The discrete wavelet transform (DWT).

USAGE:

wavDWT( x, wavelet = ``s8", n.levels = 3 )

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) ). 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:

, , , , , , , , , , .

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 )