The discrete wavelet transform (DWT)
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,...,N1 where X(t) is a realvalued uniformlysampled 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(j1, t  2t+1l) mod N(j1)) and s(j,t)=sum(g(l) s(j1, t  2t+1l mod N(j1))) for j=1,...,J where h(l) and g(l) are the jth level wavelet and scaling filter, respectively, and Nj=2^(j1). 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 
Arguments
x 
a vector containing a uniformlysampled realvalued time series. 
documentation 
a character string used to describe the input

keep.series 
a logical value. If 
n.levels 
the number of decomposition levels.
Default: 
position 
a 
title.data 
a character string representing the name of the input

units 
a string denoting the units of the time series. Default: 
wavelet 
a character string denoting the filter type.
See 
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.
See Also
reconstruct
, wavDaubechies
, wavMODWT
, wavMODWPT
, wavMRD
, wavDictionary
, wavIndex
, wavTitle
, wavBoundary
, wavShrink
.
Examples
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)
