Autonomous Dynamic Spectrum Management for Coexistence of Multiple Cognitive Tactical Radio Networks

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1 Autonomous Dynamc Spectrum Management for Coexstence of Multple Cogntve Tactcal Rado Networks Vncent Le Nr, Bart Scheers Abstract In ths paper, dynamc spectrum management s studed for multple cogntve tactcal rado networks coexstng n the same area A tactcal rado network s composed of a transmtter whch broadcasts the same nformaton to ts multple recevers Frst, we consder the problem of power mnmzaton subject to a mnmum rate constrant and a spectral mask constrant for a sngle tactcal rado network wth multple recevers over parallel channels (parallel multcast channels Then, we extend the teratve waterfllng algorthm to multple recevers for the coexstence of multple cogntve tactcal rado networks, meanng that there s no cooperaton between the dfferent networks The power allocaton s performed autonomously at the transmt sde assumng knowledge of the nose varances and channel varatons of the network Smulaton results show that the proposed algorthm s very robust n satsfyng these constrants whle mnmzng the overall power n varous scenaros Index Terms Cogntve tactcal rado networks, dynamc spectrum management, teratve water-fllng I INTRODUCTION The objectve of ths paper s to provde a dstrbuted power allocaton of multple cogntve tactcal rado networks coexstng n the same area The transmtter of each tactcal rado network broadcasts the same nformaton to ts group (voce, data Ths objectve calls for a synergy between dfferent areas: Cogntve rado [], []: A wreless node or network can adapt to the envronment by changng ts transmsson parameters (frequency, power, modulaton strategy Broadcast channel wth only common nformaton [3], [4], [5]: A tactcal rado network s a network n whch nformaton s conveyed from one transmtter to multple recevers Most of the lterature on broadcast channels covers the transmsson of separate nformaton to the dfferent recevers or the transmsson of both separate and common nformaton to the dfferent recevers over parallel channels [6], [7], [8], [9], [0], [] Dstrbuted mult-user power control [], [3], [4]: Autonomous power allocaton n the frequency doman by teratve waterfllng for nterference channels By consderng the nterference of the other users as nose, teratve updates of the power allocaton for each user reach an equlbrum The waterfllng strategy has been ntally desgned for a sngle transmtter and a sngle recever over multple subchannels [5] The waterfllng strategy can maxmze the rate of the lnk subject to a power constrant (nner loop, but can also mnmze the power subject to a rate constrant (outer loop In the frst part of the paper (Secton II, we extend the waterfllng strategy to multple recevers by consderng parallel broadcast channels wth only common nformaton (parallel multcast channels and assumng perfect channel state nformaton (CSI at the transmt sde In ths case, the extended waterfllng strategy maxmzes the mnmum rate subject to a power constrant (nner loop or mnmzes the power subject to a mnmum rate constrant (outer loop However, fndng a soluton to these problems for parallel multcast channels s not straghtforward Moreover, standardzaton often defnes a spectral mask that each transmtter has to satsfy Therefore, we propose to use an utlty functon based on the weghted sum of the possble achevable rates to the recevers for the nner loop and to fnd the best set of weghts that mnmzes the power subject to a mnmum rate constrant for all recevers and a spectral mask constrant In the second part of the paper (Secton III, captalzng on the prevous results, we ntroduce an autonomous dynamc spectrum management algorthm based on teratve waterfllng [] for multple cogntve tactcal rado networks In the teratve waterfllng algorthm, each network consders the nterference of all other networks as nose and performs a waterfllng strategy The power spectrum of the network modfes the nterference caused to all other networks Ths process s performed teratvely untl the power spectra of all networks converge The man novelty of ths paper s the extenson of the teratve waterfllng algorthm to multple recevers for the coexstence of multple cogntve tactcal rado networks The transmtter of each tactcal rado network takes nto account the spectrum sensed by all ts recevers and teratvely updates ts power spectrum untl all the constrants are satsfed n each network, e mnmum rate and a spectral mask constrants Smulaton results compare our strategy wth the worst recever strategy n Secton IV II SINGLE TACTICAL RADIO NETWORK A Mnmzaton of power subject to a mnmum rate constrant and a spectral mask constrant Consder a T -recever N c parallel fadng Gaussan broadcast channel as shown n Fgure : y t = h t x n t t = T, = N c ( where x s the transmtted sgnal, n t represents a complex nose wth varance σ t and h t corresponds to the channel on recever t and tone The prmal problem for power mnmzaton of a T -recever N c parallel fadng Gaussan broadcast channel wth only common nformaton subject to a mnmum rate constrant for all recevers R mn and a spectral mask constrant s V Le Nr and B Scheers are wth the Royal Mltary Academy, Dept Communcaton, Informaton Systems & Sensors (CISS, 30, Avenue de la Renassance B-000 Brussels BELGIUM E-mal: vncentlenr@rmaacbe bartscheers@rmaacbe Ths research work was carred out n the frame of the Belgan Defense Scentfc Research & Technology Study C4/9 funded by the Mnstry of Defense (MoD The scentfc responsblty s assumed by ts authors N c mn φ (φ Nc log ( ht φ R mn t Γσt φ φ mask (

2 Fg x h h h T n n n T T -recever N c parallel fadng Gaussan broadcast channel wth φ = E[ x ] the varance of the transmtted sgnal on channel, φ mask the mask constrant on sub-channel, and Γ the SNR gap whch measures the loss wth respect to theoretcally optmum performance [6] The dervaton of the modfed Lagrangan functon leads to a sngle varable search wth T Lagrange multplers [7] Therefore, the optmal power allocaton has an nfnte set of solutons and the problem s ntractable for T > We propose a soluton to ths problem by defnng an utlty functon whch takes nto account the possble achevable rates to the ndvdual recevers In the followng, the weghted sum rate s chosen for ths utlty functon as t allows to consder the achevable rates to the recevers wth a certan flexblty owng to the weghtng parameters Therefore, an nner loop determnes the power allocaton maxmzng the weghted sum rate subject to a total power constrant and a spectral mask constrant for a fxed set of weghts The mnmum rate s then selected amongst the possble achevable rates to the recevers Then, an outer loop mnmzes the power such that a mnmum rate constrant s acheved Ths process s repeated for all set of weghts and the set of weghts exhbtng the least power determnes the power allocaton for power mnmzaton subject to a mnmum rate constrant The prmal problem for weghted sum rate maxmzaton subject to a power constrant tot and a spectral mask constrant s: wth T N c max (φ Nc φ φ mask T w t log ( ht φ Γσt φ = tot y y y T (3 w t = As the objectve functon s concave, the power allocaton can be derved by the standard Karush-Kuhn- Tucker (KKT condton [7] By takng the dervatve of the modfed Lagrangan functon wth respect to φ, we can solve the KKT system of the optmzaton problem The dervatve wth respect to φ s gven by L(λ, (β, φ Nc = ln T w t (λβ Γσt (4 h t φ wth λ the Lagrange multpler assocated wth the total power constrant, and β the Lagrange multplers correspondng to the spectral mask constrant Nullng the dervatve gves L(λ, (φ Nc = 0 T w t = Γσ }{{} λln β ln h t φ }{{} λ β (5 From the prevous formula, one can see that the power allocaton depends on the number of recevers T Let us derve the power allocaton for dfferent number of recevers: For a sngle recever T =, the power allocaton corresponds to Gallager s water-fllng strategy for sngle-user parallel Gaussan channels [5] wth addtonal spectral mask constrant gven by: L(λ, (φ Nc = 0 φ = [ λ β Γσ h (6 For two recevers T =, the power allocaton s a type of water-fllng strategy gven by the soluton of a quadratc equaton L(λ, (β, φ Nc = 0 w w Γσ h }{{} a φ The quadratc equaton to be solved s ] Γσ = λ β (7 h }{{ φ } b ( λ β φ (( λ β (a b (w w φ ( λ β a b (w b w a = 0 (8 The dscrmnant s gven by = ( λ β (a b (w w ( λ β (a b (w w (9 The power allocaton s gven by the postve root [ φ = ( λ β ] (ww (a b(w w (a b 4( λ β ( λ β 4 ab (0 In ths formula, the power allocaton for weghted sum rate subject to a power constrant and a spectral mask constrant takes nto account the dfference between the water-fll functons and the weghts of the dfferent recevers For three recevers T = 3 and four recevers T = 4, the power allocaton s a type of water-fllng strategy gven by the soluton of a cubc and quartc equaton respectvely Therefore, the power allocaton can also be found analytcally (the soluton s not gven n ths paper due to space lmtatons Wth T > 4, the power allocaton s gven by the soluton of a polynomal equaton wth degree T In general, the roots can t be expressed analytcally but can be solved numercally The algorthm uses the weghts to mnmze the power subject to a mnmum rate constrant and a spectral mask constrant

3 3 Algorthm Mnmzaton of the power subject to a mnmum rate constrant n=0 for all w,,w T, wth T w t = 3 n=n 4 nt = nt pstep = 6 nt p = 0 7 nt R t = 0 t 8 whle mn(r,, R T R mn > ǫ 9 nt λ = nt step = nt b = 0 nt φ = 0 3 whle Nc φ > ǫ 4 Calculate φ accordng to (4 s root 5 f Nc φ < 0 6 b = b 7 λ = λ/step 8 step = step / b 9 end f 0 λ = λ step end whle Indvdual rates R t = log ( h t φ t Γσ t 3 f mn(r,, R T R mn > 0 4 p = p 5 = /pstep 6 pstep = pstep / p 7 end f 8 = pstep 9 end whle 30 n = 3 end for 3 mn = mn( n Algorthm provdes the power allocaton for power mnmzaton subject to a mnmum rate constrant R mn of a T - recever N c parallel fadng Gaussan broadcast channel wth only common nformaton (β = 0 The nner loop and the outer loop correspond to lnes 3- and 8-9 respectvely To nclude a spectral mask constrant, we need to replace the lne 4 wth the modfcatons gven n Algorthm III MULTILE COGNITIVE TACTICAL RADIO NETWORKS In ths Secton, we consder the scenaro n whch N dfferent cogntve rado networks can t cooperate wth each other and wsh to broadcast a common nformaton to ther network by sharng the same N c parallel sub-channels Ths scenaro s partcularly adapted to tactcal rado networks n whch N dfferent networks coexst n a gven area and broadcast a common nformaton (voce, data to ther group Wth current technologes, f the legacy rados of the coalton natons share the same parallel sub-channels, the nterference would ncrease and lead to a bad transmsson Cogntve rado enables the adaptaton of the transmsson parameters (transmt power, carrer frequency, modulaton strategy to these scenaros Based on the results of Secton II, we propose Algorthm Modfcatons to Algorthm to take nto account a spectral mask constrant for = to N c nt β = nt mstep = 4 nt m = 0 5 for teraton = to 0 6 Calculate φ accordng to (4 s root 7 f φ > φ mask 8 φ = φ mask 9 end f 0 f φ φ mask < 0 m = m β = β/mstep 3 mstep = mstep / m 4 end f 5 β = β mstep 6 end for 7 end for a completely autonomous dstrbuted power allocaton Consderng N dfferent networks and assumng that each network j has T j recevers, the receved data can be modeled as y j,t = h jj,t x j N h jk,t x k n j,t = N c, j = N, t = T j ( where n j,t represents a complex nose wth varance σ j,t and h jk,t corresponds to the channel from network k to j on recever t and tone Smlarly to Secton II n whch the ntal problem of power mnmzaton subject to mnmum rate constrant s ntractable for T >, we propose a way to solve the ntal problem by defnng an utlty functon (the weghted sum rate whch takes nto account all the achevable rates of the recevers and to select the mnmum rate n each network The prmal problem for the weghted sum rate maxmzaton subject to a total power constrant and a spectral mask constrant per network (nner loop s gven by: N c max (φ j j=n Nc N T j j= φ j tot j w jt log ( j h jj,t φ j Γ(σj,t h jk,t φ k φ j φ mask j, j ( Ths problem s hghly non-convex and no closed-from soluton can be derved Even f a centralzed cogntve manager was able to collect all the channel state nformaton (CSI wthn and between the dfferent networks, t would requre an exhaustve search over all possble φ j s, or a more effcent genetc algorthm To solve ths problem, we propose a sub-optmal dstrbuted algorthm based on the teratve water-fllng algorthm ntally derved for dynamc spectrum management n dgtal subscrber lne (DSL [] The teratve water-fllng prncple s extended to multple cogntve tactcal rado networks, n whch each network consders the

4 4 nterference of the other networks as nose and performs waterfllng on ts parallel multcast channels Each update of one network s water-fllng affects the nterference of the other networks and ths process s repeated teratvely between the networks untl the power allocaton of all networks converge and reach a Nash equlbrum As the power updates between networks can be performed asynchronously, an teratve waterfllng based algorthm for the coexstence between multple cogntve tactcal rado networks Let us derve the modfed Lagrangan functon: L((λ ( j j=n, (β j, φ j j=n N c = N c N T j w jt log ( j= N (λ j β j φ j N λ j j tot j= h jj,t φ j Γ(σj,t h jk,t φ k j= Nc N j= β j φ mask j (3 n whch the λ j s and β j s are the Lagrange multplers We assume that each transmtter has the knowledge of the nose varances and the channel varatons n ts own network j { σ j,t h jk,t φ k, t (4 k = j,, t h jk,t Ths knowledge can be acqured through a feedback channel from the recevers to the transmtter of each network assumng that the acquston tme s much lower than the coherence tme of the channel fadng To ths end, each termnal must be equpped wth a spectrum sensng functon to estmate the nose varances and a channel estmaton functon to estmate ts channel varatons Then, by takng the dervatve of the modfed Lagrangan functon wth respect to φ j, we can solve the KKT system of the optmzaton problem: L((λ j j=n, (β j, φ j j=n N c = j ln T j w jt Γ( σ j,t h t (λ j β j h jk,t h t φ k φ (5 For transmtter j, the power allocaton s the soluton gven by the roots of (4 wth the nterference terms estmated at each recever wthn the network j For nstance, wth two recevers T j =, the power allocaton wthn the network j s gven by (0 wth the followng modfcatons: a = Γ(σj, h jk, φ k b = Γ(σj, h jk, φ k (6 Therefore, a dstrbuted power allocaton of N dfferent networks can be obtaned by updatng teratvely the powers of the dfferent transmtters usng the sngle-transmtter power allocaton for mnmzng the power subject to a mnmum rate constrant and a spectral mask constrant However, n Algorthm, the weght loop encompasses the outer loop to fnd whch set of weghts corresponds to the global mnmum power satsfyng a mnmum rate constrant R mn As the algorthm should be dstrbuted and autonomous, the set of weghts mnmzng the power have to be determned for each network ndependently To ths end, we have to move the weght loop nsde the outer loop by ntroducng a rule based on the rates An adequate rule s to ntroduce a devaton metrc (DM whch measures the dsperson of the rates The DM must be computed wthn each network j for each set of weghts n over the T j recevers The rule s gven by the followng formula: DM j (n = T j T j [(R jt (n T j T j R jt (n T j R jt (n ] (7 wth R jt (n the rate for the network j, recever t and the set of weghts n Ths rule allows to acheve the global mnmum power although the decson has to be taken nsde the outer loop It bascally means that for a gven power the closer the rates of the dfferent recevers wthn a network, the less power wll be needed to acheve the mnmum rate constrant Ths algorthm s referred to as Algorthm 3 n the smulaton results IV RESULTS In the frst set of smulatons, we compare the algorthm for a sngle tactcal rado network wth the trval case where the waterfllng s performed on the recever wth the worst channel condtons, e the worst recever strategy Note that the worst recever strategy can be seen as a specal case of the presented algorthm n whch w t = 0 t except for the worst recever The log-dstance path loss model s used to measure the path loss between the transmtter and the recevers [8], wth bandwdth f = 5 khz, N c = 4 sub-channels, carrer frequency f c = 80 MHz, path loss exponent n = 4, reference dstance d 0 = 0 meters and thermal nose σ n = 0 6 For the smulatons, we use a square area of km n whch the transmtter and T = recevers are placed randomly usng Monte Carlo trals The SNR gap for an uncoded quadrature ampltude modulaton (QAM to operate at a symbol error rate 0 7 s Γ = 98 db The scenaro consders a very strong nose (σ n = 0 9 seen on the 4 th sub-channel by the frst recever and on the st sub-channel by the second recever The dfferent noses seen by the dfferent recevers can be thought as sub-channel varatons dependng on the locaton, a subchannel occuped by a prmary or a secondary transmtter, a jammer etc The left part of Fgure shows the results of the power mnmzaton subject to a mnmum rate constrant per recever rangng from R mn = kbps to R mn = 5 kbps over 0 3 Monte Carlo trals for the locatons of the transmtter and the recevers Algorthm provdes a substantal gan compared to the worst recever strategy The rght part of Fgure shows that the algorthm converges wthn 30 teratons (the number of teratons for convergence manly depend on the startng pont, n ths case = 0 Snce t s based on closedform expressons, the algorthm has reasonable complexty for a low number of recevers as the search for the best set of weghts requre an exhaustve search over all possble weghts

5 5 (W Algorthm Worst Recever Mnmum Rate (kbps (W Algorthm Worst Recever Iteratons (Mnmum Rate=5 kbps (W Algorthm 3 Worst Recever Mnmum Rate (kbps Devaton Metrc (DM Algorthm Iteratons (Mnmum Rate=5 kbps Fg Results on the power mnmzaton subject to a mnmum rate constrant for a sngle tactcal rado network Fg 3 Results on the averaged power mnmzaton subject to a mnmum rate constrant averaged for the coexstence of two tactcal rado networks In the second set of smulatons, we compare the teratve waterfllng based algorthm developed n Secton III for N networks wth the worst recever strategy extended to multple networks Smulaton results are performed wth N = networks, T j = recevers j and N c = 4 sub-channels We consder a scenaro n whch all recevers see a dfferent nose σ n on ther N c = 4 sub-channels (smlar to the frst set of smulatons In the frst network, a very strong nose (σ n = 0 9 s seen on the 4 th sub-channel by the frst recever and on the st sub-channel by the second recever In the second network, a very strong nose (σ n = 0 9 s seen on the 3 th sub-channel by the frst recever and the nd sub-channel by the second recever The left part of Fgure 3 shows the results of the power mnmzaton subject to a mnmum rate constrant rangng from R mn = kbps to R mn = 5 kbps over 0 3 Monte Carlo trals for the locatons of the transmtter and the recevers In ths case, Algorthm 3 s the only strategy whch provdes a vable soluton because the worst recever strategy tends to utlze the maxmum avalable power of Watt Although desgned for the coexstence of multple tactcal rado networks, the convergence of Algorthm 3 s smlar to Algorthm for both networks (rght part of Fgure The rght part of Fgure 3 shows that the devaton metrc (DM reduces as the algorthm converges It can be seen that n practcal scenaros n whch the nterference temperature vares along the subchannel and the recever locatons, Algorthm 3 provdes an effcent dstrbuted strategy to fnd the power allocaton of multple networks n whch each transmtter has to broadcast a common nformaton to ts recevers V CONCLUSION In ths paper, dynamc spectrum management was studed for multple cogntve tactcal rado networks coexstng n the same area Frst, we have consdered the problem of power mnmzaton subject to a mnmum rate constrant and a spectral mask constrant for a sngle tactcal rado network wth multple recevers over parallel channels (parallel multcast channels Then, we have extended the teratve waterfllng algorthm to multple recevers for the coexstence of multple cogntve tactcal rado networks assumng knowledge of the nose varances and channel varatons of the network Smulaton results have shown that the proposed algorthm s very robust n satsfyng these constrants whle mnmzng the overall power n varous scenaros REFERENCES [] I Joseph Mtola and J Gerald Q Magure, Cogntve Rado: Makng Software Rados More ersonal, IEEE ersonal Communcatons, vol 6, no 4, pp 3 8, Aug 999 [] S Haykn, Cogntve Rado: Bran-Empowered Wreless Communcatons, IEEE Journal on Selected Areas n Communcatons, vol 3, no, pp 0 0, Feb 005 [3] T M Cover and J A Thomas, Elements of Informaton Theory, New York: John Wley & Sons, 99 [4] N Jndal and Z Q Luo, Capacty Lmts of Multple Antenna Multcast, n roc IEEE Int Symp Informaton Theory ISIT 06, Seatle, USA, Jul 006 [5] N D Sdropoulos, T N Davdson, and Z Q Luo, Transmt Beamformng for hyscal-layer Multcastng, IEEE Transactons on Sgnal rocessng, vol 54, no 6, pp 39 5, Jun 006 [6] A E Gamal, The Capacty of the roduct and Sum of Two nconsstently Degraded Broadcast Channels, roblemy erdach Informats, vol 6, no, pp 3 3, January-March 980 [7] A J Goldsmth and M Effros, The Capacty Regon of Broadcast Channels wth Intersymbol Interference and Colored Gaussan Nose, IEEE Transactons on Informaton Theory, vol 47, no, pp 9 40, Jan 00 [8] N Jndal, W Rhee, S Vshwanath, S Jafar, and A Goldsmth, Sum ower Iteratve Water-fllng for Mult-Antenna Gaussan Broadcast Channels, IEEE Transactons on Informaton Theory, vol 5, no 4, pp , Apr 005 [9] H Wengarten, Y Stenbergand, and S Shama, The Capacty Regon of the Gaussan Multple-Input Multple-Output Broadcast Channel, IEEE Transactons on Informaton Theory, vol 5, no 9, pp , Sep 006 [0] V Le Nr, M Moonen, J Verlnden, and M Guenach, Optmal power allocaton for downstream xdsl wth per-modem total power constrants : Broadcast Channel Optmal Spectrum Balancng (BC-OSB, IEEE Transactons on Sgnal rocessng, vol 57, no 5, pp , Feb 009 [] R H Gohary and T N Davdson, On ower Allocaton for arallel Gaussan Broadcast Channels wth Common Informaton, EURASI Journal on Wreless Communcatons and Networkng, Artcle ID 4850, 009 [] W Yu, Competton and Cooperaton n Mult-user Communcaton Envronments, hd Dssertaton, Jun 00 [3] Setoodeh and S Haykn, Robust Transmt ower Control for Cogntve Rado, roceedngs of the IEEE, vol 97, no 5, pp , May 009 [4] R H Gohary and T J Wllnk, Robust IWFA for open-spectrum communcatons, IEEE Transactons on Sgnal rocessng, vol 57, no, pp , Dec 009 [5] R G Gallager, Informaton Theory and Relable Communcaton, New York: John Wley & Sons, 968 [6] J M Coff, A Multcarrer rmer, ANSI Contrbuton TE4/9-57, Nov 99 [7] S Boyd and L Vandenberghe, Convex Optmzaton, Cambrdge Unversty ress, 004 [8] T S Rappaport, Wreless Communcatons: prncples and practce, rentce Hall, 996

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