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Based on the theoretical principle and simulation results of MATLAB spherical decoding, compare 2norm spherical decoding, infinite norm spherical decoding, ML detection

2022-07-27 00:27:00 【I love c programming】

Catalog

1. Algorithm description

The basic idea of sphere decoding is to use a vector x The radius of the center is d Search grid points in the multidimensional sphere , Reduce the number of search points by limiting or reducing the search radius , Thus, the calculation time is reduced . The advantage of sphere decoding algorithm is that it does not need to search all lattice points in the whole lattice like the traditional maximum likelihood decoding algorithm , And you only need to search in a pre-set limited spherical area , If the number of points contained in this area is quite small relative to the total number of points in the whole cell , Search time will be greatly reduced .

The key problems affecting sphere decoding are ：(1) How to choose search radius d. If d Too big , Then the ball will contain too many points , The complexity will approach or reach the exponential complexity of maximum likelihood decoding . If d Too small , Then the ball may not contain any grid points , Then the spherical decoding algorithm will not get a reasonable solution .(2) How to judge whether a point is in the ball . If this judgment needs to be judged by the distance between each grid point and the vector , Then this method is not ideal , Because we need to examine all points , The amount of computation generated is also exponential .

Spherical decoding solves the problem of 2 A question , Here, it is considered that the signal is a real number , Because the complex number can be increased by doubling the dimension , Separate the real part from the imaginary part , To judge whether a point is at the radius d Of m It's difficult inside the ball . If the m Turn into 1, Then it degenerates from a ball to a spacing , This point is equivalent to the real part or imaginary part of the signal sent by an antenna , This makes the operation much simpler , You can know whether this point is within this distance . The real and imaginary parts of signals on multiple transmitting antennas are divided into many dimensions , It is possible to take values on each dimension . The spherical decoding algorithm is equivalent to building a tree , The first of the trees k The layer node corresponds to falling on a radius of d, Dimension is k Grid point in the ball of .

If the performance is required to fully meet the performance of maximum likelihood detection , Then the initial radius must be a series of values , The selected initial value is d, If the range is d Can't find the right point , You need to increase d Value , The expansion multiple is 2 times ; If the required performance is close to the performance of maximum likelihood detection , Then the initial radius is compared with the above , Take the larger one , There must be some redundancy .

For the two norm spherical decoding which only requires the performance close to the performance of maximum likelihood detection , The formula for selecting the initial radius is as follows ：

For the two norm spherical decoding whose performance is required to achieve the performance of maximum likelihood detection , The formula for selecting the initial radius is as follows ：

2. Partial procedure

        msg=randint(log2(mod_num),Nt*Number/2);
        h=modem.qammod('M',mod_num,'InputType','bit','symbolorder','Gray');%qam modulation 
        s=modulate(h,msg);
        constelPoints=[1+j 1-j -1+j -1-j];                      %$ for ML detect
  %     constelPoints=[-3-3*j -3-j -3+3*j -3+j -1-3*j -1-j -1+3*j -1+j 3-3*j 3-j 3+3*j 3+j 1-3*j 1-j 1+3*j 1+j]; 
        s1=[real(s);imag(s)];                                    %s1 Is the real matrix of the signal 
        s2=reshape(s,Nt,Number/2);                               %s2 For signal Nt Complex matrix of rows 
        if mod_num==4
           codebook=[-1 1];                                       % Indicates that after becoming a real matrix ,4qam There are only two cases 1 and -1 
        end
        if mod_num==16                                          % Indicates that after becoming a real matrix ,16qam Yes 4 In this case 
           codebook=[-3 -1 1 3]; 
        end   
        ss=reshape(s1,2*Nt,Number/2);                            %ss For signal 2*Nt Real matrix of rows   
%       H=[real(H1(:,:,round)) -imag(H1(:,:,round));imag(H1(:,:,round)) real(H1(:,:,round))];                % Change the channel complex matrix into the channel real matrix  


%/%
       r1=H1(:,:,round)*s2;                                 
       for ii=1:Nt                                              % Add noise 
           rr(ii,:)=awgn(r1(ii,:),snr_1(ISNR),'measured');
       end                                                      %rr To receive complex signals  
     
       C1=3*(2*Nt)*2*(2/(10^(snr(ISNR)/10)));                                   % Initial radius   2 norm 4qam
       C2=16*2*(2/(10^(snr(ISNR)/10)));                                      % The initial radius is infinite norm 

       rev1=[real(rr);imag(rr)];                                      %rr Turn into a real signal 
       H=[real(H1(:,:,round)) -imag(H1(:,:,round));imag(H1(:,:,round)) real(H1(:,:,round))];              % Channel real matrix 
     
     
       for tmp=1:Number/2                                            % Two norm spherical decoding 
           rev=rev1(:,tmp);   
           tic
           y_2norm_temp = spheredecode(rev, C1, H, codebook);
           for jj=1:Nt
               y_2norm(Nt*tmp+jj-Nt)=y_2norm_temp(jj)+j*y_2norm_temp(jj+Nt);                 % Store in plural 
           end
           t1(ISNR)=toc+t1(ISNR); 
       end   
     
       H=[real(H1(:,:,round)) -imag(H1(:,:,round));imag(H1(:,:,round)) real(H1(:,:,round))];              % Channel real matrix 
       for tmp=1:Number/2                                            % Two norm spherical decoding 
           rev=rev1(:,tmp);   
           tic
           y_infnorm_temp = spheredecodeinf(rev, C2, H, codebook);    
           for jj=1:Nt
               y_infnorm(Nt*tmp+jj-Nt)= y_infnorm_temp(jj)+j* y_infnorm_temp(jj+Nt);                 % Store in plural 
           end
           t2(ISNR)=toc+t2(ISNR); 
       end   
     
     
     
       for tmp=1:Number/2
           min=100000;
           tic
           for flag1=1:mod_num
               for flag2=1:mod_num
                   for flag3=1:mod_num
                       for flag4=1:mod_num
                           stmp=[constelPoints(flag1);constelPoints(flag2);constelPoints(flag3);constelPoints(flag4)];
                           rtmp=rr(:,tmp);
                           Maximum=norm(rtmp-H1(:,:,round)*stmp); 
                           if Maximum<min
                              min=Maximum;
                              slast=stmp;
                           end    
                       end
                   end
               end
           end  
           for jj=1:Nt        
               y_ML(Nt*tmp+jj-Nt)=slast(jj);
           end     
           t3(ISNR)=toc+t3(ISNR); 
        end  
  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%    
       h=modem.qamdemod('M',mod_num,'OutputType','bit','symbolorder','Gray');   % The second norm 
       DM1_msg=demodulate(h,y_2norm);
       DM_msg=reshape(DM1_msg,log2(mod_num),Nt*Number/2);
       [number,ratio] = biterr(msg,DM_msg);  

       h=modem.qamdemod('M',mod_num,'OutputType','bit','symbolorder','Gray');
       DM3_msg=demodulate(h,y_infnorm);
       DM2_msg=reshape(DM3_msg,log2(mod_num),Nt*Number/2);
       [number1,ratio1] = biterr(msg,DM2_msg);                           % The infinite norm   

       h=modem.qamdemod('M',mod_num,'OutputType','bit','symbolorder','Gray');
       DM5_msg=demodulate(h,y_ML);
       DM4_msg=reshape(DM5_msg,log2(mod_num),Nt*Number/2);
       [number2,ratio2] = biterr(msg,DM4_msg);                           %ML 
   
       sumber=sumber+ratio;  
       sumber1=sumber1+ratio1;
       sumber2=sumber2+ratio2;   
    end %round=1：SymTime
    avrgber(ISNR)=sumber/SymTime;
    avrgber1(ISNR)=sumber1/SymTime;
    avrgber2(ISNR)=sumber2/SymTime;