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) ). 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.
REQUIRED ARGUMENTS:
x
A vector containing a uniformly-sampled real-valued time series.
OPTIONAL ARGUMENTS:
wavelet
A character string denoting the filter type. See wavDaubechies for details.
Default: ``s8".
n.levels
The number of decomposition levels. Default: the maximum level at which there
exists at least one interior wavelet coefficient.
VALUE:
result
An object of class WaveletTransform.
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.
REFERENCES:
(1) D. B. Percival and A. T. Walden, ``Wavelet Methods for Time Series Analysis'',
Cambridge University Press, 2000.
SEE ALSO:
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EXAMPLES:
## calculate the DWT of an electrocardiogram
## sequence out to 4 levels
result <- wavDWT( ecg, wavelet = "s8", n.levels = 4 )
## plot the transform
plot( result )
## plot summary
eda.plot( result )
## summarize the transform
summary( result )