hello @Dakota
FYI, a simple code that does not need any special toolbox :
load('Data.mat')
Time = mat(1,:);
Data = mat(2,:);
%% fit data
[b,yf] = exp_decay_sinus_fit(Time,Data);
% b array contains 5 coefficients according to this equation (model)
% y = b(1).*exp(b(2).*x).*(sin(2*pi*b(3)*x + b(4))) + b(5)
eq_str = [ ' y = ' num2str(b(1),'%.2f'),'* exp(-' num2str(-b(2),'%.2f') ' t) * sin(2*pi*' num2str(b(3),'%.2f') ' t + ' num2str(b(4),'%.2f') ' ) + (' num2str(b(5),'%.2f') ')'];
figure
plot(Time,Data,'*',Time,yf,'r','linewidth',2);
legend('data',eq_str);
%%%%%%%%%%%%%%% functions %%%%%%%%%%%%%%%%%%%%%%
function [B,yf] = exp_decay_sinus_fit(t,y)
[yu,indm] = max(y);
yl = min(y);
yr = (yu-yl); % Range of y
yz = y-yu+(yr/2);
% zero crossing (period / freq estimation)
zt = t(yz(:) .* circshift(yz(:),[1 0]) <= 0); % Find zero-crossings
per = 2*mean(diff(zt)); % Estimate period
freq = 1/per; % Estimate frequency
% initial phase estimate
tmax = t(indm);
phase_init = mod(-2*pi*freq*tmax + pi/2,2*pi); % initial phase estimate
ym = mean(y); % Estimate DC value (offset)
fit = @(b,x) b(1).*exp(b(2).*x).*(sin(2*pi*b(3)*x + b(4))) + b(5); % Objective Function to fit
fcn = @(b) norm(fit(b,t) - y); % Least-Squares cost function
B = fminsearch(fcn, [yr; -0.1; freq; phase_init; ym]); % Minimise Least-Squares
if B(4)<0 % complement negative phase with 2pi
B(4) = B(4) + 2*pi;
end
yf = fit(B,t);
end
