信号检测实验4误码率分析-参考答案.doc

实验四
误码率的分析

1、假定QPSK信号具有如下的星座图模式：

1) 试利用二元数字通信系统的误码性能精确推导QPSK系统的误码率。
2) 假设系统中的噪声是高斯白噪声，利用matlab对该系统在工作区间（信噪比0~10dB）下的误码性能进行仿真。
简答：

（1）、试利用二元数字通信系统的误码性能精确推导QPSK系统的误码率。
解：
令

表示每比特信号的能量

对于二元数字通信系统：二进制相移键控BPSK：

对于QPSK系统：其比特错误率等于BPSK的比特错误率，而误码率则是

当性噪比较高时，

(2) 、
说明：
Eb = Energy-per-bit
Es = Energy-per-symbol = nEb with n bits per symbol
Tb = Bit duration , Rb = Bit Rate, the bit transmission time Tb = 1/Rb
Ts = Symbol duration
N0 / 2 = Noise power spectral density (W/Hz)
Pb = Probability of bit-error
Ps = Probability of symbol-error
Eb/N0 = The energy per bit to noise power spectral density ratio。It is a normalized signal-to-noise ratio (SNR) measure, also known as the "SNR per bit".

程序如下：
clear all
SNR_DB=[0:1:12]; %Signal-to-noise ratio gradually improve
sum=1000000;
data= randsrc(sum,2,[0 1]); %generate a 1000000*2 random matrix, using [0 1]
[a1,b1]=find(data(:,1)==0&data(:,2)==0); %returns the row and column indices of
the evaluated expression which are TRUE.
message(a1)=-1-j; % map [ 0 0] to 225°
[a2,b2]=find(data(:,1)==0&data(:,2)==1);
message(a2)=-1+j; % map [ 0 1] to 135°
[a3,b3]=find(data(:,1)==1&data(:,2)==0);
message(a3)=1-j; % map [ 1 0] to 275°
[a4,b4]=find(data(:,1)==1&data(:,2)==1);
message(a4)=1+j;% map [ 0 0] to 45°
scatterplot(message)
title('B点信号的星座图')
A=1;
Tb=1;
Eb=A*A*Tb;
P_signal=Eb/Tb;
NO=Eb./(10.^(SNR_DB/10)); %SNR_DB=10.*log10(Eb./NO)
P_noise=NO; %noise power 单边功率谱密度(N0)主要用在复数信号中，双边功率谱密度(N0/2)主要用在实信号中。
sigma=sqrt(P_noise);
for Eb_NO_id=1:length(sigma)
noise1=sigma(Eb_NO_id)*randn(1,sum);
noise2=sigma(Eb_NO_id)*randn(1,sum);
receive=message+noise1+noise2*j; %previously unconsidered: how to add gaussian noise to the original signal
resum=0;
total=0;
m1=find(angle(receive)<=pi/2&angle(receive)>0); %demodulate the [ 1 1]
patter
remessage(1,m1)=1+j;
redata(m1,1)=1;
redata(m1,2)=1;
m2= find( angle(receive)>pi/2&angle(receive)<=pi); %demodulate the [ 0 1] pattern
remessage(1,m2)=-1+j;
Redata(m2,1)=0;
redata(m2,2)=1;
m3=find( angle(receive)>-pi&angle(receive)<=-pi/2); %demodulate the [ 0 0] pattern
remessage(1,m3)=-1-j;
redata(m3,1)=0;
redata(m3,2)=0;
m4=find( angle(receive)>-pi/2&angle(receive)<=0); %demodulate the [ 1 0] pattern
remessage(1,m4)=1-j;
redata(m4,1)=1;
redata(m4,2)=0;
[resum,ratio1]=symerr(data,redata); % 'symerr' Compute number of symbol errors and symbol error rate
pbit(Eb_NO_id)=resum/(sum*2); %1000000 symbols -->2000000 bits QPSK: 2bits per symbol

[total,ratio2]=symerr(message,remessage); %Compute number of symbol errors and symbol error rate
pe(Eb_NO_id)=total/sum; % Calculated according to the definition
end
scatterplot(receive)
title('C点信号的星座图')
Pe=1-(1-1/2*erfc(sqrt(10.^(SNR_DB/10)/2))).^2;
Pbit=1/2*erfc(sqrt(10.^(SNR_DB/10)/2));
figure(3)
semilogy(SNR_DB,pe,':s',SNR_DB,Pe,'-*',SNR_DB,pbit,'-o',SNR_DB,Pbit,':+')
legend('QPSK仿真误码率','QPSK理论误码率','QPSK仿真误比特率','QPSK理论误比特率',1)
xlabel('信噪比/dB')
ylabel('概率P')
grid on

实验结果如下：

（3） 若信号为16PSK，重做1。

当信噪比较大时，

仿真结果：