OFDM系统各种调制阶数的QAM误码率（Symbol Error Rate）与 误比特率（Bit Error Rate）仿真结果

本文是OFDM系统的不同QAM调制阶数的误码率与误比特率仿真，仅考虑在高斯白噪声信道下的情景，着重分析不同信噪比下的误码（符号）率性能曲线，不关心具体的调制与解调方案，仿真结果与理论的误码率曲线进行了对比。

考虑一个简单的OFDM系统，每个频域子载波承载一个QAM调制符号，在经过不同信噪比白噪声信道之后，每个QAM调制符号的解调性能如何，每个符号对应的比特解码性能如何？理论的误码性能如何？

代码

clc;close all;clear

%% Seting parameters
EbN0_list = 0:1:10;
Q_order_list = 2:2:10;
loopNumber = 10;
fprintf('Qm\t EbN0 \t \t EsN0 \t \t SNR_Cal \t \t ser \t\t ser_theory\t\t\t ber\t\t nloop \t\t \n');
for iQorder = 1 : length(Q_order_list)
for iEbN0 = 1 : length(EbN0_list)

%% Frame structure
N_Frame = 10;
N_Symbol = 14;
N_RB = 106;
N_SC_perRB = 12;
N_SC = N_RB * N_SC_perRB;
N_Ant = 1;
N_fft_order = floor(log2(N_RB * N_SC_perRB));
N_fft = 2^(N_fft_order+1);
N_cp = N_fft/8;
EbN0 = EbN0_list(iEbN0);

%% Modulation
Q_order = Q_order_list(iQorder);
Qm = 2^Q_order;
N_bit = N_Frame * N_Symbol * N_RB * N_SC_perRB * Q_order;

%% Noise Calculation
SNR =  EbN0 + 10 * log10(Q_order);

%% Loop
for iloop = 1 :loopNumber
data_bit_in = randi([0 1], 1, N_bit);
dataSymbolsIn = bi2de(reshape(data_bit_in, Q_order, N_bit/Q_order).', 'left-msb'); 
dataMod = qammod(dataSymbolsIn, Qm,'UnitAveragePower', true); 

%% Show Constellation
%scatterplotme(dataMod)

%% Resource Mapping
RE_Grid = zeros(N_RB * N_SC_perRB,N_Symbol * N_Frame);
dataMod_tmp = reshape(dataMod,N_RB * N_SC_perRB,[]); %only data
Power_Scale = 1;
RE_Grid_all = Power_Scale * dataMod_tmp;

%% IFFT add CP
frame_mod_shift = ifftshift(RE_Grid_all); 
ifft_data = ifft(frame_mod_shift,N_fft)*sqrt(N_fft); 
%ifft_data = ifft(frame_mod_shift)*sqrt(1272); 
Tx_cd = [ifft_data(N_fft-N_cp+1:end,:);ifft_data];
time_signal = reshape(Tx_cd,[],1);

%% Channel
power_RE = sum(sum(abs(RE_Grid_all).^2)) / N_RB / N_SC_perRB / N_Symbol / N_Frame;
power_tp = sum(sum(abs(ifft_data).^2)) / N_RB / N_SC_perRB / N_Symbol / N_Frame;  %IFFT zero padding averages the true RE Power
N0 = power_RE .* 10.^(-SNR / 10);
white_noise_starand = 1/sqrt(2)*(randn(size(time_signal)) + 1j * randn(size(time_signal)));
TransmittedSignal = time_signal + sqrt(N0) * white_noise_starand;

%% Receive and Sys
ReceivedSignal = TransmittedSignal;

%% FFT and Frame   
frame_recieved_parallel = reshape(ReceivedSignal, N_fft + N_cp, []);
frame_Received = frame_recieved_parallel(N_cp + 1:end,:);    
frame_Grid_Received = fft(frame_Received,N_fft) / sqrt(N_fft);
RE_Grid_all_Received = fftshift(frame_Grid_Received(1 : N_SC,:));

%% Demodulation
RE_PreDeMod = reshape(RE_Grid_all_Received,[],1);
dataSymbolsOut = qamdemod(RE_PreDeMod, Qm,'UnitAveragePower', true); 
data_bit_out = reshape((de2bi(dataSymbolsOut, 'left-msb')).',1,[]); 
power_RE_receid = sum(sum(abs(RE_PreDeMod).^2)) / N_RB / N_SC_perRB / N_Symbol / N_Frame;
snr_all(iQorder,iEbN0,iloop) = 10*log10(power_RE/(power_RE_receid - power_RE));
%% Result: Ser and Ber
%Ser
sym_err = length(find(dataSymbolsOut - dataSymbolsIn));
ser_all(iQorder,iEbN0,iloop) = sym_err / length(dataSymbolsOut);
%Ber
bit_error = sum(abs(data_bit_out - data_bit_in));
ber_all(iQorder,iEbN0,iloop) = bit_error / length(data_bit_out);
end
sers = mean(ser_all,3);
snrs = mean(snr_all,3);
bers = mean(ber_all,3);
sers_theory(iQorder,iEbN0) = QAM_SER_Theory(Qm,EbN0);

    fprintf('%dQAM\t%f\t %f\t %f\t %e\t\t%e\t\t%e\t\t%d\t\n', Qm, EbN0, SNR,snrs(iQorder,iEbN0),sers(iQorder,iEbN0),sers_theory(iQorder,iEbN0),bers(iQorder,iEbN0),loopNumber);
    end
end

figure(1)
semilogy(EbN0_list, bers(1,:), 'k--+');
hold on 
grid on
semilogy(EbN0_list, bers(2,:), 'r--o');
semilogy(EbN0_list, bers(3,:), 'b--x');
semilogy(EbN0_list, bers(4,:), 'g--s');
xlabel('Eb/N0,dB');
ylabel('BER');
title('BER VERS SNR');
legend('QPSK','16QAM','256QAM','1024QAM');


figure(2)
semilogy(EbN0_list, sers(1,:), 'k--+');
hold on 
grid on
semilogy(EbN0_list, sers_theory(1,:), 'k-');
semilogy(EbN0_list, sers(2,:), 'r--o');
semilogy(EbN0_list, sers_theory(2,:), 'r-');
semilogy(EbN0_list, sers(3,:), 'b--x');
semilogy(EbN0_list, sers_theory(3,:), 'b-');
semilogy(EbN0_list, sers(4,:), 'g--s');
semilogy(EbN0_list, sers_theory(4,:), 'g-');
xlabel('Eb/N0,dB');
ylabel('SER');
title('SER VERS SNR');
%SML =  simulation, THR = theory
legend('QPSK-SML','QPSK-THR','16QAM-SML','16QAM-THR','256QAM-SML','256QAM-THR','1024QAM-SML','1024QAM-THR');

计算理论误比特率的函数需要参考文献，不过观察误码率与误比特率曲线，大体趋势相同，也许仅相差一个和调制阶数相关的常数（后来验证并非如此简单）。

%% Theory Symbol Error Rate

function SER = QAM_SER_Theory(Qm,EbN0)
   %Reference https://dsplog.com/2012/01/01/symbol-error-rate-16qam-64qam-256qam/
   Q_order = log2(Qm);
   EsN0_DB =  EbN0 + 10 * log10(Q_order);
   EsN0 = 10.^( EsN0_DB/ 10);
   k = sqrt(3 / (2*(Qm - 1)));
   k_snr = k * sqrt(EsN0);
   cer = erfc(k_snr);
   SER = 2*(1 - 1/sqrt(Qm))*cer - (1 - 2/sqrt(Qm) + 1/Qm) * (cer.^2);
%    cer = erfc(sqrt(EsN0/2));
%    SER = cer - 1/4*cer.^2;

 

 仿真结果

SER VERS SNR（该图理论（THR）误符号率曲线和实际仿真（SML）理论误符号率曲线基本重合）

 

BER VERS SNR（未画出理论误码率曲线）

 

SER VERS SNR（该图的横坐标信噪比是每个QAM符号的信噪比而不是每bit的信噪比）

BER VERS SNR（未画出理论误码率曲线）

 

分析结论

本仿真中应该重点关注信噪比的换算，包括Eb/N0(每bit的信噪比)到Es/N0(每QAM符号的信噪比)，频域通过IFFT到时域前后计算SNR，特别是子载波个数与IFFT的点数不相同时，如何在时域加噪声，每个时域采样点的噪声功率N0应该加多大。

反思

1.仅白噪声下的仿真结果，那么在多径信道下的仿真曲线如何呢？如何利用信道均衡来对抗多径带来的频率选择性衰落。
2.在调制阶数越来越高的情况下，误码率与误比特率都随之升高，那么通信中是如何通过调制阶数的升高来提升系统的吞吐量的呢？信道编码的作用。
3.如何利用多个天线MIMO技术来提高通信系统的有效性与可靠性？信道预编码与均衡。