use prony method for damped exponentials
you have C and U, and C can be written as C_i = (A_i/2)*exp(j*phase)
U_i can be written as exp(damp_i + j*2*pi*f_i)*T
its should be easy to extract the info from that. T is sampling period.

%to get damping factor i and frequency i use something like this 
%assume damping factor at sample 1 is 0.3, frequency at 1 is 20 
%and out sampling frequency is 500 Hz
damp1 = [0.3 0.4 0.5];
f1 = [20 30 40];
T = 1/500;
%assume this is your ui vector from th ealgorithm
ui = exp((damp1 + j*2*pi*f1)*T);
%get the damping and frequencies
lv1 = log(ui);
r_damp1 = real(lv1)/T;
r_f1 =imag(lv1)/(T*2*pi);

%assume this is your Ci vector from algorithm
Ai = [0.1 0.2 0.3];
phasei=[pi/8 pi/4 pi/2];
Ci = (Ai/2).*exp(j*phasei);
%extract phase and amplitude
r_Amp = abs(Ci)*2;
r_phase = angle(Ci);