# Kolmogorov-Zurbenko Fourier Transform Kolmogorov-Zurbenko Fourier Transform is an iterated Fourier transform. ## Details Kolmogorov-Zurbenko Fourier Transform (KZFT) is the Fourier transform applied over every segment of length m iterated k times. The argument alg="F" will use Fast Fourier Transforms written in C (fftw library). The alg="C" is a slow Fourier Transform but has the advantage of being able to handle missing values. It currently works in one dimension. The alg="R" is an R version of KZFT for experimental purposes.The coeff function generates the coefficients for the KZFT function. You will introduce a phase shift and decrease the fidelity of the signal if the product of f*m is not an integer. ```r kzft(x, f=0, m=1, k=1) coeff(m, k) transfer_function(m, k, lamda = seq(-0.5,0.5,by=0.01), omega = 0 ) ``` ## Arguments - `x`: The raw data - `f`: The frequency that KZFT is applied at. - `m`: The window size for transform - `k`: The number of iterations for applying the KZFT - `lamda`: The frequencies used for the calculating the transfer function. - `omega`: The frequency that KZFT is applied at. ## References I. G. Zurbenko, The spectral Analysis of Time Series. North-Holland, 1986. I. G. Zurbenko, P. S. Porter, Construction of high-resolution wavelets, Signal Processing 65: 315-327, 1998. R. Neagu, I. G. Zurbenko, Tracking and separating non-stationary multi-component chirp signals, J. Franklin Inst., 499-520, 2002. R. H. Shumway, D. S. Stoffer, Time Series Analysis and Its Applications: With R Examples, Springer, 2006. Wei Yang and Igor Zurbenko, kzft: Kolmogorov-Zurbenko Fourier Transform and Applications, R-Project 2007. Igor G. Zurbenko, Amy L. Potrzeba, Tidal Waves in Atmosphere and Their Effects, Acta Geophysica Volume 58, Number 2, 356-373 ## See Also `kzp`, `kztp`, ## Examples ```r # example taken from Wei Yang's KZFT package # coefficients of kzft(201,5) # function to calculate polnomial coefficients for kzft ## Not run: a<-coeff(201,5); t<-seq(1:1001)-501; z<-cos(2*pi*0.025*t); plot(z*a,type="l",xlab="Time", ylab="Coefficient", main="Coefficients of the kzft"); lines(a); lines(-1*a); ## End(Not run) # example taken from Wei Yang's KZFT package # transfer function of the kzft(201,5) at frequency 0.025 lamda<-seq(-0.1,0.1,by=0.001) tf1<-transfer_function(201,1,lamda,0.025) tf2<-transfer_function(201,5,lamda,0.025) matplot(lamda,cbind(log(tf1),log(tf2)),type="l",ylim=c(-15,0), ylab="Natural log transformation of the coefficients", xlab="Frequency (cycles/time unit)", main="Transfer function of kzft(201,5) at frequency 0.025") # example with missing values set.seed(2) period=101 f<-1/period t<-1:2000 s<-1*sin(2*pi*f*t) x<-s noise<-3*rnorm(length(t)) x<-s+noise m=101 rand_idx <- sample(t,100,replace=FALSE) x[rand_idx]<-NA x<-as.vector(na.omit(x)) system.time(z1<-kzft(x, m=m, k=1, f=f)) system.time(z2<-kzft(x, m=m, k=2, f=f)) system.time(z3<-kzft(x, m=m, k=3, f=f)) par(mfrow=c(2,2)) plot(x,type="l",main="Original time series",xlab="t", ylab="y") lines(s,col="blue") plot(2*Re(z1),type="l",main="kzft(101,1)",xlab="t", ylab="y", ylim=c(-6,6)) lines(s,col="blue") plot(2*Re(z2),type="l",main="kzft(101,2)",xlab="t", ylab="y", ylim=c(-6,6)) lines(s,col="blue") plot(2*Re(z3),type="l",main="kzft(101,3)",xlab="t", ylab="y", ylim=c(-6,6)) lines(s,col="blue") par(mfrow=c(1,1)) ```

kzft function