Study on symmetric extension methods in Mallat algorithm of finite length signal

Boundary extension is necessary when processing the signal which is usually time-limited in applications. On the basis of analyzing Mallat algorithm, two kinds of boundary extension methods, point-symmetric extension (PSE) and edge-symmetric extension (ESE) commonly used in Mallat algorithm of finite length signal are studied. The general process of wavelet transform with two symmetric extension methods is derived in detail. The implementation of wavelet decomposition and perfect reconstruction are studied in-depth respectively, whatever the length of the signal or the length of the filter is odd or even. In addition, in order to achieve multi-level decomposition and reconstruction, binary indicator series is adopted. And this makes the wavelet transform easily implemented by programming. Examples given show that they preserve the perfect reconstruction while keeping the signal length unchanged.