Finding peaks and valleys, and associated indexes, of time-shifted noisy sinosidal waves

I want to automate a calibration process, but I'm really struggling with the code.

I have a signal that I give my instrument (second column of attached sample data), and then two probes which record the instrument's response, independently of each other (last two columns of attached sample data). The calibration signal is a series of sine waves of varying amplitudes and periods, see below:

I need to find the following for my calibrations:

-The signal value at each peak and trough

-Whether each probe produces sinosidal waves at each period/amplitude combination

-If so, the response value for each probe at its peaks and troughs (both as individual datapoints and as the mean value) and ideally the corresponding periods and amplitudes

The first is quite easy to do and the second can be done manually without too much trouble. Where I'm running into trouble with is the third step.

I tried the findpeaks function and it's quite noisy:

I did attempt to use the MinPeakHeight argument, use a high-pass filter, and smooth my probe data using moving averages, but none of those attempts worked very well. I also tried to use a Button Down Callback Function that would store values and indexes, but that ended up being super prone to user error in terms of the exact point to click on and not very fast. Solutions like this one don't work because the two sinusoidal response aren't quite fully phase shifted, they're time-shifted by an amount of time that's not consistent either across probes or over time.

Does anyone have any suggestions for functions to look into which might help me do what I want? I'm quite stuck.

Answer 1

[William Rose]

@Oshani,

You write that you want (and I have added numbers to your three desires),

1. The signal value at each peak and trough

2. Whether each probe produces sinosidal waves at each period/amplitude combination

3. If so, the response value for each probe at its peaks and troughs (both as individual datapoints and as the mean value) and ideally the corresponding periods and amplitudes

And you write "The first is quite easy to do and the second can be done manually without too much trouble. Where I'm running into trouble with is the third step."

I am glad that you can determine manually and without much trouble whether the waves are sinusoidal. That is beyond me. I would have to use function thd() or something like it to quantify sinusoidal-ness.

The interval between samples varies, which will complicate the analysis. The plots below show the three signals (top) and the sampling interval (middle). The input signal is labelled "x" and the outputs are "y1" and "y2". I will discuss the bottom panel later.

data=load('data.mat');
t=data.M(:,1); x=data.M(:,2); y1=data.M(:,3); y2=data.M(:,4);
figure
subplot(311), plot(t,x,'-r.',t,y1,'-g.',t,y2,'-b.'); 
grid on; xlabel('Time (s)'); legend('x','y1','y2'); 
subplot(312), plot(t(2:end),diff(t),'-k.'); 
grid on; xlabel('Time (s)'); ylabel('Sampling Interval (s)')

If the long intervals only occur between the pulses, then it would not be a big problem, but that is not the case. For example, if you zoom in on the time period 2020<=t<=2060, you will see that the long time intervals sometime occur in the middle of a sinusoidal pulse.

The bottom panel is a zoomed-in view of the first pulse. The plot shows that the signal is not sampled rapidly enough to determine if it is sinusoidal or not.

subplot(313)
plot(t,x,'-r.',t,y1,'-g.',t,y2,'-b.'); 
grid on; xlabel('Time'); legend('x','y1','y2'); 
xlim([73,76]); ylim([16.6,17.4])

The bottom panel shows 10 data points for 4 cycles of oscillation, i.e. the frerquency is 0.4 cycles per sample. If this wave is non-sinusoidal, i.e. if it has power at harmonics of the fundamental frequency, then those harmonics will be above the Nyquist frequency (0.5 cycles/sample), so they will be aliased down to lower frequencies, in the range 0 to 0.5 cycles/sample, and you will not be able to un-alias them or do a reliable analysis of the signal.

Can you resample the signals faster and at a constant rate?

To accomplish your third aim, which is to find the period and amplitude of the signal and probe responses for the sinusoidal pulses, I recommend fitting a sinusoid to the signal pulse and to the pulses recorded by the probes. There is no doubt a clever way to determine where the pulses start and end*, but for the sake of simlicity I will simply determine the pulse start and end times by inspecting the plot. Then I wil fit the data in those windows. Start and end times of first four pulses and last four pulses:

% Enter pulse start, end times, determined by inspecting the plots
Ts=[73.8,112,152,193.9,2213.8,2257.3,2308.2,2362.7];  
% pulse start times (<= time of first point in each pulse)
Te=[75.6,115.6,157.4,201.1,2221,2271.7,2326.4,2391.4];  
% pulse end times (>= time of last point in each pulse)

The next chunk of code is a funciton that fits a sinusoid to x,y data:

function [f,gof] = fitSinusoid(x,y)
% FITSINUSOID Fit sinusoid to data
%   Model: y=a*sin(b*x+c)+d
%   Make a good initial guess for parameters in order to increase
%   chance of getting a good fit.
a0=(max(y)-min(y))/2;           % amplitude initial guess
nzc=length(find(diff(sign(y-mean(y)))==-2)); 
% number of negative-going zero crossings
b0=2*pi*nzc/(max(x)-min(x));    % radian freq initial guess
c0=0;                           % phase initial guess
d0=mean(y);                     % offset initial guess
myModel=@(a,b,c,d,x) a*sin(b*x+c)+d;
[f1,gof1]=fit(x,y,myModel,...
    'StartPoint', [a0,b0,c0,d0], ...
    'Lower', [0, 0, 0, -Inf], ...
    'Upper', [Inf, Inf, max(x)-min(x), Inf]);
[f2,gof2]=fit(x,y,myModel,...
    'StartPoint', [2*a0,b0,c0,d0], ...
    'Lower', [0, 0, 0, -Inf], ...
    'Upper', [Inf, Inf, max(x)-min(x), Inf]);
if gof1.sse<=gof2.sse
    f=f1; gof=gof1;
else
    f=f2; gof=gof2;
end
end

You may be wondering why I didn't just use the built-in fittype 'sin1'. The reason is that 'sin1' does not include a mean offset, as far as I can tell.

The next chunk of code fits the signal and probe repsonses during each pulse, and displays the best-fit amplitude and frequency and r^2 on the console. It also displays plots of the data and the fits.

% Analyze each pulse
Np=length(Ts);  % number of pulses
for i=1:Np
    [fx,gofx]  =fitSinusoid(t(t>=Ts(i) & t<=Te(i)),x(t>=Ts(i)  & t<=Te(i)));
    [fy1,gofy1]=fitSinusoid(t(t>=Ts(i) & t<=Te(i)),y1(t>=Ts(i) & t<=Te(i)));
    [fy2,gofy2]=fitSinusoid(t(t>=Ts(i) & t<=Te(i)),y2(t>=Ts(i) & t<=Te(i)));
    fprintf('Pulse %d:\n',i);
    fprintf('   Signal: ampl=%.3f, freq=%.3f, r^2=%.2f.\n',...
            fx.a,fx.b/(2*pi),gofx.rsquare)
    fprintf('   Probe1: ampl=%.3f, freq=%.3f, r^2=%.2f.\n',...
            fy1.a,fy1.b/(2*pi),gofy1.rsquare)
    fprintf('   Probe2: ampl=%.3f, freq=%.3f, r^2=%.2f.\n',...
            fy2.a,fy2.b/(2*pi),gofy2.rsquare)
    figure
    subplot(311) 
    plot(fx,t(t>=Ts(i) & t<=Te(i)),x(t>=Ts(i) & t<=Te(i)))
    xlabel('Time'); grid on
    subplot(312) 
    plot(fy1,t(t>=Ts(i) & t<=Te(i)),y1(t>=Ts(i) & t<=Te(i)))
    xlabel('Time'); grid on
    subplot(313) 
    plot(fy2,t(t>=Ts(i) & t<=Te(i)),y2(t>=Ts(i) & t<=Te(i)))
    xlabel('Time'); grid on
end

Pulse 1:

   Signal: ampl=0.252, freq=2.407, r^2=1.00.

   Probe1: ampl=0.000, freq=0.000, r^2=NaN.

   Probe2: ampl=0.000, freq=0.000, r^2=0.00.

Pulse 2:

   Signal: ampl=0.249, freq=1.151, r^2=1.00.

   Probe1: ampl=0.016, freq=0.291, r^2=0.66.

   Probe2: ampl=0.030, freq=0.000, r^2=0.00.

Pulse 3:

   Signal: ampl=0.250, freq=0.756, r^2=1.00.

   Probe1: ampl=0.012, freq=0.379, r^2=0.23.

   Probe2: ampl=0.050, freq=0.000, r^2=0.00.

Pulse 4:

   Signal: ampl=0.250, freq=0.562, r^2=1.00.

   Probe1: ampl=0.003, freq=0.421, r^2=0.01.

   Probe2: ampl=0.002, freq=0.278, r^2=0.03.

Pulse 5:

   Signal: ampl=3.999, freq=0.567, r^2=1.00.

   Probe1: ampl=0.125, freq=0.424, r^2=0.03.

   Probe2: ampl=0.516, freq=0.000, r^2=0.00.

Pulse 6:

   Signal: ampl=4.000, freq=0.280, r^2=1.00.

   Probe1: ampl=1.551, freq=0.279, r^2=0.86.

   Probe2: ampl=0.875, freq=0.140, r^2=-1.08.

Pulse 7:

   Signal: ampl=3.995, freq=0.220, r^2=0.99.

   Probe1: ampl=1.859, freq=0.219, r^2=0.87.

   Probe2: ampl=0.164, freq=0.220, r^2=0.22.

Pulse 8:

   Signal: ampl=0.282, freq=0.139, r^2=0.14.

   Probe1: ampl=2.449, freq=0.138, r^2=0.93.

   Probe2: ampl=1.385, freq=0.138, r^2=0.76.

The plots show that the fits are not always the best, but maybe with some tweaking, you can get this approach to be useful. Maybe use fmincon() instead of fit(). Also, the script reports r^2=-1.08 for pulse 6. That shouldn't happen, so it merits further investigation.

Good luck with your analysis.

[Mathieu NOE]

dear all

IMHO the test signals have a too high frequency , the probe will not keep with the firsts sine bursts that are obviously beyond their frequency bandwidth

maybe you should redo the test with lower freq bursts (and that would also increase the number of samples per period )

to test the sinus like shape maybe we could also use first some bandpass filtering of the probe signals , based on the frequency measured on the input (test) signal. frequency can be measured from the "zero" crossing rate of the signal (using the mean of the signal as the threshold value) - see second subplot here

data=load('data.mat');
t=data.M(:,1); x=data.M(:,2); y1=data.M(:,3); y2=data.M(:,4);
figure
subplot(211), plot(t,x,'-r.',t,y1,'-g.',t,y2,'-b.'); 
grid on; xlabel('Time (s)'); legend('x','y1','y2'); 
xlim([min(t) max(t)])


% counts signal crossing 
threshold = mean(x)+0.1; % 

[t0_pos,t0_neg] = find_zc(t,x,threshold);

freq_input = [0;1./diff(t0_pos)];
% discard values below 0.05 Hz (generated by distant t0_pos belonging to separate burst)
ind = (freq_input<0.05);
freq_input(ind) = [];
t0_pos(ind) = [];

subplot(212), plot(t0_pos,freq_input,'*'); 
grid on; xlabel('Time (s)'); ylabel('Signal freq (Hz)')
xlim([min(t) max(t)])

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [ZxP,ZxN] = find_zc(x,y,threshold)

% put data in rows
x = x(:);
y = y(:);

% positive slope "zero" crossing detection, using linear interpolation
y = y - threshold;
zci = @(data) find(diff(sign(data))>0);  %define function: returns indices of +ZCs
ix=zci(y);                      %find indices of + zero crossings of x
ZeroX = @(x0,y0,x1,y1) x0 - (y0.*(x0 - x1))./(y0 - y1); % Interpolated x value for Zero-Crossing 
ZxP = ZeroX(x(ix),y(ix),x(ix+1),y(ix+1));

% negative slope "zero" crossing detection, using linear interpolation
zci = @(data) find(diff(sign(data))<0);  %define function: returns indices of +ZCs
ix=zci(y);                      %find indices of + zero crossings of x
ZeroX = @(x0,y0,x1,y1) x0 - (y0.*(x0 - x1))./(y0 - y1); % Interpolated x value for Zero-Crossing 
ZxN = ZeroX(x(ix),y(ix),x(ix+1),y(ix+1));

end

[Oshani]

thank you! this ended up being the bulk of what I used, along with some of what @Mathieu NOE did

[William Rose]

@Oshani, you're welcome. Good luck with your work.

Answer 2

[Mathieu NOE]

Following my first comment above , this could be another approach

• extract blocks of data (you need to find the start and stop times , here I used islocalmax to find peaks
• bandpass filter your probe signals , once you have measured the test signal frequency (to adapt the low and high corner frequencies of the filter
• you could easily add a simple test based on the filtered probe signals :
• amplitude ratio vs test signal amplitude
• if the ratio is very low => this is not really a valid output
• if the ratio is above 10% (for example) , again you can try to fit a sinus and extract the parameters (done in William's answer)

data=load('data.mat');
t=data.M(:,1); x=data.M(:,2); y1=data.M(:,3); y2=data.M(:,4);

samples = numel(t);
% figure
% subplot(211), plot(t,x,'-r.',t,y1,'-g.',t,y2,'-b.'); 
% grid on; xlabel('Time (s)'); legend('x','y1','y2'); 
% xlim([min(t) max(t)])


Fs = 1./mean(diff(t));

% define start/end time points of blocks of data 
tmp = abs(detrend(x,1));
tmp = tmp -min(tmp); % meake tmp min equals 0
tf = islocalmax(tmp); % option 3 with islocalmax (you can also try with find peaks)
tt = t(tf);
env = tmp(tf);
figure
plot(t,tmp,tt,env,'*')

% find indexes of distant blocks
ind = find(diff(tt)>20);
tt_ind = tt(ind);
n_blocks = numel(ind);

for k= 1:n_blocks
    tstop(k,1) = tt_ind(k); % end of k th block 
    tbefore = tt(tt<=tstop(k,1));
    if k<2
        tstart(k,1) = min(tbefore);  % start of 1st block 
    else
        tstart(k,1) = min(tbefore(tbefore>tstop(k-1,1))); % start of k th block
    end
    % add some time before and after (several seconds)
    extra_time = 4; % seconds
    tstop(k,1) = min(samples,tstop(k,1) + extra_time);
    tstart(k,1) = max(0,tstart(k,1) - extra_time);
    

    % now extract portion of data (k th block)
    ind = (t>=tstart(k,1) & t<=tstop(k,1) );
    ti = t(ind);
    xx = x(ind);
    yy1 = y1(ind);
    yy2 = y2(ind);

    % counts (input) test signal threshold crossing to get its frequency
    threshold = mean(xx)+0.02; % 
    [t_pos,t_neg] = find_zc(ti,xx,threshold);
    t_pos = unique(t_pos);
    freq_input = mean(1./diff(t_pos))
    
    % now create a bandpass filter
    if freq_input<Fs/2
    a = 0.5;
    flow = a*freq_input;
    fhigh = min(1/a*freq_input,Fs/2.56);
    else
    flow = 0;
    fhigh = Fs/2.56;
    end

    [B,A] = butter(2,2/Fs*[flow fhigh]);

    yy1m = mean(yy1);
    yy1f = filtfilt(B,A,detrend(yy1,1)); 
    yy1f = yy1f + yy1m;
    yy2m = mean(yy2);
    yy2f = filtfilt(B,A,detrend(yy2,1)); 
    yy2f = yy2f + yy2m;

    figure
    plot(ti,xx,'-r.',ti,yy1f,'-g.',ti,yy2f,'-b.'); 
    grid on; xlabel('Time (s)'); legend('x','y1','y2');
    title(['Signal Freq = ' num2str(freq_input) ' Hz'])

end

freq_input = 2.3968

freq_input = 1.1509

freq_input = 0.7577

freq_input = 0.5639

freq_input = 0.2797

freq_input = 0.2189

freq_input = 0.1388

freq_input = 2.3914

freq_input = 1.1532

freq_input = 0.7566

freq_input = 0.4470

freq_input = 0.2796

freq_input = 0.2223

freq_input = 0.1384

freq_input = 2.3931

freq_input = 1.1468

freq_input = 0.7547

freq_input = 0.5647

freq_input = 0.2785

freq_input = 0.2224

freq_input = 0.1386

freq_input = 2.3883

freq_input = 1.1490

freq_input = 0.7582

freq_input = 0.5625

freq_input = 0.2796

freq_input = 0.2154

freq_input = 0.1387

freq_input = 2.3950

freq_input = 1.1506

freq_input = 0.7537

freq_input = 0.5666

freq_input = 0.2801

freq_input = 0.2230

freq_input = 0.1377

freq_input = 2.3951

freq_input = 1.1559

freq_input = 0.7595

freq_input = 0.5640

freq_input = 0.2798

freq_input = 0.2222

freq_input = 0.1242

freq_input = 2.3889

freq_input = 1.1559

freq_input = 0.7552

freq_input = 0.5666

freq_input = 0.2797

freq_input = 0.2196

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [ZxP,ZxN] = find_zc(x,y,threshold)

    % put data in rows
    x = x(:);
    y = y(:);
    
    % positive slope "zero" crossing detection, using linear interpolation
    y = y - threshold;
    zci = @(data) find(diff(sign(data))>0);  %define function: returns indices of +ZCs
    ix=zci(y);                      %find indices of + zero crossings of x
    ZeroX = @(x0,y0,x1,y1) x0 - (y0.*(x0 - x1))./(y0 - y1); % Interpolated x value for Zero-Crossing 
    ZxP = ZeroX(x(ix),y(ix),x(ix+1),y(ix+1));
    
    % negative slope "zero" crossing detection, using linear interpolation
    zci = @(data) find(diff(sign(data))<0);  %define function: returns indices of +ZCs
    ix=zci(y);                      %find indices of + zero crossings of x
    ZeroX = @(x0,y0,x1,y1) x0 - (y0.*(x0 - x1))./(y0 - y1); % Interpolated x value for Zero-Crossing 
    ZxN = ZeroX(x(ix),y(ix),x(ix+1),y(ix+1));

end

[Mathieu NOE]

some upgrades : added the sinus fit using from this (excellent) Fex submission (I don't have the curve fitting toolbox !)

Fminspleas - File Exchange - MATLAB Central

you can now use the results to do your tests (fit model : y = A*sin(B*x+C)+D )

hope it helps

fyi , this is the last block of data plot :

data=load('data.mat');
t=data.M(:,1); x=data.M(:,2); y1=data.M(:,3); y2=data.M(:,4);

samples = numel(t);
% figure
% subplot(211), plot(t,x,'-r.',t,y1,'-g.',t,y2,'-b.'); 
% grid on; xlabel('Time (s)'); legend('x','y1','y2'); 
% xlim([min(t) max(t)])


Fs = 1./mean(diff(t));

% define start/end time points of blocks of data 
tmp = abs(detrend(x,1));
tmp = tmp -min(tmp); % meake tmp min equals 0
tf = islocalmax(tmp); % option 3 with islocalmax (you can also try with find peaks)
tt = t(tf);
env = tmp(tf);
figure
plot(t,tmp,tt,env,'*')

% find indexes of distant blocks
ind = find(diff(tt)>20);
tt_ind = tt(ind);
n_blocks = numel(ind);

for k= 1:n_blocks
    tstop(k,1) = tt_ind(k); % end of k th block 
    tbefore = tt(tt<=tstop(k,1));
    if k<2
        tstart(k,1) = min(tbefore);  % start of 1st block 
    else
        tstart(k,1) = min(tbefore(tbefore>tstop(k-1,1))); % start of k th block
    end
    % add some time before and after (several seconds)
    extra_time = 4; % seconds
    tstop(k,1) = min(samples,tstop(k,1) + extra_time);
    tstart(k,1) = max(0,tstart(k,1) - extra_time);
    

    % now extract portion of data (k th block)
    ind = (t>=tstart(k,1) & t<=tstop(k,1) );
    ti = t(ind);
    xx = x(ind);
    yy1 = y1(ind);
    yy2 = y2(ind);

    % counts (input) test signal threshold crossing to get its frequency
    threshold = mean(xx)+0.02; % 
    [t_pos,t_neg] = find_zc(ti,xx,threshold);
    t_pos = unique(t_pos);
    freq_input = mean(1./diff(t_pos))
    
    % now create a bandpass filter
    if freq_input<Fs/2
    a = 0.5;
    flow = a*freq_input;
    fhigh = min(1/a*freq_input,Fs/2.56);
    else
    flow = 0;
    fhigh = Fs/2.56;
    end

    [B,A] = butter(2,2/Fs*[flow fhigh]);

    yy1m = mean(yy1);
    yy1f = filtfilt(B,A,detrend(yy1,1)); 
    yy1f = yy1f + yy1m;
    % sinus fit for first signal
    % use a shorter time range for the sinus fit 
    ind = (ti>=tstart(k,1)+extra_time & ti<=tstop(k,1)-extra_time );
    tfit = ti(ind);

    [yfit1,A1,B1,C1,D1] = my_sinus_fit(tfit,yy1f(ind),freq_input);% fit model : y = A*sin(B*x+C)+D

    yy2m = mean(yy2);
    yy2f = filtfilt(B,A,detrend(yy2,1)); 
    yy2f = yy2f + yy2m;

    % sinus fit for 2nd signal 
    [yfit2,A2,B2,C2,D2] = my_sinus_fit(tfit,yy2f(ind),freq_input);% fit model : y = A*sin(B*x+C)+D
    figure
    plot(ti,xx,'-r',ti,yy1f,'-g',ti,yy2f,'-b',tfit,yfit1,'-*g',tfit,yfit2,'-*b'); 
    grid on; xlabel('Time (s)'); legend('x','y1','y2','y1 fit','y2 fit');
    title(['Signal Freq = ' num2str(freq_input) ' Hz'])

    % store data 
    out_data(k,:) = [freq_input A1 B1 C1 D1 A2 B2 C2 D2];

end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% sinus fit using fminspleas %% 
function [yfit,A,B,C,D] = my_sinus_fit(t,y,freq)
%% Solution based on fminspleas
% fit model : y = A*sin(B*x+C)+D
% There is a tool on the FEX, called fminspleas . It does all of the hard work for you.
% http://www.mathworks.com/matlabcentral/fileexchange/10093-fminspleas
% All you will need to do is define the function to be used. Here they would be something like...
funslist = {1, @(coef,xdata) cos(coef*xdata), @(coef,xdata) sin(coef*xdata)};
% Fminspleas will return 4 coefficients, in two different return variables. 
% Here, Bstart is the starting value for B. It is the ONLY one you need to provide.

%%%%%%%%%%%%% main code %%%%%%%%%%%%%%%%%
yu = max(y);
yl = min(y);
% ym = mean(y);                                             % Estimate offset
ym = (yl + yu)/2;                                             % Estimate offset
yr = (yu-yl)/2;                                             % Estimate Amplitude of 'y'
yz = y-ym;
% yz = fastsmooth(yz,4,2,1);                       % smooth data to remove noise artifacts (adjust factors)

% % Find zero-crossings
% [ZxP,ZxN] = find_zc(t,yz,0);
% per = 2*abs(ZxP(1) - ZxN(1)); % Estimate period
% freq = 1/per                                 % Estimate FREQUENCY

%%%%%%%%%%
Bstart = 2*pi*freq;
[B,ILP] = fminspleas(funslist,Bstart,t,y);
D = ILP(1);
C = atan2(ILP(2),ILP(3));
A = ILP(2)/sin( C );

yfit = A*sin(B*t+C)+D;

end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [ZxP,ZxN] = find_zc(x,y,threshold)

    % put data in rows
    x = x(:);
    y = y(:);
    
    % positive slope "zero" crossing detection, using linear interpolation
    y = y - threshold;
    zci = @(data) find(diff(sign(data))>0);  %define function: returns indices of +ZCs
    ix=zci(y);                      %find indices of + zero crossings of x
    ZeroX = @(x0,y0,x1,y1) x0 - (y0.*(x0 - x1))./(y0 - y1); % Interpolated x value for Zero-Crossing 
    ZxP = ZeroX(x(ix),y(ix),x(ix+1),y(ix+1));
    
    % negative slope "zero" crossing detection, using linear interpolation
    zci = @(data) find(diff(sign(data))<0);  %define function: returns indices of +ZCs
    ix=zci(y);                      %find indices of + zero crossings of x
    ZeroX = @(x0,y0,x1,y1) x0 - (y0.*(x0 - x1))./(y0 - y1); % Interpolated x value for Zero-Crossing 
    ZxN = ZeroX(x(ix),y(ix),x(ix+1),y(ix+1));

end

[William Rose]

Very nice answer by @Mathieu NOE.

[Mathieu NOE]

tx @William Rose

[Oshani]

thank you! ended up going with a combination of this answer & @William Rose 's, wish I could accept both!

[Mathieu NOE]

hello @Oshani

no problem - glad you have now a working solution

you cannot accept multiple answers , but you can still vote for the other answers that have contributed to the team work