# kzft: Kolmogorov-Zurbenko Fourier Transform In kza: Kolmogorov-Zurbenko Adaptive Filters

## Description

Kolmogorov-Zurbenko Fourier Transform is an iterated Fourier transform.

## Usage

 ```1 2 3``` ```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.

## 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.

## 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

`kzp`, `kztp`,

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53``` ```# 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)) ```

### Example output   ```   user  system elapsed
0.012   0.000   0.012
user  system elapsed
0.051   0.000   0.050
user  system elapsed
0.051   0.000   0.052
```

kza documentation built on March 26, 2020, 6:27 p.m.