Design And Implementation Of Wavelet Packet

Design and Implementation of Wavelet Packet-Based Filter Bank Trees for Multiple Access Communications Mike Sablatash Communications Research Centre Todor Cooklev The University of Toronto John Lodge Communications Research Centre ABSTRACT: 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

0-7803-3925-8197 $ 1 0.00 0 1 997 IEEE

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,

176

~

2L'2yo(2Lt - k ) ,

at highest resolution, 2 L , which constitute the basis functions for this subspace, derived in [16,19], is AL Qo = ~ l o s , ( ~ , { 2 ~ ' ~ ~ ~ (E2Z~} t=V,. - k ) (1) lk 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)

to scaling functions which are at resolution 2L-', i.e., at half the resolution of the scaling functions used at the root of the tree. The time-shifted versions of these scaling functions are the basis functions of the subspace denoted by AL-'Ro = VL-' . The coordinates f ( o , L ) are also digitally filtered by a highpass filter with transfer function G(z) and then downsampled by 2 , as again denoted by the arr0.w pointing down, as illustrated in the upper branch of Fig. 1, leaving the root of the tree. The operator 4 is analytically described by

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.

4 {s(k)}(n) =

c

W g * ( k - 2n)

(3)

Y

k

where the unit pulse response of the highpass filter is g * ( k ). The resulting coordinates obtained as a result of applying this operator to the coordinates denoted by

f('3L-'),

f(OSL)

are

and they are coordinates with respect

to wavelets which are at resolution 2L-'. The time-shifted versions of these wavelets are the basis functions of the subspace denoted by AL-'Ql = WL-l, which is the orthogonal complement in Hilbert space of the subspace AL-'Qo = VL-', in the subspace ALQo=V,. The coordinates f ( 0 3 L - 1 ) are then filtered by the filters with transfer functions H ( z ) and G(z), and downsampled by a factor of 2 , corresponding to applying the operators in (2) and (3), respectively, to f ( o * L - ' ) , resulting in coordinates

f(o,L-2)

with respect to time-shifted scaling

functions at resolution 2L-2 as basis functions for the subspace AL-2Qo = VL-2,and coordinates f(',L-2)with respect to time-shifted wavelets at resolution 2L-2 as basis functions for the subspace AL-2Q, = WL-2,which is the orthogonal complement of the subspace AL-2Qo= VL-2 in the subspace AL-'Q0 = VL-', in

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

Hilbert space. The filtering of the coordinates f(l*L-')bY the lowpass and highpass filters, and then downsampling their outputs by 2 results in the coordinates f(2,L-2)and

f ( 3 * L - 2with ) respect to wavelet packets which are basis functions at resolution 2L-2 for subspaces AL-2S22 and

The operator Fo is analytically described by

Fo{s(k)}(n) =

c

S(k)h*(k - 2n)

AL-2Q2,,respectively.

k

orthogonal complements in the Hilbert subspace AL-'Ql = WL-,. The pattern of the symbolism and

where the unit pulse response of the lowpass filter is h*( k ) . The resulting coordinates obtained as a result of applying this operator to the coordinates denoted by

f(07L-'),

f('*')

The latter two subspaces are

operations in Fig. 1 and their meanings is now clear by

are

iteration of that which has now been established. For the synthesis filter banks the operators (2) and (3) are

and they are coordinates with respect

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replaced by operators which represent upsampling by 2 followed by lowpass and highpass filters whose impulse responses are the complex conjugates of those in (2) and (3), respectively. Using the Gray code ordering for the signals [13, 161, dropping the conjugates for ease of notation, and using multi-rate analysis by the z transform, results in Fig.2 for a synthesis filter bank with 8 input ports in which all the prototype quadrature mirror filter banks are the same, followed by a wideband vestigial sideband filter which passes all positive frequencies, and attenuates all negative frequencies, except for vestiges around 0 and 76. Another possibility is to make all the filter banks at each level the same. The signals along paths taken from each leaf of the tree to the root are orthogonal, thus enabling the non-interfering multiplexing of the signals into a single channel, 'and the frequency responses of the transfer functions along each of the paths, and then through the wideband VSB filter, are translated VSB responses which, as shown in [ 181, are equivalent to OQPSK. An analysis filter bank at the other end of the channel demultiplexes and perfectly reconstructs this signal. The possibility of obtaining bandwidth on demand by factors of two is also shown in Fig. 2. Two Designs for the Filter Banks Two design methods for the wavelet packet-based filter banks have been described in [17, 191. In the first design method the prototype filter transition bandwidth (defined as the frequency difference between the frequency at which the 3-dB down point occurs and that at which the attenuation first reaches the minimum stopband attenuation) for a bank with M = 2K users is given by

6=

76

.

In the second design method, because of

( M + 2) the decrease, by a divisive factor of two from level to level, from the transition bandwidth 6, of the prototype filters at the leaves at level K to the transition bandwidth 6, of the QMF pair at level l next to the root of the tree, the relationship

6, 6 , =2K 1 -

(4)

must hold. The two stopband edges must meet at the minimum attenuation, so equating the expressions for their locations and using (7) yields

n / 2 + 6 1 = 7G/2+7G/2K - 6 K / 2 K - ' n/2+7r/2K -a,,

=

This is the key design equation which determines the transition bandwidth, 6 , , of the QMF pair at level 1 , next to the root of the tree, for a tree withK levels and M = 2K users (leaves). The transition bandwidth at level y2 is given by

(7) Thus, for M = 16 users (leaves), K = 4 , and (6) gives a transition bandwidth of 6, = 76/32 for the filters

I

User 1

I

Fig. 2: Wavelet Packet-based transmit multiplexer (synthesis filter bank) consisting of linear time-invariant filters preceded by interpolation by a factor of 2, followed by a wideband VSB filter centred on the positive frequency axis, set up for analysis in the z-domain with input leaves ordered according to the Gray code so that the centres of the bands of the magnitudes of the frequency responses for the paths from the input leaves to the output (input to the channel) increase monotonically from bottom to top. This multiplexer allows 2 of the 4 users, users 2 and 3, to each transmit at the lowest rate of r kb/s, user 1 to transmit at the rate of 4r kb/s, and user 4 to transmit at the intermediate rate of 2r kb/s. A user could also transmit at the highest of 8r kb/s directly into the input of the wideband VSB filter.

(5)

constituting the QMF pair next to the root of the tree. Using (4)or (7) the transition bandwidth for the prototype filter to be designed at level 4 is 6, = 76/4. Using this in (7) yields the transition bandwidths 6, = n/8 at level

from which 1')

/L/ L

6 , =-.

2K

178

3 and 6, = n/16 at level 2 for the prototype filters to be designed for these levels. Spectral Factorization of High-Order Polynomials In the design of the filters used in the quadrature mirror filter pairs for the wavelet packet-based filter banks spectral factorization must be performed to obtain H ( z ) from a designed H ( z ) H ( l / z ). For the number of leaves in the tree equal to 16 the order of the polynomial which must be factored is 162, and for more leaves (which represent the maximum number of users) the order will approximately double with a doubling in the number of users. Cooklev [20] has investigated direct factorization, the cepstrum method and the Bauer method for the accurate spectral factorization of high-order polynomials with zeros on the unit circle. These zeros occur because of the requirement for some regularity to obtain smoothness of the scaling functions, wavelets and wavelet packets which are basis funtions for the coordinates discussed earlier. He has found the cepstrum method to be by far the best, although the factors obtained are minimum and maximum phase. The steps in the cepstrum method are as follows. The first step is to define the causal sequence

go(n)= p-"ho(n - N )

(8)

9

n=-N

where 1 < p < mink l/lbk with b, the zeros of the filter inside the unit circle. Since the b, are unknown a priori the value of p must be set by using experience and good judgement. This is the tricky part. For optimum results the value of p must be adjusted according to the length of the polynomial. For polynomials of order up to 150 p = 1.02 is appropriate, and is recommended by other authors, as noted in [20]. For polynomials of higher order a smaller value, still greater than one, is necessary. Examples are given in [20]. The second step is to compute the complex cepstrum of go(n). The complex cepstrum can be computed using the Fourier transform: -

+ j$Go( ~ r ) )

minimum phase factor. The fourth step in this process is to find the minimum phase factor, H , ( z ) , from its complex cepstrum (cepstral inversion). Both We can write

R,(z) and H,(z)

converge for

IzI > 1.

we obtain h,(O) = eh,('). We note that H,(z) = eHm(')

where now we denote H ( z ) H ( l / z )= Z h , ( n ) z - " , and

Go(e'") = lnlGo(e'")I

two to obtain the complex cepstrum h,(n) of the

By equating the constant coefficients on both sides of -(10) -

N

1,

wrapped. The MATLAB routine cceps was not found to be useful because it assumes that the cepstrum has the same length as the polynomial being factored, which leads to aliasing. A MATLAB program is provided in [20] which computes the complex cepstrum for any number of points. Other MATLAB programs, further details and examples can be found in [20]. The third step is as follows. The complex cepstrum is noncausal. The values for negative n must be identified. If an FFT with length L is used then g(L) = g(-l),g(L- 1) = g(-2),... . After identifying the anticausal part it is made causal and added to the causal part, g(L) is added to g(l),g(L - 1) is added to g ( 2 ) , and so on. The resulting sequence is divided by

(9)

which, in turn, is always computed using the FFT, because of the efficiency of computing the DFT by using the FFT. To avoid aliasing the length of the FFT should be higher than the length of g,(n). In the computation of the complex cepstrum the unwrapped phase is necessary because the complex cepstrum of a product will not be the sum of the individual complex cepstra if the phase is

179

HA(z) = H,(z)l?;(z). Therefore, nh,(n) = h,(n) * nh,(n). The other samples of h, (n)

implies

that

can be computed by recursion from

kh,(n) = z-h,(k)h,(n

-k),

IZ

> 0.

(1 1)

k=l

In MATLAB direct factorization cannot be used with any degree of confidence for factorization of polynomials of order greater than about 100. The cepstrum method has been used to successfully factor a polynomial with 699 coefficients, and can probably be used to factor much higher order polynomials. The spectral method of Bauer is slower and less accurate. Therefore, the cepstrum method is recommended. Low-pass minimum phase filters for a synthesis filter bank (from which the analysis filter bank is easily computed) with 32 users (leaves), in which the widths of the transition bands of the prototype filters at each level were increased by 2 from level to level, from the root to the leaves of the tree, and the attenuation in the stopband was equal-mininum, was designed using the cepstrum method in [20]. Quantization Effects in the Implementation of Filter

Banks The round-off error due to finite word length has been

~

modelled in the usual way as a zero-mean uniformly distributed random process over [- Q/2, Q/2] , where Q represents the quantization step size. Each error is considered to be statistically independent of the others. Their effect is to reduce the precision. The cumulative effect of this error can be controlled by adjusting the filter's word length. The effects of quantization were studied through simulation. The lattice structure was used in the implementation of the filter banks for the quantization study because such a structure has two properties important for the filter bank design: (1) without sacrificing computational efficiency it preserves the perfect-reconstruction property even under the constraints of finite-word-length arithmetic, and (2) it is general, since every paraunitary filter bank can be implemented using the lattice structure. The frequency responses of a full-tree filter bank with four leaves, and no VSB filter following the root of a tree, with all QMF filter pairs used in the tree the same, with filter implementation such that the impulse response coefficients are the multiplier coefficients (the direct form implementation), and with about 47 dB minimum stopband attenuation in the unquantized case, were simulated using 8, 10, 12 an infinite number of bits. The minimum stopband attenuation decreased to about 37 dB using 8-bit coefficient quantization, to about 44 dB using 10-bit quantization, and was approximately at 47 dB using 12-bit quantization. With a lattice implementation and 8-bit quantization of the lattice coefficients, the minimum stopband attenuation improved to 40 dB, and with 10-bit quantization it improved to 45 dB over a small frequency support, to 46 dB over a still small support, and to 47 dB or more over most of the support. For 8 users and no VSB filter, all QMF filter pairs the same throughout the filter bank, direct form implementation, and 40 dB minimum stopband attenuation in the infinite wordlength case, the minimum stopband attenuation drops to 36 dB for 8 bits of quantization, to 39 dB for 10 bits of quantization, and to essentially 40 dB for 12 bits of quantization. With 16 users, different filter designs at each level of the tree, a VSB filter as described earlier after the root of the tree, lattice structure implementation, and unquantized stopband minimum attenuation of 40 dB, the minimum stopband attenuation dropped to 21 dB with 6-bit quantization, 29 dB with 8-bit quantization, 38.5 dB with 10-bit quantization, and 40 dB with 12-bit quantization, as shown for the last of these in Fig. 3. Concluding Remarks It was shown that the cepstrum method for spectral factorization can be used to overcome the problem of spectral factorization of high order polynomials with

-40 -50

-60

-70 -80 -90 ,

"V

0

200

400

600

800

1000

1200

Fig. 3: Frequency Response of a Filter Bank Tree for 16 Users, where the Filters Vary from Level to Level, with 12-bit Implementation Using the Lattice Structure, and VSB Filter Following the Root of the Tree. hundreds, and probably thousands, of coefficients, encountered in the design of the filters for constructing filter banks for wavelet packet-based multiple access communications. A half-band filter polynomial with 699 coefficients, for example was factored with no difficulty. Lattice realizations were used to study the effects of quantization errors on the frequency responses of the filter banks due to finite word lengths for the coefficients by using computer simulations. A number of filter banks were simulated and it was found that with lattice implementions, a word length of 10 or 12 bits was adequate to obtain very nearly the ideal frequency responses for filter banks with ideal (infinite word length) stopband attenuation of 47 dB minimum stopband attenuation, and 4 users, and with 40 dB minimum stopband attenuation and 8 users, and no VSB filter following the root of the tree, and for 16 users, 40 dB minimum stopband attenuation, and a VSB filter following the root of the tree. The effects of quantization depend on the constraints, such as stopband attenuation. Recent research at CRC has resulted in the realization of multiple access communications systems in the downstream direction with bandwidths on demand which are multiples of 2 using a combination of wavelet packetbased trees followed by a DFT polyphase synthesis filter bank at the transmitter end, and a corresponding matched receiver resulting in perfect reconstruction. Further research into efficient filter bank structures with minimum delays is important. Smart joint detection to mitigate adjacent and co-channel interferences, and to enable the use of more users than the number of leaves of the tree,

180

and performance studies of multiple access schemes with bandwidth on demand, and incorporating smart joint detection schemes, under realistic channel conditions, and with synchronization phase and timing errors, are topics to be investigated on the road to the realization of a practical multiple access communication system using these wavelet packet-based filter bank techniques. Due to the asynchronous transmission these techniques seem to be particularly appropriate for upstream transmission.

References 1. R.E. Learned, H. Krim, B. Claus, A.S. Willsky and W.C. Karl, “Wavelet-packet-based multiple access communication”, Proc. of SPIE International Symuosium on Optics. I m a L Imaging: Wavelet AD -plications in Signal and Image Processing, vol. 2303, San Diego, California, July 27-29, 1994, pp.246-259. 2. R.E. Learned and A.S. Willsky, “Low complexity optimal multiple access joint detection for linearly dependent user sets”, Procs. 1996 International g& JICASSP ‘961, vol. 2, Atlanta, Georgia, May 6-11, 1996, pp. 1089-1092. 3. R.E. Learned, H. Krim and A.S. Willsky, “Examination of wavelet packet signal sets for over-saturated multiple access communications”, Procs.of the IEEE-SP Scale Analvsis, Paris, France, June 18-21, 1996, pp. 401404. 4. W.W. Jones, “A unified approach to orthogonally multiplexed communications using wavelet bases and digital filter banks”, Ph.D. thesis, Faculty of the Russ College of Engineering and Technology , Ohio University, Aug. 1994. 5. A.R. Lindsey, “Supersymbol tuning algorithms for wavelet packet modulation”, Proc.of the Twentv-Ninth Annual Conference on Information Sciences and Svstems, Department of Electrical and Computer Eng. The Johns Hopkins University, Baltimore , Maryland, U.S.A., March 22-24, 1995, pp. 737-743. 6. A.R. Lindsey, “Wavelet packet modulation: a generalized method for orthogonally multiplexed communications”, Proc. of the 27th IEEE Southeastern Svmuosium on Svstem Theorv, Starkville, Mississippi, March 12-14, 1995. 7. A.R. Lindsey, “Multidimesional signaling via wavelet packets”, Proc. of the 1995 SPIE Conference-Wavelet Auplications for Dual Use, vol. 2491-29, Orlando, Florida, April 17-21, 1995. 8. A.R. Lindsey, “Generalized Orthogonally Multiplexed Communication via Wavelet Packet Basis”, Ph.D. and Technology, Ohio University, June 9, 1995.

9. J. Wu, K.M. Wong and Q. Jin, “Multiplexing based on wavelet packets”, Proc.SPIE International Svmuosium on AEROSENSE, Orlando, FL, April 1995. 10. J. Wu , K.M. Wong and Q. Jin, “Optimum wavelet packets for multiple signal transmission in communication systems”, submitted to ~, Oct. 1995. 11. M. Sablatash, “Multiple access communication based on wavelet packet signal decompositions and smart detection”, Abstract of Papers of the 1995 Canadian ~) WIT ., compiled by P. Fortier and J.-Y. Chouinard, Manoir du Lac Delage, Quebec, May 28-31, 1995, p.14. 12. C.K. Chui, Introduction to Wavelets. New York: Academic Press, 1992. 13. M.V. Wickerhauser, Adapted Wavelet Analysis from Theory to Software. Wellesley, Massachusetts: A.K. Peters, 1994. 14. R. R. Coifman and M. V. Wickerhauser, “Entropybased algorithms for best basis selection”, IEEE Trans. Inform. Theory, vol. IT-38, pp.713-718, March 1992. 15. M.V. Wickerhauser, “Lectures on wavelet packet algorithms”, Notes, Dept. of Mathematics, Yale University, New Haven, Conn., Nov.18, 1991. 75 pp. 16. C.J. Zarowski, “Notes on Orthogonal Wavelets and Weavelet Packets”, Report No. 1-95, Department of Electrical and Computer Engineering, Kingston, Ontario, Canada, Nov. 3, 1995. 17. M. Sablatash, J.H. Lodge and W.F. McGee, “The design of filter banks with specified minimum stopband attenuation for wavelet packet-based multiple access communications”, Procs. 18th Biennial Svmposium on Communications, Queen’s University, Kingston, June 2-5, 1996. 18. M. Sablatash, J.H. Lodge and W.F. McGee, “Equivalence between vestigial sideband (VSB) and offset quadrature phase shift (OQPSK) modulations and relationships to wavelet packet-based multiplexing”, Procs. 18th Biennial Svmposium on CommUnications, Queen’s University, Kingston, June 2-5, 1996. 19. M. Sablatash, J.H. Lodge and C.J. Zarowski, “Theory and Design of Communication Systems Based on Scaling Functions, Wavelets, Wavelet Packets and Filter Banks”, J Wireless Communications , vol. 2, Coast Plaza Hotel, Calgary, Alberta, Canada, 8-10 July 1996, pp. 640-659. 20. T. Cooklev, “On the Design and Implementation of Filter Bank Trees for Multiple Access Communications”, Report on Contract 67CRC-5-3315 for Communications Research Centre, Ottawa, Canada, 1996.

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