7.13 Finite-length sequences and the dwt matrix

The wavelet transform, viewed from a filterbank perspective, consists of iterated 2-channel analysis stages like the one in .

First consider a very long (i.e., practically infinite-length) sequence $\{c_k(m)\colon m\in \mathbb{Z}\}$ . For every pair of input samples $\{c_k(2n), c_k(2n-1)\}$ that enter the $k^{\mathrm{th}}$ filterbank stage, exactly one pair of output samples $\{c_{k+1}(n), d_{k+1}(n)\}$ are generated. In other words, the number of output equals the number of input during a fixed time interval. Thisproperty is convenient from a real-time processing perspective.

For a short sequence $\{c_k(m)\colon m\in \{0, \dots , M-1\}\}$ , however, linear convolution requires that we make an assumption about the tails of our finite-length sequence.One assumption could be

A more convenient assumption regarding the tails of $\{c_k(m)\colon m\in \{0, \dots , M-1\}\}$ is that the data outside of the time window $\{0, \dots , M-1\}$ is a cyclic extension of data inside the time window. In other words, given alength- $M$ sequence, the points outside the sequence are related to points inside the sequences via

It is instructive to write the circular-convolution analysis fiterbank operation in matrix form. In we give an example for filter length $N=4$ , sequence length $N=8$ , and causal synthesis filters $H(z)$ and $G(z)$ .

gives perfect reconstruction, and since the cascade of matrix operations $T_M^T{T}_M$ also corresponds to perfect reconstruction, we expect that the matrix operation $T_M^T$ describes the action of the synthesis filterbank. This is readily confirmed by writing the upsampled circularconvolutions in matrix form:

The two-stage analysis operation (assuming circular convolution) can be expressed in matrix form as