Design and Implementation of Wavelet Packet-Based Filter Bank Trees for Multiple Access Communications

The relationship of filter bank trees to wavelet packet trees for subspaces, decimation operators, and coordinates of the projections of a function onto the subspaces is briefly explained. Two designs for the filter banks are summarized and the equations for the designs are provided. For one of these the prototype filters are all the same, and for the other they are the same at each level of the tree. The best method for spectral factorization of high-order polynomials, needed for the filter designs, is selected. Lattice structures preserving perfect reconstruction with finite word lengths are used in computer simulations to study quantization effects. Introduction and Brief Survey of Previous Work A more detailed survey is given in [19]. The groundbreaking work by Rachel Learned at MIT [l-31 appears to be the first systematic and comprehensive approach to the subject. Learned has offered a theory based on the selection of wavelet packets, together with the use of joint detection for multiple access communications. Substantial original research on orthogonally multiplexed communications via wavelet packets was pursued by Lindsey[S-81, who generalized the work of Jones [4]. Some relatively recent work [9, lo] has provided further analysis, insights and performance results. The first and third authors have also designed othonormal filter banks corresponding to a complete wavelet packet transform such that the attenuation along all paths from the leaves to the root is above a specified minimum value [11, 17, 181. The filter banks described in [17] are of two types. The first replicates the same quadrature mirror filter (QMF) pair throughout the analysis filter bank, and the usual related QMF pair throughout the synthesis filter bank, so only one set of filter coefficients need be designed, i.e., only one prototype filter need be designed, and that is traditionally a lowpass prototype. This design of uniform filter banks is the one that has been assumed in all the works on using wavelet packet-based filter banks for communications known to the authors. The second, more interesting, design uses different QMF pair designs at each level of the tree. In [ 181 the equivalence of digital VSB and OQPSK is shown, and the frequency responses along the paths through a wavelet packet-based synthesis filter bank followed by a wideband VSB filter filtering out either all negative or all positive frequencies, are demonstrated to be shifted versions of VSB (and, because of the equivalence, OQPSK) responses, and various details discussed. In [ 191 the theory of wavelet packets provided by Zarowski [ 161 is used as a “core” to provide the same tree structure for the wavelet packets, operators (where an operator is a filter followed by a downsampler in the analysis filter bank, or an upsampler followed by a filter in the synthesis filter bank), and coordinates using the wavelet packets as basis functions. Then it is easily seen, and shown in that paper, how the previously described works are related to this theory and follow from it, with scale-based coding, wavelet-based coding, M-band wavelet modulation, orthogonal frequency division multiplexing (OFDM), wavelet packet multiplexing and multiple access communications, and the filter bank design approach to wavelet packet-based multiple access communications taken by the first and third authors, being special cases. The results recently published in conference papers [ 17191 are briefly reviewed in this paper. Then new results are presented which address issues in the design of wavelet packet-based filter banks for multiplexing, demultiplexing and multiple access communications. These are the spectral factorization of high-order for the design of the quadrature mirror filters used in the filter bank construction, lattice structures and coefficients for implementation of the filter banks, and quantization effects due to the use of finite word lengths for the coefficients. Some future directions for research are given in the concluding remarks. Brief Review of Relationships of Filter Banks to Wavelet Packet Trees More detailed treatments with developments of the equations are given in [16,19]. Here the explanations of the basic ideas depend more on an understanding of the symbols given on the analysis tree in Fig. 1, and explanations of their meanings with a minimum number of equations. The ALn, = V, is the subspace at the root of the tree with L levels (with the root being considered to be at the L + 1st level), which has scaling functions at the highest resolution (this highest resolution often being arbitrarily assigned the value 1) as basis functions. The relationship to the time-shifted set of scaling functions, 0-7803-3925-8197 $1 0.00 0 1 997 IEEE 176 2L'2 yo (2L tk ) , at highest resolution, 2 L , which constitute the basis functions for this subspace, derived in [16,19], is A L Qo = ~ l o s , ( ~ , { 2 ~ ' ~ ~ ~ ( 2 ~ t k ) l k E Z } =V,. (1) Thus, the exponent L is also an index for the resolution. The V denotes the use of scaling functions as basis functions. The projection of f ( t ) onto ALQo=V, has coordinates { f ( o * L ) ( p ) } ~~~ , abbreviated to f ( o ,L ) in Fig.1. These coordinates are digitally filtered by a lowpass filter with transfer function H ( z ) and then downsampled by 2 , as denoted by the arrow pointing down, as illustrated in the lower branch of Fig. 1, leaving the root of the tree. Fig. 1: Tree Sructure for Complete Wavelet Packet Decomposition of the Subspace at the Root of the Tree into Subspaces, Decimation Operators, and the Coordinates of the Projection of a Function f in Hilbert Space onto the Subspaces The operator Fo is analytically described by