Can you give a MATLAB example of a multi carrier system based on CMFB

Sure, here's an example of a multi-carrier system based on Constant Modulus Feedback (CMFB) in MATLAB:

%% Define parameters
N = 1024; % Number of subcarriers
M = 16; % Modulation order
P = 4; % Number of pilot subcarriers
L = 8; % Length of equalizer
sigma2 = 0.1; % Noise variance

%% Generate symbols
x = randi([0 M-1], N-P, 1); % Random data symbols
p = randi([0 M-1], P, 1); % Pilot symbols

%% Modulate symbols
x_mod = qammod(x, M); % QAM modulation
p_mod = qammod(p, M); % QAM modulation

%% Add pilot symbols
x_with_pilot = zeros(N, 1);
x_with_pilot(1:P) = p_mod;
x_with_pilot(P+1:end) = x_mod;

%% Generate channel coefficients
h = randn(L, 1) + randn(L, 1)*1i; % Complex channel coefficients

%% Generate received signal
y = conv(h, x_with_pilot); % Channel convolution
y = y(P+1:end-L+1); % Remove pilot and excess convolution samples
y = awgn(y, sigma2); % Add noise

%% CMFB equalization
H = toeplitz(h, zeros(1, N-L));
e = zeros(N, 1);
mu = 0.1; % Step size
w = zeros(L, 1); % Initial weights
for i = 1:N
    z = H(:,i)' * w; % Estimate of symbol
    e(i) = y(i) - z; % Error
    w = w + mu * e(i) * H(:,i); % Update weights
end

%% Calculate BER
x_hat = qamdemod(H' * w, M); % Demodulate received symbols
ber = sum(x ~= x_hat) / length(x);
disp(['BER: ' num2str(ber)]);

This code generates random QAM symbols, adds pilot symbols, convolves with a random channel impulse response, adds noise, and then uses CMFB equalization to estimate the transmitted symbols. Finally, it calculates the bit error rate (BER) to evaluate the performance of the system.