Wavelet synthesis

The wavelet synthesis provides the inverse operation to the wavelet transfrom. That is, it reconstructs the original time-series from the wavelet coefficients m(b,a) as

\[x(t) = \int \frac{da}{a} \int db\,m(b,a)\,\frac{1}{a}h\left(\frac{b-t}{a}\right)\]

where h(⋅) is the synthesis wavelet associated with the analysis wavelet g(⋅) used to obtain the wavelet coefficients m(b,a).

The synthesis() method implements this inverse transform or synthesis. Please note that the wavelet synthesis above will generate the so-called Hilbert transformed of the original signal. Therefore, the synthesis() method accepts an optional keyword argument hilbert to choose whether the complex Hilbert transform or the real signal gets returned (the latter is the default).

ContinuousWavelet.synthesis — Method

synthesis(wt::WaveletTransformed)

Implements the continuous wavelet synthesis on some WaveletTransformed object. That is, it reconstructs the signal, wt was created from. The optional keyword argument hilbert specifies whether the complex Hilbert transformed of the original signal or the real signal is returned. By default the real reconstruction is returned.

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