find fitting equation and parameters to fit damped oscillations curves from set of data

I am trying to fit my data (bellow) to a damped oscillations curve, I can't determine the fitting equation, because it's not easy.

I don't know if it's even possible to do it, without knowing the fitting equation, and the most important is finding the parameters for this fitting.

x30=	[1.40E+04	1.40E+04	1.41E+04	1.41E+04	1.42E+04	1.42E+04	1.42E+04	1.43E+04	1.43E+04	1.43E+04	1.44E+04	1.44E+04	1.45E+04	1.45E+04	1.45E+04	1.46E+04	1.46E+04	1.46E+04	1.47E+04	1.47E+04	1.48E+04	1.48E+04	1.48E+04	1.49E+04	1.49E+04	1.49E+04	1.50E+04	1.50E+04	1.51E+04	1.51E+04	1.51E+04	1.52E+04	1.52E+04	1.53E+04	1.53E+04	1.53E+04	1.54E+04	1.54E+04	1.54E+04	1.55E+04	15521.13777	15559.45668	15597.79092	15636.14029	1.57E+04	1.57E+04	1.58E+04	1.58E+04	1.58E+04	1.59E+04	1.59E+04	1.59E+04	1.60E+04	1.60E+04	1.61E+04	1.61E+04	1.61E+04	1.62E+04	1.62E+04	1.63E+04	1.63E+04	1.63E+04	1.64E+04	1.64E+04	1.64E+04	1.65E+04	1.65E+04	1.66E+04	1.66E+04	1.66E+04	1.67E+04	1.68E+04	1.69E+04	1.70E+04	1.71E+04	1.72E+04	1.73E+04	1.74E+04	1.75E+04	1.76E+04	1.77E+04	1.78E+04	1.79E+04	1.80E+04	1.81E+04	1.82E+04	1.83E+04	1.84E+04	1.85E+04	1.86E+04	1.87E+04	1.88E+04	1.89E+04	1.90E+04	1.91E+04	1.91E+04	1.92E+04	1.93E+04	1.94E+04	1.95E+04	1.96E+04	1.97E+04	1.98E+04	1.99E+04	2.00E+04	2.01E+04	2.02E+04	2.03E+04	2.04E+04	20512.09298	20609.50662	20706.9258	20804.34834	20901.7768	20999.22015	21096.67455	21194.13379	21291.59834	21389.06623	21486.53808	21584.02386	21681.52239	21779.02615	21876.53569	21974.04901	22071.56586	22169.09515	22266.63918	22364.18951	22461.7455	22559.30577	22656.8689	22754.44281	22852.03143	22949.62642	23047.2266	23144.83138	23242.43867	23340.05445	23437.6868	23535.32846	23632.97578	23730.6286	23828.28439	23925.94739	24023.62551	24121.31208	24219.00331	24316.69957	24414.39881	24512.10282	24609.82226	24707.55504	24805.29389	24903.03895	]																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																								
Slope30 = [	-3.55E-06	-5.52E-06	-3.96E-06	-2.94E-06	-4.54E-06	-3.28E-06	-3.00E-06	-4.32E-06	-4.08E-06	-3.59E-06	-3.82E-06	-3.70E-06	-3.55E-06	-3.23E-06	-2.03E-06	-2.52E-06	-3.89E-06	-4.25E-06	-3.95E-06	-3.83E-06	-3.42E-06	-3.01E-06	-2.50E-06	-2.12E-06	-2.07E-06	-2.34E-06	-2.97E-06	-3.52E-06	-3.87E-06	-3.85E-06	-3.34E-06	-2.93E-06	-2.83E-06	-2.33E-06	-1.74E-06	-2.00E-06	-2.61E-06	-2.32E-06	-2.05E-06	-2.90E-06	-3.83E-06	-4.08E-06	-3.48E-06	-3.20E-06	-1.89E-06	-7.26E-07	-1.88E-06	-2.48E-06	-2.59E-06	-1.27E-06	-7.86E-07	-3.29E-06	-4.32E-06	-2.73E-06	-2.27E-06	-3.85E-06	-3.93E-06	-1.62E-06	-1.50E-06	-3.07E-06	-1.13E-06	-4.50E-07	-2.52E-06	-1.27E-06	-1.04E-06	-3.51E-06	-1.98E-06	-1.51E-06	4.72E-05	0.000298027	-0.000468594	-0.001470947	-0.001476212	-0.001213067	-0.000780846	-0.000501233	-0.000356343	-0.000248914	-0.000182324	-0.000150969	-0.00013269	-0.000102512	-8.25E-05	-8.26E-05	-6.45E-05	-5.90E-05	-5.18E-05	-3.23E-05	-4.96E-05	-4.07E-05	-1.29E-05	-2.82E-05	-3.70E-05	-1.20E-05	7.40E-06	-5.13E-06	-2.71E-05	-2.28E-05	-4.30E-06	8.62E-06	3.01E-06	-1.38E-05	-1.55E-05	-4.66E-06	7.66E-06	1.62E-05	6.76E-06	-1.25E-05	-1.61E-05	-5.46E-06	6.31E-06	1.36E-05	5.88E-06	-1.10E-05	-1.35E-05	-4.49E-06	5.74E-06	1.42E-05	9.33E-06	-8.89E-06	-1.53E-05	-6.31E-06	4.68E-06	1.37E-05	1.26E-05	-3.62E-06	-1.46E-05	-9.11E-06	1.07E-06	1.06E-05	1.14E-05	-2.25E-06	-1.53E-05	-1.08E-05	1.53E-06	1.19E-05	1.76E-05	4.64E-06	-1.31E-05	-1.33E-05	-3.67E-06	6.79E-06	1.21E-05	8.66E-07	-1.38E-05	-1.23E-05	-1.32E-06	1.00E-05	1.82E-05	1.19E-05	-7.78E-06	-1.51E-05	-7.22E-06	2.52E-06	1.00E-05]																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																																									

Thanks in advance for your help !

[Sam Chak]

Can you clarify if you really want to fit this signal?

I'm not if there is an analytical equation exists for this curve.

I'm unsure, but the signal looks like a response from a non-minimum phase system.

x30     = [1.40E+04	1.40E+04	1.41E+04	1.41E+04	1.42E+04	1.42E+04	1.42E+04	1.43E+04	1.43E+04	1.43E+04	1.44E+04	1.44E+04	1.45E+04	1.45E+04	1.45E+04	1.46E+04	1.46E+04	1.46E+04	1.47E+04	1.47E+04	1.48E+04	1.48E+04	1.48E+04	1.49E+04	1.49E+04	1.49E+04	1.50E+04	1.50E+04	1.51E+04	1.51E+04	1.51E+04	1.52E+04	1.52E+04	1.53E+04	1.53E+04	1.53E+04	1.54E+04	1.54E+04	1.54E+04	1.55E+04	15521.13777	15559.45668	15597.79092	15636.14029	1.57E+04	1.57E+04	1.58E+04	1.58E+04	1.58E+04	1.59E+04	1.59E+04	1.59E+04	1.60E+04	1.60E+04	1.61E+04	1.61E+04	1.61E+04	1.62E+04	1.62E+04	1.63E+04	1.63E+04	1.63E+04	1.64E+04	1.64E+04	1.64E+04	1.65E+04	1.65E+04	1.66E+04	1.66E+04	1.66E+04	1.67E+04	1.68E+04	1.69E+04	1.70E+04	1.71E+04	1.72E+04	1.73E+04	1.74E+04	1.75E+04	1.76E+04	1.77E+04	1.78E+04	1.79E+04	1.80E+04	1.81E+04	1.82E+04	1.83E+04	1.84E+04	1.85E+04	1.86E+04	1.87E+04	1.88E+04	1.89E+04	1.90E+04	1.91E+04	1.91E+04	1.92E+04	1.93E+04	1.94E+04	1.95E+04	1.96E+04	1.97E+04	1.98E+04	1.99E+04	2.00E+04	2.01E+04	2.02E+04	2.03E+04	2.04E+04	20512.09298	20609.50662	20706.9258	20804.34834	20901.7768	20999.22015	21096.67455	21194.13379	21291.59834	21389.06623	21486.53808	21584.02386	21681.52239	21779.02615	21876.53569	21974.04901	22071.56586	22169.09515	22266.63918	22364.18951	22461.7455	22559.30577	22656.8689	22754.44281	22852.03143	22949.62642	23047.2266	23144.83138	23242.43867	23340.05445	23437.6868	23535.32846	23632.97578	23730.6286	23828.28439	23925.94739	24023.62551	24121.31208	24219.00331	24316.69957	24414.39881	24512.10282	24609.82226	24707.55504	24805.29389	24903.03895];
Slope30 = [	-3.55E-06	-5.52E-06	-3.96E-06	-2.94E-06	-4.54E-06	-3.28E-06	-3.00E-06	-4.32E-06	-4.08E-06	-3.59E-06	-3.82E-06	-3.70E-06	-3.55E-06	-3.23E-06	-2.03E-06	-2.52E-06	-3.89E-06	-4.25E-06	-3.95E-06	-3.83E-06	-3.42E-06	-3.01E-06	-2.50E-06	-2.12E-06	-2.07E-06	-2.34E-06	-2.97E-06	-3.52E-06	-3.87E-06	-3.85E-06	-3.34E-06	-2.93E-06	-2.83E-06	-2.33E-06	-1.74E-06	-2.00E-06	-2.61E-06	-2.32E-06	-2.05E-06	-2.90E-06	-3.83E-06	-4.08E-06	-3.48E-06	-3.20E-06	-1.89E-06	-7.26E-07	-1.88E-06	-2.48E-06	-2.59E-06	-1.27E-06	-7.86E-07	-3.29E-06	-4.32E-06	-2.73E-06	-2.27E-06	-3.85E-06	-3.93E-06	-1.62E-06	-1.50E-06	-3.07E-06	-1.13E-06	-4.50E-07	-2.52E-06	-1.27E-06	-1.04E-06	-3.51E-06	-1.98E-06	-1.51E-06	4.72E-05	0.000298027	-0.000468594	-0.001470947	-0.001476212	-0.001213067	-0.000780846	-0.000501233	-0.000356343	-0.000248914	-0.000182324	-0.000150969	-0.00013269	-0.000102512	-8.25E-05	-8.26E-05	-6.45E-05	-5.90E-05	-5.18E-05	-3.23E-05	-4.96E-05	-4.07E-05	-1.29E-05	-2.82E-05	-3.70E-05	-1.20E-05	7.40E-06	-5.13E-06	-2.71E-05	-2.28E-05	-4.30E-06	8.62E-06	3.01E-06	-1.38E-05	-1.55E-05	-4.66E-06	7.66E-06	1.62E-05	6.76E-06	-1.25E-05	-1.61E-05	-5.46E-06	6.31E-06	1.36E-05	5.88E-06	-1.10E-05	-1.35E-05	-4.49E-06	5.74E-06	1.42E-05	9.33E-06	-8.89E-06	-1.53E-05	-6.31E-06	4.68E-06	1.37E-05	1.26E-05	-3.62E-06	-1.46E-05	-9.11E-06	1.07E-06	1.06E-05	1.14E-05	-2.25E-06	-1.53E-05	-1.08E-05	1.53E-06	1.19E-05	1.76E-05	4.64E-06	-1.31E-05	-1.33E-05	-3.67E-06	6.79E-06	1.21E-05	8.66E-07	-1.38E-05	-1.23E-05	-1.32E-06	1.00E-05	1.82E-05	1.19E-05	-7.78E-06	-1.51E-05	-7.22E-06	2.52E-06	1.00E-05];
plot(x30, Slope30)

[Tesla]

Yes exactly I want to fit this signal.

Answer 1

[Star Strider]

I can approximate the fit, however not to the oscillations.

The best model is a mathematical model of the process that created those data, not necessarily the one I created (for no reason other than it seems to be a logical choice).

Try this —

x30=	[1.40E+04	1.40E+04	1.41E+04	1.41E+04	1.42E+04	1.42E+04	1.42E+04	1.43E+04	1.43E+04	1.43E+04	1.44E+04	1.44E+04	1.45E+04	1.45E+04	1.45E+04	1.46E+04	1.46E+04	1.46E+04	1.47E+04	1.47E+04	1.48E+04	1.48E+04	1.48E+04	1.49E+04	1.49E+04	1.49E+04	1.50E+04	1.50E+04	1.51E+04	1.51E+04	1.51E+04	1.52E+04	1.52E+04	1.53E+04	1.53E+04	1.53E+04	1.54E+04	1.54E+04	1.54E+04	1.55E+04	15521.13777	15559.45668	15597.79092	15636.14029	1.57E+04	1.57E+04	1.58E+04	1.58E+04	1.58E+04	1.59E+04	1.59E+04	1.59E+04	1.60E+04	1.60E+04	1.61E+04	1.61E+04	1.61E+04	1.62E+04	1.62E+04	1.63E+04	1.63E+04	1.63E+04	1.64E+04	1.64E+04	1.64E+04	1.65E+04	1.65E+04	1.66E+04	1.66E+04	1.66E+04	1.67E+04	1.68E+04	1.69E+04	1.70E+04	1.71E+04	1.72E+04	1.73E+04	1.74E+04	1.75E+04	1.76E+04	1.77E+04	1.78E+04	1.79E+04	1.80E+04	1.81E+04	1.82E+04	1.83E+04	1.84E+04	1.85E+04	1.86E+04	1.87E+04	1.88E+04	1.89E+04	1.90E+04	1.91E+04	1.91E+04	1.92E+04	1.93E+04	1.94E+04	1.95E+04	1.96E+04	1.97E+04	1.98E+04	1.99E+04	2.00E+04	2.01E+04	2.02E+04	2.03E+04	2.04E+04	20512.09298	20609.50662	20706.9258	20804.34834	20901.7768	20999.22015	21096.67455	21194.13379	21291.59834	21389.06623	21486.53808	21584.02386	21681.52239	21779.02615	21876.53569	21974.04901	22071.56586	22169.09515	22266.63918	22364.18951	22461.7455	22559.30577	22656.8689	22754.44281	22852.03143	22949.62642	23047.2266	23144.83138	23242.43867	23340.05445	23437.6868	23535.32846	23632.97578	23730.6286	23828.28439	23925.94739	24023.62551	24121.31208	24219.00331	24316.69957	24414.39881	24512.10282	24609.82226	24707.55504	24805.29389	24903.03895	];
Slope30 = [	-3.55E-06	-5.52E-06	-3.96E-06	-2.94E-06	-4.54E-06	-3.28E-06	-3.00E-06	-4.32E-06	-4.08E-06	-3.59E-06	-3.82E-06	-3.70E-06	-3.55E-06	-3.23E-06	-2.03E-06	-2.52E-06	-3.89E-06	-4.25E-06	-3.95E-06	-3.83E-06	-3.42E-06	-3.01E-06	-2.50E-06	-2.12E-06	-2.07E-06	-2.34E-06	-2.97E-06	-3.52E-06	-3.87E-06	-3.85E-06	-3.34E-06	-2.93E-06	-2.83E-06	-2.33E-06	-1.74E-06	-2.00E-06	-2.61E-06	-2.32E-06	-2.05E-06	-2.90E-06	-3.83E-06	-4.08E-06	-3.48E-06	-3.20E-06	-1.89E-06	-7.26E-07	-1.88E-06	-2.48E-06	-2.59E-06	-1.27E-06	-7.86E-07	-3.29E-06	-4.32E-06	-2.73E-06	-2.27E-06	-3.85E-06	-3.93E-06	-1.62E-06	-1.50E-06	-3.07E-06	-1.13E-06	-4.50E-07	-2.52E-06	-1.27E-06	-1.04E-06	-3.51E-06	-1.98E-06	-1.51E-06	4.72E-05	0.000298027	-0.000468594	-0.001470947	-0.001476212	-0.001213067	-0.000780846	-0.000501233	-0.000356343	-0.000248914	-0.000182324	-0.000150969	-0.00013269	-0.000102512	-8.25E-05	-8.26E-05	-6.45E-05	-5.90E-05	-5.18E-05	-3.23E-05	-4.96E-05	-4.07E-05	-1.29E-05	-2.82E-05	-3.70E-05	-1.20E-05	7.40E-06	-5.13E-06	-2.71E-05	-2.28E-05	-4.30E-06	8.62E-06	3.01E-06	-1.38E-05	-1.55E-05	-4.66E-06	7.66E-06	1.62E-05	6.76E-06	-1.25E-05	-1.61E-05	-5.46E-06	6.31E-06	1.36E-05	5.88E-06	-1.10E-05	-1.35E-05	-4.49E-06	5.74E-06	1.42E-05	9.33E-06	-8.89E-06	-1.53E-05	-6.31E-06	4.68E-06	1.37E-05	1.26E-05	-3.62E-06	-1.46E-05	-9.11E-06	1.07E-06	1.06E-05	1.14E-05	-2.25E-06	-1.53E-05	-1.08E-05	1.53E-06	1.19E-05	1.76E-05	4.64E-06	-1.31E-05	-1.33E-05	-3.67E-06	6.79E-06	1.21E-05	8.66E-07	-1.38E-05	-1.23E-05	-1.32E-06	1.00E-05	1.82E-05	1.19E-05	-7.78E-06	-1.51E-05	-7.22E-06	2.52E-06	1.00E-05];

% Q1 = numel(x30);

[Ux30, ix1, ix2] = unique(x30, 'stable');
USlope30 = Slope30(ix1);

Fs = 1/mean(diff(Ux30));                                            % Sampling Frequency
Fn = Fs/2;                                                          % Nyquist Frequency

[Lv,m,b] = ischange(USlope30,'linear', 'Threshold',1E-6);           % Detect Changes

idx = find(Lv, 1, 'first')-2;                                       % Find Index Of First Chjange (Adjust)
Ux30e = Ux30(idx:end)-Ux30(idx);                                    % Edited Independent Variable Vector
USlope30e = USlope30(idx:end);                                      % Edited Dependent Variable Vector

% figure                                                              % Test Plot
% plot(Ux30, [m; b; USlope30], '.-')
% ylim([-1 1]*1E-5)

ixs = find(diff(sign(USlope30e)));                                  % Zero Crossing Indices
per = mean(diff(Ux30e(ixs(end-20:end))));                           % Period (Units Of Independent Variable)
Freq = 1/per;                                                       % Oscillation Frequency

objfcn = @(b,x) b(1) .* b(2).*x.* exp(b(3).*x) .* exp(b(4).*x) .* sin(2*pi.*b(5) + 2*pi.*b(6)) + b(7);
B0 = [min(USlope30e)*[1;1]; -0.001; -0.0005; Freq; 1E-4; 1E-6];

% objfcn = @(b,x) b(1) .* b(2).*x .* exp(b(3).*x) .* sin(2*pi.*b(4) + 2*pi.*b(5)) + b(6);
% B0 = [min(USlope30e)*[1;1]; -0.001; Freq*1.5; 1E-4; 1E-6];

[B,fv] = fminsearch(@(b) norm(USlope30e - objfcn(b,Ux30e)), B0)

B = 7×1

-0.0201 0.0387 -0.0073 0.0029 0.0031 0.0000 -0.0000

fv = 6.9397e-04

figure
plot(Ux30e, USlope30e, '.')
hold on
plot(Ux30e, objfcn(B,Ux30e), '-r')
hold off
grid

These two models provide the best approximations to the data. (I experimented with others as well.)

The ‘B0’ vector is a vector of the initial parameter estimates.

.

[Tesla]

Thank you for your suggestion, its good but there are 7 variables which is a lot indeed for comparision with other signals.

otherwise, is possible to calculate the relaxtion time from those data?

[Star Strider]

This does not fit the curve at all well, because I have no idea what process created your data. The best model is one that mathematically describes that process. (The model I use here is simply an empirical guess.)

If you want to get the relaxation time, I would do something like this instead of fitting the data —

x30=	[1.40E+04	1.40E+04	1.41E+04	1.41E+04	1.42E+04	1.42E+04	1.42E+04	1.43E+04	1.43E+04	1.43E+04	1.44E+04	1.44E+04	1.45E+04	1.45E+04	1.45E+04	1.46E+04	1.46E+04	1.46E+04	1.47E+04	1.47E+04	1.48E+04	1.48E+04	1.48E+04	1.49E+04	1.49E+04	1.49E+04	1.50E+04	1.50E+04	1.51E+04	1.51E+04	1.51E+04	1.52E+04	1.52E+04	1.53E+04	1.53E+04	1.53E+04	1.54E+04	1.54E+04	1.54E+04	1.55E+04	15521.13777	15559.45668	15597.79092	15636.14029	1.57E+04	1.57E+04	1.58E+04	1.58E+04	1.58E+04	1.59E+04	1.59E+04	1.59E+04	1.60E+04	1.60E+04	1.61E+04	1.61E+04	1.61E+04	1.62E+04	1.62E+04	1.63E+04	1.63E+04	1.63E+04	1.64E+04	1.64E+04	1.64E+04	1.65E+04	1.65E+04	1.66E+04	1.66E+04	1.66E+04	1.67E+04	1.68E+04	1.69E+04	1.70E+04	1.71E+04	1.72E+04	1.73E+04	1.74E+04	1.75E+04	1.76E+04	1.77E+04	1.78E+04	1.79E+04	1.80E+04	1.81E+04	1.82E+04	1.83E+04	1.84E+04	1.85E+04	1.86E+04	1.87E+04	1.88E+04	1.89E+04	1.90E+04	1.91E+04	1.91E+04	1.92E+04	1.93E+04	1.94E+04	1.95E+04	1.96E+04	1.97E+04	1.98E+04	1.99E+04	2.00E+04	2.01E+04	2.02E+04	2.03E+04	2.04E+04	20512.09298	20609.50662	20706.9258	20804.34834	20901.7768	20999.22015	21096.67455	21194.13379	21291.59834	21389.06623	21486.53808	21584.02386	21681.52239	21779.02615	21876.53569	21974.04901	22071.56586	22169.09515	22266.63918	22364.18951	22461.7455	22559.30577	22656.8689	22754.44281	22852.03143	22949.62642	23047.2266	23144.83138	23242.43867	23340.05445	23437.6868	23535.32846	23632.97578	23730.6286	23828.28439	23925.94739	24023.62551	24121.31208	24219.00331	24316.69957	24414.39881	24512.10282	24609.82226	24707.55504	24805.29389	24903.03895	];
Slope30 = [	-3.55E-06	-5.52E-06	-3.96E-06	-2.94E-06	-4.54E-06	-3.28E-06	-3.00E-06	-4.32E-06	-4.08E-06	-3.59E-06	-3.82E-06	-3.70E-06	-3.55E-06	-3.23E-06	-2.03E-06	-2.52E-06	-3.89E-06	-4.25E-06	-3.95E-06	-3.83E-06	-3.42E-06	-3.01E-06	-2.50E-06	-2.12E-06	-2.07E-06	-2.34E-06	-2.97E-06	-3.52E-06	-3.87E-06	-3.85E-06	-3.34E-06	-2.93E-06	-2.83E-06	-2.33E-06	-1.74E-06	-2.00E-06	-2.61E-06	-2.32E-06	-2.05E-06	-2.90E-06	-3.83E-06	-4.08E-06	-3.48E-06	-3.20E-06	-1.89E-06	-7.26E-07	-1.88E-06	-2.48E-06	-2.59E-06	-1.27E-06	-7.86E-07	-3.29E-06	-4.32E-06	-2.73E-06	-2.27E-06	-3.85E-06	-3.93E-06	-1.62E-06	-1.50E-06	-3.07E-06	-1.13E-06	-4.50E-07	-2.52E-06	-1.27E-06	-1.04E-06	-3.51E-06	-1.98E-06	-1.51E-06	4.72E-05	0.000298027	-0.000468594	-0.001470947	-0.001476212	-0.001213067	-0.000780846	-0.000501233	-0.000356343	-0.000248914	-0.000182324	-0.000150969	-0.00013269	-0.000102512	-8.25E-05	-8.26E-05	-6.45E-05	-5.90E-05	-5.18E-05	-3.23E-05	-4.96E-05	-4.07E-05	-1.29E-05	-2.82E-05	-3.70E-05	-1.20E-05	7.40E-06	-5.13E-06	-2.71E-05	-2.28E-05	-4.30E-06	8.62E-06	3.01E-06	-1.38E-05	-1.55E-05	-4.66E-06	7.66E-06	1.62E-05	6.76E-06	-1.25E-05	-1.61E-05	-5.46E-06	6.31E-06	1.36E-05	5.88E-06	-1.10E-05	-1.35E-05	-4.49E-06	5.74E-06	1.42E-05	9.33E-06	-8.89E-06	-1.53E-05	-6.31E-06	4.68E-06	1.37E-05	1.26E-05	-3.62E-06	-1.46E-05	-9.11E-06	1.07E-06	1.06E-05	1.14E-05	-2.25E-06	-1.53E-05	-1.08E-05	1.53E-06	1.19E-05	1.76E-05	4.64E-06	-1.31E-05	-1.33E-05	-3.67E-06	6.79E-06	1.21E-05	8.66E-07	-1.38E-05	-1.23E-05	-1.32E-06	1.00E-05	1.82E-05	1.19E-05	-7.78E-06	-1.51E-05	-7.22E-06	2.52E-06	1.00E-05];

[Ux30, ix1, ix2] = unique(x30, 'stable');
USlope30 = Slope30(ix1);
Ux30 = x30(ix1);

[minv,idx1] = min(USlope30);                                                % Minimum & First Index

minv = -0.0015

idx1 = 33

idx2 = find(USlope30(idx1:end) < -1E-4, 1, 'last') + idx1;                  % Second Index

idx2 = 43

rlxv = exp(-1)                                                              % Relaxation Constant

rlxv = 0.3679

rlxt = interp1(USlope30(idx1:idx2), Ux30(idx1:idx2), rlxv*minv, 'pchip')    % Relaxation Time

rlxt = 1.7181e+04

figure
plot(Ux30, USlope30, '.-b', 'DisplayName','Unique Data')
hold on
plot(rlxt, rlxv*minv, 'sr', 'DisplayName','Relaxation Time')
hold off
grid
legend('Location','best')

The interp1 function requires that the query points (second argument) have unique values, and since the data are noisy and oscillate, it is necessary to restrict the region for the interpolation, therefore the two indices limiting it. The relaxation time is about 17181 time units.

.

[Tesla]

Thank you very much

[Star Strider]

As always, my pleasure!