Holger Nobach - nambis.de

Signal- und Messdatenverarbeitung

Codes für stochastisch und unabhängig abgetastete, reelle Leistungssignalpaare, ohne Gewichtung

		Python	Matlab/Octave
Vorbereitung (Laden von Modulen/Paketen)		`from numpy import * from numpy.random import * from matplotlib.pyplot import *`
Generieren von Wertepaaren $(t_{x, i}, x_{i})$ mit den Messzeitpunkten $t_{x, i}$ und den Messwerten $x_{i}$ mit $i = 0 \dots N_{x} - 1$ sowie den Wertepaaren $(t_{y, i}, y_{i})$ mit den Messzeitpunkten $t_{y, i}$ und den Messwerten $y_{i}$ mit $i = 0 \dots N_{y} - 1$ (zwei korrelierte und gegeneinander verschobene AR1-Prozess als Beispiel, mit Erwartungswerten verschieden von null)		Tx=10000.0 Ty=10100.0 Ti=10.0 drx=1.0 dry=1.5 mxs=3.0 mys=2.0 vxs=1.0 vys=2.0 cc=0.5 c1=sqrt(0.5+0.5sqrt(1-cc2/(vxsvys))) c2=cc/sqrt(vxsvys)/(2c1) c11=sqrt(vxs)c1 c12=sqrt(vxs)c2 c21=sqrt(vys)c2 c22=sqrt(vys)c1 tx=[] ty=[] x=[] y=[] te=0.0 ue=standard_normal() ve=standard_normal() while te<maximum(Tx,Ty): tp=exponential(1.0/(drx+dry)) te+=tp if te<maximum(Tx,Ty): phi=exp(-tp/Ti) theta=sqrt(1-phi*2) ue=uephi+normal(0,theta) ve=vephi+normal(0,theta) if random()<drx/float(drx+dry): if te<Tx: tx.append(te) x.append(c11ue+c12ve+mxs) else: if te<Ty: ty.append(te) y.append(c21ue+c22*ve+mys) tx=array(tx) ty=array(ty) x=array(x) y=array(y) plot(tx,x,'o',ty,y,'o') show()	Tx=10000; Ty=10100; Ti=10; drx=1; dry=1.5; mxs=3; mys=2; vxs=1; vys=2; cc=0.5; c1=sqrt(0.5+0.5sqrt(1-cc^2/(vxsvys))); c2=cc/sqrt(vxsvys)/(2c1); c11=sqrt(vxs)c1; c12=sqrt(vxs)c2; c21=sqrt(vys)c2; c22=sqrt(vys)c1; tx=[]; ty=[]; x=[]; y=[]; te=0; ue=randn(); ve=randn(); while te<max(Tx,Ty) tp=-log(1-rand())/(drx+dry); te=te+tp; if te<max(Tx,Ty) phi=exp(-tp/Ti); theta=sqrt(1-phi^2); ue=uephi+thetarandn(); ve=vephi+thetarandn(); if rand<drx/(drx+dry) if te<Tx tx(end+1)=te; x(end+1)=c11ue+c12ve+mxs; end else if te<Ty ty(end+1)=te; y(end+1)=c21ue+c22ve+mys; end end end end plot(tx,x,'o',ty,y,'o')
Mittelwerte	$\overline{x} = \frac{1}{N_{x}} \cdot \sum_{i = 0}^{N_{x} - 1} x_{i}$ $\overline{y} = \frac{1}{N_{y}} \cdot \sum_{i = 0}^{N_{y} - 1} y_{i}$	`mxe=mean(x) mye=mean(y)`	`mxe=mean(x) mye=mean(y)`
Varianzen (ohne Bessel-Korrektur, asymptotisch erwartungstreu)	$s_{x}^{2} = \frac{1}{N_{x}} \cdot \sum_{i = 0}^{N_{x} - 1} {(x_{i} - \overline{x})}^{2} = (\frac{1}{N_{x}} \cdot \sum_{i = 0}^{N_{x} - 1} x_{i}^{2}) - {\overline{x}}^{2}$ $s_{y}^{2} = \frac{1}{N_{y}} \cdot \sum_{i = 0}^{N_{y} - 1} {(y_{i} - \overline{y})}^{2} = (\frac{1}{N_{y}} \cdot \sum_{i = 0}^{N_{y} - 1} y_{i}^{2}) - {\overline{y}}^{2}$	`vxe=var(x) vye=var(y)`	`vxe=var(x,1) vye=var(y,1)`
Kreuzkorrelationsfunktion und Kreuzleistungsdichtespektrum über Slotkorrelation	$R_{y x, k} = R_{y x} (τ_{k}) = R_{x y} (- τ_{k}) = \frac{\sum_{i = 0}^{N_{x} - 1} \sum_{j = 0}^{N_{y} - 1} b_{k} (t_{j} - t_{i}) x_{i} y_{j}}{\sum_{i = 0}^{N_{x} - 1} \sum_{j = 0}^{N_{y} - 1} b_{k} (t_{j} - t_{i})}$ $b_{k} (t) = {\begin{matrix} 1 & falls 0 < \| t \| < Δ t / 2 \\ 0 & sonst \end{matrix}$ $τ_{k} = k Δ t$ $k = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$ $K < min (2 T_{x}, 2 T_{y}) / Δ t$ $S_{y x, j} = S_{y x} (f_{j}) = Δ t \cdot F F T {R_{y x, k}} = Δ t \cdot \sum_{k = ⌊ K / 2 ⌋}^{⌊ (K - 1) / 2 ⌋} R_{y x, k} 𝐞^{- 2 π 𝐢 j k / K}$ (imaginäre Einheit $𝐢$ ) $f_{j} = \frac{j}{K Δ t}$ $j = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$	from numpy.fft import * Nx=len(tx) Ny=len(ty) dt=1.0 K=200 K=minimum(int(ceil(2minimum(Tx,Ty)/float(dt)))-1,K) tau=roll(arange(-(K//2),(K+1)//2)dt,(K+1)//2) R1=zeros(K) R0=zeros(K) i=0 jb=i while i<Nx: j=jb while (jb<Ny) and (ty[jb]<tx[i]+((K-1)//2+0.5)dt): jb+=1 j=jb-1 while (j>=0) and (ty[j]>tx[i]-(K//2+0.5)dt): k=int(round((ty[j]-tx[i])/float(dt))) R1[k]+=x[i]y[j] R0[k]+=1.0 j-=1 i+=1 Ryx=R1/R0 plot(tau,Ryx,'o',arange(-5K,5K)dt/10.0,ccsqrt(vxsvys)exp(-abs(arange(-5K,5K)dt/10.0/float(Ti)))+mxsmys) show() f=roll(arange(-(K//2),(K+1)//2)/float(Kdt),(K+1)//2) Syx=dtfft(Ryx) p1=1-dt/float(Ti) loglog(f,real(Syx),'o',arange(1,5K)/float(10Kdt),ccsqrt(vxsvys)(1-p12)dt/(1+p1*2-2p1cos(2piarange(1,5K)/float(10Kdt)*dt)),f,imag(Syx),'o') show()	Nx=length(tx); Ny=length(ty); dt=1.0; K=200; K=min(ceil(2min(Tx,Ty)/dt)-1,K); tau=circshift((-fix(K/2):fix((K-1)/2))dt,[0;fix((K+1)/2)]); R1=zeros(1,K); R0=zeros(1,K); i=1; jb=i; while i<=Nx j=jb; while (jb<=Ny) && (ty(jb)<tx(i)+(fix((K-1)/2)+0.5)dt) jb=jb+1; end j=jb-1; while (j>0) && (ty(j)>tx(i)-(fix(K/2)+0.5)dt) k=round((ty(j)-tx(i))/dt); R1(mod(K+k,K)+1)=R1(mod(K+k,K)+1)+x(i)y(j); R0(mod(K+k,K)+1)=R0(mod(K+k,K)+1)+1; j=j-1; end i=i+1; end Ryx=R1./R0; plot(tau,Ryx,'o',(-5K:5K)dt/10,ccsqrt(vxsvys)exp(-abs((-5K:5K)dt/10/Ti))+mxsmys) f=circshift((-fix(K/2):fix((K-1)/2))/(Kdt),[0;fix((K+1)/2)]); Syx=dtfft(Ryx); p1=1-dt/Ti; loglog(f,real(Syx),'o',(1:5K)/(10Kdt),ccsqrt(vxsvys)(1-p1^2)dt./(1+p1^2-2p1cos(2pi(1:5K)/(10Kdt)dt)),f,imag(Syx),'o')
Kreuzkorrelationsfunktion und Kreuzleistungsdichtespektrum über Slotkorrelation (mit lokaler Normierung)	$R_{y x, k} = R_{y x} (τ_{k}) = R_{x y} (- τ_{k}) = \frac{[\sum_{i = 0}^{N_{x} - 1} \sum_{j = 0}^{N_{y} - 1} b_{k} (t_{j} - t_{i}) (x_{i} - \overline{x}) (y_{j} - \overline{y})] \cdot \sqrt{s_{x}^{2} s_{y}^{2}}}{\sqrt{[\sum_{i = 0}^{N_{x} - 1} \sum_{j = 0}^{N_{y} - 1} b_{k} (t_{j} - t_{i}) {(x_{i} - \overline{x})}^{2}] [\sum_{i = 0}^{N_{x} - 1} \sum_{j = 0}^{N_{y} - 1} b_{k} (t_{j} - t_{i}) {(y_{j} - \overline{y})}^{2}]}} + \overline{x} \cdot \overline{y}$ $b_{k} (t) = {\begin{matrix} 1 & falls 0 < \| t \| < Δ t / 2 \\ 0 & sonst \end{matrix}$ $τ_{k} = k Δ t$ $k = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$ $K < min (2 T_{x}, 2 T_{y}) / Δ t$ $S_{y x, j} = S_{y x} (f_{j}) = Δ t \cdot F F T {R_{y x, k}} = Δ t \cdot \sum_{k = ⌊ K / 2 ⌋}^{⌊ (K - 1) / 2 ⌋} R_{y x, k} 𝐞^{- 2 π 𝐢 j k / K}$ (imaginäre Einheit $𝐢$ ) $f_{j} = \frac{j}{K Δ t}$ $j = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$	from numpy.fft import * Nx=len(tx) Ny=len(ty) dt=1.0 K=200 K=minimum(int(ceil(2minimum(Tx,Ty)/float(dt)))-1,K) tau=roll(arange(-(K//2),(K+1)//2)dt,(K+1)//2) mxe=mean(x) mye=mean(y) vxe=var(x) vye=var(y) R1=zeros(K) R2=zeros(K) R3=zeros(K) i=0 jb=i while i<Nx: j=jb while (jb<Ny) and (ty[jb]<tx[i]+((K-1)//2+0.5)dt): jb+=1 j=jb-1 while (j>=0) and (ty[j]>tx[i]-(K//2+0.5)dt): k=int(round((ty[j]-tx[i])/float(dt))) R1[k]+=(x[i]-mxe)(y[j]-mye) R2[k]+=(x[i]-mxe)2 R3[k]+=(y[j]-mye)2 j-=1 i+=1 Ryx=sqrt(vxevye)R1/sqrt(R2R3)+mxemye plot(tau,Ryx,'o',arange(-5K,5K)dt/10.0,ccsqrt(vxsvys)exp(-abs(arange(-5K,5K)dt/10.0/float(Ti)))+mxsmys) show() f=roll(arange(-(K//2),(K+1)//2)/float(Kdt),(K+1)//2) Syx=dtfft(Ryx) p1=1-dt/float(Ti) loglog(f,real(Syx),'o',arange(1,5K)/float(10Kdt),ccsqrt(vxsvys)(1-p12)dt/(1+p1*2-2p1cos(2piarange(1,5K)/float(10Kdt)*dt)),f,imag(Syx),'o') show()	Nx=length(tx); Ny=length(ty); dt=1.0; K=200; K=min(ceil(2min(Tx,Ty)/dt)-1,K); tau=circshift((-fix(K/2):fix((K-1)/2))dt,[0;fix((K+1)/2)]); mxe=mean(x); mye=mean(y); vxe=var(x,1); vye=var(y,1); R1=zeros(1,K); R2=zeros(1,K); R3=zeros(1,K); i=1; jb=i; while i<=Nx j=jb; while (jb<=Ny) && (ty(jb)<tx(i)+(fix((K-1)/2)+0.5)dt) jb=jb+1; end j=jb-1; while (j>0) && (ty(j)>tx(i)-(fix(K/2)+0.5)dt) k=round((ty(j)-tx(i))/dt); R1(mod(K+k,K)+1)=R1(mod(K+k,K)+1)+(x(i)-mxe)(y(j)-mye); R2(mod(K+k,K)+1)=R2(mod(K+k,K)+1)+(x(i)-mxe)^2; R3(mod(K+k,K)+1)=R3(mod(K+k,K)+1)+(y(j)-mye)^2; j=j-1; end i=i+1; end Ryx=sqrt(vxevye)R1./sqrt(R2.R3)+mxemye; plot(tau,Ryx,'o',(-5K:5K)dt/10,ccsqrt(vxsvys)exp(-abs((-5K:5K)dt/10/Ti))+mxsmys) f=circshift((-fix(K/2):fix((K-1)/2))/(Kdt),[0;fix((K+1)/2)]); Syx=dtfft(Ryx); p1=1-dt/Ti; loglog(f,real(Syx),'o',(1:5K)/(10Kdt),ccsqrt(vxsvys)(1-p1^2)dt./(1+p1^2-2p1cos(2pi(1:5K)/(10Kdt)dt)),f,imag(Syx),'o')
Kreuzkorrelationsfunktion und Kreuzleistungsdichtespektrum über direkte Spektralschätzung (Fourier-Transformation, ohne Normierung)	$E_{y x, j} = E_{y x} (f_{j}) = \frac{T_{x} T_{y} X_{j}^{*} Y_{j}}{N_{x} N_{y}}$ $X_{j} = \sum_{i = 0}^{N_{x} - 1} x_{i} 𝐞^{- 2 π 𝐢 f_{j} t_{x, i}}$ $Y_{j} = \sum_{i = 0}^{N_{y} - 1} y_{i} 𝐞^{- 2 π 𝐢 f_{j} t_{y, i}}$ (imaginäre Einheit $𝐢$ ) $f_{j} = \frac{j}{J Δ t}$ $j = - ⌊ J / 2 ⌋ \dots ⌊ (J - 1) / 2 ⌋$ $J \geq ⌈ (T_{x} + T_{y}) / Δ t ⌉$ $R_{E, y x, k} = R_{E, y x} (τ_{k}) = \frac{1}{Δ t} \cdot I F F T {E_{y x, j}} = \frac{1}{J Δ t} \cdot \sum_{j = - ⌊ J / 2 ⌋}^{⌊ (J - 1) / 2 ⌋} E_{y x, j} 𝐞^{2 π 𝐢 f_{j} τ_{k} / J}$ $R_{y x, k} = R_{y x} (τ_{k}) = R_{x y} (- τ_{k}) = \frac{R_{E, y x, k}}{min (T_{x}, T_{y}, T_{x} + τ_{k}, T_{y} - τ_{k})}$ $τ_{k} = k Δ t$ $k = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$ $K < min (2 T_{x}, 2 T_{y}) / Δ t$ $S_{y x, j} = S_{y x} (f_{j}) = Δ t \cdot F F T {R_{y x, k}} = Δ t \cdot \sum_{k = ⌊ K / 2 ⌋}^{⌊ (K - 1) / 2 ⌋} R_{y x, k} 𝐞^{- 2 π 𝐢 j k / K}$ $f_{j} = \frac{j}{K Δ t}$ $j = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$	from numpy.fft import * Nx=len(tx) Ny=len(ty) dt=1.0 J=int(ceil((Tx+Ty)/float(dt))) fp=roll(arange(-(J//2),(J+1)//2)/float(Jdt),(J+1)//2) X=zeros(size(fp))+0j Y=zeros(size(fp))+0j for i in range(0,Nx): X+=x[i]exp(-2jpifptx[i]) for i in range(0,Ny): Y+=y[i]exp(-2jpifpty[i]) Eyx=TxTyconj(X)Y/float(NxNy) REyx=real(ifft(Eyx))/float(dt) K=200 K=minimum(int(ceil(2minimum(Tx,Ty)/float(dt)))-1,K) tau=roll(arange(-(K//2),(K+1)//2)dt,(K+1)//2) Ryx=zeros(K) for k in range(-(K//2),(K+1)//2): Ryx[k]=REyx[k]/float(minimum(Tx,minimum(Ty,minimum(Tx+tau[k],Ty-tau[k])))) plot(tau,Ryx,'o',arange(-5K,5K)dt/10.0,ccsqrt(vxsvys)exp(-abs(arange(-5K,5K)dt/10.0/float(Ti)))+mxsmys) show() f=roll(arange(-(K//2),(K+1)//2)/float(Kdt),(K+1)//2) Syx=dtfft(Ryx) p1=1-dt/float(Ti) loglog(f,real(Syx),'o',arange(1,5K)/float(10Kdt),ccsqrt(vxsvys)(1-p12)dt/(1+p1*2-2p1cos(2piarange(1,5K)/float(10Kdt)*dt)),f,imag(Syx),'o') show()	Nx=length(tx); Ny=length(ty); dt=1.0; J=ceil((Tx+Ty)/dt); fp=circshift((-fix(J/2):fix((J-1)/2))/(Jdt),[0;fix((J+1)/2)]); X=zeros(size(fp))+0j; Y=zeros(size(fp))+0j; for i=1:Nx X=X+x(i)exp(-2jpifptx(i)); end for i=1:Ny Y=Y+y(i)exp(-2jpifpty(i)); end Eyx=TxTyconj(X).Y/(NxNy); REyx=real(ifft(Eyx))/dt; K=200; K=min(ceil(2min(Tx,Ty)/dt)-1,K); tau=circshift((-fix(K/2):fix((K-1)/2))dt,[0;fix((K+1)/2)]); Ryx=zeros([1,K]); for k=-fix(K/2):fix((K-1)/2) Ryx(mod(k+K,K)+1)=REyx(mod(k+length(REyx),length(REyx))+1)/min(min(Tx,Ty),min(Tx+tau(mod(k+K,K)+1),Ty-tau(mod(k+K,K)+1))); end plot(tau,Ryx,'o',(-5K:5K)dt/10,ccsqrt(vxsvys)exp(-abs((-5K:5K)dt/10/Ti))+mxsmys) f=circshift((-fix(K/2):fix((K-1)/2))/(Kdt),[0;fix((K+1)/2)]); Syx=dtfft(Ryx); p1=1-dt/Ti; loglog(f,real(Syx),'o',(1:5K)/(10Kdt),ccsqrt(vxsvys)(1-p1^2)dt./(1+p1^2-2p1cos(2pi(1:5K)/(10Kdt)dt)),f,imag(Syx),'o')
Kreuzkorrelationsfunktion und Kreuzleistungsdichtespektrum über direkte Spektralschätzung (Fourier-Transformation, mit Normierung)	$E_{y x, j} = E_{y x} (f_{j}) = \frac{T_{x} T_{y} X_{j}^{} Y_{j}}{N_{x} N_{y}}$ $E_{y x, j}^{'} = E_{y x}^{'} (f_{j}) = \frac{T_{x} T_{y} X_{j}^{' } Y_{j}^{'}}{N_{x} N_{y}}$ $X_{j} = \sum_{i = 0}^{N_{x} - 1} x_{i} 𝐞^{- 2 π 𝐢 f_{j} t_{x, i}}$ $Y_{j} = \sum_{i = 0}^{N_{y} - 1} y_{i} 𝐞^{- 2 π 𝐢 f_{j} t_{y, i}}$ $X_{j}^{'} = \sum_{i = 0}^{N_{x} - 1} 𝐞^{- 2 π 𝐢 f_{j} t_{x, i}}$ $Y_{j}^{'} = \sum_{i = 0}^{N_{y} - 1} 𝐞^{- 2 π 𝐢 f_{j} t_{y, i}}$ (imaginäre Einheit $𝐢$ ) $f_{j} = \frac{j}{J Δ t}$ $j = - ⌊ J / 2 ⌋ \dots ⌊ (J - 1) / 2 ⌋$ $J \geq ⌈ (T_{x} + T_{y}) / Δ t ⌉$ $R_{E, y x, k} = R_{E, y x} (τ_{k}) = \frac{1}{Δ t} \cdot I F F T {E_{y x, j}} = \frac{1}{J Δ t} \cdot \sum_{j = - ⌊ J / 2 ⌋}^{⌊ (J - 1) / 2 ⌋} E_{y x, j} 𝐞^{2 π 𝐢 f_{j} τ_{k} / J}$ $R_{E, y x, k}^{'} = R_{E, y x}^{'} (τ_{k}) = \frac{1}{Δ t} \cdot I F F T {E_{y x, j}^{'}} = \frac{1}{J Δ t} \cdot \sum_{j = - ⌊ J / 2 ⌋}^{⌊ (J - 1) / 2 ⌋} E_{y x, j}^{'} 𝐞^{2 π 𝐢 f_{j} τ_{k} / J}$ $R_{y x, k} = R_{y x} (τ_{k}) = R_{x y} (- τ_{k}) = \frac{R_{E, y x, k}}{R_{E, y x, k}^{'}}$ $τ_{k} = k Δ t$ $k = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$ $K < min (2 T_{x}, 2 T_{y}) / Δ t$ $S_{y x, j} = S_{y x} (f_{j}) = Δ t \cdot F F T {R_{y x, k}} = Δ t \cdot \sum_{k = ⌊ K / 2 ⌋}^{⌊ (K - 1) / 2 ⌋} R_{y x, k} 𝐞^{- 2 π 𝐢 j k / K}$ $f_{j} = \frac{j}{K Δ t}$ $j = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$	from numpy.fft import * Nx=len(tx) Ny=len(ty) dt=1.0 J=int(ceil((Tx+Ty)/float(dt))) fp=roll(arange(-(J//2),(J+1)//2)/float(Jdt),(J+1)//2) X=zeros(size(fp))+0j Y=zeros(size(fp))+0j Xp=zeros(size(fp))+0j Yp=zeros(size(fp))+0j for i in range(0,Nx): X+=x[i]exp(-2jpifptx[i]) Xp+=exp(-2jpifptx[i]) for i in range(0,Ny): Y+=y[i]exp(-2jpifpty[i]) Yp+=exp(-2jpifpty[i]) Eyx=TxTyconj(X)Y/float(NxNy) Eyxp=TxTyconj(Xp)Yp/float(NxNy) REyx=real(ifft(Eyx))/float(dt) REyxp=real(ifft(Eyxp))/float(dt) K=200 K=minimum(int(ceil(2minimum(Tx,Ty)/float(dt)))-1,K) tau=roll(arange(-(K//2),(K+1)//2)dt,(K+1)//2) Ryx=zeros(K) for k in range(-(K//2),(K+1)//2): Ryx[k]=REyx[k]/float(REyxp[k]) plot(tau,Ryx,'o',arange(-5K,5K)dt/10.0,ccsqrt(vxsvys)exp(-abs(arange(-5K,5K)dt/10.0/float(Ti)))+mxsmys) show() f=roll(arange(-(K//2),(K+1)//2)/float(Kdt),(K+1)//2) Syx=dtfft(Ryx) p1=1-dt/float(Ti) loglog(f,real(Syx),'o',arange(1,5K)/float(10Kdt),ccsqrt(vxsvys)(1-p12)dt/(1+p1*2-2p1cos(2piarange(1,5K)/float(10Kdt)*dt)),f,imag(Syx),'o') show()	Nx=length(tx); Ny=length(ty); dt=1.0; J=ceil((Tx+Ty)/dt); fp=circshift((-fix(J/2):fix((J-1)/2))/(Jdt),[0;fix((J+1)/2)]); X=zeros(size(fp))+0j; Y=zeros(size(fp))+0j; Xp=zeros(size(fp))+0j; Yp=zeros(size(fp))+0j; for i=1:Nx X=X+x(i)exp(-2jpifptx(i)); Xp=Xp+exp(-2jpifptx(i)); end for i=1:Ny Y=Y+y(i)exp(-2jpifpty(i)); Yp=Yp+exp(-2jpifpty(i)); end Eyx=TxTyconj(X).Y/(NxNy); Eyxp=TxTyconj(Xp).Yp/(NxNy); REyx=real(ifft(Eyx))/dt; REyxp=real(ifft(Eyxp))/dt; K=200; K=min(ceil(2min(Tx,Ty)/dt)-1,K); tau=circshift((-fix(K/2):fix((K-1)/2))dt,[0;fix((K+1)/2)]); Ryx=zeros([1,K]); for k=-fix(K/2):fix((K-1)/2) Ryx(mod(k+K,K)+1)=REyx(mod(k+length(REyx),length(REyx))+1)/REyxp(mod(k+length(REyxp),length(REyxp))+1); end plot(tau,Ryx,'o',(-5K:5K)dt/10,ccsqrt(vxsvys)exp(-abs((-5K:5K)dt/10/Ti))+mxsmys) f=circshift((-fix(K/2):fix((K-1)/2))/(Kdt),[0;fix((K+1)/2)]); Syx=dtfft(Ryx); p1=1-dt/Ti; loglog(f,real(Syx),'o',(1:5K)/(10Kdt),ccsqrt(vxsvys)(1-p1^2)dt./(1+p1^2-2p1cos(2pi(1:5K)/(10Kdt)dt)),f,imag(Syx),'o')
Kreuzkorrelationsfunktion und Kreuzleistungsdichtespektrum über direkte Spektralschätzung (Fourier-Transformation, mit lokaler Normierung)	$E_{y x, j} = E_{y x} (f_{j}) = \frac{T_{x} T_{y} X_{j}^{} Y_{j}}{N_{x} N_{y}}$ $E_{y x, j}^{'} = E_{y x}^{'} (f_{j}) = \frac{T_{x} T_{y} X_{j}^{'' } Y_{j}^{'}}{N_{x} N_{y}}$ $E_{y x, j}^{''} = E_{y x}^{''} (f_{j}) = \frac{T_{x} T_{y} X_{j}^{' *} Y_{j}^{''}}{N_{x} N_{y}}$ $X_{j} = \sum_{i = 0}^{N_{x} - 1} (x_{i} - \overline{x}) 𝐞^{- 2 π 𝐢 f_{j} t_{x, i}}$ $Y_{j} = \sum_{i = 0}^{N_{y} - 1} (y_{i} - \overline{y}) 𝐞^{- 2 π 𝐢 f_{j} t_{y, i}}$ $X_{j}^{'} = \sum_{i = 0}^{N_{x} - 1} 𝐞^{- 2 π 𝐢 f_{j} t_{x, i}}$ $Y_{j}^{'} = \sum_{i = 0}^{N_{y} - 1} 𝐞^{- 2 π 𝐢 f_{j} t_{y, i}}$ $X_{j}^{''} = \sum_{i = 0}^{N_{x} - 1} {(x_{i} - \overline{x})}^{2} 𝐞^{- 2 π 𝐢 f_{j} t_{x, i}}$ $Y_{j}^{''} = \sum_{i = 0}^{N_{y} - 1} {(y_{i} - \overline{y})}^{2} 𝐞^{- 2 π 𝐢 f_{j} t_{y, i}}$ (imaginäre Einheit $𝐢$ ) $f_{j} = \frac{j}{J Δ t}$ $j = - ⌊ J / 2 ⌋ \dots ⌊ (J - 1) / 2 ⌋$ $J \geq ⌈ (T_{x} + T_{y}) / Δ t ⌉$ $R_{E, y x, k} = R_{E, y x} (τ_{k}) = \frac{1}{Δ t} \cdot I F F T {E_{y x, j}} = \frac{1}{J Δ t} \cdot \sum_{j = - ⌊ J / 2 ⌋}^{⌊ (J - 1) / 2 ⌋} E_{y x, j} 𝐞^{2 π 𝐢 f_{j} τ_{k} / J}$ $R_{E, y x, k}^{'} = R_{E, y x}^{'} (τ_{k}) = \frac{1}{Δ t} \cdot I F F T {E_{y x, j}^{'}} = \frac{1}{J Δ t} \cdot \sum_{j = - ⌊ J / 2 ⌋}^{⌊ (J - 1) / 2 ⌋} E_{y x, j}^{'} 𝐞^{2 π 𝐢 f_{j} τ_{k} / J}$ $R_{E, y x, k}^{''} = R_{E, y x}^{''} (τ_{k}) = \frac{1}{Δ t} \cdot I F F T {E_{y x, j}^{''}} = \frac{1}{J Δ t} \cdot \sum_{j = - ⌊ J / 2 ⌋}^{⌊ (J - 1) / 2 ⌋} E_{y x, j}^{''} 𝐞^{2 π 𝐢 f_{j} τ_{k} / J}$ $R_{y x, k} = R_{y x} (τ_{k}) = R_{x y} (- τ_{k}) = \frac{R_{E, y x, k} \sqrt{s_{x}^{2} s_{y}^{2}}}{\sqrt{R_{E, y x, k}^{'} R_{E, y x, k}^{''}}} + \overline{x} \cdot \overline{y}$ $τ_{k} = k Δ t$ $k = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$ $K < min (2 T_{x}, 2 T_{y}) / Δ t$ $S_{y x, j} = S_{y x} (f_{j}) = Δ t \cdot F F T {R_{y x, k}} = Δ t \cdot \sum_{k = ⌊ K / 2 ⌋}^{⌊ (K - 1) / 2 ⌋} R_{y x, k} 𝐞^{- 2 π 𝐢 j k / K}$ $f_{j} = \frac{j}{K Δ t}$ $j = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$	from numpy.fft import * Nx=len(tx) Ny=len(ty) dt=1.0 mxe=mean(x) mye=mean(y) vxe=var(x) vye=var(y) J=int(ceil((Tx+Ty)/float(dt))) fp=roll(arange(-(J//2),(J+1)//2)/float(Jdt),(J+1)//2) X=zeros(size(fp))+0j Y=zeros(size(fp))+0j Xp=zeros(size(fp))+0j Yp=zeros(size(fp))+0j Xpp=zeros(size(fp))+0j Ypp=zeros(size(fp))+0j for i in range(0,Nx): X+=(x[i]-mxe)exp(-2jpifptx[i]) Xp+=exp(-2jpifptx[i]) Xpp+=(x[i]-mxe)*2exp(-2jpifptx[i]) for i in range(0,Ny): Y+=(y[i]-mye)exp(-2jpifpty[i]) Yp+=exp(-2jpifpty[i]) Ypp+=(y[i]-mye)*2exp(-2jpifpty[i]) Eyx=TxTyconj(X)Y/float(NxNy) Eyxp=TxTyconj(Xpp)Yp/float(NxNy) Eyxpp=TxTyconj(Xp)Ypp/float(NxNy) REyx=real(ifft(Eyx))/float(dt) REyxp=real(ifft(Eyxp))/float(dt) REyxpp=real(ifft(Eyxpp))/float(dt) K=200 K=minimum(int(ceil(2minimum(Tx,Ty)/float(dt)))-1,K) tau=roll(arange(-(K//2),(K+1)//2)dt,(K+1)//2) Ryx=zeros(K) for k in range(-(K//2),(K+1)//2): Ryx[k]=sqrt(vxevye)REyx[k]/sqrt(REyxp[k]REyxpp[k])+mxemye plot(tau,Ryx,'o',arange(-5K,5K)dt/10.0,ccsqrt(vxsvys)exp(-abs(arange(-5K,5K)dt/10.0/float(Ti)))+mxsmys) show() f=roll(arange(-(K//2),(K+1)//2)/float(Kdt),(K+1)//2) Syx=dtfft(Ryx) p1=1-dt/float(Ti) loglog(f,real(Syx),'o',arange(1,5K)/float(10Kdt),ccsqrt(vxsvys)(1-p12)dt/(1+p1*2-2p1cos(2piarange(1,5K)/float(10Kdt)*dt)),f,imag(Syx),'o') show()	Nx=length(tx); Ny=length(ty); dt=1.0; mxe=mean(x); mye=mean(y); vxe=var(x,1); vye=var(y,1); J=ceil((Tx+Ty)/dt); fp=circshift((-fix(J/2):fix((J-1)/2))/(Jdt),[0;fix((J+1)/2)]); X=zeros(size(fp))+0j; Y=zeros(size(fp))+0j; Xp=zeros(size(fp))+0j; Yp=zeros(size(fp))+0j; Xpp=zeros(size(fp))+0j; Ypp=zeros(size(fp))+0j; for i=1:Nx X=X+(x(i)-mxe)exp(-2jpifptx(i)); Xp=Xp+exp(-2jpifptx(i)); Xpp=Xpp+(x(i)-mxe)^2exp(-2jpifptx(i)); end for i=1:Ny Y=Y+(y(i)-mye)exp(-2jpifpty(i)); Yp=Yp+exp(-2jpifpty(i)); Ypp=Ypp+(y(i)-mye)^2exp(-2jpifpty(i)); end Eyx=TxTyconj(X).Y/(NxNy); Eyxp=TxTyconj(Xpp).Yp/(NxNy); Eyxpp=TxTyconj(Xp).Ypp/(NxNy); REyx=real(ifft(Eyx))/dt; REyxp=real(ifft(Eyxp))/dt; REyxpp=real(ifft(Eyxpp))/dt; K=200; K=min(ceil(2min(Tx,Ty)/dt)-1,K); tau=circshift((-fix(K/2):fix((K-1)/2))dt,[0;fix((K+1)/2)]); Ryx=zeros([1,K]); for k=-fix(K/2):fix((K-1)/2) Ryx(mod(k+K,K)+1)=sqrt(vxevye)REyx(mod(k+length(REyx),length(REyx))+1)/sqrt(REyxp(mod(k+length(REyxp),length(REyxp))+1)REyxpp(mod(k+length(REyxpp),length(REyxpp))+1))+mxemye; end plot(tau,Ryx,'o',(-5K:5K)dt/10,ccsqrt(vxsvys)exp(-abs((-5K:5K)dt/10/Ti))+mxsmys) f=circshift((-fix(K/2):fix((K-1)/2))/(Kdt),[0;fix((K+1)/2)]); Syx=dtfft(Ryx); p1=1-dt/Ti; loglog(f,real(Syx),'o',(1:5K)/(10Kdt),ccsqrt(vxsvys)(1-p1^2)dt./(1+p1^2-2p1cos(2pi(1:5K)/(10Kdt)dt)),f,imag(Syx),'o')
(keine Lomb-Scargle-Methode für Kreuzspektren in Matlab und Python)
Kreuzkorrelationsfunktion und Kreuzleistungsdichtespektrum über Zeitquantisierung (ohne Normierung)		from numpy.fft import * Nx=len(tx) Ny=len(ty) dt=1.0 J=int(ceil((Tx+Ty)/float(dt))) x1=zeros(J) y1=zeros(J) for i in range(0,Nx): j=int(floor((tx[i]-Tfloor(tx[i]/float(T)))/float(dt))) x1[j]+=x[i]; for i in range(0,Ny): j=int(floor((ty[i]-Tfloor(ty[i]/float(T)))/float(dt))) y1[j]+=y[i]; X=fft(x1) Y=fft(y1) Eyx=TxTyconj(X)Y/float(NxNy) REyx=real(ifft(Eyx))/float(dt) K=200 K=minimum(int(ceil(2minminimum((Tx,Ty)/float(dt)))-1,K) tau=roll(arange(-(K//2),(K+1)//2)dt,(K+1)//2) Ryx=zeros(K) for k in range(-(K//2),(K+1)//2): Ryx[k]=REyx[k]/float(minimum(Tx,minimum(Ty,minimum(Tx+tau[k],Ty-tau[k])))) plot(tau,Ryx,'o',arange(-5K,5K)dt/10.0,ccsqrt(vxsvys)exp(-abs(arange(-5K,5K)dt/10.0/float(Ti)))+mxsmys) show() f=roll(arange(-(K//2),(K+1)//2)/float(Kdt),(K+1)//2) Syx=dtfft(Ryx) p1=1-dt/float(Ti) loglog(f,real(Syx),'o',arange(1,5K)/float(10Kdt),ccsqrt(vxsvys)(1-p1*2)dt/(1+p1*2-2p1cos(2piarange(1,5K)/float(10Kdt)*dt)),f,imag(Syx),'o') show()	Nx=length(tx); Ny=length(ty); dt=1.0; J=ceil((Tx+Ty)/dt); x1=zeros([1,J]); y1=zeros([1,J]); for i=1:Nx j=fix((tx(i)-Tfix(tx(i)/T))/dt)+1; x1(j)=x1(j)+x(i); end for i=1:Ny j=fix((ty(i)-Tfix(ty(i)/T))/dt)+1; y1(j)=y1(j)+y(i); end X=fft(x1); Y=fft(y1); Eyx=TxTyconj(X).Y/(NxNy); REyx=real(ifft(Eyx))/dt; K=200; K=min(ceil(2min(Tx,Ty)/dt)-1,K); tau=circshift((-fix(K/2):fix((K-1)/2))dt,[0;fix((K+1)/2)]); Ryx=zeros([1,K]); for k=-fix(K/2):fix((K-1)/2) Ryx(mod(k+K,K)+1)=REyx(mod(k+length(REyx),length(REyx))+1)/min(min(Tx,Ty),min(Tx+tau(mod(k+K,K)+1),Ty-tau(mod(k+K,K)+1))); end plot(tau,Ryx,'o',(-5K:5K)dt/10,ccsqrt(vxsvys)exp(-abs((-5K:5K)dt/10/Ti))+mxsmys) f=circshift((-fix(K/2):fix((K-1)/2))/(Kdt),[0;fix((K+1)/2)]); Syx=dtfft(Ryx); p1=1-dt/Ti; loglog(f,real(Syx),'o',(1:5K)/(10Kdt),ccsqrt(vxsvys)(1-p1^2)dt./(1+p1^2-2p1cos(2pi(1:5K)/(10Kdt)*dt)),f,imag(Syx),'o')
Kreuzkorrelationsfunktion und Kreuzleistungsdichtespektrum über Zeitquantisierung (mit Normierung)		from numpy.fft import * Nx=len(tx) Ny=len(ty) dt=1.0 J=int(ceil((Tx+Ty)/float(dt))) x0=zeros(J) y0=zeros(J) x1=zeros(J) y1=zeros(J) for i in range(0,Nx): j=int(floor((tx[i]-Tfloor(tx[i]/float(T)))/float(dt))) x0[j]+=1.0; x1[j]+=x[i]; for i in range(0,Ny): j=int(floor((ty[i]-Tfloor(ty[i]/float(T)))/float(dt))) y0[j]+=1.0; y1[j]+=y[i]; X=fft(x1) Y=fft(y1) Xp=fft(x0) Yp=fft(y0) Eyx=TxTyconj(X)Y/float(NxNy) Eyxp=TxTyconj(Xp)Yp/float(NxNy) REyx=real(ifft(Eyx))/float(dt) REyxp=real(ifft(Eyxp))/float(dt) K=200 K=minimum(int(ceil(2minminimum((Tx,Ty)/float(dt)))-1,K) tau=roll(arange(-(K//2),(K+1)//2)dt,(K+1)//2) Ryx=zeros(K) for k in range(-(K//2),(K+1)//2): Ryx[k]=REyx[k]/float(REyxp[k]) plot(tau,Ryx,'o',arange(-5K,5K)dt/10.0,ccsqrt(vxsvys)exp(-abs(arange(-5K,5K)dt/10.0/float(Ti)))+mxsmys) show() f=roll(arange(-(K//2),(K+1)//2)/float(Kdt),(K+1)//2) Syx=dtfft(Ryx) p1=1-dt/float(Ti) loglog(f,real(Syx),'o',arange(1,5K)/float(10Kdt),ccsqrt(vxsvys)(1-p1*2)dt/(1+p1*2-2p1cos(2piarange(1,5K)/float(10Kdt)*dt)),f,imag(Syx),'o') show()	Nx=length(tx); Ny=length(ty); dt=1.0; J=ceil((Tx+Ty)/dt); x0=zeros([1,J]); y0=zeros([1,J]); x1=zeros([1,J]); y1=zeros([1,J]); for i=1:Nx j=fix((tx(i)-Tfix(tx(i)/T))/dt)+1; x0(j)=x0(j)+1; x1(j)=x1(j)+x(i); end for i=1:Ny j=fix((ty(i)-Tfix(ty(i)/T))/dt)+1; y0(j)=y0(j)+1; y1(j)=y1(j)+y(i); end X=fft(x1); Y=fft(y1); Xp=fft(x0); Yp=fft(y0); Eyx=TxTyconj(X).Y/(NxNy); Eyxp=TxTyconj(Xp).Yp/(NxNy); REyx=real(ifft(Eyx))/dt; REyxp=real(ifft(Eyxp))/dt; K=200; K=min(ceil(2min(Tx,Ty)/dt)-1,K); tau=circshift((-fix(K/2):fix((K-1)/2))dt,[0;fix((K+1)/2)]); Ryx=zeros([1,K]); for k=-fix(K/2):fix((K-1)/2) Ryx(mod(k+K,K)+1)=REyx(mod(k+length(REyx),length(REyx))+1)/REyxp(mod(k+length(REyxp),length(REyxp))+1); end plot(tau,Ryx,'o',(-5K:5K)dt/10,ccsqrt(vxsvys)exp(-abs((-5K:5K)dt/10/Ti))+mxsmys) f=circshift((-fix(K/2):fix((K-1)/2))/(Kdt),[0;fix((K+1)/2)]); Syx=dtfft(Ryx); p1=1-dt/Ti; loglog(f,real(Syx),'o',(1:5K)/(10Kdt),ccsqrt(vxsvys)(1-p1^2)dt./(1+p1^2-2p1cos(2pi(1:5K)/(10Kdt)*dt)),f,imag(Syx),'o')
Kreuzkorrelationsfunktion und Kreuzleistungsdichtespektrum über Zeitquantisierung (mit lokaler Normierung)		from numpy.fft import * Nx=len(tx) Ny=len(ty) dt=1.0 mxe=mean(x) mye=mean(y) vxe=var(x) vye=var(y) J=int(ceil((Tx+Ty)/float(dt))) x0=zeros(J) y0=zeros(J) x1=zeros(J) y1=zeros(J) x2=zeros(J) y2=zeros(J) for i in range(0,Nx): j=int(floor((tx[i]-Tfloor(tx[i]/float(T)))/float(dt))) x0[j]+=1.0; x1[j]+=x[i]-mxe; x2[j]+=(x[i]-mxe)2; for i in range(0,Ny): j=int(floor((ty[i]-Tfloor(ty[i]/float(T)))/float(dt))) y0[j]+=1.0; y1[j]+=y[i]-mye; y2[j]+=(y[i]-mye)*2; X=fft(x1) Y=fft(y1) Xp=fft(x0) Yp=fft(y0) Xpp=fft(x2) Ypp=fft(y2) Eyx=TxTyconj(X)Y/float(NxNy) Eyxp=TxTyconj(Xpp)Yp/float(NxNy) Eyxpp=TxTyconj(Xp)Ypp/float(NxNy) REyx=real(ifft(Eyx))/float(dt) REyxp=real(ifft(Eyxp))/float(dt) REyxpp=real(ifft(Eyxpp))/float(dt) K=200 K=minimum(int(ceil(2minminimum((Tx,Ty)/float(dt)))-1,K) tau=roll(arange(-(K//2),(K+1)//2)dt,(K+1)//2) Ryx=zeros(K) for k in range(-(K//2),(K+1)//2): Ryx[k]=sqrt(vxevye)REyx[k]/sqrt(REyxp[k]REyxpp[k])+mxemye plot(tau,Ryx,'o',arange(-5K,5K)dt/10.0,ccsqrt(vxsvys)exp(-abs(arange(-5K,5K)dt/10.0/float(Ti)))+mxsmys) show() f=roll(arange(-(K//2),(K+1)//2)/float(Kdt),(K+1)//2) Syx=dtfft(Ryx) p1=1-dt/float(Ti) loglog(f,real(Syx),'o',arange(1,5K)/float(10Kdt),ccsqrt(vxsvys)(1-p12)dt/(1+p1*2-2p1cos(2piarange(1,5K)/float(10Kdt)*dt)),f,imag(Syx),'o') show()	Nx=length(tx); Ny=length(ty); dt=1.0; mxe=mean(x); mye=mean(y); vxe=var(x,1); vye=var(y,1); J=ceil((Tx+Ty)/dt); x0=zeros([1,J]); y0=zeros([1,J]); x1=zeros([1,J]); y1=zeros([1,J]); x2=zeros([1,J]); y2=zeros([1,J]); for i=1:Nx j=fix((tx(i)-Tfix(tx(i)/T))/dt)+1; x0(j)=x0(j)+1; x1(j)=x1(j)+x(i)-mxe; x2(j)=x2(j)+(x(i)-mxe)^2; end for i=1:Ny j=fix((ty(i)-Tfix(ty(i)/T))/dt)+1; y0(j)=y0(j)+1; y1(j)=y1(j)+y(i)-mye; y2(j)=y2(j)+(y(i)-mye)^2; end X=fft(x1); Y=fft(y1); Xp=fft(x0); Yp=fft(y0); Xpp=fft(x2); Ypp=fft(y2); Eyx=TxTyconj(X).Y/(NxNy); Eyxp=TxTyconj(Xpp).Yp/(NxNy); Eyxpp=TxTyconj(Xp).Ypp/(NxNy); REyx=real(ifft(Eyx))/dt; REyxp=real(ifft(Eyxp))/dt; REyxpp=real(ifft(Eyxpp))/dt; K=200; K=min(ceil(2min(Tx,Ty)/dt)-1,K); tau=circshift((-fix(K/2):fix((K-1)/2))dt,[0;fix((K+1)/2)]); Ryx=zeros([1,K]); for k=-fix(K/2):fix((K-1)/2) Ryx(mod(k+K,K)+1)=sqrt(vxevye)REyx(mod(k+length(REyx),length(REyx))+1)/sqrt(REyxp(mod(k+length(REyxp),length(REyxp))+1)REyxpp(mod(k+length(REyxpp),length(REyxpp))+1))+mxemye; end plot(tau,Ryx,'o',(-5K:5K)dt/10,ccsqrt(vxsvys)exp(-abs((-5K:5K)dt/10/Ti))+mxsmys) f=circshift((-fix(K/2):fix((K-1)/2))/(Kdt),[0;fix((K+1)/2)]); Syx=dtfft(Ryx); p1=1-dt/Ti; loglog(f,real(Syx),'o',(1:5K)/(10Kdt),ccsqrt(vxsvys)(1-p1^2)dt./(1+p1^2-2p1cos(2pi(1:5K)/(10Kdt)*dt)),f,imag(Syx),'o')
Kreuzkorrelationsfunktion und Kreuzleistungsdichtespektrum über Interpolation (Sample-and-Hold, mit Filterkorrektur, ohne Normierung)		from numpy.fft import * Nx=len(tx) Ny=len(ty) dt=1.0 J=int(ceil((Tx+Ty)/float(dt))) xp=zeros(J) yp=zeros(J) for i in range(0,Nx): for j in range(int(floor(tx[i]/float(dt)))+1,int(floor((tx[(i+1)%Nx]+Tx((i+1)//Nx))/float(dt)))+1): xp[j]=x[i] for i in range(0,Ny): for j in range(int(floor(ty[i]/float(dt)))+1,int(floor((ty[(i+1)%Ny]+Ty((i+1)//Ny))/float(dt)))+1): yp[j]=y[i] Xp=fft(xp) Yp=fft(yp) Eyxp=dt*2conj(Xp)Yp REyxp=real(ifft(Eyxp))/float(dt) K=200 K=minimum(int(ceil(2minimum(Tx,Ty)/float(dt)))-1,K) tau=roll(arange(-(K//2),(K+1)//2)dt,(K+1)//2) Ryx=zeros(K) cx=exp(-ddx/float(dt))/((1-exp(-ddx/float(dt)))(1-exp(-ddy/float(dt)))) cy=exp(-ddy/float(dt))/((1-exp(-ddx/float(dt)))(1-exp(-ddy/float(dt)))) for k in range(-(K//2),(K+1)//2): Ryx[k]=((cx+cy+1)REyxp[k]-cyREyxp[k-1]-cxREyxp[k+1])/float(minimum(Tx,minimum(Ty,minimum(Tx+tau[k],Ty-tau[k])))) plot(tau,Ryx,'o',arange(-5K,5K)dt/10.0,ccsqrt(vxsvys)exp(-abs(arange(-5K,5K)dt/10.0/float(Ti)))+mxsmys) show() f=roll(arange(-(K//2),(K+1)//2)/float(Kdt),(K+1)//2) Syx=dtfft(Ryx) p1=1-dt/float(Ti) loglog(f,real(Syx),'o',arange(1,5K)/float(10Kdt),ccsqrt(vxsvys)(1-p1*2)dt/(1+p1*2-2p1cos(2piarange(1,5K)/float(10Kdt)*dt)),f,imag(Syx),'o') show()	Nx=length(tx); Ny=length(ty); dt=1.0; J=ceil((Tx+Ty)/dt); xp=zeros([1,J]); yp=zeros([1,J]); for i=1:Nx for j=fix(tx(i)/dt)+2:fix((tx(mod(i,Nx)+1)+Txfix(i/Nx))/dt)+1 xp(j)=x(i); end end for i=1:Ny for j=fix(ty(i)/dt)+2:fix((ty(mod(i,Ny)+1)+Tyfix(i/Ny))/dt)+1 yp(j)=y(i); end end Xp=fft(xp); Yp=fft(yp); Eyxp=dt^2conj(Xp).Yp; REyxp=real(ifft(Eyxp))/dt; K=200; K=min(ceil(2min(Tx,Ty)/dt)-1,K); tau=circshift((-fix(K/2):fix((K-1)/2))dt,[0;fix((K+1)/2)]); Ryx=zeros([1,K]); cx=exp(-ddx/dt)/((1-exp(-ddx/dt))(1-exp(-ddy/dt))); cy=exp(-ddy/dt)/((1-exp(-ddx/dt))(1-exp(-ddy/dt))); for k=-fix(K/2):fix((K-1)/2) Ryx(mod(k+K,K)+1)=((cx+cy+1)REyxp(mod(k+length(REyxp),length(REyxp))+1)-cyREyxp(mod(k-1+length(REyxp),length(REyxp))+1)-cxREyxp(mod(k+1+length(REyxp),length(REyxp))+1))/min(min(Tx,Ty),min(Tx+tau(mod(k+K,K)+1),Ty-tau(mod(k+K,K)+1))); end plot(tau,Ryx,'o',(-5K:5K)dt/10,ccsqrt(vxsvys)exp(-abs((-5K:5K)dt/10/Ti))+mxsmys) f=circshift((-fix(K/2):fix((K-1)/2))/(Kdt),[0;fix((K+1)/2)]); Syx=dtfft(Ryx); p1=1-dt/Ti; loglog(f,real(Syx),'o',(1:5K)/(10Kdt),ccsqrt(vxsvys)(1-p1^2)dt./(1+p1^2-2p1cos(2pi(1:5K)/(10Kdt)dt)),f,imag(Syx),'o')