Estimating a(t) from the output y(t) in the presence of noise can be done using system identification techniques. One approach you can consider is the Least-Squares (LS) method, which aims to minimize the sum of the squared errors between the model prediction and the measured output. This approach can be used with windowing to handle time-varying systems.
Here is a simple code.
% Given system equation: y(t) = s(t) + a(t)
% where a(t) = vd * (C(t) * h(t)) / (Ci + Cc(t)) + D
% and H(jw) = jw / (k(t) + jw)
% Define the system model
% y(t) = s(t) + vd * (C(t) * h(t)) / (Ci + Cc(t)) + D
% Define the range of k(t) and other constants
k_min = 0.1; % Minimum value of k(t)
k_max = 1.0; % Maximum value of k(t)
Ci = 2.0; % Constant Ci
D = 1.0; % Constant D
vd = 0.5; % Constant vd
% Generate sample data for s(t) and y(t)
t = 0:0.1:10; % Time vector
s = sin(t); % Sample input signal s(t)
a = vd * (cos(t) ./ (Ci + t)); % True noise signal a(t)
y = s + a; % Measured output y(t)
% Windowing and LS Parameter Estimation
block_size = 5; % Block size for windowing
num_blocks = length(t) / block_size;
estimated_a = zeros(size(t));
for i = 1:num_blocks
block_start = (i - 1) * block_size + 1;
block_end = i * block_size;
block_t = t(block_start:block_end);
block_y = y(block_start:block_end);
% Define the model for the block
model = @(params, t) params(1) * (cos(t) ./ (params(2) + t)) + params(3);
% Set up the LS problem
ls_problem = @(params) block_y - (s(block_start:block_end) + model(params, block_t));
initial_params = [vd, k_min, D]; % Initial guess for parameters
% Solve the LS problem using lsqnonlin
estimated_params = lsqnonlin(ls_problem, initial_params);
% Store the estimated a(t) for this block
estimated_a(block_start:block_end) = model(estimated_params, block_t);
end
% Plot the results
figure;
plot(t, a, 'b', 'LineWidth', 2); % True noise signal a(t)
hold on;
plot(t, estimated_a, 'r', 'LineWidth', 2); % Estimated noise signal a(t)
legend('True a(t)', 'Estimated a(t)');
xlabel('Time t');
ylabel('a(t)');
title('True vs. Estimated Noise Signal a(t)');
% Note: This is a simplified example, and you may need to adjust the model and
% LS problem based on the specific characteristics of your system and data.
