# wavDWT: The discrete wavelet transform (DWT) In wmtsa: Wavelet Methods for Time Series Analysis

## Description

The discrete wavelet transform using convolution style filtering and periodic extension.

Let j, t be the decomposition level, and time index, respectively, and s(0,t)=X(t) for t=0,...,N-1 where X(t) is a real-valued uniformly-sampled time series. The jth level DWT wavelet coefficients (d(j,t)) and scaling coefficients (s(j,t)) are defined as d(j,t)=sum(h(l) s(j-1, t - 2t+1-l) mod N(j-1)) and s(j,t)=sum(g(l) s(j-1, t - 2t+1-l mod N(j-1))) for j=1,...,J where h(l) and g(l) are the jth level wavelet and scaling filter, respectively, and Nj=2^(j-1). The DWT is a collection of all wavelet coefficients and the scaling coefficients at the last level: d(1),d(2),...,d(J),s(J) where d(j) and s(j) denote a collection of wavelet and scaling coefficients, respectively, at level j.

## Usage

 ```1 2 3``` ```wavDWT(x, n.levels=ilogb(length(x), base=2), wavelet="s8", position=list(from=1,by=1,units=character()), units=character(), title.data=character(), documentation=character(), keep.series=FALSE) ```

## Arguments

 `x` a vector containing a uniformly-sampled real-valued time series. `documentation` a character string used to describe the input `data`. Default: `character()`. `keep.series` a logical value. If `TRUE`, the original series is preserved in the output object. Default: `FALSE`. `n.levels` the number of decomposition levels. Default: `as.integer(floor(logb(length(x),base=2)))`. `position` a `list` containing the arguments `from, by` and `to` which describe the position(s) of the input `data`. All position arguments need not be specified as missing members will be filled in by their default values. Default: `list(from=1, by=1, units=character())`. `title.data` a character string representing the name of the input `data`. Default: `character()`. `units` a string denoting the units of the time series. Default: `character()` (no units). `wavelet` a character string denoting the filter type. See `wavDaubechies` for details. Default: `"s8"`.

## Details

This DWT imposes an ad hoc storage sytem for odd length scaling coefficient crystals: if the length of a scaling coefficient crystal is odd, the last coefficient is "stored" in the extra crystal. During reconstruction, any extra scaling coefficients are returned to their proper location. Such as system imposes no spurious energy in the transform coefficients at the cost of a little bookkeeping.

## Value

an object of class `wavTransform`.

## References

D. B. Percival and A. T. Walden, Wavelet Methods for Time Series Analysis, Cambridge University Press, 2000.

`reconstruct`, `wavDaubechies`, `wavMODWT`, `wavMODWPT`, `wavMRD`, `wavDictionary`, `wavIndex`, `wavTitle`, `wavBoundary`, `wavShrink`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```## calculate the DWT of linear chirp linchirp <- make.signal("linchirp", n=1024) result <- wavDWT(linchirp, wavelet="s8", n.levels=5, keep.series=TRUE) ## plot the transform shifted for approximate zero ## phase alignment plot(wavShift(result)) ## plot summary eda.plot(result) ## summarize the transform summary(result) ```