Holger Nobach - nambis.de

Signal- und Messdatenverarbeitung

Codes für stochastisch und gemeinsam 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 Wertetripeln $(t_{i}, x_{i}, y_{i})$ mit den Messzeitpunkten $t_{i}$ und den Messwerten $x_{i}$ und $y_{i}$ (zwei korrelierte AR1-Prozess als Beispiel, mit Erwartungswerten verschieden von null)		T=10000.0 Ti=10.0 dr=1.0 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 t=[] x=[] y=[] te=0.0 ue=standard_normal() ve=standard_normal() while te<T: tp=exponential(1.0/dr) te+=tp if te<T: t.append(te) phi=exp(-tp/Ti) theta=sqrt(1-phi*2) ue=uephi+normal(0,theta) ve=vephi+normal(0,theta) x.append(c11ue+c12ve+mxs) y.append(c21ue+c22*ve+mys) t=array(t) x=array(x) y=array(y) plot(t,x,'o',t,y,'o') show()	`T=10000; Ti=10; dr=1; 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; t=[]; x=[]; y=[]; te=0; ue=randn(); ve=randn(); while te<T tp=-log(1-rand())/dr; te=te+tp; if te<T t(end+1)=te; phi=exp(-tp/Ti); theta=sqrt(1-phi^2); ue=uephi+thetarandn(); ve=vephi+thetarandn(); x(end+1)=c11ue+c12ve+mxs; y(end+1)=c21ue+c22ve+mys; end end plot(t,x,'o',t,y,'o')`
Mittelwerte	$\overline{x} = \frac{1}{N} \cdot \sum_{i = 0}^{N - 1} x_{i}$ $\overline{y} = \frac{1}{N} \cdot \sum_{i = 0}^{N - 1} y_{i}$	`mxe=mean(x) mye=mean(y)`	`mxe=mean(x) mye=mean(y)`
Varianzen (ohne Bessel-Korrektur, asymptotisch erwartungstreu)	$s_{x}^{2} = \frac{1}{N} \cdot \sum_{i = 0}^{N - 1} {(x_{i} - \overline{x})}^{2} = (\frac{1}{N} \cdot \sum_{i = 0}^{N - 1} x_{i}^{2}) - {\overline{x}}^{2}$ $s_{y}^{2} = \frac{1}{N} \cdot \sum_{i = 0}^{N - 1} {(y_{i} - \overline{y})}^{2} = (\frac{1}{N} \cdot \sum_{i = 0}^{N - 1} y_{i}^{2}) - {\overline{y}}^{2}$	`vxe=var(x) vye=var(y)`	`vxe=var(x,1) vye=var(y,1)`
Kreuzkorrelationsfunktion und Kreuzleistungsdichtespektrum über Slotkorrelation (ohne koinzidente Produkte)	$R_{y x, k} = R_{y x} (τ_{k}) = R_{x y} (- τ_{k}) = \frac{\sum_{i = 0}^{N - 1} \sum_{j = 0}^{N - 1} b_{k} (t_{j} - t_{i}) x_{i} y_{j}}{\sum_{i = 0}^{N - 1} \sum_{j = 0}^{N - 1} b_{k} (t_{j} - t_{i})}$ $b_{k} (t) = {\begin{matrix} 1 & falls 0 < \| t \| < Δ t / 2 \\ 0 & sonst \end{matrix}$ $τ_{k} = k Δ t$ $k = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$ $K < 2 T / Δ t$ $S_{y x, j} = S_{y x} (f_{j}) = Δ t \cdot F F T {R_{y x, k}} = Δ t \cdot \sum_{k = ⌊ K / 2 ⌋}^{⌊ (K - 1) / 2 ⌋} R_{y x, k} 𝐞^{- 2 π 𝐢 j k / K}$ (imaginäre Einheit $𝐢$ ) $f_{j} = \frac{j}{K Δ t}$ $j = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$	from numpy.fft import * N=len(t) dt=1.0 K=200 K=minimum(int(ceil(2T/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<N: j=jb while (jb<N) and (t[jb]<t[i]+((K-1)//2+0.5)dt): jb+=1 j=jb-1 while (j>=0) and (t[j]>t[i]-(K//2+0.5)dt): k=int(round((t[j]-t[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()	N=length(t); dt=1.0; K=200; K=min(ceil(2T/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=2; jb=i; while i<=N j=jb; while (jb<=N) && (t(jb)<t(i)+(fix((K-1)/2)+0.5)dt) jb=jb+1; end j=jb-1; while (j>0) && (t(j)>t(i)-(fix(K/2)+0.5)dt) k=round((t(j)-t(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 (ohne koinzidente Produkte, mit lokaler Normierung)	$R_{y x, k} = R_{y x} (τ_{k}) = R_{x y} (- τ_{k}) = \frac{[\sum_{i = 0}^{N - 1} \sum_{j = 0}^{N - 1} b_{k} (t_{j} - t_{i}) (x_{i} - \overline{x}) (y_{j} - \overline{y})] \cdot \sqrt{s_{x}^{2} s_{y}^{2}}}{\sqrt{[\sum_{i = 0}^{N - 1} \sum_{j = 0}^{N - 1} b_{k} (t_{j} - t_{i}) {(x_{i} - \overline{x})}^{2}] [\sum_{i = 0}^{N - 1} \sum_{j = 0}^{N - 1} b_{k} (t_{j} - t_{i}) {(y_{j} - \overline{y})}^{2}]}} + \overline{x} \cdot \overline{y}$ $b_{k} (t) = {\begin{matrix} 1 & falls 0 < \| t \| < Δ t / 2 \\ 0 & sonst \end{matrix}$ $τ_{k} = k Δ t$ $k = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$ $K < 2 T / Δ t$ $S_{y x, j} = S_{y x} (f_{j}) = Δ t \cdot F F T {R_{y x, k}} = Δ t \cdot \sum_{k = ⌊ K / 2 ⌋}^{⌊ (K - 1) / 2 ⌋} R_{y x, k} 𝐞^{- 2 π 𝐢 j k / K}$ (imaginäre Einheit $𝐢$ ) $f_{j} = \frac{j}{K Δ t}$ $j = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$	from numpy.fft import * N=len(t) dt=1.0 K=200 K=minimum(int(ceil(2T/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<N: j=jb while (jb<N) and (t[jb]<t[i]+((K-1)//2+0.5)dt): jb+=1 j=jb-1 while (j>=0) and (t[j]>t[i]-(K//2+0.5)dt): k=int(round((t[j]-t[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()	N=length(t); dt=1.0; K=200; K=min(ceil(2T/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=2; jb=i; while i<=N j=jb; while (jb<=N) && (t(jb)<t(i)+(fix((K-1)/2)+0.5)dt) jb=jb+1; end j=jb-1; while (j>0) && (t(j)>t(i)-(fix(K/2)+0.5)dt) k=round((t(j)-t(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, mit Abzug der koinzidenten Produkte, ohne Normierung)	$E_{y x, j} = E_{y x} (f_{j}) = \frac{T^{2}}{N (N - 1)} {X_{j}^{*} Y_{j} - \sum_{i = 0}^{N - 1} x_{i} y_{i}}$ $X_{j} = \sum_{i = 0}^{N - 1} x_{i} 𝐞^{- 2 π 𝐢 f_{j} t_{i}}$ $Y_{j} = \sum_{i = 0}^{N - 1} y_{i} 𝐞^{- 2 π 𝐢 f_{j} t_{i}}$ (imaginäre Einheit $𝐢$ ) $f_{j} = \frac{j}{J Δ t}$ $j = - ⌊ J / 2 ⌋ \dots ⌊ (J - 1) / 2 ⌋$ $J \geq ⌈ 2 T / Δ t ⌉$ $R_{E, y x, k} = R_{E, y x} (τ_{k}) = \frac{1}{Δ t} \cdot I F F T {E_{y x, j}} = \frac{1}{J Δ t} \cdot \sum_{j = - ⌊ J / 2 ⌋}^{⌊ (J - 1) / 2 ⌋} E_{y x, j} 𝐞^{2 π 𝐢 f_{j} τ_{k} / J}$ $R_{y x, k} = R_{y x} (τ_{k}) = R_{x y} (- τ_{k}) = \frac{R_{E, y x, k}}{T - \| τ_{k} \|}$ $τ_{k} = k Δ t$ $k = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$ $K < 2 T / Δ t$ $S_{y x, j} = S_{y x} (f_{j}) = Δ t \cdot F F T {R_{y x, k}} = Δ t \cdot \sum_{k = ⌊ K / 2 ⌋}^{⌊ (K - 1) / 2 ⌋} R_{y x, k} 𝐞^{- 2 π 𝐢 j k / K}$ $f_{j} = \frac{j}{K Δ t}$ $j = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$	from numpy.fft import * N=len(t) dt=1.0 J=int(ceil(2T/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,N): X+=x[i]exp(-2jpifpt[i]) Y+=y[i]exp(-2jpifpt[i]) Eyx=T*2(conj(X)Y-sum(xy))/float(N*2-N) REyx=real(ifft(Eyx))/float(dt) K=200 K=minimum(int(ceil(2T/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(T-abs(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()	N=length(t); dt=1.0; J=ceil(2T/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:N X=X+x(i)exp(-2jpifpt(i)); Y=Y+y(i)exp(-2jpifpt(i)); end Eyx=T^2(conj(X).Y-sum(x.y))/(N^2-N); REyx=real(ifft(Eyx))/dt; K=200; K=min(ceil(2T/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)/(T-abs(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 Abzug der koinzidenten Produkte, mit Normierung)	$E_{y x, j} = E_{y x} (f_{j}) = \frac{T^{2}}{N (N - 1)} {X_{j}^{*} Y_{j} - \sum_{i = 0}^{N - 1} x_{i} y_{i}}$ $E_{y x, j}^{'} = E_{y x}^{'} (f_{j}) = \frac{T^{2}}{N (N - 1)} {{\| X_{j}^{'} \|}^{2} - N}$ $X_{j} = \sum_{i = 0}^{N - 1} x_{i} 𝐞^{- 2 π 𝐢 f_{j} t_{i}}$ $Y_{j} = \sum_{i = 0}^{N - 1} y_{i} 𝐞^{- 2 π 𝐢 f_{j} t_{i}}$ $X_{j}^{'} = \sum_{i = 0}^{N - 1} 𝐞^{- 2 π 𝐢 f_{j} t_{i}}$ (imaginäre Einheit $𝐢$ ) $f_{j} = \frac{j}{J Δ t}$ $j = - ⌊ J / 2 ⌋ \dots ⌊ (J - 1) / 2 ⌋$ $J \geq ⌈ 2 T / Δ t ⌉$ $R_{E, y x, k} = R_{E, y x} (τ_{k}) = \frac{1}{Δ t} \cdot I F F T {E_{y x, j}} = \frac{1}{J Δ t} \cdot \sum_{j = - ⌊ J / 2 ⌋}^{⌊ (J - 1) / 2 ⌋} E_{y x, j} 𝐞^{2 π 𝐢 f_{j} τ_{k} / J}$ $R_{E, y x, k}^{'} = R_{E, y x}^{'} (τ_{k}) = \frac{1}{Δ t} \cdot I F F T {E_{y x, j}^{'}} = \frac{1}{J Δ t} \cdot \sum_{j = - ⌊ J / 2 ⌋}^{⌊ (J - 1) / 2 ⌋} E_{y x, j}^{'} 𝐞^{2 π 𝐢 f_{j} τ_{k} / J}$ $R_{y x, k} = R_{y x} (τ_{k}) = R_{x y} (- τ_{k}) = \frac{R_{E, y x, k}}{R_{E, y x, k}^{'}}$ $τ_{k} = k Δ t$ $k = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$ $K < 2 T / Δ t$ $S_{y x, j} = S_{y x} (f_{j}) = Δ t \cdot F F T {R_{y x, k}} = Δ t \cdot \sum_{k = ⌊ K / 2 ⌋}^{⌊ (K - 1) / 2 ⌋} R_{y x, k} 𝐞^{- 2 π 𝐢 j k / K}$ $f_{j} = \frac{j}{K Δ t}$ $j = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$	from numpy.fft import * N=len(t) dt=1.0 J=int(ceil(2T/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 for i in range(0,N): X+=x[i]exp(-2jpifpt[i]) Y+=y[i]exp(-2jpifpt[i]) Xp+=exp(-2jpifpt[i]) Eyx=T2(conj(X)Y-sum(xy))/float(N2-N) Eyxp=T2(abs(Xp)2-N)/float(N2-N) REyx=real(ifft(Eyx))/float(dt) REyxp=real(ifft(Eyxp))/float(dt) K=200 K=minimum(int(ceil(2T/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()	N=length(t); dt=1.0; J=ceil(2T/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; for i=1:N X=X+x(i)exp(-2jpifpt(i)); Y=Y+y(i)exp(-2jpifpt(i)); Xp=Xp+exp(-2jpifpt(i)); end Eyx=T^2(conj(X).Y-sum(x.y))/(N^2-N); Eyxp=T^2(abs(Xp).^2-N)/(N^2-N); REyx=real(ifft(Eyx))/dt; REyxp=real(ifft(Eyxp))/dt; K=200; K=min(ceil(2T/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 Abzug der koinzidenten Produkte, mit lokaler Normierung)	$E_{y x, j} = E_{y x} (f_{j}) = \frac{T^{2}}{N (N - 1)} {X_{j}^{} Y_{j} - \sum_{i = 0}^{N - 1} (x_{i} - \overline{x}) (y_{i} - \overline{y})}$ $E_{y x, j}^{'} = E_{y x}^{'} (f_{j}) = \frac{T^{2}}{N (N - 1)} {X_{j}^{'' } X_{j}^{'} - \sum_{i = 0}^{N - 1} {(x_{i} - \overline{x})}^{2}}$ $E_{y x, j}^{''} = E_{y x}^{''} (f_{j}) = \frac{T^{2}}{N (N - 1)} {X_{j}^{' *} Y_{j}^{''} - \sum_{i = 0}^{N - 1} {(y_{i} - \overline{y})}^{2}}$ $X_{j} = \sum_{i = 0}^{N - 1} (x_{i} - \overline{x}) 𝐞^{- 2 π 𝐢 f_{j} t_{i}}$ $Y_{j} = \sum_{i = 0}^{N - 1} (y_{i} - \overline{y}) 𝐞^{- 2 π 𝐢 f_{j} t_{i}}$ $X_{j}^{'} = \sum_{i = 0}^{N - 1} 𝐞^{- 2 π 𝐢 f_{j} t_{i}}$ $X_{j}^{''} = \sum_{i = 0}^{N - 1} {(x_{i} - \overline{x})}^{2} 𝐞^{- 2 π 𝐢 f_{j} t_{i}}$ $Y_{j}^{''} = \sum_{i = 0}^{N - 1} {(y_{i} - \overline{y})}^{2} 𝐞^{- 2 π 𝐢 f_{j} t_{i}}$ (imaginäre Einheit $𝐢$ ) $f_{j} = \frac{j}{J Δ t}$ $j = - ⌊ J / 2 ⌋ \dots ⌊ (J - 1) / 2 ⌋$ $J \geq ⌈ 2 T / Δ t ⌉$ $R_{E, y x, k} = R_{E, y x} (τ_{k}) = \frac{1}{Δ t} \cdot I F F T {E_{y x, j}} = \frac{1}{J Δ t} \cdot \sum_{j = - ⌊ J / 2 ⌋}^{⌊ (J - 1) / 2 ⌋} E_{y x, j} 𝐞^{2 π 𝐢 f_{j} τ_{k} / J}$ $R_{E, y x, k}^{'} = R_{E, y x}^{'} (τ_{k}) = \frac{1}{Δ t} \cdot I F F T {E_{y x, j}^{'}} = \frac{1}{J Δ t} \cdot \sum_{j = - ⌊ J / 2 ⌋}^{⌊ (J - 1) / 2 ⌋} E_{y x, j}^{'} 𝐞^{2 π 𝐢 f_{j} τ_{k} / J}$ $R_{E, y x, k}^{''} = R_{E, y x}^{''} (τ_{k}) = \frac{1}{Δ t} \cdot I F F T {E_{y x, j}^{''}} = \frac{1}{J Δ t} \cdot \sum_{j = - ⌊ J / 2 ⌋}^{⌊ (J - 1) / 2 ⌋} E_{y x, j}^{''} 𝐞^{2 π 𝐢 f_{j} τ_{k} / J}$ $R_{y x, k} = R_{y x} (τ_{k}) = R_{x y} (- τ_{k}) = \frac{R_{E, y x, k} \sqrt{s_{x}^{2} s_{y}^{2}}}{\sqrt{R_{E, y x, k}^{'} R_{E, y x, k}^{''}}} + \overline{x} \cdot \overline{y}$ $τ_{k} = k Δ t$ $k = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$ $K < 2 T / Δ t$ $S_{y x, j} = S_{y x} (f_{j}) = Δ t \cdot F F T {R_{y x, k}} = Δ t \cdot \sum_{k = ⌊ K / 2 ⌋}^{⌊ (K - 1) / 2 ⌋} R_{y x, k} 𝐞^{- 2 π 𝐢 j k / K}$ $f_{j} = \frac{j}{K Δ t}$ $j = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$	from numpy.fft import * N=len(t) dt=1.0 mxe=mean(x) mye=mean(y) vxe=var(x) vye=var(y) J=int(ceil(2T/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 Xpp=zeros(size(fp))+0j Ypp=zeros(size(fp))+0j for i in range(0,N): X+=(x[i]-mxe)exp(-2jpifpt[i]) Y+=(y[i]-mye)exp(-2jpifpt[i]) Xp+=exp(-2jpifpt[i]) Xpp+=(x[i]-mxe)2exp(-2jpifpt[i]) Ypp+=(y[i]-mye)2exp(-2jpifpt[i]) Eyx=T2(conj(X)Y-sum((x-mxe)(y-mye)))/float(N2-N) Eyxp=T2(conj(Xpp)Xp-sum((x-mxe)2))/float(N2-N) Eyxpp=T*2(conj(Xp)Ypp-sum((y-mye)2))/float(N2-N) REyx=real(ifft(Eyx))/float(dt) REyxp=real(ifft(Eyxp))/float(dt) REyxpp=real(ifft(Eyxpp))/float(dt) K=200 K=minimum(int(ceil(2T/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()	N=length(t); dt=1.0; mxe=mean(x); mye=mean(y); vxe=var(x,1); vye=var(y,1); J=ceil(2T/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; Xpp=zeros(size(fp))+0j; Ypp=zeros(size(fp))+0j; for i=1:N X=X+(x(i)-mxe)exp(-2jpifpt(i)); Y=Y+(y(i)-mye)exp(-2jpifpt(i)); Xp=Xp+exp(-2jpifpt(i)); Xpp=Xpp+(x(i)-mxe)^2exp(-2jpifpt(i)); Ypp=Ypp+(y(i)-mye)^2exp(-2jpifpt(i)); end Eyx=T^2(conj(X).Y-sum((x-mxe).(y-mye)))/(N^2-N); Eyxp=T^2(conj(Xpp).Xp-sum((x-mxe).^2))/(N^2-N); Eyxpp=T^2(conj(Xp).Ypp-sum((y-mye).^2))/(N^2-N); REyx=real(ifft(Eyx))/dt; REyxp=real(ifft(Eyxp))/dt; REyxpp=real(ifft(Eyxpp))/dt; K=200; K=min(ceil(2T/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 (mit Abzug der koinzidenten Produkte, ohne Normierung)		from numpy.fft import * N=len(t) dt=1.0 J=int(ceil(2T/float(dt))) x1=zeros(J) y1=zeros(J) for i in range(0,N): j=int(floor((t[i]-Tfloor(t[i]/float(T)))/float(dt))) x1[j]+=x[i]; y1[j]+=y[i]; X=fft(x1) Y=fft(y1) Eyx=T*2(conj(X)Y-sum(xy))/float(N*2-N) REyx=real(ifft(Eyx))/float(dt) K=200 K=minimum(int(ceil(2T/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(T-abs(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()	N=length(t); dt=1.0; J=ceil(2T/dt); x1=zeros([1,J]); y1=zeros([1,J]); for i=1:N j=fix((t(i)-Tfix(t(i)/T))/dt)+1; x1(j)=x1(j)+x(i); y1(j)=y1(j)+y(i); end X=fft(x1); Y=fft(y1); Eyx=T^2(conj(X).Y-sum(x.y))/(N^2-N); REyx=real(ifft(Eyx))/dt; K=200; K=min(ceil(2T/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)/(T-abs(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 Abzug der koinzidenten Produkte, mit Normierung)		from numpy.fft import * N=len(t) dt=1.0 J=int(ceil(2T/float(dt))) x0=zeros(J) x1=zeros(J) y1=zeros(J) for i in range(0,N): j=int(floor((t[i]-Tfloor(t[i]/float(T)))/float(dt))) x0[j]+=1.0; x1[j]+=x[i]; y1[j]+=y[i]; X=fft(x1) Y=fft(y1) Xp=fft(x0) Eyx=T*2(conj(X)Y-sum(xy))/float(N2-N) Eyxp=T2(abs(Xp)2-N)/float(N2-N) REyx=real(ifft(Eyx))/float(dt) REyxp=real(ifft(Eyxp))/float(dt) K=200 K=minimum(int(ceil(2T/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()	N=length(t); dt=1.0; J=ceil(2T/dt); x0=zeros([1,J]); x1=zeros([1,J]); y1=zeros([1,J]); for i=1:N j=fix((t(i)-Tfix(t(i)/T))/dt)+1; x0(j)=x0(j)+1; x1(j)=x1(j)+x(i); y1(j)=y1(j)+y(i); end X=fft(x1); Y=fft(y1); Xp=fft(x0); Eyx=T^2(conj(X).Y-sum(x.y))/(N^2-N); Eyxp=T^2(abs(Xp).^2-N)/(N^2-N); REyx=real(ifft(Eyx))/dt; REyxp=real(ifft(Eyxp))/dt; K=200; K=min(ceil(2T/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 Abzug der koinzidenten Produkte, mit lokaler Normierung)		from numpy.fft import * N=len(t) dt=1.0 mxe=mean(x) mye=mean(y) vxe=var(x) vye=var(y) J=int(ceil(2T/float(dt))) x0=zeros(J) x1=zeros(J) y1=zeros(J) x2=zeros(J) y2=zeros(J) for i in range(0,N): j=int(floor((t[i]-Tfloor(t[i]/float(T)))/float(dt))) x0[j]+=1.0; x1[j]+=x[i]-mxe; y1[j]+=y[i]-mye; x2[j]+=(x[i]-mxe)2; y2[j]+=(y[i]-mye)2; X=fft(x1) Y=fft(y1) Xp=fft(x0) Xpp=fft(x2) Ypp=fft(y2) Eyx=T*2(conj(X)Y-sum((x-mxe)(y-mye)))/float(N2-N) Eyxp=T2(conj(Xpp)Xp-sum((x-mxe)2))/float(N2-N) Eyxpp=T*2(conj(Xp)Ypp-sum((y-mye)2))/float(N2-N) REyx=real(ifft(Eyx))/float(dt) REyxp=real(ifft(Eyxp))/float(dt) REyxpp=real(ifft(Eyxpp))/float(dt) K=200 K=minimum(int(ceil(2T/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()	N=length(t); dt=1.0; mxe=mean(x); mye=mean(y); vxe=var(x,1); vye=var(y,1); J=ceil(2T/dt); x0=zeros([1,J]); x1=zeros([1,J]); y1=zeros([1,J]); x2=zeros([1,J]); y2=zeros([1,J]); for i=1:N j=fix((t(i)-Tfix(t(i)/T))/dt)+1; x0(j)=x0(j)+1; x1(j)=x1(j)+x(i)-mxe; y1(j)=y1(j)+y(i)-mye; x2(j)=x2(j)+(x(i)-mxe)^2; y2(j)=y2(j)+(y(i)-mye)^2; end X=fft(x1); Y=fft(y1); Xp=fft(x0); Xpp=fft(x2); Ypp=fft(y2); Eyx=T^2(conj(X).Y-sum((x-mxe).(y-mye)))/(N^2-N); Eyxp=T^2(conj(Xpp).Xp-sum((x-mxe).^2))/(N^2-N); Eyxpp=T^2(conj(Xp).Ypp-sum((y-mye).^2))/(N^2-N); REyx=real(ifft(Eyx))/dt; REyxp=real(ifft(Eyxp))/dt; REyxpp=real(ifft(Eyxpp))/dt; K=200; K=min(ceil(2T/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 * N=len(t) dt=1.0 J=int(ceil(2T/float(dt))) xp=zeros(J) yp=zeros(J) for i in range(0,N): for j in range(int(floor(t[i]/float(dt)))+1,int(floor((t[(i+1)%N]+T((i+1)//N))/float(dt)))+1): xp[j]=x[i] 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(2T/float(dt)))-1,K) tau=roll(arange(-(K//2),(K+1)//2)dt,(K+1)//2) Ryx=zeros(K) c=exp(-dd/float(dt))/(1-exp(-dd/float(dt)))2 for k in range(-(K//2),(K+1)//2): if k==0: Ryx[k]=REyxp[k]/float(T) else: Ryx[k]=((2c+1)REyxp[k]-cREyxp[k-1]-cREyxp[k+1])/float(T-abs(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()	N=length(t); dt=1.0; J=ceil(2T/dt); xp=zeros([1,J]); yp=zeros([1,J]); for i=1:N for j=fix(t(i)/dt)+2:fix((t(mod(i,N)+1)+Tfix(i/N))/dt)+1 xp(j)=x(i); 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(2T/dt)-1,K); tau=circshift((-fix(K/2):fix((K-1)/2))dt,[0;fix((K+1)/2)]); Ryx=zeros([1,K]); c=exp(-dd/dt)/(1-exp(-dd/dt))^2; for k=-fix(K/2):fix((K-1)/2) if k==0 Ryx(mod(k+K,K)+1)=REyxp(mod(k+length(REyxp),length(REyxp))+1)/T; else Ryx(mod(k+K,K)+1)=((2c+1)REyxp(mod(k+length(REyxp),length(REyxp))+1)-cREyxp(mod(k-1+length(REyxp),length(REyxp))+1)-cREyxp(mod(k+1+length(REyxp),length(REyxp))+1))/(T-abs(tau(mod(k+K,K)+1))); end 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')

		Python	Matlab/Octave
Vorbereitung (Laden von Modulen/Paketen)		`from numpy import * from numpy.random import * from matplotlib.pyplot import *`
Generieren von Wertetripeln $(t_{i}, x_{i}, y_{i})$ mit den Messzeitpunkten $t_{i}$ und den Messwerten $x_{i}$ und $y_{i}$ (zwei korrelierte AR1-Prozess als Beispiel, mit Erwartungswerten verschieden von null)		T=10000.0 Ti=10.0 dr=1.0 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 t=[] x=[] y=[] te=0.0 ue=standard_normal() ve=standard_normal() while te<T: tp=exponential(1.0/dr) te+=tp if te<T: t.append(te) phi=exp(-tp/Ti) theta=sqrt(1-phi*2) ue=uephi+normal(0,theta) ve=vephi+normal(0,theta) x.append(c11ue+c12ve+mxs) y.append(c21ue+c22*ve+mys) t=array(t) x=array(x) y=array(y) plot(t,x,'o',t,y,'o') show()	`T=10000; Ti=10; dr=1; 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; t=[]; x=[]; y=[]; te=0; ue=randn(); ve=randn(); while te<T tp=-log(1-rand())/dr; te=te+tp; if te<T t(end+1)=te; phi=exp(-tp/Ti); theta=sqrt(1-phi^2); ue=uephi+thetarandn(); ve=vephi+thetarandn(); x(end+1)=c11ue+c12ve+mxs; y(end+1)=c21ue+c22ve+mys; end end plot(t,x,'o',t,y,'o')`
Mittelwerte	$\overline{x} = \frac{1}{N} \cdot \sum_{i = 0}^{N - 1} x_{i}$ $\overline{y} = \frac{1}{N} \cdot \sum_{i = 0}^{N - 1} y_{i}$	`mxe=mean(x) mye=mean(y)`	`mxe=mean(x) mye=mean(y)`
Varianzen (ohne Bessel-Korrektur, asymptotisch erwartungstreu)	$s_{x}^{2} = \frac{1}{N} \cdot \sum_{i = 0}^{N - 1} {(x_{i} - \overline{x})}^{2} = (\frac{1}{N} \cdot \sum_{i = 0}^{N - 1} x_{i}^{2}) - {\overline{x}}^{2}$ $s_{y}^{2} = \frac{1}{N} \cdot \sum_{i = 0}^{N - 1} {(y_{i} - \overline{y})}^{2} = (\frac{1}{N} \cdot \sum_{i = 0}^{N - 1} y_{i}^{2}) - {\overline{y}}^{2}$	`vxe=var(x) vye=var(y)`	`vxe=var(x,1) vye=var(y,1)`
Kreuzkorrelationsfunktion und Kreuzleistungsdichtespektrum über Slotkorrelation (ohne koinzidente Produkte)	$R_{y x, k} = R_{y x} (τ_{k}) = R_{x y} (- τ_{k}) = \frac{\sum_{i = 0}^{N - 1} \sum_{j = 0}^{N - 1} b_{k} (t_{j} - t_{i}) x_{i} y_{j}}{\sum_{i = 0}^{N - 1} \sum_{j = 0}^{N - 1} b_{k} (t_{j} - t_{i})}$ $b_{k} (t) = {\begin{matrix} 1 & falls 0 < \| t \| < Δ t / 2 \\ 0 & sonst \end{matrix}$ $τ_{k} = k Δ t$ $k = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$ $K < 2 T / Δ t$ $S_{y x, j} = S_{y x} (f_{j}) = Δ t \cdot F F T {R_{y x, k}} = Δ t \cdot \sum_{k = ⌊ K / 2 ⌋}^{⌊ (K - 1) / 2 ⌋} R_{y x, k} 𝐞^{- 2 π 𝐢 j k / K}$ (imaginäre Einheit $𝐢$ ) $f_{j} = \frac{j}{K Δ t}$ $j = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$	from numpy.fft import * N=len(t) dt=1.0 K=200 K=minimum(int(ceil(2T/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<N: j=jb while (jb<N) and (t[jb]<t[i]+((K-1)//2+0.5)dt): jb+=1 j=jb-1 while (j>=0) and (t[j]>t[i]-(K//2+0.5)dt): k=int(round((t[j]-t[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()	N=length(t); dt=1.0; K=200; K=min(ceil(2T/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=2; jb=i; while i<=N j=jb; while (jb<=N) && (t(jb)<t(i)+(fix((K-1)/2)+0.5)dt) jb=jb+1; end j=jb-1; while (j>0) && (t(j)>t(i)-(fix(K/2)+0.5)dt) k=round((t(j)-t(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 (ohne koinzidente Produkte, mit lokaler Normierung)	$R_{y x, k} = R_{y x} (τ_{k}) = R_{x y} (- τ_{k}) = \frac{[\sum_{i = 0}^{N - 1} \sum_{j = 0}^{N - 1} b_{k} (t_{j} - t_{i}) (x_{i} - \overline{x}) (y_{j} - \overline{y})] \cdot \sqrt{s_{x}^{2} s_{y}^{2}}}{\sqrt{[\sum_{i = 0}^{N - 1} \sum_{j = 0}^{N - 1} b_{k} (t_{j} - t_{i}) {(x_{i} - \overline{x})}^{2}] [\sum_{i = 0}^{N - 1} \sum_{j = 0}^{N - 1} b_{k} (t_{j} - t_{i}) {(y_{j} - \overline{y})}^{2}]}} + \overline{x} \cdot \overline{y}$ $b_{k} (t) = {\begin{matrix} 1 & falls 0 < \| t \| < Δ t / 2 \\ 0 & sonst \end{matrix}$ $τ_{k} = k Δ t$ $k = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$ $K < 2 T / Δ t$ $S_{y x, j} = S_{y x} (f_{j}) = Δ t \cdot F F T {R_{y x, k}} = Δ t \cdot \sum_{k = ⌊ K / 2 ⌋}^{⌊ (K - 1) / 2 ⌋} R_{y x, k} 𝐞^{- 2 π 𝐢 j k / K}$ (imaginäre Einheit $𝐢$ ) $f_{j} = \frac{j}{K Δ t}$ $j = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$	from numpy.fft import * N=len(t) dt=1.0 K=200 K=minimum(int(ceil(2T/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<N: j=jb while (jb<N) and (t[jb]<t[i]+((K-1)//2+0.5)dt): jb+=1 j=jb-1 while (j>=0) and (t[j]>t[i]-(K//2+0.5)dt): k=int(round((t[j]-t[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()	N=length(t); dt=1.0; K=200; K=min(ceil(2T/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=2; jb=i; while i<=N j=jb; while (jb<=N) && (t(jb)<t(i)+(fix((K-1)/2)+0.5)dt) jb=jb+1; end j=jb-1; while (j>0) && (t(j)>t(i)-(fix(K/2)+0.5)dt) k=round((t(j)-t(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, mit Abzug der koinzidenten Produkte, ohne Normierung)	$E_{y x, j} = E_{y x} (f_{j}) = \frac{T^{2}}{N (N - 1)} {X_{j}^{*} Y_{j} - \sum_{i = 0}^{N - 1} x_{i} y_{i}}$ $X_{j} = \sum_{i = 0}^{N - 1} x_{i} 𝐞^{- 2 π 𝐢 f_{j} t_{i}}$ $Y_{j} = \sum_{i = 0}^{N - 1} y_{i} 𝐞^{- 2 π 𝐢 f_{j} t_{i}}$ (imaginäre Einheit $𝐢$ ) $f_{j} = \frac{j}{J Δ t}$ $j = - ⌊ J / 2 ⌋ \dots ⌊ (J - 1) / 2 ⌋$ $J \geq ⌈ 2 T / Δ t ⌉$ $R_{E, y x, k} = R_{E, y x} (τ_{k}) = \frac{1}{Δ t} \cdot I F F T {E_{y x, j}} = \frac{1}{J Δ t} \cdot \sum_{j = - ⌊ J / 2 ⌋}^{⌊ (J - 1) / 2 ⌋} E_{y x, j} 𝐞^{2 π 𝐢 f_{j} τ_{k} / J}$ $R_{y x, k} = R_{y x} (τ_{k}) = R_{x y} (- τ_{k}) = \frac{R_{E, y x, k}}{T - \| τ_{k} \|}$ $τ_{k} = k Δ t$ $k = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$ $K < 2 T / Δ t$ $S_{y x, j} = S_{y x} (f_{j}) = Δ t \cdot F F T {R_{y x, k}} = Δ t \cdot \sum_{k = ⌊ K / 2 ⌋}^{⌊ (K - 1) / 2 ⌋} R_{y x, k} 𝐞^{- 2 π 𝐢 j k / K}$ $f_{j} = \frac{j}{K Δ t}$ $j = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$	from numpy.fft import * N=len(t) dt=1.0 J=int(ceil(2T/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,N): X+=x[i]exp(-2jpifpt[i]) Y+=y[i]exp(-2jpifpt[i]) Eyx=T*2(conj(X)Y-sum(xy))/float(N*2-N) REyx=real(ifft(Eyx))/float(dt) K=200 K=minimum(int(ceil(2T/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(T-abs(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()	N=length(t); dt=1.0; J=ceil(2T/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:N X=X+x(i)exp(-2jpifpt(i)); Y=Y+y(i)exp(-2jpifpt(i)); end Eyx=T^2(conj(X).Y-sum(x.y))/(N^2-N); REyx=real(ifft(Eyx))/dt; K=200; K=min(ceil(2T/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)/(T-abs(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 Abzug der koinzidenten Produkte, mit Normierung)	$E_{y x, j} = E_{y x} (f_{j}) = \frac{T^{2}}{N (N - 1)} {X_{j}^{*} Y_{j} - \sum_{i = 0}^{N - 1} x_{i} y_{i}}$ $E_{y x, j}^{'} = E_{y x}^{'} (f_{j}) = \frac{T^{2}}{N (N - 1)} {{\| X_{j}^{'} \|}^{2} - N}$ $X_{j} = \sum_{i = 0}^{N - 1} x_{i} 𝐞^{- 2 π 𝐢 f_{j} t_{i}}$ $Y_{j} = \sum_{i = 0}^{N - 1} y_{i} 𝐞^{- 2 π 𝐢 f_{j} t_{i}}$ $X_{j}^{'} = \sum_{i = 0}^{N - 1} 𝐞^{- 2 π 𝐢 f_{j} t_{i}}$ (imaginäre Einheit $𝐢$ ) $f_{j} = \frac{j}{J Δ t}$ $j = - ⌊ J / 2 ⌋ \dots ⌊ (J - 1) / 2 ⌋$ $J \geq ⌈ 2 T / Δ t ⌉$ $R_{E, y x, k} = R_{E, y x} (τ_{k}) = \frac{1}{Δ t} \cdot I F F T {E_{y x, j}} = \frac{1}{J Δ t} \cdot \sum_{j = - ⌊ J / 2 ⌋}^{⌊ (J - 1) / 2 ⌋} E_{y x, j} 𝐞^{2 π 𝐢 f_{j} τ_{k} / J}$ $R_{E, y x, k}^{'} = R_{E, y x}^{'} (τ_{k}) = \frac{1}{Δ t} \cdot I F F T {E_{y x, j}^{'}} = \frac{1}{J Δ t} \cdot \sum_{j = - ⌊ J / 2 ⌋}^{⌊ (J - 1) / 2 ⌋} E_{y x, j}^{'} 𝐞^{2 π 𝐢 f_{j} τ_{k} / J}$ $R_{y x, k} = R_{y x} (τ_{k}) = R_{x y} (- τ_{k}) = \frac{R_{E, y x, k}}{R_{E, y x, k}^{'}}$ $τ_{k} = k Δ t$ $k = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$ $K < 2 T / Δ t$ $S_{y x, j} = S_{y x} (f_{j}) = Δ t \cdot F F T {R_{y x, k}} = Δ t \cdot \sum_{k = ⌊ K / 2 ⌋}^{⌊ (K - 1) / 2 ⌋} R_{y x, k} 𝐞^{- 2 π 𝐢 j k / K}$ $f_{j} = \frac{j}{K Δ t}$ $j = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$	from numpy.fft import * N=len(t) dt=1.0 J=int(ceil(2T/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 for i in range(0,N): X+=x[i]exp(-2jpifpt[i]) Y+=y[i]exp(-2jpifpt[i]) Xp+=exp(-2jpifpt[i]) Eyx=T2(conj(X)Y-sum(xy))/float(N2-N) Eyxp=T2(abs(Xp)2-N)/float(N2-N) REyx=real(ifft(Eyx))/float(dt) REyxp=real(ifft(Eyxp))/float(dt) K=200 K=minimum(int(ceil(2T/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()	N=length(t); dt=1.0; J=ceil(2T/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; for i=1:N X=X+x(i)exp(-2jpifpt(i)); Y=Y+y(i)exp(-2jpifpt(i)); Xp=Xp+exp(-2jpifpt(i)); end Eyx=T^2(conj(X).Y-sum(x.y))/(N^2-N); Eyxp=T^2(abs(Xp).^2-N)/(N^2-N); REyx=real(ifft(Eyx))/dt; REyxp=real(ifft(Eyxp))/dt; K=200; K=min(ceil(2T/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 Abzug der koinzidenten Produkte, mit lokaler Normierung)	$E_{y x, j} = E_{y x} (f_{j}) = \frac{T^{2}}{N (N - 1)} {X_{j}^{} Y_{j} - \sum_{i = 0}^{N - 1} (x_{i} - \overline{x}) (y_{i} - \overline{y})}$ $E_{y x, j}^{'} = E_{y x}^{'} (f_{j}) = \frac{T^{2}}{N (N - 1)} {X_{j}^{'' } X_{j}^{'} - \sum_{i = 0}^{N - 1} {(x_{i} - \overline{x})}^{2}}$ $E_{y x, j}^{''} = E_{y x}^{''} (f_{j}) = \frac{T^{2}}{N (N - 1)} {X_{j}^{' *} Y_{j}^{''} - \sum_{i = 0}^{N - 1} {(y_{i} - \overline{y})}^{2}}$ $X_{j} = \sum_{i = 0}^{N - 1} (x_{i} - \overline{x}) 𝐞^{- 2 π 𝐢 f_{j} t_{i}}$ $Y_{j} = \sum_{i = 0}^{N - 1} (y_{i} - \overline{y}) 𝐞^{- 2 π 𝐢 f_{j} t_{i}}$ $X_{j}^{'} = \sum_{i = 0}^{N - 1} 𝐞^{- 2 π 𝐢 f_{j} t_{i}}$ $X_{j}^{''} = \sum_{i = 0}^{N - 1} {(x_{i} - \overline{x})}^{2} 𝐞^{- 2 π 𝐢 f_{j} t_{i}}$ $Y_{j}^{''} = \sum_{i = 0}^{N - 1} {(y_{i} - \overline{y})}^{2} 𝐞^{- 2 π 𝐢 f_{j} t_{i}}$ (imaginäre Einheit $𝐢$ ) $f_{j} = \frac{j}{J Δ t}$ $j = - ⌊ J / 2 ⌋ \dots ⌊ (J - 1) / 2 ⌋$ $J \geq ⌈ 2 T / Δ t ⌉$ $R_{E, y x, k} = R_{E, y x} (τ_{k}) = \frac{1}{Δ t} \cdot I F F T {E_{y x, j}} = \frac{1}{J Δ t} \cdot \sum_{j = - ⌊ J / 2 ⌋}^{⌊ (J - 1) / 2 ⌋} E_{y x, j} 𝐞^{2 π 𝐢 f_{j} τ_{k} / J}$ $R_{E, y x, k}^{'} = R_{E, y x}^{'} (τ_{k}) = \frac{1}{Δ t} \cdot I F F T {E_{y x, j}^{'}} = \frac{1}{J Δ t} \cdot \sum_{j = - ⌊ J / 2 ⌋}^{⌊ (J - 1) / 2 ⌋} E_{y x, j}^{'} 𝐞^{2 π 𝐢 f_{j} τ_{k} / J}$ $R_{E, y x, k}^{''} = R_{E, y x}^{''} (τ_{k}) = \frac{1}{Δ t} \cdot I F F T {E_{y x, j}^{''}} = \frac{1}{J Δ t} \cdot \sum_{j = - ⌊ J / 2 ⌋}^{⌊ (J - 1) / 2 ⌋} E_{y x, j}^{''} 𝐞^{2 π 𝐢 f_{j} τ_{k} / J}$ $R_{y x, k} = R_{y x} (τ_{k}) = R_{x y} (- τ_{k}) = \frac{R_{E, y x, k} \sqrt{s_{x}^{2} s_{y}^{2}}}{\sqrt{R_{E, y x, k}^{'} R_{E, y x, k}^{''}}} + \overline{x} \cdot \overline{y}$ $τ_{k} = k Δ t$ $k = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$ $K < 2 T / Δ t$ $S_{y x, j} = S_{y x} (f_{j}) = Δ t \cdot F F T {R_{y x, k}} = Δ t \cdot \sum_{k = ⌊ K / 2 ⌋}^{⌊ (K - 1) / 2 ⌋} R_{y x, k} 𝐞^{- 2 π 𝐢 j k / K}$ $f_{j} = \frac{j}{K Δ t}$ $j = - ⌊ K / 2 ⌋ \dots ⌊ (K - 1) / 2 ⌋$	from numpy.fft import * N=len(t) dt=1.0 mxe=mean(x) mye=mean(y) vxe=var(x) vye=var(y) J=int(ceil(2T/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 Xpp=zeros(size(fp))+0j Ypp=zeros(size(fp))+0j for i in range(0,N): X+=(x[i]-mxe)exp(-2jpifpt[i]) Y+=(y[i]-mye)exp(-2jpifpt[i]) Xp+=exp(-2jpifpt[i]) Xpp+=(x[i]-mxe)2exp(-2jpifpt[i]) Ypp+=(y[i]-mye)2exp(-2jpifpt[i]) Eyx=T2(conj(X)Y-sum((x-mxe)(y-mye)))/float(N2-N) Eyxp=T2(conj(Xpp)Xp-sum((x-mxe)2))/float(N2-N) Eyxpp=T*2(conj(Xp)Ypp-sum((y-mye)2))/float(N2-N) REyx=real(ifft(Eyx))/float(dt) REyxp=real(ifft(Eyxp))/float(dt) REyxpp=real(ifft(Eyxpp))/float(dt) K=200 K=minimum(int(ceil(2T/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()	N=length(t); dt=1.0; mxe=mean(x); mye=mean(y); vxe=var(x,1); vye=var(y,1); J=ceil(2T/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; Xpp=zeros(size(fp))+0j; Ypp=zeros(size(fp))+0j; for i=1:N X=X+(x(i)-mxe)exp(-2jpifpt(i)); Y=Y+(y(i)-mye)exp(-2jpifpt(i)); Xp=Xp+exp(-2jpifpt(i)); Xpp=Xpp+(x(i)-mxe)^2exp(-2jpifpt(i)); Ypp=Ypp+(y(i)-mye)^2exp(-2jpifpt(i)); end Eyx=T^2(conj(X).Y-sum((x-mxe).(y-mye)))/(N^2-N); Eyxp=T^2(conj(Xpp).Xp-sum((x-mxe).^2))/(N^2-N); Eyxpp=T^2(conj(Xp).Ypp-sum((y-mye).^2))/(N^2-N); REyx=real(ifft(Eyx))/dt; REyxp=real(ifft(Eyxp))/dt; REyxpp=real(ifft(Eyxpp))/dt; K=200; K=min(ceil(2T/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 (mit Abzug der koinzidenten Produkte, ohne Normierung)		from numpy.fft import * N=len(t) dt=1.0 J=int(ceil(2T/float(dt))) x1=zeros(J) y1=zeros(J) for i in range(0,N): j=int(floor((t[i]-Tfloor(t[i]/float(T)))/float(dt))) x1[j]+=x[i]; y1[j]+=y[i]; X=fft(x1) Y=fft(y1) Eyx=T*2(conj(X)Y-sum(xy))/float(N*2-N) REyx=real(ifft(Eyx))/float(dt) K=200 K=minimum(int(ceil(2T/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(T-abs(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()	N=length(t); dt=1.0; J=ceil(2T/dt); x1=zeros([1,J]); y1=zeros([1,J]); for i=1:N j=fix((t(i)-Tfix(t(i)/T))/dt)+1; x1(j)=x1(j)+x(i); y1(j)=y1(j)+y(i); end X=fft(x1); Y=fft(y1); Eyx=T^2(conj(X).Y-sum(x.y))/(N^2-N); REyx=real(ifft(Eyx))/dt; K=200; K=min(ceil(2T/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)/(T-abs(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 Abzug der koinzidenten Produkte, mit Normierung)		from numpy.fft import * N=len(t) dt=1.0 J=int(ceil(2T/float(dt))) x0=zeros(J) x1=zeros(J) y1=zeros(J) for i in range(0,N): j=int(floor((t[i]-Tfloor(t[i]/float(T)))/float(dt))) x0[j]+=1.0; x1[j]+=x[i]; y1[j]+=y[i]; X=fft(x1) Y=fft(y1) Xp=fft(x0) Eyx=T*2(conj(X)Y-sum(xy))/float(N2-N) Eyxp=T2(abs(Xp)2-N)/float(N2-N) REyx=real(ifft(Eyx))/float(dt) REyxp=real(ifft(Eyxp))/float(dt) K=200 K=minimum(int(ceil(2T/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()	N=length(t); dt=1.0; J=ceil(2T/dt); x0=zeros([1,J]); x1=zeros([1,J]); y1=zeros([1,J]); for i=1:N j=fix((t(i)-Tfix(t(i)/T))/dt)+1; x0(j)=x0(j)+1; x1(j)=x1(j)+x(i); y1(j)=y1(j)+y(i); end X=fft(x1); Y=fft(y1); Xp=fft(x0); Eyx=T^2(conj(X).Y-sum(x.y))/(N^2-N); Eyxp=T^2(abs(Xp).^2-N)/(N^2-N); REyx=real(ifft(Eyx))/dt; REyxp=real(ifft(Eyxp))/dt; K=200; K=min(ceil(2T/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 Abzug der koinzidenten Produkte, mit lokaler Normierung)		from numpy.fft import * N=len(t) dt=1.0 mxe=mean(x) mye=mean(y) vxe=var(x) vye=var(y) J=int(ceil(2T/float(dt))) x0=zeros(J) x1=zeros(J) y1=zeros(J) x2=zeros(J) y2=zeros(J) for i in range(0,N): j=int(floor((t[i]-Tfloor(t[i]/float(T)))/float(dt))) x0[j]+=1.0; x1[j]+=x[i]-mxe; y1[j]+=y[i]-mye; x2[j]+=(x[i]-mxe)2; y2[j]+=(y[i]-mye)2; X=fft(x1) Y=fft(y1) Xp=fft(x0) Xpp=fft(x2) Ypp=fft(y2) Eyx=T*2(conj(X)Y-sum((x-mxe)(y-mye)))/float(N2-N) Eyxp=T2(conj(Xpp)Xp-sum((x-mxe)2))/float(N2-N) Eyxpp=T*2(conj(Xp)Ypp-sum((y-mye)2))/float(N2-N) REyx=real(ifft(Eyx))/float(dt) REyxp=real(ifft(Eyxp))/float(dt) REyxpp=real(ifft(Eyxpp))/float(dt) K=200 K=minimum(int(ceil(2T/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()	N=length(t); dt=1.0; mxe=mean(x); mye=mean(y); vxe=var(x,1); vye=var(y,1); J=ceil(2T/dt); x0=zeros([1,J]); x1=zeros([1,J]); y1=zeros([1,J]); x2=zeros([1,J]); y2=zeros([1,J]); for i=1:N j=fix((t(i)-Tfix(t(i)/T))/dt)+1; x0(j)=x0(j)+1; x1(j)=x1(j)+x(i)-mxe; y1(j)=y1(j)+y(i)-mye; x2(j)=x2(j)+(x(i)-mxe)^2; y2(j)=y2(j)+(y(i)-mye)^2; end X=fft(x1); Y=fft(y1); Xp=fft(x0); Xpp=fft(x2); Ypp=fft(y2); Eyx=T^2(conj(X).Y-sum((x-mxe).(y-mye)))/(N^2-N); Eyxp=T^2(conj(Xpp).Xp-sum((x-mxe).^2))/(N^2-N); Eyxpp=T^2(conj(Xp).Ypp-sum((y-mye).^2))/(N^2-N); REyx=real(ifft(Eyx))/dt; REyxp=real(ifft(Eyxp))/dt; REyxpp=real(ifft(Eyxpp))/dt; K=200; K=min(ceil(2T/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 * N=len(t) dt=1.0 J=int(ceil(2T/float(dt))) xp=zeros(J) yp=zeros(J) for i in range(0,N): for j in range(int(floor(t[i]/float(dt)))+1,int(floor((t[(i+1)%N]+T((i+1)//N))/float(dt)))+1): xp[j]=x[i] 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(2T/float(dt)))-1,K) tau=roll(arange(-(K//2),(K+1)//2)dt,(K+1)//2) Ryx=zeros(K) c=exp(-dd/float(dt))/(1-exp(-dd/float(dt)))2 for k in range(-(K//2),(K+1)//2): if k==0: Ryx[k]=REyxp[k]/float(T) else: Ryx[k]=((2c+1)REyxp[k]-cREyxp[k-1]-cREyxp[k+1])/float(T-abs(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()	N=length(t); dt=1.0; J=ceil(2T/dt); xp=zeros([1,J]); yp=zeros([1,J]); for i=1:N for j=fix(t(i)/dt)+2:fix((t(mod(i,N)+1)+Tfix(i/N))/dt)+1 xp(j)=x(i); 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(2T/dt)-1,K); tau=circshift((-fix(K/2):fix((K-1)/2))dt,[0;fix((K+1)/2)]); Ryx=zeros([1,K]); c=exp(-dd/dt)/(1-exp(-dd/dt))^2; for k=-fix(K/2):fix((K-1)/2) if k==0 Ryx(mod(k+K,K)+1)=REyxp(mod(k+length(REyxp),length(REyxp))+1)/T; else Ryx(mod(k+K,K)+1)=((2c+1)REyxp(mod(k+length(REyxp),length(REyxp))+1)-cREyxp(mod(k-1+length(REyxp),length(REyxp))+1)-cREyxp(mod(k+1+length(REyxp),length(REyxp))+1))/(T-abs(tau(mod(k+K,K)+1))); end 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')