谱校正方法

谱校正方法

采样间隔归一化成1T ?=,采样长度为N .这样FFT 离散谱线为0,1)i X i N =-（,相应的频率分辨率2/(1/)N f N ωπ?=?=. 设FFT 离散谱线局部极高谱线为m (为了数学上简洁,假定从0开始,注意在MATLAB 环境下数组实际操作的是从1开始),记频偏量δωδω=?. 我们需要使用谱线m 和与之相邻一条次高谱线，记这连续两条谱线中左边一条序号为M (当次高谱线在m 左侧时1M m =-,反之M m =).

下面列出若干算法的δ计算公式

1. 加矩形窗的精确谱校正[1]

i i i X U jV =+

111()sin()()cos()M M M M opt M M

V V M U U M K U U ωω+++-?+-?=- 1211cos()sin()cos()sin()opt M M opt M M K M Z V U M K M Z V U M ωωωωωω++-???=+?????-?+???=+???+??? 2121

cos()cos()()Z M Z M M m Z Z ωωωδ?+?-?=+-- 2. 加矩形窗情形,采用解析单频模型的幅值比校正[1, 2]

11||()||||

M M M X M m X X δ++=+-+ 3. 加汉宁窗情形,采用解析单频模型的幅值比校正[1, 2]

112||||()||||

M M M M X X M m X X δ++-=+-+ 4. 加矩形窗情形,采用解析单频模型的复比值校正[3]

1

1Re ()M M M X M m X X δ++??=+- ?-??

5. 加汉宁窗情形,采用解析单频模型的复比值校正[3]

112()M M M M

X X M m X X δ+++=+-- 6. 加矩形窗情形,采用解析单频模型的复合复比值校正[3]

11Re ()M m M M X M m X X δ++??=+- ?-?? 11m R m m X X X δ++=-,111

1m m L m m m m X X X X X X δ---=-=-- 0.5)0.5)m L m R δδδδδ=-++（（

7. 加汉宁窗情形,采用解析单频模型的复合复比值校正[3]

112Re ()M M m M M X X M m X X δ++??+=+-

?-?? 112m m R m m X X X X δ+++=-,1111

221m m m m L m m m m X X X X X X X X δ----++=-=-- 0.5)0.5)m L m R δδδδδ=-++（（

8. 加矩形窗,Quin 校正[4]

11Re()Re(),Re()Re()m m L R m m X X X X αα-+=

= 11L R L R L R

ααδδαα==---，, 00 R R L R δδδδδ>>?=??当且其它

9. 加汉宁窗,Quin 校正[4]

11Re()Re(),Re()Re()m m L R m m X X X X αα-+=

= 212111L R

L R L R

ααδδαα++==---，,

00 R R L R δδδδδ>>?=??当且其它

References

1. Schoukens, J., R. Pintelon,H. Van Hamme. The interpolated fast Fourier transform: A

comparative study . IEEE Transactions on Instrumentation and Measurement. 1992, 41(2):

226-232.

2. 谢明,丁康. 频谱分析的校正方法.振动工程学报. 1994, 7(2): 172-179.

3. 陈奎孚, 王建立,张森文. 频谱校正的复比值法.振动工程学报(已投). 2007.

4. Quinn, B.G. Estimating frequency by interpolation using Fourier coefficients.IEEE

Transactions on Signal Processing. 1994, 42(5): 1264-1268.

%========================这是调用调试==================

DT=1;

N=1024;

PHI=pi/3;

Ampl=1;

CiR=11.9; %cycles in record

Freq=CiR/(DT*N); %frequency

TV=[0:N-1];

DatVec=Ampl*cos(Freq*TV*2*pi+PHI);

FV=fft(DatVec);

figure

subplot(2,1,1);plot(TV,DatV ec);

subplot(2,1,2);plot(abs(FV(1:round(N/2.56))));grid on

[MV,MI]=max(abs(FV));

%加矩形窗的解析校正--1

FreqShift=SpecCorr(FV,MI,N,1);

%加矩形窗的解析单频模型校正--2

FreqShift=SpecCorr(FV,MI,N,2);

%加汉宁窗的解析单频模型校正--3

HanDat=DatVec.*hanning(N,'periodic')';

FV=fft(HanDat);

FreqShift=SpecCorr(FV,MI,N,3);

%加矩形窗的解析单频模型校正+复比值法--4

FV=fft(DatVec);

FreqShift=SpecCorr(FV,MI,N,4);

%加汉宁窗的解析单频模型校正+复比值法--5

HanDat=DatVec.*hanning(N,'periodic')';

FV=fft(HanDat);

FreqShift=SpecCorr(FV,MI,N,5);

%加矩形窗的解析单频模型校正+复比值法+左右平均--6

FV=fft(DatVec);

FreqShift=SpecCorr(FV,MI,N,6);

%加汉宁窗的解析单频模型校正+复比值法+左右平均--7

HanDat=DatVec.*hanning(N,'periodic')';

FV=fft(HanDat);

FreqShift=SpecCorr(FV,MI,N,7);

%加矩形窗的解析单频模型校正+Quinn算法--8

FV=fft(DatVec);

FreqShift=SpecCorr(FV,MI,N,8);

%加汉宁窗的解析单频模型校正+Quinn算法--9

HanDat=DatVec.*hanning(N,'periodic')';

FV=fft(HanDat);

FreqShift=SpecCorr(FV,MI,N,9);

===========这是子程序======================

%spectrum correction assemble

% the sampling interval is 1 s (or unitary)

%Input: SpecVec--Discrte Fourier Spectrum from FFT

% PeakIdx--the peak index, noting the matrix in MatLab start from 1

% TL--the length (or the point number) of the FFT

% method--correction method

% output: PeakShift-- the corrected peak shifting from the peak in discrete

% spectrum

function PeakShift=SpecCorr(SpecVec,PeakIdx,TL,method)

% picking up the second highest spectrum line

if(abs(SpecVec(PeakIdx-1))>abs(SpecVec(PeakIdx+1)))

IP=[PeakIdx-1,PeakIdx];

ShiftCorr=-1; %shift aligning with the PeakIdx

else

IP=[PeakIdx,PeakIdx+1];

ShiftCorr=0; %shift aligning with the PeakIdx

end

II=IP(1)-1; % noting that the index of a matrix in MATLAB starts from 1, not zero if(method==1) %an analyitic solution for rectangular window

U=real(SpecVec(IP));

V=imag(SpecVec(IP));

DW=2*pi/TL;

KOPT=(sin(II*DW)*(V(2)-V(1))+cos(II*DW)*(U(2)-U(1)))/(U(2)-U(1));

Z=V.*(KOPT-cos((IP-1)*DW))./(sin(DW*(IP-1)))+U;

Tmp1=(Z(2)*cos(DW*(II+1))-Z(1)*cos(DW*II))/(Z(2)-Z(1));

PeakPos=acos(Tmp1)/DW;

PeakShift=PeakPos-(PeakIdx-1);

elseif(method==2) %based on the analytical-single-tone model for rectangular window PeakShift=abs(SpecVec(IP(2)))/(abs(SpecVec(IP(2)))+abs(SpecVec(IP(1))));

PeakShift=PeakShift+ShiftCorr; %shift aligning with the PeakIdx

elseif(method==3) %based on the analytical-single-tone model for Hanning window

PeakShift=(2*abs(SpecVec(IP(2)))-abs(SpecV ec(IP(1))))/(abs(SpecV ec(IP(2)))+abs(SpecVec(IP(1 ))));

PeakShift=PeakShift+ShiftCorr; %shift aligning with the PeakIdx

elseif(method==4) %based on the analytical-single-tone model for rectangular window with complex correction

PeakShift=real(SpecVec(IP(2))/(SpecVec(IP(2))-SpecVec(IP(1))));

PeakShift=PeakShift+ShiftCorr; %shift aligning with the PeakIdx

elseif(method==5) %based on the analytical-single-tone model for Hanning window with complex correction

PeakShift=(2*SpecVec(IP(2))+SpecVec(IP(1)))/(SpecVec(IP(2))-SpecVec(IP(1)));

PeakShift=real(PeakShift)+ShiftCorr; %shift aligning with the PeakIdx elseif(method==6) %based on the analytical-single-tone model for rectangular window with complex correction+average

PeakShift=real(SpecVec(IP(2))/(SpecVec(IP(2))-SpecVec(IP(1))));

MaxPeakShift=PeakShift+ShiftCorr; %shift aligning with the PeakIdx

LeftShift=real(SpecVec(PeakIdx)/(SpecVec(PeakIdx)-SpecVec(PeakIdx-1)))-1;

RightShift=real(SpecVec(PeakIdx+1)/(SpecVec(PeakIdx+1)-SpecVec(PeakIdx)));

%average

PeakShift=(0.5-MaxPeakShift)*LeftShift+(0.5+MaxPeakShift)*RightShift;

elseif(method==7) %based on the analytical-single-tone model for Hanning window with complex correction+average,????

PeakShift=(2*SpecVec(IP(2))+SpecVec(IP(1)))/(SpecVec(IP(2))-SpecVec(IP(1)));

MaxPeakShift=real(PeakShift)+ShiftCorr; %shift aligning with the PeakIdx

LeftShift=(2*SpecVec(PeakIdx)+SpecV ec(PeakIdx-1))/(SpecVec(PeakIdx)-SpecVec(PeakIdx-1))-1;

RightShift=(2*SpecVec(PeakIdx+1)+SpecV ec(PeakIdx))/(SpecVec(PeakIdx+1)-SpecVec(PeakIdx ));

%average

PeakShift=(0.5-MaxPeakShift)*LeftShift+(0.5+MaxPeakShift)*RightShift;

elseif(method==8) %Quinn method for the rectangular window

a1 = real(SpecV ec(PeakIdx-1)/SpecVec(PeakIdx)); %left

a2 = real(SpecV ec(PeakIdx+1)/SpecVec(PeakIdx)); %right

LeftShift = a1/(1-a1);

RightShift = -a2/(1-a2);

if (LeftShift>0 & RightShift>0)

PeakShift = RightShift;

else

PeakShift = LeftShift;

end

elseif(method==9) %Quinn method for the Hanning window

a1 = real(SpecV ec(PeakIdx-1)/SpecVec(PeakIdx)); %left

a2 = real(SpecV ec(PeakIdx+1)/SpecVec(PeakIdx)); %right

LeftShift = (2*a1+1)/(1-a1);

RightShift = -(2*a2+1)/(1-a2);

if (LeftShift>0 & RightShift>0)

PeakShift = RightShift;

else

PeakShift = LeftShift;

end

end

end