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DWTexact <- function(x,filter){
## Computes the scaling function and the wavelet function (for compactly supported wavelet) using the
## cascade algorithm on the grid of dyadic integer $2^{-\text{J}}$
##
## INPUT x Vector of signal values to
## filter Filter of the wavelet transform
##
## OUTPUT dwt Vector of wavelet coeffcients
## indmaxband Vector containing the maximal index of wavelet coefficients at each scale
## Jmax Maximal scale
##
## based on the paper of Fay, Moulines, Roueff, Taqqu, 2009
## Achard & Gannaz (2014)
##________________________________________________________________________________________________
n <- length(x)
N <- length(filter)
h<-filter
g<-rev(filter*(-1)^(0:(N-1)))
tmp <- compute_nj(n,N)
nscale <- tmp$nj
Jmax <- tmp$J
indmaxband <- cumsum(nscale)
# Pyramidal algorithm
dwt <- rep(0,indmaxband[(length(indmaxband))])
aold <- x
indmin <- 1
for(iscale in 1:Jmax){
acur <- rep(0,nscale[iscale])
d <- rep(0,nscale[iscale])
offset <- 1
for(p in 1:nscale[iscale]){
acur[p] <- sum(h*aold[offset:(offset+N-1)])
dwt[indmin+p-1] <- sum(g*aold[offset:(offset+N-1)])
offset <- offset+2
}
aold <- acur
indmin <- indmaxband[iscale]+1
}
list(dwt=dwt,indmaxband=indmaxband,Jmax=Jmax)
}
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