Rife算法的一种Matlab实现
Rife算法原理及介绍
信号频谱的最大两根谱线进行插值对频率进行估计,即Rife算法。当信号频率位于离散傅里叶变换(DFT)两个相邻量化频率点的中心区域时，Rife算法精度很高,均方根误差接近克拉美罗限,且计算量小于牛顿迭代。Rife算法常用于单频信号频率估计。
加性高斯白噪声污染的正弦波信号可以表示为
x ( n ) = a e j H 0 e j 2 π f c n Δ t + w ( n ) n = 0 , 1 , … N − 1 x(n)=a e^{j H_{0}} e^{j 2 \pi f_{c} n \Delta t}+w(n) \quad n=0,1, \ldots N-1 x(n)=aejH0ej2πfcnΔt+w(n)n=0,1,…N−1
式中a , f c , H 0 f_{c}, \mathrm{H}_{0} fc,H0分别为振幅，频率和初相；∆t 为采样间隔，N 为样本数；w(t)为实部和虚部相互独立的高斯白噪声。
对x(n)作FFT,取其中的最大谱线值|X(K_0)|。给出利用两根谱线进行正弦波频率估计的计算公式(即Rife算法)
f ^ c = 1 T ( k 0 + r ∣ X ( K 0 + r ) ∣ ∣ X ( K 0 ) ∣ + ∣ X ( K 0 + r ) ∣ ) \widehat{f}_{c}=\frac{1}{T}\left(k_{0}+r \frac{\left|X\left(K_{0}+r\right)\right|}{\left|X\left(K_{0}\right)\right|+\left|X\left(K_{0}+r\right)\right|}\right) f c=T1(k0+r∣X(K0)∣+∣X(K0+r)∣∣X(K0+r)∣)
式中： 当
∣ X ( K 0 + 1 ) ∣ ≤ ∣ X ( K 0 − 1 ) ∣ \left|X\left(K_{0}+1\right)\right| \leq\left|X\left(K_{0}-1\right)\right| ∣X(K0+1)∣≤∣X(K0−1)∣时，r = -1；
当 ∣ X ( K 0 − 1 ) ∣ ≤ ∣ X ( K 0 + 1 ) ∣ \left|X\left(K_{0}-1\right)\right| \leq\left|X\left(K_{0}+1\right)\right| ∣X(K0−1)∣≤∣X(K0+1)∣时，r = 1；T = NΔt。[1]
Matlab示例程序
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%%
clc;clear;close all;
%% 参数设置
N = 1000;
n = 1:N;
fs = 16;
f = 4.9; %可以在此选择模拟信号频率
T = N/fs;
dt = 1/fs;
SNR_n = 20;
SNR = -1*SNR_n+2:2:SNR_n-2;SNR = SNR+8;
%% 信号生成
signal_0 = cos(2*pi*f/fs*n)';
signal = zeros(N,SNR_n);
for k = 1:SNR_n-1
signal(:,k+1) = awgn(signal_0,SNR(k)); %高斯白噪声
end
signal(:,1) = signal_0;
%% 窗函数
rect = rectwin(N);
hn = hann(N);
hm = hamming(N);
bk = blackman(N);
%% 加窗
signalRectWin = repmat(rect,1,SNR_n).*signal; %将窗函数复制，与信号向量做点积
signalHann = repmat(hn,1,SNR_n).*signal;
signalHamming = repmat(hm,1,SNR_n).*signal;
signalBlackman = repmat(bk,1,SNR_n).*signal;
%% FFT
wRectWin = real(fftshift(fft(signalRectWin),1));
wHann = real(fftshift(fft(signalHann),1));
wHamming = real(fftshift(fft(signalHamming),1));
wBlackman = real(fftshift(fft(signalBlackman),1));
x = [-N+1:2:N]/N*fs/2;
%plot(x,db(wRectWin));
%% Rife算法
%%%%%%%%%%%%%%%%%%%%%%%%%%
[valueMaxRectWin,posMaxRectWin] = max(wRectWin(N/2+1:N,:));
r = zeros(1,SNR_n);
for k = 1:SNR_n
r(k)= 2*((wRectWin(posMaxRectWin(k)+N/2+1,k) > wRectWin(posMaxRectWin(k)+N/2-1,k))-0.5);
rat(k) = valueMaxRectWin(k)/(valueMaxRectWin(k)+wRectWin(posMaxRectWin(k)+N/2+r(k),k));
fcRectWin(k) = 1/T*(posMaxRectWin(k)+r(k)*rat(k)-1);
end
rmseRectWin = sqrt(abs(sum(fcRectWin-f)));
%%%%%%%%%%%%%%%%%%%%%%%%%%%
[valueMaxHann,posMaxHann] = max(wHann(N/2+1:N,:));
r = zeros(1,SNR_n);
for k = 1:SNR_n
r(k)= 2*((wHann(posMaxHann(k)+N/2+1,k) > wHann(posMaxHann(k)+N/2-1,k))-0.5);
rat(k) = valueMaxHann(k)/(valueMaxHann(k)+wHann(posMaxHann(k)+N/2+r(k),k));
fcHann(k) = 1/T*(posMaxHann(k)+r(k)*rat(k)-1);
end
rmseHann = sqrt(abs(sum(fcHann-f)));
%%%%%%%%%%%%%%%%%%%%%%%%%%%
[valueMaxHamming,posMaxHamming] = max(wHamming(N/2+1:N,:));
r = zeros(1,SNR_n);
for k = 1:SNR_n
r(k)= 2*((wHamming(posMaxHamming(k)+N/2+1,k) > wHamming(posMaxHamming(k)+N/2-1,k))-0.5);
rat(k) = valueMaxHamming(k)/(valueMaxHamming(k)+wHamming(posMaxHamming(k)+N/2+r(k),k));
fcHamming(k) = 1/T*(posMaxHamming(k)+r(k)*rat(k)-1);
end
rmseHamming = sqrt(abs(sum(fcHamming-f)));
%%%%%%%%%%%%%%%%%%%%%%%%%%%
[valueMaxBlackman,posMaxBlackman] = max(wBlackman(N/2+1:N,:));
r = zeros(1,SNR_n);
for k = 1:SNR_n
r(k)= 2*((wBlackman(posMaxBlackman(k)+N/2+1,k) > wBlackman(posMaxBlackman(k)+N/2-1,k))-0.5);
rat(k) = valueMaxBlackman(k)/(valueMaxBlackman(k)+wBlackman(posMaxBlackman(k)+N/2+r(k),k));
fcBlackman(k) = 1/T*(posMaxBlackman(k)+r(k)*rat(k)-1);
end
rmseBlackman = sqrt(abs(sum(fcBlackman-f)));
%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 误差分析
RMSE = [rmseRectWin;rmseHann;rmseHamming;rmseBlackman];
errRW = fcRectWin-f;
errHN = fcHann-f;
errHM = fcHamming-f;
errBM = fcBlackman-f;
err = [errRW;errHN;errHM;errBM]/f*100;
plot(SNR,abs(err(:,2:SNR_n)));
grid on;
xlabel('SNR/dB');
ylabel('Relative Error/%');
leg = {'RectWin','Hann','Hamming','Blackman'};
legend(leg);
str = strcat('信号真实频率为',num2str(f),'Hz');
title(str);
为方便使用，将Rife算法独立编写成函数，代码如下：
function fc = rife(spec,fs)
%Rife算法matlab实现
% spec:信号频谱列向量或以列向量叠加的矩阵
% fs:信号采样率
% 输出fc为Rife算法估计出的频率
[length,SignalNum] = size(spec);
[valueMax,posMax] = max(spec(length/2+1:length,:));
T = length/fs;
for k = 1:SignalNum
r= 2*((spec(posMax(k)+length/2+1,k) > spec(posMax(k)+length/2-1,k))-0.5);
rat = valueMax(k)/(valueMax(k)+spec(posMax(k)+length/2+r,k));
fc(k) = 1/T*(posMax(k)+r*rat-1);
end
end
另附试验用生成信号的函数SignalFormFFT
function [wOUT,SNR] = signalFormFFT(N,f,fs,SNR_n)
%生成模拟信号以及加窗与FFT
% N为采样点数，f为信号频率，fs为采样率，SNR_n为信噪比个数，对应计算出的信噪比序列为
% -1*SNR_n+2:2:SNR_n-2;SNR = SNR+8;
% 输出元胞的顺序分别是加矩形窗，汉宁窗，海明窗和布拉克曼窗
%% 参数设置
n = 1:N;
T = N/fs;
dt = 1/fs;
SNR = -1*SNR_n+2:2:SNR_n-2;SNR = SNR+8;
%% 信号生成
signal_0 = cos(2*pi*f/fs*n)';
signal = zeros(N,SNR_n);
for k = 1:SNR_n-1
signal(:,k+1) = awgn(signal_0,SNR(k)); %高斯白噪声
end
signal(:,1) = signal_0;
%% 窗函数
rect = rectwin(N);
hn = hann(N);
hm = hamming(N);
bk = blackman(N);
%% 加窗
signalRectWin = repmat(rect,1,SNR_n).*signal; %将窗函数复制，与信号向量做点积
signalHann = repmat(hn,1,SNR_n).*signal;
signalHamming = repmat(hm,1,SNR_n).*signal;
signalBlackman = repmat(bk,1,SNR_n).*signal;
%% FFT
wRectWin = real(fftshift(fft(signalRectWin),1));
wHann = real(fftshift(fft(signalHann),1));
wHamming = real(fftshift(fft(signalHamming),1));
wBlackman = real(fftshift(fft(signalBlackman),1));
wOUT = {wRectWin,wHann,wHamming,wBlackman};
end
示例程序：
clc;clear;close all;
%% 原始数据输入
N = 500;
f = 4.08;
fs = 16;
SNR_n = 20;
%% 生成信号
[W,SNR] = signalFormFFT(N,f,fs,SNR_n);
wr = W{1};
whn = W{2};
whm = W{3};
wb = W{4};
%% Rife算法估算频率
fcr = rife(wr,fs);
fchn = rife(whn,fs);
fchm = rife(whm,fs);
fcb = rife(wb,fs);
fc = [fcr;fchn;fchm;fcb];
%% 分析误差
err = (fc-f)/f*100;
plot(SNR,abs(err(:,2:SNR_n)));
grid on;
xlabel('SNR/dB');
ylabel('Relative Error/%');
leg = {'RectWin','Hann','Hamming','Blackman'};
legend(leg);
str = strcat('信号真实频率为',num2str(f),'Hz');
title(str);
disp(str);
附下载链接
Rife.m
参考文献
[1]邓振淼,刘渝,王志忠.正弦波频率估计的修正Rife算法[J].数据采集与处理,2006,{4}(04):473-477.