function [ x_mf_norm, x_cs_norm ] = yAxn_recovery( A, SNR, Target_index, lambda )

    % matrix parameters
    [M, N] = size(A);
    mul = A(:,1)'*A(:,1);
    
    % H0 sample
    L = round(Target_index + N * 0.1);
    R = round(N * 0.9);
    Lambda_C = L: R;
    
    % parameters
    sigma_n = 0.1;
    Target_amplitude = sqrt(10^(SNR/10) * sigma_n^2 / mul);

    
    %% CS setting
    % CS-parameters
%     lambda = 0.0005;
    alpha = 1/4;
    delta = 1e-8*alpha;
    
    % CS-normalization
    J1 = A*A';
    lambda_J=eig(J1);
    A_norm = A / sqrt(lambda_J(end));
    Hp = A_norm'*A_norm;
    [~, D] = eig(Hp);
    d = diag(D);
    sigma_n_norm = sigma_n / sqrt(lambda_J(end));
    
    
    %% generate x
    x = zeros(N, 1);
    x(Target_index) = Target_amplitude;
    
    
    %% generate y
    noise = random('Normal', 0, sigma_n/sqrt(2), M, 1) + 1j * random('Normal', 0, sigma_n/sqrt(2), M, 1);
    y = A*x + noise;
    
    
    %% recover
    % MF
    x_mf = A' * y ./ mul;
    sigma_MF = sqrt(var(x_mf(Lambda_C)));
    x_mf_norm = x_mf ./ sigma_MF;
    
    % CS
    y_norm = y / sqrt(lambda_J(end));
    x_FISTA = FISTA_v1(y_norm, A_norm, lambda, delta, Hp);
    [x_d_cal_f, hat_Q1_cal_f, sigma_d_cal_f] = cal_debiased_LASSO_v1(x_FISTA, A_norm, y_norm, lambda, sigma_n_norm, d);
    sigma_CS = sqrt(var(x_d_cal_f(Lambda_C)));
    x_cs_norm = x_d_cal_f ./ sigma_CS;
    
end