I wrote a LDPC decoding code based on DVB-S2, but it also decodes well even under EbNo < -1.6dB

Below is the communication system model code written based on DVB-S2 standard.

The code runs well, but even when EbNo < -1.6dB, BER converges to 10^(-5) for low code rates.

I realized that the data should not be decoded well for low EbNo.

So can somebody help me with this?

The code might look little bit complicated, but I guess that the problem occurs at % RIcian channel fading part.

Thank you in advance.

CN = []; % For data collecting
list_G = [0 0; 1 0; 1 1; 0 1];
list_IQ = [cos(pi/4)+1i*sin(pi/4); cos(3*pi/4)+1i*sin(3*pi/4); cos(5*pi/4)+1i*sin(5*pi/4); cos(7*pi/4)+1i*sin(7*pi/4)];

for R = 3
if R == 1
    r = 1/3; msglen = 5400; TIMES = 20; EbNo = [2,3,4];
elseif R == 2
    r = 1/2; msglen = 7200; TIMES = 3; EbNo = [2];
elseif R == 3
    r = 3/5; msglen = 9720; TIMES = 3; EbNo = [-0.5];%EbNo = 0.5:0.25:2.5;
elseif R == 4
    r = 2/3; msglen = 10800; TIMES = 3; EbNo = [0 0.5];%EbNo = [1, 1.25, 1.5, 1.75, 2, 2.5, 3];
elseif R == 5
    r = 3/4; msglen = 11880; TIMES = 2; EbNo = [2 3];%EbNo = 1.5:0.25:3.5;
elseif R == 6
    r = 4/5; msglen = 12600; TIMES = 2; EbNo = [2.5 3.5];%EbNo = 2:0.25:4;
elseif R == 7
    r = 5/6; msglen = 13320; TIMES = 2; EbNo = [2.5 3.5];%EbNo = [2, 2.25, 2.5, 2.75, 3, 3.5, 4];
elseif R == 8
    r = 8/9; msglen = 14400; TIMES = 2; EbNo = [3.5 4.5];%EbNo = [3, 3.25, 3.5, 3.75, 4, 4.5, 5];
end
Hchk = DVBS2LDPC_H_short(r); [Hrow, Hcol] = find(Hchk ~= 0); % Sparse parity check matrix for DVB-S2

for TOT = 1:TIMES
    %% Inner encoding: LDPC
msg = randi([0,1], 1, msglen);
Tx = DVBS2LDPC_short(msg, r); % Refer to DVBS2_QPSK_LDPC.m file
    %% QPSK modulation
Tx4 = zeros(1,16200/2);
for i = 1:length(Tx)/2
    if Tx(2*(i-1)+1:2*(i-1)+2) == [0 0]
        Tx4(i) = list_IQ(1);
    elseif Tx(2*(i-1)+1:2*(i-1)+2) == [1 0]
        Tx4(i) = list_IQ(2);
    elseif Tx(2*(i-1)+1:2*(i-1)+2) == [1 1]
        Tx4(i) = list_IQ(3);
    else
        Tx4(i) = list_IQ(4);
    end
end
    %% Rician channel fading (I think the problem is around here!)
K = 10;
Es = mean(abs(Tx4).^2);
No = Es./((10.^(EbNo/10))*log2(4));
for jj = 1:length(EbNo)
    tic
    h = sqrt(K/(K+1)) + sqrt(1/(K+1))*(1/sqrt(2))*(randn(size(Tx4)) + 1j*randn(size(Tx4)));
    ric_Tx = Tx4.*h;

    AWGN = sqrt(No(jj)/2) * (randn(size(Tx4)) + 1j*randn(size(Tx4)));
    ric_Rx_noise = ric_Tx + AWGN;
    ric_Rx = ric_Rx_noise ./ h;
        %% QPSK demodulation
    Rx4 = [];
    for i = 1:length(ric_Rx)
        temp = abs(list_IQ - ric_Rx(i));
        [m, idx] = min(temp);
        Rx4 = [Rx4, list_G(idx,:)];
    end
        %% LDPC decode
    % 0) Parity check: if checked good, no need to decode. Start from 1) if not
    % 1) Initialize u_n
    U = zeros(1,16200); count = 0;
    for i = 1:length(ric_Rx)
        U(2*(i-1)+1) = -(1/No(jj))*(real(ric_Rx(i))-(1/sqrt(2)))^2 + (1/No(jj))*(real(ric_Rx(i))-(-1/sqrt(2)))^2; % a priori log-likelihood ratio for the transmitted bit
        U(2*(i-1)+2) = -(1/No(jj))*(imag(ric_Rx(i))-(1/sqrt(2)))^2 + (1/No(jj))*(imag(ric_Rx(i))-(-1/sqrt(2)))^2; % Remind: No = 2v^2 for Gaussian distribution
    end
    % 2) Initialization of outgoing messages from bit nodes
    V_msg = [transpose(Hcol); transpose(Hrow)]; % 1st row: v-node, 2nd row: c-node, 3rd row: update from v-node to c-node
    for i = 1:size(Hrow,1)
        V_msg(3,i) = U(V_msg(1,i));
    end

    chk = mod(Hchk*Rx4',2); % Initialize chk to start LDPC iteration
    while any(chk) && count < 50 && sum(chk==1)>=15
        % 3) Check node update
        C_re = []; % 1st row: c-node, 2nd row: v-node, 3rd row: respond from c-node to v-node
        for i = 1:length(U)-msglen
            tempC = []; % k번째 c-node가 연결되어 있는 v-node 및 받은 데이터 저장
            for j = 1:size(V_msg,2)
                if V_msg(2,j) == i
                    tempC = [tempC, V_msg(:,j)];
                else
                end
            end
            for j = 1:size(tempC,2)
                temp = tempC(3,:); temp(:,j) = [];
                temp = g(temp);
                C_re = [C_re, [tempC(2,j);tempC(1,j);temp]]; % k번째 c-node가 연결되어 있는 v-node 및 보내는 데이터 저장
            end
        end
        % 4) Bit node update
        for i = 1:length(U)
            tempC = []; % n번째 v-node가 연결되어 있는 c-node 및 받은 데이터 저장
            for j = 1:size(C_re,2)
                if C_re(2,j) == i
                    tempC = [tempC, C_re(:,j)];
                else
                end
            end
            for j = 1:size(tempC,2)
                temp = tempC(3,:); temp(:,j) = [];
                temp = sum(temp);
                for n = 1:size(V_msg,2)
                    if V_msg(1,n) == tempC(2,j) && V_msg(2,n) == tempC(1,j)
                        V_msg(3,n) = U(V_msg(1,n)) + temp;
                        break
                    else
                    end
                end
            end
        end
        % 5) Hard decision making
        for i = 1:length(U)/2
            for j = 1:size(V_msg,2) % 1st bit of symbol
                if C_re(2,j) == 2*i-1
                    temp = C_re(:,j);
                    break
                end
            end
            list = [];
            for j = 1:size(V_msg,2)
                if V_msg(1,j) == temp(2)
                    list = [list, V_msg(:,j)];
                end
            end
            for j = 1:size(list,2)
                if list(2,j) == temp(1)
                    list = list(:,j);
                    break
                end
            end
            TEMP = temp(3) + list(3); % temp(3) = w, list(3) = v
            if TEMP >= 0
                Rx4(2*i-1) = 0;
            else
                Rx4(2*i-1) = 1;
            end
            for j = 1:size(V_msg,2) % 2nd bit of symbol
                if C_re(2,j) == 2*i
                    temp = C_re(:,j);
                    break
                end
            end
            list = [];
            for j = 1:size(V_msg,2)
                if V_msg(1,j) == temp(2)
                    list = [list, V_msg(:,j)];
                end
            end
            for j = 1:size(list,2)
                if list(2,j) == temp(1)
                    list = list(:,j);
                    break
                end
            end
            TEMP = temp(3) + list(3); % temp(3) = w, list(3) = v
            if TEMP >= 0
                Rx4(2*i) = 0;
            else
                Rx4(2*i) = 1;
            end
        end
        count = count +1;
        chk = mod(Hchk*Rx4',2);

        [number,BER] = biterr(Tx(1:msglen),Rx4(1:msglen));
        CN = [CN; r, TOT, EbNo(jj), count, number, sum(chk==1)];
    end
    if number <= 8
        CN = [CN; r, TOT, EbNo(jj), count, 0, 0]; % Assume that BCH decoding is done well
    end
    toc
end
end
end

Answer 1

[Riya]

Hello Geewon,

As per my understanding, you wrote a LDPC decoding code based on DVB-S2, but it should decode well even under EbNo < -1.6dB

Please note that based on the provided code, here's a breakdown of the potential areas to focus on for addressing the convergence issue, along with some suggestions for improvement:

• Rician Channel Fading

The Rician channel model seems to be implemented correctly. However, ensure that the parameters such as K factor, channel estimation, and noise variance are appropriate for the given scenario. You might consider testing different K factor values to see if the convergence behavior changes.

% Rician Channel Fading 
K = 10; 
Es = mean(abs(Tx4).^2); 
No = Es./((10.^(EbNo/10))*log2(4)); 

for jj = 1:length(EbNo) 
    h = sqrt(K/(K+1)) + sqrt(1/(K+1))*(1/sqrt(2))*(randn(size(Tx4)) + 1j*randn(size(Tx4))); 
    ric_Tx = Tx4.*h; 
    AWGN = sqrt(No(jj)/2) * (randn(size(Tx4)) + 1j*randn(size(Tx4))); 
    ric_Rx_noise = ric_Tx + AWGN; 
    ric_Rx = ric_Rx_noise ./ h; 
    % ...
end 

• LDPC Decoding

Ensure that the LDPC decoder is functioning correctly, especially for low EbNo values. You may want to verify the LDPC decoder's performance at low SNR values.

For more information regarding fading channels you can refer the following document:

https://www.mathworks.com/help/comm/ug/fading-channels.html#bsw99hk

I hope it helps!