Change proper gaussian signal to improper gaussian signal

Hello everyone, I am having trouble trying to change proper gaussian signal to improper gaussian signal, would like to have your opinion on how I can proceed with this. Thanks

%Joint Iterative Decoding and Detection using polar coded SCMA-OFDM with
%channel estimation at PHY layer
%% Initilization
tic
EbN0 = 0:1:10; 
polar_N = 256;
polar_K = 128;
polar_n = log2(polar_N);
construction_method = 0;%0-BA,1-MC,2-GA
design_snr_dB = 0;%BA and MC construction method will use it
sigma = 0.9;%GA construction method will use it
crc_size = 0;
global pilot_loc Xp
[FZlookup,bitreversedindices,F_kron_n] = initPC(polar_N,polar_K,polar_n,construction_method,design_snr_dB,sigma,crc_size);

Unrecognized function or variable 'initPC'.

alpha = 0.6;
iter_num = 5;
isInterleaver = 1;

%% load codebook
load('codebook_6users_4chips_qpsk.mat','CB');
%load('channel_custom_complex.mat','channel');

K = size(CB, 1); % number of orthogonal resources
M = size(CB, 2); % number of codewords in each codebook
V = size(CB, 3); % number of users (layers)
%polar initial and encoding



SCAN_ITER_NUM = 1;
N = polar_N/log2(M); %Number of scma symbols of each user
SNR  = EbN0 + 10*log10(polar_K/polar_N*log2(M)*V/K);
N0 = 1./10.^(SNR/10); % Noise variance

Nerr = zeros(1,length(EbN0));
Nbits = zeros(1,length(EbN0));
BER   = zeros(1, length(EbN0));

%maxNumErrs = 10000;
maxNumBits = 1e7; %total numer of bits
minNumBits = 50000;
minNumErrs = 50;


% maxNumPEs = 10; % The maximum number of packet errors at an SNR point
% maxNumPackets = 100; % Maximum number of packets at an SNR point
% S = numel(SNR);
% packetErrorRate = zeros(S,1);
% 
% 
% % Loop to simulate multiple packets
% numPacketErrors = 0;
% numPkt = 1; % Index of packet transmitted

%% polar encoder
    for iter_ebn0 = 1:length(EbN0)
        
        
        while ((min(Nerr(:,iter_ebn0)) < minNumErrs) && (Nbits(1,iter_ebn0) < maxNumBits) || (Nbits(1,iter_ebn0) <minNumBits) )%100 010 000
            
            
   %% generate bits         
            infobits = randi([0 1],V,polar_K);
            c = zeros(V,polar_N);
            for user = 1:V
                c(user,:) = pencode(infobits(user,:),FZlookup,crc_size,bitreversedindices,F_kron_n);
            end
   %% interleaver         
            if isInterleaver ~= 0
                interleaver = zeros(V,polar_N);
                interleavered_bits = zeros(size(c));
                for ii = 1:V
                    interleaver(ii,:) = randperm(polar_N);
                    interleavered_bits(ii,:) = c(ii, interleaver(ii,:));
                end
                
            else
                interleavered_bits = c;
            end
            
            temp1 = reshape(interleavered_bits',polar_N*V,1);
            temp2 = reshape(temp1,log2(M),N*V);
            x_temp = bi2de(temp2',log2(M),'left-msb');
            x = reshape(x_temp,N,V);
            x = x';
            %% Channel 
            %h = channel;
            %h = 1/sqrt(2)*(repmat(randn(1, V, N), K, 1)+1j*repmat(randn(1, V, N), K, 1)); %UL Rayleigh W/o Diversity
            %h = 1/sqrt(2)*(randn(K, V, N)+1j*randn(K, V, N)); % Rayleigh channel
            h = ones(K, V, N); % perfect CSI
            %h = 1/sqrt(2)*(repmat(randn(1, V, N), K, 1)+1j*repmat(randn(1, V, N), K, 1));
            %h = 1/sqrt(2)*(repmat(randn(K, 1, N), 1, V)+1j*repmat(randn(K, 1, N), 1, V)); % DL with Diversity
            %h = 1/sqrt(2)*(repmat(repmat(randn(1, 1, N),K, 1), 1,...
            %V)+1j*(repmat(repmat(randn(1, 1, N), K, 1), 1, V))); %DL Rayleigh
            %w/o Diversity
            
            s = scmaenc(x, CB, h);
            %% Add Pilot
            
            s_pilot = add_pilot(s);
            
            %% IFFT
            
            s_pilot_ifft = ifft(s_pilot);
            
            %% Add Cyclic Prefix
            
            s_pilot_ifft_cp = add_CP(s_pilot_ifft);
            
            
            %% parallel to serial
            
            S_pilot_ifft_cp_serial = s_pilot_ifft_cp.';
            
            %% AWGN
            
            y = awgn(S_pilot_ifft_cp_serial, SNR(iter_ebn0),'measured');
            
            %% Serial to parallel
            
            y_parallel = y.';
            %% Remove Cyclic Prefix
            
            y_pilot_ifft = remove_CP(y_parallel);
            
            %% FFT
            
            y_pilot = fft(y_pilot_ifft);
                        for snr = 0:2:30
            %% Channel estimation
            
            %Least Square
            
            Nfft=640;
            Nps=4;
            %H_est = LS_CE(y_pilot,Xp,pilot_loc,Nfft,Nps,'linear');
            %                 %err=(H-H_est)*(H-H_est)';
            %                 %z=[z err/(Nfft*Nsym)];
            %                 %y_eq = LS_CE2(y);
            
            %MMSE
            
                            H_est = MMSE_CE(y_pilot,Xp,pilot_loc,Nfft,Nps,h,snr);
                        end
            %% Equalization
            
            y_eq = y_pilot./H_est;
            
            %% Remove pilot
            
            y1 = remove_pilot(y_eq);
            
            %% Joint Decoding and Detection (SCMA + Polar + Deinterleaver)
            
            mhat_llr = JIDD(y1,polar_N,polar_K,FZlookup,K,V,M,N,CB,N0(iter_ebn0),h,iter_num,isInterleaver,interleaver,alpha);
            
            %**********************************************************
            llr = reshape(mhat_llr',1,V*polar_K);
            m_reshape = reshape(infobits', 1, polar_K*V);
            m_hat = llr<0;
            err = sum(m_hat~=m_reshape);
            Nerr(iter_ebn0) = Nerr(iter_ebn0) + err;
            Nbits(iter_ebn0) = Nbits(iter_ebn0) + length(m_reshape);
            %             Pb(iter_ebn0)=mean(abs((m_reshape)-Nbits(iter_ebn0)));
            fprintf('.')
        end
        %     figure(1);
        fprintf('\n')
        BER(iter_ebn0) = Nerr(iter_ebn0)/Nbits(iter_ebn0);
        
        
        
        
        fprintf('EbN0 is %d, have runned %d bits, found %d errors, BER=%.7f \n',EbN0(iter_ebn0),Nbits(iter_ebn0),Nerr(iter_ebn0),BER(iter_ebn0));
        
        
        
        
    end%Joint Iterative Decoding and Detection using polar coded SCMA-OFDM with
%channel estimation at PHY layer
%% Initilization
tic
EbN0 = 0:1:10; 
polar_N = 256;
polar_K = 128;
polar_n = log2(polar_N);
construction_method = 0;%0-BA,1-MC,2-GA
design_snr_dB = 0;%BA and MC construction method will use it
sigma = 0.9;%GA construction method will use it
crc_size = 0;
global pilot_loc Xp
[FZlookup,bitreversedindices,F_kron_n] = initPC(polar_N,polar_K,polar_n,construction_method,design_snr_dB,sigma,crc_size);
alpha = 0.6;
iter_num = 5;
isInterleaver = 1;

%% load codebook
load('codebook_6users_4chips_qpsk.mat','CB');
%load('channel_custom_complex.mat','channel');

K = size(CB, 1); % number of orthogonal resources
M = size(CB, 2); % number of codewords in each codebook
V = size(CB, 3); % number of users (layers)
%polar initial and encoding



SCAN_ITER_NUM = 1;
N = polar_N/log2(M); %Number of scma symbols of each user
SNR  = EbN0 + 10*log10(polar_K/polar_N*log2(M)*V/K);
N0 = 1./10.^(SNR/10); % Noise variance

Nerr = zeros(1,length(EbN0));
Nbits = zeros(1,length(EbN0));
BER   = zeros(1, length(EbN0));

%maxNumErrs = 10000;
maxNumBits = 1e7; %total numer of bits
minNumBits = 50000;
minNumErrs = 50;


% maxNumPEs = 10; % The maximum number of packet errors at an SNR point
% maxNumPackets = 100; % Maximum number of packets at an SNR point
% S = numel(SNR);
% packetErrorRate = zeros(S,1);
% 
% 
% % Loop to simulate multiple packets
% numPacketErrors = 0;
% numPkt = 1; % Index of packet transmitted

%% polar encoder
    for iter_ebn0 = 1:length(EbN0)
        
        
        while ((min(Nerr(:,iter_ebn0)) < minNumErrs) && (Nbits(1,iter_ebn0) < maxNumBits) || (Nbits(1,iter_ebn0) <minNumBits) )%100 010 000
            
            
   %% generate bits         
            infobits = randi([0 1],V,polar_K);
            c = zeros(V,polar_N);
            for user = 1:V
                c(user,:) = pencode(infobits(user,:),FZlookup,crc_size,bitreversedindices,F_kron_n);
            end
   %% interleaver         
            if isInterleaver ~= 0
                interleaver = zeros(V,polar_N);
                interleavered_bits = zeros(size(c));
                for ii = 1:V
                    interleaver(ii,:) = randperm(polar_N);
                    interleavered_bits(ii,:) = c(ii, interleaver(ii,:));
                end
                
            else
                interleavered_bits = c;
            end
            
            temp1 = reshape(interleavered_bits',polar_N*V,1);
            temp2 = reshape(temp1,log2(M),N*V);
            x_temp = bi2de(temp2',log2(M),'left-msb');
            x = reshape(x_temp,N,V);
            x = x';
            %% Channel 
            %h = channel;
            %h = 1/sqrt(2)*(repmat(randn(1, V, N), K, 1)+1j*repmat(randn(1, V, N), K, 1)); %UL Rayleigh W/o Diversity
            %h = 1/sqrt(2)*(randn(K, V, N)+1j*randn(K, V, N)); % Rayleigh channel
            h = ones(K, V, N); % perfect CSI
            %h = 1/sqrt(2)*(repmat(randn(1, V, N), K, 1)+1j*repmat(randn(1, V, N), K, 1));
            %h = 1/sqrt(2)*(repmat(randn(K, 1, N), 1, V)+1j*repmat(randn(K, 1, N), 1, V)); % DL with Diversity
            %h = 1/sqrt(2)*(repmat(repmat(randn(1, 1, N),K, 1), 1,...
            %V)+1j*(repmat(repmat(randn(1, 1, N), K, 1), 1, V))); %DL Rayleigh
            %w/o Diversity
            
            s = scmaenc(x, CB, h);
            %% Add Pilot
            
            s_pilot = add_pilot(s);
            
            %% IFFT
            
            s_pilot_ifft = ifft(s_pilot);
            
            %% Add Cyclic Prefix
            
            s_pilot_ifft_cp = add_CP(s_pilot_ifft);
            
            
            %% parallel to serial
            
            S_pilot_ifft_cp_serial = s_pilot_ifft_cp.';
            
            %% AWGN
            
            y = awgn(S_pilot_ifft_cp_serial, SNR(iter_ebn0),'measured');
            
            %% Serial to parallel
            
            y_parallel = y.';
            %% Remove Cyclic Prefix
            
            y_pilot_ifft = remove_CP(y_parallel);
            
            %% FFT
            
            y_pilot = fft(y_pilot_ifft);
                        for snr = 0:2:30
            %% Channel estimation
            
            %Least Square
            
            Nfft=640;
            Nps=4;
            %H_est = LS_CE(y_pilot,Xp,pilot_loc,Nfft,Nps,'linear');
            %                 %err=(H-H_est)*(H-H_est)';
            %                 %z=[z err/(Nfft*Nsym)];
            %                 %y_eq = LS_CE2(y);
            
            %MMSE
            
                            H_est = MMSE_CE(y_pilot,Xp,pilot_loc,Nfft,Nps,h,snr);
                        end
            %% Equalization
            
            y_eq = y_pilot./H_est;
            
            %% Remove pilot
            
            y1 = remove_pilot(y_eq);
            
            %% Joint Decoding and Detection (SCMA + Polar + Deinterleaver)
            
            mhat_llr = JIDD(y1,polar_N,polar_K,FZlookup,K,V,M,N,CB,N0(iter_ebn0),h,iter_num,isInterleaver,interleaver,alpha);
            
            %**********************************************************
            llr = reshape(mhat_llr',1,V*polar_K);
            m_reshape = reshape(infobits', 1, polar_K*V);
            m_hat = llr<0;
            err = sum(m_hat~=m_reshape);
            Nerr(iter_ebn0) = Nerr(iter_ebn0) + err;
            Nbits(iter_ebn0) = Nbits(iter_ebn0) + length(m_reshape);
            %             Pb(iter_ebn0)=mean(abs((m_reshape)-Nbits(iter_ebn0)));
            fprintf('.')
        end
        %     figure(1);
        fprintf('\n')
        BER(iter_ebn0) = Nerr(iter_ebn0)/Nbits(iter_ebn0);
        
        
        
        
        fprintf('EbN0 is %d, have runned %d bits, found %d errors, BER=%.7f \n',EbN0(iter_ebn0),Nbits(iter_ebn0),Nerr(iter_ebn0),BER(iter_ebn0));
        
        
        
        
    end

[Image Analyst]

What is initPC?

[navid seif]

If s(t) is a guassian signal, then the jth column of Ψi corresponds to πj that πj is a 3D vector [xj; yj; zj] shown above:

How can I implement this formula in MATLAB?

Answer 1

[Pooja Kumari]

Dear Bryan,

It is my understanding that you are facing issues with changing proper guassian signal to improper guassian signal. Proper guassian signal using "awgn" function can be converted to improper guassian signal by two methods:

1. By adding the improper component using amplitde shift :

%% AWGN
y = awgn(S_pilot_ifft_cp_serial, SNR(iter_ebn0),'measured');
            
% Generate random amplitude variations
amplitudeVariations = randn(size(y));

% Scale the proper signal with the amplitude variations
improperSignal = y .* amplitudeVariations;

2. By adding random phase shift to the signal:

% Generate random phase shift
phaseShift = 2 * pi * rand(size(y));

% Apply phase shift to the proper signal
improperSignal = y .* exp(1j * phaseShift);

Generate a random phase shift using "rand" and multiply it by "2 * pi". Apply the phase shift to the proper guassian signal by element-wise multiplication with the complex exponential "exp(1j * phaseShift)". The resulting signal will have the same statistical properties as the original proper Gaussian signal but with a random phase shift, making it an improper Gaussian signal.

Regards,

Pooja Kumari