The discrete wavelet transform using convolution style filtering
and periodic extension.
Let \(j, t\) be the decomposition level,
and time index, respectively, and
\(s_{0,t}=X_{t=0}^{N-1}\) where
\(X_t\) is a real-valued uniformly-sampled time series. The
\(j^{th}\) level DWT wavelet
coefficients (\(d_{j,t}\))
and scaling coefficients (\(s_{j,t}\))
are defined as \(d_{j,t} \equiv \sum_{l=0}^{L-1} h_l s_{j-1,2t+1-l \bmod N_{j-1}},
\quad t=0,\ldots, N_j -1\) and
\(s_{j,t} \equiv \sum_{l=0}^{L-1} g_l s_{j-1,2t+1-l \bmod N_{j-1}},
\quad t=0,\ldots, N_j -1\)
for \(j=1,\ldots,J\) where \(\{ h_l \}\) and \(\{ g_l \}\) are the \(j^{th}\) level wavelet and scaling filter, respectively, and
\(N_j \equiv N / 2^j\). The DWT is a collection of all wavelet coefficients and the
scaling coefficients at the last level:
\(\mathbf{d_1,d_2},\ldots,\mathbf{d_J,s_J}\) where
\(\mathbf{d_j}\) and
\(\mathbf{s_j}\) denote a collection of wavelet
and scaling coefficients, respectively, at level \(j\).