Computation

Here are some numerical computing strategies:

• The algorithm only does conversion between wavelet coefficients and sqeuence in frequency domain. (I)FFT are used to transform between time and frequency domain.

• Outside the transition band, value of Meyer kernel is either \(1\) or \(0\). So, calculate the transition band only.

Transform

\[w_{mn}=A\frac{(-1)^{mn}}2\sum_{k=0}^{N-1}\left(\tilde{x}[k-mN_t]\pm\tilde{x}[k+mN_t]\right) \tilde{\phi}[k]\mathrm{e}^{\mathrm{i}\pi nk/N_t}.\]

Summation regarding \(k\) contains \(2N_t\) items actually and can be calculated with IFFT. For real sequence \(x[j]\), the above formula is reduced to

\[\begin{split}\begin{gather} w_{mn} = A(-1)^{mn} \mathcal{O}_{mn}\sum_{k}\tilde{x}[k-mN_t]\tilde{\phi}[k]\mathrm{e}^{\mathrm{i}\pi nk/N_t}. \\ O_{mn} = \mathcal{R},\qquad\text{if}\; m+n\;\text{even}\quad \text{else}\; \mathcal{I}. \end{gather}\end{split}\]

Inverse transform

\[\tilde{x}[k] = \frac12 \sum_{m}\left(\tilde{\phi}[k-mN_t]\pm\tilde{\phi}[k+mN_t]\right) \sum_{n}A_{mn}w_{mn}\mathrm{e}^{-\mathrm{i}\pi nk/N_t}.\]

Summation regarding \(n\) can be calculated with FFT. For real sequence \(x[j]\), items with negative \(k\) are redudant. Therefore, \(\tilde{\phi}[k+mN_t]\) is ignored.

Attention

Special treatment for \(m=0\) and \(N_f/2\).