DESIGN OF BRANCH-HOPPED WAVELET PACKET DIVISION MULTIPLEXING SCHEMES

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1 DESIGN OF BRANCHHOPPED WAVELET PACKET DIVISION MULTIPLEXING SCHEMES T. N. Davidso, Z.Q. Luo ad A.J. Schott Commuicatios Research Laboratory, McMaster Uiversity, Hamilto, Otario, L8S 4K, Caada. ABSTRACT Wavelet Packet Divisio Multiplexig (WPDM) is a highcapacity, flexible ad robust orthogoal multiplexig techique i which wavelet packet basis fuctios are chose as the codig waveforms. BrachHopped WPDM (BH WPDM) is a extesio of WPDM which icorporates hoppig strategies aalogous to those of frequecyhopped schemes. It is based o a efficiet modular switched trasmultiplexer structure which provides the advatages of hoppig whilst retaiig may of the desirable features of WPDM. I previouswork we have idetified classes of slow ad fast BHWPDM schemes ad have evaluated a umber of switchig strategies. I the preset paper we provide a method for redesigig the filters withi the trasmultiplexer modules to provide further robustess to arrow frequecyselective fadig chaels uder a give switchig strategy. Key Words: multiplexig, wavelet packets, hoppig, frequecyselective fadig.. INTRODUCTION Wavelet Packet Divisio Multiplexig (WPDM) [] is a emergig orthogoal multiplexig techique i which wavelet packet basis fuctios [] are chose as the codig waveforms. I cotrast to the covetioal time divisio (TDM), frequecy divisio (FDM) ad code divisio (CDM) multiplexig schemes, the waveforms used to represet the data symbols of each user overlap i both time ad frequecy. The fact that the waveforms are of fiite duratio ad overlap i time ad frequecy provides a substatial icrease i capacity over TDM ad FDM [] ad robustess to certai adverse chael eviromets [, 3], whilst their close relatioships with multirate filter baks (trasmultiplexers) provide particularly simple trasmitter ad receiver structures []. davidso@suspot.crl.mcmaster.ca. Fax: Orthogoal multiplexig schemes are sesitive to the effects of frequecyselective chaels. Whilst the frequecy overlappig of the WPDM waveforms provides some robustess to these effects, the commoly used wavelet packet basis fuctios are still localized i frequecy, ad hece are susceptible to the perturbatio of frequecyselective chaels. I frequecyhopped commuicatio schemes, the susceptibility of a arrowbad commuicatio scheme to a ukow frequecyselective chael is reduced by hoppig the carrier frequecy betwee several frequecies i a patter which is kow by the receiver. The Brach Hopped WPDM (BHWPDM) scheme employs a efficiet modular switched trasmultiplexer structure to achieve a aalogous hoppig effect for WPDM. I previous work [4, 5] we evaluated a umber of differet switchig strategies for BHWPDM ad idetified hoppig schemes with performace advatages which are aalogous to those of slow ad (coheretly combied) fast frequecy hoppig. I this paper we provide a method for redesigig the filters withi the trasmultiplexer modules to provide further robustess to arrow frequecyselective fadig chaels uder a give switchig strategy.. WAVELET PACKET DIVISION MULTIPLEXING We begi with a brief review of the WPDM scheme []. (See [] ad refereces therei for some related work.) To defie the wavelet packet basis fuctios, let g be a uiteergy real causal FIR filter of legth L which is orthogoal to its eve traslates; i.e., g []g [ ; m] =δ[m] where δ[m] is the Kroecker delta, ad let g be the (cojugate) quadrature mirror filter, g [] =(;) g [L ; ; ]. If g satisfies some mild techical coditios [], we ca use a iterative algorithm to fid the fuctio φ (t) = p g []φ (t ; T ) for a arbitrary iterval T.

2 Subsequetly, we ca defie the family of fuctios φ, `, m ` i the followig (biary) treestructured maer: φ`+ m;(t) = φ`+ m(t) = g []φ(t ; T`) g []φ(t ; T`) (a) (b) where T` = `T. For ay give tree structure, the fuctios at the termials ofthetreeformawavelet packet []. They have a fiite duratio, (L ; )T`, ad are self ad mutuallyorthogoal at iteger multiples of dyadic itervals, ad hece they are a atural choice for multiplexig applicatios. More precisely, if T deotes the set of termial idex pairs, the for (` m) (λ μ) T hφ(t ; T`) φ λμ (t ; kt λ )i = δ[` ; λ]δ[m ; μ]δ[ ; k]: () I the WPDM scheme, the (biary) message data at the (` m)th termial, d[], are waveform coded by pulse amplitude modulatio (PAM) of the fuctio at that termial. Hece the WPDM composite sigal is s(t) = (` m)t d[]φ(t ; T`): (Note that the termials o differet levels have differet symbol rates, =T`.) Due to the orthogoality relatioship i Eq. () the data ca be extracted from the trasmitted sigal without itersymbol iterferece or crosstalk usig a simple matched filter receiver for each termial. By exploitig the structure i Eq. () we obtai a alterative trasmitter structure usig a treestructured multirate sythesis filter bak ad a sigle PAM modulator (see Fig. ) where σ [k] = s(t) = k (` m)t σ [k]φ (t ; kt ) (3) f[k ; `]d[] (4) ad f is the equivalet filter from the (` m)th termial to the root ode, which ca be foud recursively usig Eq. () ad f[k]=hφ(t) φ (t ;kt )i. The orthogoality property of the equivalet filters, h f[k ; `] f λμ [k ; λ i]i = δ[` ; λ]δ[m ; μ]δ[ ; i], for (` m) (λ μ) T,cofirms that a alterative receiver structure cosistig of a sigle matched filter ad a treestructured multirate aalysis filter bak is available (see Fig. ). By substitutig Eq. (4) ito Eq. (3) we ca view the WPDM scheme as a member of a class of geeralized orthogoal CDM schemes i which the codes are the equivalet filters f ad the chip waveform is φ. This geeralized class icludes the covetioal orthogoal CDM schemes (i.e., WalshHadamard schemes), but exteds those schemes to allow for realvalued orthogoal codes which overlap i time, ad orthogoal chip waveforms which have a duratio loger tha the chip iterval. 3. BRANCHHOPPED WPDM The BHWPDM scheme is based o a modular switched trasmultiplexer structure i which a twoiput twooutput memoryless switchig uit is attached to the iput of each merge module at the trasmitter ad to the output of each correspodig split module at the receiver, as illustrated i Fig.. If we toggle the state of each switch at the trasmitter i a patter which is kow at the receiver, we hop the braches of the treestructured filter baks. Log or short itervals betwee switch state chages lead to schemes with aalogies to the slow ad fast frequecyhopped schemes, respectively. Of course, the BH WPDM hoppig schemes require sychroizatio of the switches, but that is o more arduous tha the sychroizatio of frequecyhopped schemes. The BHWPDM composite sigal ca be writte i a form aalogous to Eq. (3) as (see Fig. ) s(t) = σ [k]φ (t ; kt ) (5) k where σ is obtaied from the switched sythesis filter bak, σ [k] = (` m)t f[k ]d[] () ad f[k ] is the equivalet filter from the (` m)th termial to the root ode see by a uit sample at istat at the (` m)th termial. If we defie x λμ [i] to be the state of the switch at the (λ μ)th ode at the ith istat, with zero represetig a parallel coectio ad oe represetig a cross coectio the f[k ] ca be foud recursively as A alterative matrixbased otatio for the filter baks i Eqs (4) ad () has also bee developed [4].

3 d [] d [] d 3 [] d 4 [] U U f " g [i] x []? + e " f g [i] f " g [i] x []? + e " f g [i] σ [ j] σ [ j] U f " g [i] x [ j]? + e " f g [i] σ [k]? PAM Modulator φ (t ; kt ) s(t) r(t) Chael φ (;t) t = kt σ g [;i] f x [ j] # g [;i] f # g [;i] f U σ [ j] σ [ j] g [;i] f # x [] # g [;i] f x [] # g [;i] f U # U d [] d [] d 3 [] d 4 [] Fig. : A fouruser BHWPDM scheme. The dashed boxes represet switchig uits which provide either a parallel or a cross coectio at each istat. If the switches remai i the parallel state we have a WPDM scheme. d[] "4 f f[k ] d[] "4 f f[k ] d3[] "4 f f3[k ] d4[] "4 f f4[k ] e? +? σ[k] + e + e φ(t ; kt) t = kt? s(t) PAM r(t) Chael φ(;t) EQUIVALENT DISCRETETIME CHANNEL σ[k] Fig. : A equivalet model for Fig.. d[] f[;k ] #4 f d[] f[;k ] #4 f d3[] f3[;k ] #4 f d4[] f4[;k ] #4 f f`+ m;[k ] = ; g [i ; ]( ; x[]) + g [i ; ]x[] f[k i] i f`+ m[k ] = ; g [i ; ]x[]+g [i ; ]( ; x[]) f[k i] i (7a) (7b) with f [k ] =δ[k ; ]. (If all the switches are i the parallel state the f[k ]= f[k;`] ad we retur to the uderlyig WPDM scheme.) The equivalet filters retai the orthogoality property of those i the WPDM scheme, h f[k ] f λμ [k i]i = δ[` ; λ]δ[m ; μ]δ[ ;i], for(` m) (λ μ) T, ad hece the receiver cosists of a sigle matched filter ad a switched aalysis filter bak. The properties of f[k ] esure that (with white biary data) the BHWPDM composite sigal, s, has the same cyclic spectra as that of the uderlyig WPDM scheme. Hece BHWPDM retais the capacity advatages of WPDM. What the BH WPDM scheme does chage is the way i which the spectra of s are allocated to the user at each termial. By substitutig Eq. () ito Eq. (5) the BHWPDM scheme ca be viewed as a codehopped extesio to the class of geeralized orthogoal CDM schemes discussed at the ed of Sectio i which the orthogoal codes allocated to the data symbols at a give termial may vary from symbol to symbol, as illustrated i Fig.. 4. SWITCHING STRATEGIES Oce the uderlyig tree structure of a BHWPDM scheme has bee chose, its performace depeds o the switchig strategy ad the filter g. Cetral to the aalysis of the effects of the switchig strategies at each ode are the followig observatios [5], obtaied by careful ispectio of Eq. (7):. The equivalet filter f[k ] depeds o switch states at all the odes alog the path from the (` m)th termial to the ( ) ode. Those odes are (λ μ), where λ ` ;, μ = d λ;e, ad dwe deotes the least iteger w.. The umber of states of the switch at the (λ μ)th ode (alog the above metioed path) which affect f[k ] is the legth of the equivalet filter from the (` m)th termial to the (λ + d λ+;e)th ode. The legth of that filter is L λ+,wherel, (`;λ λ` ; )(L ; )+. ` Combiig these two observatios, we fid that f[k ] depeds o x λμ [i] for λ ` ;, μ = d λ;e ad i [`;λ; `;λ; + L λ+ ; ]. We ` collect these states, i a particular arragemet, i the vector x[]. ThereareK`, `; λ= Lλ+ = ` L` ; `L ; elemets i x[], wherel`, L`. Therefore, there are K` possible values of x[], each geeratig a distict equivalet filter. Distict equivalet filters are, i geeral, distict i the sese that they are ot shifted versios of each other. The role of the switchig strategy is to carve out a subset of those K` distict equivalet filters ad hop f[k ] amogst (shifted versios of) that set. To capture that otio, let x (i) deote the ith possible value of x[], adlet [k] deote the (timeivariat) equivalet filter f[k ] whe x[]=x (i).

4 With those defiitios, if x[] =x (i) the f[k ] = [k ; `]: (8) For a give switchig strategy we assess the relative frequecy of the assigmet i Eq. (8) by associatig aweight,w (i), with each x(i) (ad hece with each [k]). For a give iterval of iterest N ` of legth N` symbols, W (i), N (i) =N`, wheren (i) is the umber of istats i N ` for which the switchig strategy makes the assigmet x[] =x (i) We collect the filters. [k] with ozero weights i the set of distictly geerated equivalet filters for othe give switchig strategy, F = W [k] (i) = ad let U K` deote the umber of elemets i that set. I order to classify BHWPDM schemes, we observe that if x (i) cotais oly oe state of each switch, the [k] = f[k] for some m `; i.e., the filter [k] is a equivalet filter from a uderlyig WPDM scheme. Otherwise, [k] depeds o both filters i the merge uit at at least oe ode. If the itervals betwee switch state chages are log, the oly elemets of F with substatial weights are equivalet filters from a uderlyig WPDM scheme. I that case, the scheme will be said to be a slow BH WPDM scheme. If the itervals betwee switch state chages are short, the scheme will be said to be a fast BHWPDM scheme. I that case, the elemets of F with substatial weights will ot be filters from a uderlyig scheme, but will be filters which deped o both filters i the merge uit at at least oe ode. Sice g ad g ted to be lowpass ad highpass, respectively, such filters ted to have a broader frequecy respose (as a fractio of the badwidth of the whole multiplexig scheme) tha the correspodig WPDM filters. I previous work [4, 5] we have show that i a slowly varyig frequecyselective chael, slow ad fast BHWPDM schemes provide performace averagig aalogous to that of slow ad fast frequecy hoppig schemes, respectively. Slow BH WPDM schemes ted to provide performace averagig amogst idividual users, but they ted ot to pro By log we mea that for the switch at the (λ μ)th ode the itervals betwee switch state chages are log with respect to L λ+,where ax is the maximum value of ` over the termials, ax (` m), affected by the switch at the (λ μ)th ode. Magitude (a) Magitude Fig. 3: Spectra of the equivalet filters for the fouruser system illustrated i Fig. based o a stadard Daubechies filter [] of legth four: (a) the WPDM scheme (o switchig), with a sapshot of the otch chael used to produce Fig. 4 (asterisks); (b) a fast BHWPDM scheme i which each switch is toggled at each istat. BER 3 (b) SNR, db Fig. 4: Simulated overall BER agaist SNR for the schemes i Fig. 3, with ucoded biary data, i a slowly ad uiformly varyig otch chael with additive white Gaussia oise. A sapshot of the otch chael is give i Fig. 3(a). Leged: WPDM ad ay slow BHWPDM scheme (idistiguishable): solid; the fast BHWPDM scheme: dotdashed. vide a overall performace improvemet. I cotrast, the broader spectra of the equivalet filters i the fast schemes provide a overall performace improvemet i such chaels. This effect is illustrated i Figs 3 ad 4. (See [4, 5, 7] for further details.) 5. DESIGN OF BHWPDM FILTERS I the previous sectio, we made the (heuristic) observatio that the spectral broadeig of the equivalet filters iduced by the switchig trasiets i a fast BHWPDM scheme leads to improved performace i (ukow) slowly fadig frequecyselective chaels which are arrow with respect to the badwidth of the whole multiplexig scheme. (Related observatios have bee made for spread spectrum commuicatio

5 systems.) I this sectio we exploit that observatio to redesig the filter g i the trasmultiplexer modules for a give switchig strategy, i order to gai further improved performace i such chaels. As a measure of the breadth of the spectrum of the equivalet filter [k] (as a fractio of the badwidth of the multiplexig scheme as a whole) we defie the (RMS) deviatio from frequecy flatess of the equivalet filter [k] to be D (i), π Z π H (i) (e j ) ; d! = (9) where H (i) (e j ) is the DiscreteTime Fourier Trasform of [k]. The deviatio from frequecy flatess is simple to compute ad small values of D (i) ca be a effective guide towards good switchig strategies for arrow frequecyselective chaels. For example, for the WPDM scheme i Fig. 3(a) the deviatios are : ad : (two of each), ad for the BHWPDM scheme i Fig. 3(b), the deviatios are :3 ad :35 (agai two of each). The performace advatage of the BHWPDM scheme predicted by its smaller deviatios is clearly achieved i the sceario of Fig. 4. The derivatio of our desig method proceeds by reiteratig the observatio that for a give switchig strategy, a filter g for which all distict equivalet filters [k] i each set F have small values of D (i) ought to lead to good performace i slowly fadig arrow frequecyselective chaels. I order to simplify the expositio, we will restrict our attetio to BHWPDM schemes i which all the termials are at the same level, ad to switchig strategies for which each distict equivalet filter has the same weight. 3 A iitial desig problem ca be phrased as follows: For the give switchig strategy, fid a filter g which miimizes a particular D (i ) `m subject to D (i) for all the other equivalet filters beig o greater tha ( + ε D )D (i ) `m, for a small positive ε D,adtog beig of uit eergy ad selforthogoal at eve traslatios. Note that we have implicitly decoupled the desig of g from that of φ (t) so that o regularity costraits [] o g are required. This decouplig is kow as the splittig trick i the wavelet literature []. 3 May periodic switchig strategies satisfy the secod criterio [7]. The extesio of this work to more geeral schemes is straightforward but otatioally cumbersome. Careful ispectio of Eq. (9) reveals that if g [] = δ[] the D (i) = forall(` m) T ad i U. Therefore, such a solutio, which correspods to a (scrambled) TDM scheme, is a global optimum for this problem. However, TDM schemes ted be sesitive to timeselective effects such as impulsive oise [3], so we impose a costrait o the time localizatio of the equivalet filters. By aalogy with Eq. (9) we defie the (RMS) deviatio from ideal time spread for the equivalet filter [k] to be τ (i), L`;! = [k] ; =L` : k= We costrai the deviatio from ideal time spread of all the equivalet filters [k] to be of the same order as that obtaied by the BHWPDM scheme with the give switchig strategy ad with g beig the stadard Daubechies filter [] of the desired legth. This will esure that the robustess of the desiged BH WPDM scheme to time selective effects is of the same order as that of the BHWPDM scheme with the same switchig strategy ad the Daubechies filter. The desig problem ca ow be formally stated: For a give switchig strategy ad a give filter legth L, select a particular (` m ) T,adi, i U` m,adfid a filter g [],L;, which achieves the miimum of subject to mi D (i ) `m g [] L; g []g [ ; k]=δ[k] k d(l ; )=e () =k D (i) ( + ε D)D (i ) `m (` m) T i U τ (i) ( + ε τ)τ (` m) T i U where ε D ad ε τ are small positive costats, τ is the maximal τ (i) for the BHWPDM scheme with the same switchig strategy ad the stadard Daubechies filter [] of legth L. Note that L must be eve i order to satisfy Eq. () []. Based o our heuristic observatio, we would expect such a optimized scheme to perform better tha a scheme employig the stadard Daubechies filter of that legth i a slowly fadig arrow frequecyselective eviromet. I the followig example we demostrate that sigificat performace

6 gais are ideed achievable i this maer. Example Cosider the fouruser BHWPDM scheme illustrated i Fig., ad two switchig strategies: the WPDM scheme (with o switchig) ad the fast scheme i which each switch is toggled at each istat. A optimal g of legth four was foud for each scheme by solvig the desig problem with ε D = ε τ = :, usig a stadard Sequetial Quadratic Programmig techique. This resulted i D (i ) ` m = :8 for the WPDM scheme ad D (i ) ` m = :7 for the fast scheme. For compariso, the deviatios for the schemes based o the stadard Daubechies filter of legth 4 are. ad. (two of each) for the WPDM scheme, ad.3 ad.35 (two of each) for the fast scheme. The spectra of the equivalet filters for the optimized fast scheme are plotted i Fig. 5, ad are clearly flatter i frequecy tha those for the fast scheme with the Daubechies filter. Usig the deviatios ad the precedig discussio, we predict that each optimized scheme will perform better tha the correspodig scheme based o the Daubechies filter i a slowly fadig arrow frequecyselective chael. Furthermore, we predict that the optimized fast scheme will perform better tha the optimized WPDM scheme. Simulated BER curves for trasmissio through the slowly ad uiformly varyig otch chael used to produce Fig. 4, a sapshot of which was provided i Fig. 3, are plotted i Fig.. These figures demostrate that a sigificat improvemet i BER performace is ideed achievable. For example, at db SNR, the BER performace gai of the the optimized fast scheme over the Daubechies filter based fast scheme is aroud 5%. (Equivaletly, the SNR gai of the optimized fast scheme is more tha db at a SNR of aroud db.) Fially, ote that the predictio that the optimized fast scheme would perform better tha the optimized WPDM scheme is cofirmed. This demostrates a itrisic advatage of a optimized hopped scheme over a optimized static scheme. Numerical experimets i a umber of other scearios cofirm the treds illustrated i this example [7].. REFERENCES [] K. M. Wog, J. Wu, T. N. Davidso, ad Q. Ji. Wavelet packet divisio multiplexig ad wavelet Magitude Magitude Fig. 5: Spectra of the equivalet filters for the fast schemes i Ex.. Optimized scheme: solid; Daubechies filter based scheme: dashed. BER SNR, db (a) A B BER SNR, db Fig. : Simulated overall BER agaist SNR for the optimized schemes (solid) ad the Daubechies filter based schemes (dashed) i Ex.. (a) WPDM; (b) fast BHWPDM. packet desig uder timig error effects. IEEE Tras. Sigal Processig, 45():877 89, Dec [] I. Daubechies. Te Lectures o Wavelets. SIAM, Philadelphia, 99. [3] J. Wu, K. M. Wog, Q. Ji, ad T. N. Davidso. Performace of wavelet packet divisio multiplexig i impulsive ad Gaussia oise chaels. Proc. SPIE, 85:5 57, Oct. 99. [4] T. N. Davidso, Z.Q. Luo, ad K. M. Wog. A hoppig scheme for wavelet packet divisio multiplexig. I Proc. Symp. Wavelet Subbad Block Trasforms Commu., NJIT, Newark, NJ, Mar pages. [5] T. N. Davidso, A.J. Schott, ad K. M. Wog. Brachhopped wavelet packet divisio multiplexig. To appear i the Proceedigs of ICASSP 98. [] G. W. Worell. Emergig applicatios of multirate sigal processig ad wavelets i digital commuicatios. Proc. IEEE, 84(4):58 3, Apr. 99. [7] A.J. Schott. A hoppig scheme for wavelet paccket divisio multiplexig. M.Eg. Thesis, Dept. Elec. Comp. Eg., McMaster Uiversity, Hamilto, Otario, Caada, Ja (b)