genera wl packet mod

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Wavelet Packet Modulation:A Generalized Method for Orthogonally Multiplexed CommunicationsAlan R. Lindsey', Jeffrey C. Dill

Department of Electrical & Computer EngineeringOhio U niversity, Athens, Ohio 45701Abstract

Building on recently introduced multidimensional signalingtechniques, a multirate wavelet based modulation ormat ispresented which can utilize existing channels designed fo rconventionai QAM. Customizable wavelet packet basisfunctions are employed as novel pulse shapes upon whichindependent QAM data at lower rates are placed. Theadvantages include dimensionality in both time andfrequency for lexible channel exploitation and an efficientall-digital filter bank implementation.1. Introduction

At the beginning of the decade, a series of articlesintroducing the notion of wavelet packets were published[ l ] , [2], [3], 141. These functions provide a basis forL Z ( K ) ,and have a structure that is extremely useful indecompositions and transforms making them well suitedfor the signal design problem in communications. In fact,these functions generalize standard wavelet bases and soinherit the properties that make their predecessors soattractive. This work attempts to establish a link betweenwavelet packet functions and digitally modulatedcommunication signals, by building on the special cases ofMulti-Scale and M-Band Wavelet Modulations worked outby Jones in [6].2. Construction of Wavelet Packet Bases

Following [2],we start with a pair of quadrature mirrorfilters of length L, h, and g t =( - 1 ) ' hL- ,+ , and introducetwo operators H, G acting on the vector space of allsquare summable infinite-length sequences, I z ( Z ) . Whenconsidering signals with finite length, say N=2", one needonly consider the space 1 '(N). These operators are givenby

and correspond to the convolution-subsampling operationsshown in figure 1. These operators have adjoints (a "dual"relation) given by

corresponding to the upsampling-anticonvolution operationsalso shown in figure 1. These four operators have beenshown to satisfy the perfect reconstruction conditions

(3). HG' = G H ' = 02. H H ' + GG ' = Iwhere I is the identity operator in 1*(a

................................. .................................H i H* i+h(-n)Hlz + 4l e H h b , k:.............................. ; ................................ a

G* 1.... ...........................................................G j+ " - 9 ' H 1 2 + 4t 2 H

:.............................. : ................................ :Figure 1: Analysis and Synthesis fittersimplementing H, G, H, G' operators.Now define the following recursive sequence of

functions,

'Employed by The United States Air Force at Rome Laboratory. RUC3BB. Griffiss AFB, NY 13441-4505.

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The function p,(x) is the unique fixed point of the firstt w o - d e equation above and is exactly the scalingfunction from Multiresolution Analysis (MRA)heory [7],[5], i.e., the function which forms the basis for thesubspace VI. Similarly,p,(x) is the corresponding waveletfunction by the second equation and thus, we assign at theoutset po = Q and pI = w. Indeed, MRA and thewavelet transform (and M-band wavelets at the f i tdecomposition) are special cases of this generalconstruction. These functions have some very usefulproperties, as given by Chui [l]. The fmt one,

says that each individual wavelet packet is orthogonal toitself at all nonzero translates. Also,

meaning that pairs of packets coming from the same parentpacket are orthogonal at all translates. The same functionp"is used to generate both pz and pz,+l,nd so we refer tothe former as the "parent" of the latter "children".

We now examine in detail what happens to a waveletpacket function as it traverses one stage of a two-channelfilter bank. The result will allow us to write a generalexpression for the functions at any node of the bank.Rewriting the first equation in (4). we modify the argumentand amplitude-scale, then change variables to yield a newform in our operator H :

kk l

= 2+ h,pn(2j"x-(21+k))

= 2T hm-upm(2j+1x-m)k

j+ (7)

j+ lwhere 2Tpn(2i*'x-m) is actually a sequence indexed onm, with x and n fixed. This expression represents a oneparameter family of discrete sequences, where eachsequence depends on the value of x. That x comes froman uncountable set is not an issue since it remains fixed fora given sequence. A slight perturbation of x generates anew sequence which upon traversing the filter (operator)yields another sequence, very close to the first. Over allx, the result is a family of sequences whose element-by-element concatenation is a family of continuous functions.It is this interpretation which admits the discrete filtering

of a continuous function. We c a n apply the same steps tothe second equation in (4) to get

These equations are easily interpreted in a filter bank senseas figure 2 shows. The input "signal" gets decomposedinto two orthogonal "signals" at a smaller scale (smallerscales increase width - decreasing time resolution.) Again,these signals are actually discrete sequences for a given x,which when considered together form an orthogonal familyof functions of x, and are actually all translations of onefunction at a given scale.

Figure 2: Wavelet packet decomposition bytwo-channel filter bank.

Each translate of a wavelet packet function lives inL (89 and the whole set of translates generates a subspaceof L '(n) . We introduce the notation

where indicates closed linear span. From (4). bothp z and pz+, are expansions in the scaled function pn(2x),implying that the spaces generated by these two functionsare both subspaces of that generated by the parent. Thatis.

Furthermore, the orthogonality properties of ( 5 ) and (6) aresufficient to admit the decomposition relation,

which is a generalization of the single orthogonaldecomposition V,,, = V , @W, provided by a MRA. Theproof of (11) is provided in [11. Thus every parent spaceis decomposed into orthogonal subspaces as illustrated infigure 3. The result is a binary tree in which every noderepresents an L '(4-subspace whose member functions aresplit by a two-channel filter bank (refer again to figure 2 )and the functions produced at the outputs generate

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orthogonally complementary subspacesof the parent space.

W,:w.I

$w.Iigure 3: Subspace decomposition tree.

In [ 2 ] , it is shown that the set of wavelet packets,pn(x-k) , n&Z form an orthonormal basis for L *(W.Moreover, the following theorem establishes the set ofbasis functions for an arbitrary pruning of the infinitebinary tree.Theorem 1:

If the collect ion 8={(l .n)} s such that the dyadicinlervals 11,, = [2'n, 2'(n+l)) form a disjointcovering of [O,-), then the set of wavelet packets2mpp,(2'x-k), EZ, (1,n)E 8 , form a completeorthonormal basis for L '(W.In particular, the collection { (Ll)), k Z , gives rise to thewavelet basis for L 2(R). We refer to the collection 8 asa "partition" in that it defines the partitioning of theinterval into dyadic subintervals determining thelocalization of the basis (though not directly as might beexpected - this problem, which has a very elegant solution,is deferred to another discussion.)

From a practical viewpoint, we are only interested in asubset of L z m functions, namely those that can berepresented as expansions in the basis set of the initialspace, W:. Functions outside this space do not meet oneor more of our criterion for practical communicationsignals, in particular the bandwidth, which is intimatelyrelated to the sampling rate and defines the initial space.Thus we desire to apply the "finite" version of this theoremwhich is credited to Coifman, et. al. in [ 4 ] .Theorem 2:Let W; CL'0e equipped with orthonormal basis2NRp,(2Nx-k),k Z . If the collection 8,c8=( I , n ) }

is such that the dyadic intervals1 = [2'n, 2'(n+1))form a disjoint covering of [ 0 , 2k , then the set of

wavelet packets 2'Rp,(2'x-k), k 2, ( 1 , n ) ~ forma complete orthonormal ba sis fo r W:.This theorem can be interpreted in terms of the direct sumof the subspaces spanned by each family of packetsdefined by the elements in the relevant partition 8,. Thatis,

In particular, this interpretation allows us to see clearly thatif p,(x) is a valid scaling function in Wz(.o,hen the N=Ocase of (12) says that it can be written as a linearcombination of the basis functions for the wavelet packetsubspaces. It is precisely this nexus which allows us touse wavelet packets for communications.2.1 Example of Theorem 2

Consider a partition of [OJ) given by

It is easily seen that these ordered pairs do actually providea disjoint covering of the interval. The unique filter bankassociated with this partition follows naturally starting atthe space Wi, nd branching out as in figure 3. Fromhere, we can deduce the corresponding tiling diagram if weknow the rules, which are not in the scope of thisdiscussion. The whole process is illustrated in figure 4.

Figure 4: Example application of theorem 3.2,with partition and corresponding filter bank.

3. Waveform DevelopmentWe now come to the issue at hand, which is designing

a communication signal having desirable properties for agiven channel. A desirable signal in our case is one which

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maximizes the information transferred from sender toobserver with minimal distortion in minimal time. Thedistortion and time criterion translate into figures of merit(bandwidth, bit error rate, performance in Gaussian noise)for the signal characteristics.

To accomplish the task described, the signal must staywithin the allotted frequency band, saturating it withinformation and minimizing losses, i.e. having a relativelysmall number of erroneous symbols. A very good way todo this is to look at energy in the time-frequency plane andsee where interference in the channel shows up. Then onecan use a variety of techniques which utilize time-frequency methods to optimize communication.

One method is to "communicate around" theinterferences - that is design the signal in such a way thatit has no time-frequency components competing withinterference components. This method relies on aknowledge of the channel before transmission. Anothertechnique is to excise those components of the receivedsignal which are unusually high - indicating an interferencesource. This method relies on designinga signal that doesnot have unusually high components to start with. Variousother techniques are available as well, but in all cases wemust optimize over time and frequency to do a better job.

Jones' work 161was a substantial step in this direction,but the two methods he introduced still have theundesirable property that their time-frequencyrepresentations are fixed. Thus for MSM, a highfrequency interferer will do more damage than a lowfrequency interferer due to the longer bandwidths of thebasis functions in those subchannels. For MWM, anytime-impulsive noise is bad because all the basis functionsspread out over an entire super-symbol. Clearly, what weneed is a signal whose T-F representation can be specifiedarbitnrily (indicated in the loose sense - it must be a validrepresentation) so as to minimize the effects of a broaderclass of interference sources. Wavelet Packet Modulationis one answer to this problem.

We begin this development with a general QAM signal[SI,

S ( l ) = u,$(+-m)m--

where El is the average signal energy over the normalizedperiod of a QAM symbol am pulse-shaped by $. Thenormalization of the usual T-second symbol period is foranalytical convenience and does not affect the result. Nowsuppose $ EW i . The W; space can be decomposed, asper theorem 2. into a finite set of orthogonal subspacesdefined by the collection 8 = ( ( l l ,n , ) , (l,,nJ, ... , ( lJ ,nJ) }as

Thus we can rewrite the QAM signal in terms of the basisfunctions of these decomposed spaces, obtaining themultidimensional signal,

where a: are the complex QAM signals at scale 2'8.Since data is placed on orthogonal functions arising froma wavelet packet decomposition of the W ; space, we referto the signal defined in (16) asWavelet Packet Modulationor WPM.

The orthogonality across scales and within fixed scalesof the constituent packets insures a zero IS1 waveformwhich is also free of cross channel interference. Noticethat the partition

is the decomposition defining the Multiscale Modulationscheme of Jones [6], where 2J-1 s the number of QAMsymbols representing J dyadic frequency bands in the T-Fdiagram. For the same 2'-' coefficients, the partition

2 ' 48,, = U ( - l , i )i4

where I=J-1, is the decomposition defining the M-BandWavelet Modulation scheme also in [6], in which allsymbols occur at the super-symbol rate, but occupy smallerbandwidths. Hence these formats are special cases of themore general WPM scheme. Of course the original QAMsignal, with no decomposition performed means eachsymbol occupys the entire bandwidth, but with smallerdurations. This corresponds to the partition 8,, = (0,O).3.1 An Example

Clearly, due to the fact that MSM and MWM arespecial cases of the more general WPM, the performanceof WPM in any interference environment will be at leastas good and usually better, because of the inherentflexibility. There may be environments which areoptimally handled with MSM or MWM, and in thosecases, WPM will accomodate them. The real advantage ofthe WPM scheme is demonstrated in environments where

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other T-F decompositions would be preferred over MSMor MWM. For i n s m e , we consider the case where anarrowband jammer is operating in the high frequencyportion of our bandwidth in conjunction with an impulsivetime domain interferer.

Let J=5, providing 25-'=16 T-F atoms in thedecomposition. There will be 5 dyadic frequency bands inMSM and 16 equal-width bands in MWM, illustrated bythe top two blocks in figure 5. The normal QAMconstruction is given in the bottom left and a WaveletPacket construction is shown in the bottom right.

Notice that for MSM, the tone jammer corrupts 8 of thecoefficients and the impulse corrupts 5 . One of these isaffected by both, thus 12 of the 16 coefficients are noisy.

certain channel impairments which makes wavelet packetmodulation an attractive time-frequency method formultidimensional signaling.4. Conclusions

The wavelet packet basis functions developed in section2 have been shown to admit a very useful method ofdigital modulation. This Wavelet Packet Modulationscheme has the benefits of previously introducedwaveforms but incorporates a flexibility which allows forbetter channel exploitation over a larger set of interferenceenvironments. Through a simple example we have shownthat W M s indeed a superior method for orthogonallymultiplexed digital communication, and have layed thegroundwork for future discussions.c MWMReferences

tt:ItFigure 5: Time-Frequency comparison ofmodulation methods in a tonelimpulseinterference environment. Grey-shaded areasrepresent corrupt symbols.

The MW M case has all 16 coefficients affected by theimpulse and QAM has all 16 affected by the tone. Thusnone of these methods is ideal for these noise sources.However, the W M iagram in the bottom right, which isoptimized for this environment, isolates the interferers to5 of the 16 coefficients - a significant improvement overthe previous three. It is precisely this flexibility to isolate

C. K. Chui, An Introduction to Wavelets, San Diego:Academic Press, 1992 .R.R. Coifman and Y. Meyer, "Orthonormal Wave PacketBases." preprint, Numerical Algorithms Research G roup,Yale University, 1989.R.R. Coifman, Y. Meyer. S. Quake, and M.V.Wickerhauser, "Signal Processing w ith W ave Packets,"preprint, Numerical Algorithms Research Group, YaleUniversity, April 1990.R.R. Coifman, Y. Meyer, and V. Wickerhauser, "SizeProperties of Wavelet Packe~ s,"reprint, Yale University,1992.I. Daubechies, Ten Lectures on Wavelets, Philadelphia:SIAM, Volume 61 , 1992.W.W. Jones, "A Unifed Approach to OrthogonallyMulliplexed Communication Using Wavelet Bases andDigital Filter Banks," Ph.D. Dissertation, OhioUniversity, Athens, OH, August, 1994.S.G. Mallat, "A Theory for Multiresolution SignalDecomposition: The Wavela Representation," E E ETransactions on Pattern Analysis and Machine Intelli-gence, vol. 11, no. 7, July 1989.J.G. Proakis, Digital Communications, 2nd Edition,McGraw Hill, 1989.

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