Correcting the error in Alamouti code (2 x 1)

I trying to code Alamouti code for BPSK in Rayleigh fading channel for BER simulation.There is a matrix dimention mismatch in decode section.I tried to correct it ,but it did not work.

Need your help to correct it.Thank you

clear all;
close all;
clc;

Ps = 1;%Transmit power
N = 10^6; % number of bits or symbols
NR = 1;
num_runs = 10;
snr_dB = -3:10 ; % in dB

rand('seed',1); % initializing the rand() function so that random bits produced are same in every simulation
randn('seed',1);
% Transmitter

x_data1 = rand(1,N/2)>0.5;
x_data2 = rand(1,N/2)>0.5;

binaryData = [x_data1; x_data2];

BPSK_data1 = 2*x_data1 - 1;
BPSK_data2 = 2*x_data2 - 1;

% BPSK modulation - 2x500000 matrix
BPSK_data =[BPSK_data1; BPSK_data2];

n1 = AWGN(1);
n2 = AWGN(1);

h1 = Rayleigh(1);
h2 = Rayleigh(1);

SNR = 10.^(snr_dB/10);
Noise_power = zeros(1,length(snr_dB));


for  k = 1:length(snr_dB)
     % Add complex AWGN noise power
     Noise_power = Ps./SNR(k);

% In the first time slot, the received signal
y1 = h1.*BPSK_data1 + h2.*BPSK_data2 + n1;

% In the first time slot, the received signal
y2 = h1.*(-conj(BPSK_data2)) + h2.*(conj(BPSK_data1)) + n2;

%Rayleigh fading channel - 2x1 matrix
H = [h1  h2 ; conj(h2)  -conj(h1)];

%Decode the sysmbols - ZF decoder
x1_hat = pinv(H).*y1;
x2_hat = pinv(H).*conj(y2);


rx_bits = [x1_hat;x2_hat];
rx_bits = real(rx_bits) > 0;
      % Count errors
nErr(k) = size(find(binaryData - rx_bits),1);

end
      % counting the errors
    ber = nErr/N; % simulated ber  

    figure(1);
    semilogy(snr_dB,ber,'b.-');
    % axis([0 10 10^-4 0.6])
    grid on;
    legend( 'MIMO - BER');
    xlabel('SNR[dB]');
    ylabel('Bit Error Rate');
    title('BER for MIMO-BPSK');

%%%%% AWGN noise %%%%%
function n = AWGN(N)
    n = rand(1,N); 
end

%%%%%  Rayleigh channel %%%%%%

function h = Rayleigh(N);

   x = sqrt(1/2).*rand(1,N);
   y = sqrt(1/2).*rand(1,N);
   
   h = abs(x + i*y);
   
end

Answer 1

[Shashi Kiran]

Hi Kasun,

I see that you are implementing the Alamouti code and generating the BER curve.

I have made a few adjustments to the code to ensure it works as expected. One key point is that the noise should be complex since we are dealing with complex baseband signals.

Also, while decoding using Zero Forcing decoder, I have used the below calculations for simplicity.

On solving the equation, it can be minimized to:

 % Decode the symbols - Alamouti scheme
    x_hat1 = conj(h1).*y1 + h2.*conj(y2);
    x_hat2 = conj(h2).*y1 - h1.*conj(y2);

% Normalize by the channel power
    h_power = abs(h1).^2 + abs(h2).^2;
    x_hat1 = x_hat1 ./ h_power;
    x_hat2 = x_hat2 ./ h_power;

Here is the updated version of the code with the changes made .

clear all;
close all;
clc;

Ps = 1; % Transmit power
N = 10^6; % Number of bits or symbols
snr_dB = -8:18; % in dB

rand('seed',1); % Initializing the rand() function so that random bits produced are the same in every simulation
randn('seed',1);

% Transmitter
x_data1 = rand(1,N/2) > 0.5;
x_data2 = rand(1,N/2) > 0.5;

binaryData = [x_data1; x_data2];

BPSK_data1 = 2*x_data1 - 1;
BPSK_data2 = 2*x_data2 - 1;

% BPSK modulation - 2x500000 matrix
BPSK_data = [BPSK_data1; BPSK_data2];

SNR = 10.^(snr_dB/10);
Noise_power = zeros(1,length(snr_dB));
nErr = zeros(1,length(snr_dB)); % Initialize the error count array

for k = 1:length(snr_dB)
    % Add complex AWGN noise power
    % disp(k)
    Noise_power(k) = Ps./SNR(k);

    % Generate noise
    n1 = sqrt(Noise_power(k)/2) * AWGN(N/2);
    n2 = sqrt(Noise_power(k)/2) * AWGN(N/2);

    % Generate Rayleigh fading channel coefficients
    h1 = Rayleigh(N/2);
    h2 = Rayleigh(N/2);

    % In the first time slot, the received signal
    y1 = h1.*BPSK_data1 + h2.*BPSK_data2 + n1;

    % In the second time slot, the received signal
    y2 = h1.*(-conj(BPSK_data2)) + h2.*(conj(BPSK_data1)) + n2;

    % Decode the symbols - Alamouti scheme
    x_hat1 = conj(h1).*y1 + h2.*conj(y2);
    x_hat2 = conj(h2).*y1 - h1.*conj(y2);

    % Normalize by the channel power
    h_power = abs(h1).^2 + abs(h2).^2;
    x_hat1 = x_hat1 ./ h_power;
    x_hat2 = x_hat2 ./ h_power;
    
    rx_bits1 = real(x_hat1) > 0;
    rx_bits2 = real(x_hat2) > 0;

    rx_bits = [rx_bits1; rx_bits2];

    % Count errors
    nErr(k) = sum(sum(binaryData ~= rx_bits));
end

% Calculate BER
ber = nErr/N; % Simulated BER

%%%%% AWGN noise %%%%%
function n = AWGN(N)
    n = randn(1, N) + 1i*randn(1, N); 
end

%%%%% Rayleigh channel %%%%%%
function h = Rayleigh(N)
    x = sqrt(1/2) * randn(1, N);
    y = sqrt(1/2) * randn(1, N);
    h = x + 1i*y;
end

% Plot results
semilogy(snr_dB, ber, 'b.-');
grid on;
legend('MIMO - BER');
xlabel('SNR [dB]');
ylabel('Bit Error Rate');
title('BER for MIMO-BPSK with Alamouti Scheme');

Refer below for the working of Alamouti code.

1. http://spacetimecodes.blogspot.com/p/alamouti-code.html
2. https://in.mathworks.com/help/matlab/ref/pinv.html?searchHighlight=pinv&s_tid=srchtitle_support_results_1_pinv

Hope this solves your query.