) instead of timestamps.\n',1/avg_gap); fprintf(1, ' Continue at your own risk! (press any key to continue)\n'); pause; end if verbose >= 1 fprintf(1, 'allan: End of timestamp data: %g sec\n',dtime(end)); fprintf(1, ' Average rate: %g Hz (%g sec/measurement)\n',1/avg_gap,avg_gap); if max(diff(dtime)) ~= 1/mean(diff(dtime)) fprintf(1, ' Max. gap: %g sec at position %d\n',max(diff(dtime)),gap_pos(1)); end if max(diff(dtime)) > 5*avg_gap fprintf(1, ' WARNING: Max. gap in time record is suspiciously large (>5x the average interval).\n'); end end % find halfway point halftime = fix(dtime(end)/2); % truncate tau to appropriate values tau = tau(tau >= max(diff(dtime)) & tau <= halftime); if isempty(tau) error('allan: ERROR: no appropriate tau values (> %g s, < %g s)\n',max(diff(dtime)),halftime); end % save the freq data for the loop dfreq=data.freq; dtime=dtime(1:length(dfreq)); if verbose >= 1, fprintf(1,'allan: calculating Allan deviation...\n'); end k=0; tic; for i = tau if verbose >= 2, fprintf(1,'%d ',i); end k=k+1; fa=[]; %f=[]; km=0; % truncate data set to an even multiple of this tau value freq=dfreq(dtime <= dtime(end)-rem(dtime(end),i)); time=dtime(dtime <= dtime(end)-rem(dtime(end),i)); %freq=dfreq; %time=dtime; % break up the data into groups of tau length in sec while i*km < time(end) km=km+1; % progress bar if verbose >= 2 if rem(km,100)==0, fprintf(1,'.'); end if rem(km,1000)==0, fprintf(1,'%g/%g\n',km,round(time(end)/i)); end end f = freq(i*(km-1) < time & time <= i*km); f = f(~isnan(f)); % make sure values are valid if ~isempty(f) fa(km)=mean(f); else fa(km)=0; end if TAUBIN == 1 % WARNING: this has a significant impact on performance % save the binning points for plotting %if find(time <= i*km) > 0 fs(k,km)=max(time(time <= i*km)); %else if isempty(fs(k,km)) fs(k,km)=0; end fval{k}=fa; end % save tau bin plot points end if verbose >= 2, fprintf(1,'\n'); end % first finite difference of the averaged results fd=diff(fa); % calculate Allan deviation for this tau M=length(fa); sm(k)=sqrt(0.5/(M-1)*(sum(fd.^2))); % estimate error bars sme(k)=sm(k)/sqrt(M+1); end if verbose == 2, fprintf(1,'\n'); end calctime=toc; if verbose >= 2, fprintf(1,'allan: Elapsed time for calculation: %e seconds\n',calctime); end else error('allan: WARNING: no DATA.rate or DATA.time! Type "help allan" for more information. [err2]'); end %%%%%%%% %% Plotting if verbose >= 2 % show all data % plot the frequency data, centered on median if size(dtime,2) > size(dtime,1), dtime=dtime'; end % this should not be necessary, but dsplot 1.1 is a little bit brittle try % dsplot makes a new figure hd=dsplot(dtime,medianfreq); catch ME figure; if length(dtime) ~= length(medianfreq) fprintf(1,'allan: Warning: length of time axis (%d) is not equal to data array (%d)\n',length(dtime),length(medianfreq)); end hd=plot(dtime,medianfreq); if verbose >= 1, fprintf(1,'allan: Note: Install dsplot.m for improved plotting of large data sets (File Exchange File ID: #15850).\n'); end if verbose >= 2, fprintf(1,' (Message: %s)\n',ME.message); end end set(hd,'Marker','.','LineStyle','none','Color','b'); % equivalent to '.-' hold on; % show center (0) plot(xlim,[0 0],':k'); % show 5x Median Absolute Deviation (MAD) values hm=plot(xlim,[5*s.MAD 5*s.MAD],'-r'); plot(xlim,[-5*s.MAD -5*s.MAD],'-r'); % show linear fit line hf=plot(xlim,polyval(s.linear,xlim)-s.median,'-g'); title(['Data: ' name],'FontSize',FontSize+2,'FontName',FontName); %set(get(gca,'Title'),'Interpreter','none'); xlabel('Time [sec]','FontSize',FontSize,'FontName',FontName); if isfield(data,'units') ylabel(['data - median(data) [' data.units ']'],'FontSize',FontSize,'FontName',FontName); else ylabel('freq - median(freq)','FontSize',FontSize,'FontName',FontName); end set(gca,'FontSize',FontSize,'FontName',FontName); legend([hd hm hf],{'data (centered on median)','5x MAD outliers',['Linear Fit (' num2str(s.linear(1),'%g') ')']},'FontSize',max(10,FontSize-2)); % tighten up xlim([dtime(1) dtime(end)]); % Optional tau bin (y_k samples) plot if TAUBIN == 1 % plot the tau divisions on the data plot rfs=size(fs,1); colororder=get(gca,'ColorOrder'); axis tight; kc=2; %ap=axis; for j=1:rfs kc=kc+1; if rem(kc,length(colororder))==1, kc=2; end %for b=1:max(find(fs(j,:))); % new form of "find" in r2009a for b=1:find(fs(j,:), 1, 'last' ); % plot the tau division boundaries %plot([fs(j,b) fs(j,b)],[ap(3)*1.1 ap(4)*1.1],'-','Color',colororder(kc,:)); % plot tau group y values if b == 1 plot([dtime(1) fs(j,b)],[fval{j}(b)-s.median fval{j}(b)-s.median],'-','Color',colororder(kc,:),'LineWidth',4); else plot([fs(j,b-1) fs(j,b)],[fval{j}(b)-s.median fval{j}(b)-s.median],'-','Color',colororder(kc,:),'LineWidth',4); end end end axis auto end % End optional bin plot end % end plot raw data if verbose >= 1 % show ADEV results % plot Allan deviation results if ~isempty(sm) figure % Choose loglog or semilogx plot here #PLOTLOG %semilogx(tau,sm,'.-b','LineWidth',plotlinewidth,'MarkerSize',24); loglog(tau,sm,'.-b','LineWidth',plotlinewidth,'MarkerSize',24); % in R14SP3, there is a bug that screws up the error bars on a semilog plot. % When this is fixed in a future release, uncomment below to use normal errorbars %errorbar(tau,sm,sme,'.-b'); set(gca,'XScale','log'); % this is a hack to approximate the error bars hold on; plot([tau; tau],[sm+sme; sm-sme],'-k','LineWidth',max(plotlinewidth-1,2)); grid on; title(['Allan Deviation: ' name],'FontSize',FontSize+2,'FontName',FontName); %set(get(gca,'Title'),'Interpreter','none'); xlabel('\tau [sec]','FontSize',FontSize,'FontName',FontName); if isfield(data,'units') ylabel(['\sigma_y(\tau) [' data.units ']'],'FontSize',FontSize,'FontName',FontName); else ylabel('\sigma_y(\tau)','FontSize',FontSize,'FontName',FontName); end set(gca,'FontSize',FontSize,'FontName',FontName); % expand the x axis a little bit so that the errors bars look nice adax = axis; axis([adax(1)*0.9 adax(2)*1.1 adax(3) adax(4)]); % display the minimum value fprintf(1,'allan: Minimum ADEV value: %g at tau = %g seconds\n',min(sm),tau(sm==min(sm))); elseif verbose >= 1 fprintf(1,'allan: WARNING: no values calculated.\n'); fprintf(1,' Check that TAU > 1/DATA.rate and TAU values are divisible by 1/DATA.rate\n'); fprintf(1,'Type "help allan" for more information.\n\n'); end end % end plot ADEV data retval = sm; errorb = sme; return