多种频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 iX U jV =+111()sin()()cos()M M M M opt M MV 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++-∆⎡⎤=+⎢∆⎣⎦-∆+∆⎡⎤=+⎢⎥∆+∆⎣⎦2121cos()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]11Re ()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 MX 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++=-,1111m m L m m m m X X X X X X δ---=-=--0.5)0.5)m L m Rδδδδδ=-++（（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+++=-,1111221m m m m L m m m m X X X X X X X X δ----++=-=--0.5)0.5)m L m Rδδδδδ=-++（（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 Rααδδαα==---，, 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 Rααδδαα++==---，, 00 RR L R δδδδδ>>⎧=⎨⎩当且其它References1.Schoukens,J.,R.Pintelon,H.Van Hamme.The interpolated fast Fourier transform:Acomparative 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.IEEETransactions 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 recordFreq=CiR/(DT*N);%frequencyTV=[0:N-1];DatVec=Ampl*cos(Freq*TV*2*pi+PHI);FV=fft(DatVec);figuresubplot(2,1,1);plot(TV,DatVec);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--1FreqShift=SpecCorr(FV,MI,N,1);%加矩形窗的解析单频模型校正--2FreqShift=SpecCorr(FV,MI,N,2);%加汉宁窗的解析单频模型校正--3HanDat=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--4FV=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--5HanDat=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--6FV=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--7HanDat=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--8FV=fft(DatVec);FreqShift=SpecCorr(FV,MI,N,8);%加汉宁窗的解析单频模型校正+Quinn算法--9HanDat=DatVec.*hanning(N,'periodic')';FV=fft(HanDat);FreqShift=SpecCorr(FV,MI,N,9);===========这是子程序======================%spectrum correction assemble%the sampling interval is1s(or unitary)%Input:SpecVec--Discrte Fourier Spectrum from FFT%PeakIdx--the peak index,noting the matrix in MatLab start from1%TL--the length(or the point number)of the FFT%method--correction method%output:PeakShift--the corrected peak shifting from the peak in discrete%spectrumfunction PeakShift=SpecCorr(SpecVec,PeakIdx,TL,method)%picking up the second highest spectrum lineif(abs(SpecVec(PeakIdx-1))>abs(SpecVec(PeakIdx+1)))IP=[PeakIdx-1,PeakIdx];ShiftCorr=-1;%shift aligning with the PeakIdxelseIP=[PeakIdx,PeakIdx+1];ShiftCorr=0;%shift aligning with the PeakIdxendII=IP(1)-1;%noting that the index of a matrix in MATLAB starts from1,not zero if(method==1)%an analyitic solution for rectangular windowU=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 PeakIdxelseif(method==3)%based on the analytical-single-tone model for Hanning window PeakShift=(2*abs(SpecVec(IP(2)))-abs(SpecVec(IP(1))))/(abs(SpecVec(IP(2)))+abs(SpecVec(IP(1) )));PeakShift=PeakShift+ShiftCorr;%shift aligning with the PeakIdxelseif(method==4)%based on the analytical-single-tone model for rectangular window with complex correctionPeakShift=real(SpecVec(IP(2))/(SpecVec(IP(2))-SpecVec(IP(1))));PeakShift=PeakShift+ShiftCorr;%shift aligning with the PeakIdxelseif(method==5)%based on the analytical-single-tone model for Hanning window with complex correctionPeakShift=(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+averagePeakShift=real(SpecVec(IP(2))/(SpecVec(IP(2))-SpecVec(IP(1))));MaxPeakShift=PeakShift+ShiftCorr;%shift aligning with the PeakIdxLeftShift=real(SpecVec(PeakIdx)/(SpecVec(PeakIdx)-SpecVec(PeakIdx-1)))-1;RightShift=real(SpecVec(PeakIdx+1)/(SpecVec(PeakIdx+1)-SpecVec(PeakIdx)));%averagePeakShift=(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)+SpecVec(PeakIdx-1))/(SpecVec(PeakIdx)-SpecVec(PeakIdx-1))-1;RightShift=(2*SpecVec(PeakIdx+1)+SpecVec(PeakIdx))/(SpecVec(PeakIdx+1)-SpecVec(PeakIdx) );%averagePeakShift=(0.5-MaxPeakShift)*LeftShift+(0.5+MaxPeakShift)*RightShift;elseif(method==8)%Quinn method for the rectangular windowa1=real(SpecVec(PeakIdx-1)/SpecVec(PeakIdx));%lefta2=real(SpecVec(PeakIdx+1)/SpecVec(PeakIdx));%rightLeftShift=a1/(1-a1);RightShift=-a2/(1-a2);if(LeftShift>0&RightShift>0)PeakShift=RightShift;elsePeakShift=LeftShift;endelseif(method==9)%Quinn method for the Hanning windowa1=real(SpecVec(PeakIdx-1)/SpecVec(PeakIdx));%lefta2=real(SpecVec(PeakIdx+1)/SpecVec(PeakIdx));%rightLeftShift=(2*a1+1)/(1-a1);RightShift=-(2*a2+1)/(1-a2);if(LeftShift>0&RightShift>0)PeakShift=RightShift;elsePeakShift=LeftShift;endendend。