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目录
在白噪声下，真实正弦波的精确频率估计是一个重要的研究课题，因为白噪声会对信号的频率进行干扰，使得频率估计变得困难。下面是一些常用的方法来进行真实正弦波的精确频率估计研究：
1. 傅里叶变换：通过对信号进行傅里叶变换，可以得到信号的频谱信息，从而可以估计出频率。然而，在白噪声下，信号的频谱可能会被噪声掩盖，导致频率估计的不准确性。
2. 自相关函数：自相关函数可以用来评估信号中重复出现的模式，从而可以用来估计信号的频率。然而，在白噪声下，自相关函数的峰值可能会被噪声掩盖，导致频率估计的不准确性。
3. 峰值检测：通过寻找信号的峰值来进行频率估计。然而，在白噪声下，峰值可能会被噪声掩盖，导致频率估计的不准确性。
4. 希尔伯特变换：希尔伯特变换可以将信号从时域转换到频域，从而可以得到信号的频率信息。然而，在白噪声下，希尔伯特变换可能会受到噪声的影响，导致频率估计的不准确性。
综上所述，白噪声下真实正弦波的精确频率估计是一个复杂的问题，需要综合运用多种方法来进行研究和解决。在实际应用中，可以根据具体的情况选择合适的方法来进行频率估计。
可视化代码：
figure()
semilogy(SNRdB,CRLB/Fs*N,'--ks',SNRdB,RMSEBin(:,1),'-mo',SNRdB,RMSEBin(:,2),'-r*',...
SNRdB,RMSEBin(:,3),'-gd',SNRdB,RMSEBin(:,4),'-c.',SNRdB,RMSEBin(:,5),'-b^',...
SNRdB,RMSEBin(:,6),'-bv','LineWidth',1.2)
legend('CRLB','Proposed, second','Proposed, third','PSVD [5]','Djukanovic [9]','Ye [12],\it{T}=2','Ye [12],\it{T}=4')
xlabel('SNR(dB)')
ylabel('RMSE(in DFT bins)')
axis([-10,60,5*10^-5,20])
grid on
saveas(gcf,'..\results\Fig1.png')
%% Fig 2
Fs = 1; % Sampling frequency
N = [16 64 256 1024 4096]; % Number of samples
A = 1; % Half of Amplitude
SimTimes = 2000; % Number of simulation
SNRdB = 20; % SNR
RMSE = zeros(length(N),6); % RMSE
CRLB = sqrt(3*Fs^2./(pi^2*N.*(N.^2-1)*10.^(SNRdB/10))).'; % CRLB
for ii = 1:length(N)
ErrF = zeros(SimTimes,6); % Error of frequency estimation
for jj = 1:SimTimes
F = Fs/N(ii)*1.2; % Frequency
P = rand*2*pi; % Initial Phase
n = (0:N(ii)-1).';
s = 2*A*cos(2*pi*F/Fs*n+P); % real sinusoid
Ps = mean(s.^2); % Signal power
Pn = Ps/(10^(SNRdB/10)); % Noise Power
w = sqrt(Pn)*randn(N(ii),1);
s = w+s; % AWGN channel
EstF = ProposedInitial(s,Fs); % Initial estimate of proposed algorithm
ErrF(jj,1) = abs(EstF-F);
if ErrF(jj,1)>Fs/2
ErrF(jj,1) = Fs-ErrF(jj,1);
end
EstF = ProposedFine(s,Fs); % Fine estimate of proposed algorithm
ErrF(jj,2) = abs(EstF-F);
if ErrF(jj,2)>Fs/2
ErrF(jj,2) = Fs-ErrF(jj,2);
end
EstF = PSVD(s,Fs);
ErrF(jj,3) = abs(EstF-F);
if ErrF(jj,3)>Fs/2
ErrF(jj,3) = Fs-ErrF(jj,3);
end
EstF = Djukanovic(s,Fs);
ErrF(jj,4) = abs(EstF-F);
if ErrF(jj,4)>Fs/2
ErrF(jj,4) = Fs-ErrF(jj,4);
end
EstF = Ye(s,Fs,2); % Ye algorithm with T=2
ErrF(jj,5) = abs(EstF-F);
if ErrF(jj,5)>Fs/2
ErrF(jj,5) = Fs-ErrF(jj,5);
end
EstF = Ye(s,Fs,4); % Ye algorithm with T=4
ErrF(jj,6) = abs(EstF-F);
if ErrF(jj,6)>Fs/2
ErrF(jj,6) = Fs-ErrF(jj,6);
end
end
RMSE(ii,:) = sqrt(mean(ErrF.^2)); % RMSE in DFT bins
end
RMSEBin = RMSE/Fs.*repmat(N.',1,6); % RMSE in DFT bins
figure()
semilogy(log2(N),CRLB/Fs.*N.','--ks',log2(N),RMSEBin(:,1),'-mo',log2(N),RMSEBin(:,2),'-r*',...
log2(N),RMSEBin(:,3),'-gd',log2(N),RMSEBin(:,4),'-c.',log2(N),RMSEBin(:,5),'-b^',...
log2(N),RMSEBin(:,6),'-bv','LineWidth',1.2)
legend('CRLB','Proposed, second','Proposed, third','PSVD [5]','Djukanovic [9]','Ye [12],\it{T}=2','Ye [12],\it{T}=4')
xlabel('log_2\it{N}')
ylabel('RMSE(in DFT bins)')
grid on
saveas(gcf,'..\results\Fig2.png')
%% Fig 3
Fs = 1; % Sampling frequency
N = 1024; % Number of samples
A = 1; % Half of Amplitude
SimTimes = 1000; % Number of simulation
SNRdB = 20; % SNR
x = [0.55:0.25:2.05,3,6:N/2/9:N/2-6,N/2-3,N/2-2.05:0.25:N/2-0.55]; % Frequency in DFT bins
F = x*Fs/N; % Frequency in Hz
RMSE = zeros(length(F),6); % RMSE
CRLB = sqrt(3*Fs^2./(pi^2*N.*(N.^2-1)*10.^(SNRdB/10)))*ones(length(F),1); % CRLB
for ii = 1:length(F)
ErrF = zeros(SimTimes,6);
CRLBInst = zeros(SimTimes,1);
for jj = 1:SimTimes
P = rand*2*pi; % Initial Phase
n = (0:N-1).';
s = 2*A*cos(2*pi*F(ii)/Fs*n+P); % real sinusoid
Ps = mean(s.^2); % Signal power
Pn = Ps/(10^(SNRdB/10)); % Noise Power
w = sqrt(Pn)*randn(N,1);
s = w+s; % AWGN channel
EstF = ProposedInitial(s,Fs); % Initial estimate of proposed algorithm
ErrF(jj,1) = abs(EstF-F(ii));
if ErrF(jj,1)>Fs/2
ErrF(jj,1) = Fs-ErrF(jj,1);
end
EstF = ProposedFine(s,Fs); % Fine estimate of proposed algorithm
ErrF(jj,2) = abs(EstF-F(ii));
if ErrF(jj,2)>Fs/2
ErrF(jj,2) = Fs-ErrF(jj,2);
end
EstF = PSVD(s,Fs);
ErrF(jj,3) = abs(EstF-F(ii));
if ErrF(jj,3)>Fs/2
ErrF(jj,3) = Fs-ErrF(jj,3);
end
EstF = Djukanovic(s,Fs);
ErrF(jj,4) = abs(EstF-F(ii));
if ErrF(jj,4)>Fs/2
ErrF(jj,4) = Fs-ErrF(jj,4);
end
EstF = Ye(s,Fs,2); % Ye algorithm with T=2
ErrF(jj,5) = abs(EstF-F(ii));
if ErrF(jj,5)>Fs/2
ErrF(jj,5) = Fs-ErrF(jj,5);
end
EstF = Ye(s,Fs,4); % Ye algorithm with T=4
ErrF(jj,6) = abs(EstF-F(ii));
if ErrF(jj,6)>Fs/2
ErrF(jj,6) = Fs-ErrF(jj,6);
end
end
RMSE(ii,:) = sqrt(mean(ErrF.^2,1));
end
RMSEBin = RMSE/Fs*N; % RMSE in DFT bins
figure()
semilogy(x,CRLB/Fs*N,'--ks',x,RMSEBin(:,1),'-mo',x,RMSEBin(:,2),'-r*',...
x,RMSEBin(:,3),'-gd',x,RMSEBin(:,4),'-c.',x,RMSEBin(:,5),'-b^',...
x,RMSEBin(:,6),'-bv','LineWidth',1.2)
legend('CRLB','Proposed, second','Proposed, third','PSVD [5]','Djukanovic [9]','Ye [12],\it{T}=2','Ye [12],\it{T}=4')
xlabel('\it{Nf/fs}')
ylabel('RMSE(Hz)')
axis([0,512,1*10^-3,1])
grid on
saveas(gcf,'..\results\Fig3.png')
文章中一些内容引自网络，会注明出处或引用为参考文献，难免有未尽之处，如有不妥，请随时联系删除。
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[2] Liu S , Xu W , Wang Z .An Accurate Frequency Estimator of Single Sinusoid in Gaussian Colored Noise.[C]//International Conference on Innovative Computing.IEEE, 2006.DOI:10.1109/ICICIC.2006.227.
[4]丁志中.白噪声背景下正弦信号频率和功率估计的有效算法[J].合肥工业大学学报：自然科学版, 1995, 18(4):6.DOI:CNKI:SUN:HEFE.0.1995-04-002.
[5]吴珊珊胡国兵丁宁汤滟.正弦波频率估计结果的可靠性评估算法研究[J].现代雷达, 2015, 37(3):31-35.
