Optmal Spectrum Management n Multuser Intererence Channels Yue Zhao, and Gregory J. Potte Department o Electrcal Engneerng Unversty o Calorna, Los Angeles Los Angeles, CA, 90095, USA Emal: yuezhao@ucla.edu, potte@ee.ucla.edu Abstract In ths paper, we nvestgate the optmal spectrum management problem n multuser requency selectve ntererence channels. Frst, a smple parwse ntererence couplng condton under whch FDMA can acheve all Pareto optmal ponts o the rate regon s dscovered. Not only s ths condton sucent, we show that t s also necessary or FDMA to be always optmal at least n symmetrc channels. For the general cases where ths condton s not necessarly satsed, we rst explctly obtan the optmal soluton as the optmal combnaton o lat FDMA and lat requency sharng or the sum-rate maxmzaton problem n two user symmetrc lat channels, and then show that the general n-user weghted sum-rate maxmzaton n non-symmetrc requency selectve channels can be ormulated nto prmal doman convex optmzatons. I. INTRODUCTION We consder the scenaro o multple multcarrer communcatons systems contendng n a common requency band, n whch ntererence couplng between derent users remans a major problem that lmts the multuser perormance. We nvestgate the optmal spectrum and power allocaton that acheves any Pareto optmal pont o the achevable rate regon, under the assumpton that ntererence s treated as nose at the recevers. There are essentally two strateges or multple users to co-exst: FDMA and requency sharng (overlappng). As the cross couplng vares rom beng extremely strong to extremely weak, the preerable co-exstence strateges ntutvely sht rom complete avodance (FDMA) to pure requency sharng. We start rom the strong couplng scenaro, and nvestgate the weakest ntererence condton under whch FDMA s stll guaranteed to be optmal. In the lterature, a relatvely strong parwse couplng condton or FDMA to be optmal was proved, and t apples to all Pareto optmal ponts o the n-user rate regon [5]. By parwse we mean that whether two users should avod each other only depends on the ntererence condton between those two users. For one typcal Pareto optmal pont whch s the sum-rate maxmzaton pont, the requred couplng strengths or FDMA to be optmal are urther lowered [6]. However, ths condton s a group-wse one, meanng that the couplngs between all exstng users are requred to be strong or FDMA to be provably sum-rate optmal. We relax these condtons and obtan the weakest possble parwse condton or FDMA to be optmal: or any two users, as long as the two normalzed cross couplngs between them are both larger than or equal to /, all n-user Pareto optmal ponts are guaranteed to be achevable wth FDMA between these two users. When the ntererence couplng s less than / n symmetrc channels, we gve a precse characterzaton o the non-empty power constrant regon wthn whch requency sharng between two users leads to a hgher rate than an FDMA between them. Thus, the proposed condton or FDMA to be always optmal s not only sucent, but also necessary. Wth the ntererence couplng less than /, the weghted sum-rate maxmzaton s n the orm o a non-convex optmzaton and generally hard to solve [9]. However, the Lagrangan dual problem s decomposed n requency and easer to solve [4][0]. It s shown n the lterature that the dualty gap goes to zero when the number o sub-channels goes to nnty [0]. Ths justes the asymptotc optmalty o solvng the problem n the dual doman, and many spectrum balancng algorthms usng dual methods have been developed [3] [4] [0]. We approach ths general non-convex optmzaton rom the prmal perspectve. We start wth the sum-rate maxmzaton problem n two-user symmetrc lat channels, and obtan analytcally the optmal soluton by combnng FDMA and requency sharng n an optmal way. By generalzng ths method, we show that all the general n-user arbtrarly weghted sum-rate maxmzaton n non-symmetrc requency selectve channels can be ormulated nto equvalent prmal doman convex optmzatons. As wll be shown at the end, t also drectly mples the zero dualty gap theorem n the lterature [0]. In retrospect, the methodology we provde shares some common nsght wth the tme sharng condton dscussed n [0].
TABLE I PROBLEMS, PRIOR WORK, AND RELATED SECTIONS IN THIS PAPER Problems Pror Work Our Results Spectrum Management n Cooperatve Scenaros Strong Intererence Condtons or FDMA schemes to be optmal [5][6] Secton III Scenaros Fndng optmal schemes wth FDMA constrants [6][8] General Contnuous Prmal doman soluton: Equvalent Convex Formulaton Secton IV Intererence Frequency Scenaros Dual doman methods [0] Scenaros Dscrete Frequency Scenaros: Approxmaton Algorthms [3][4][7][0] Spectrum Sharng n Non-cooperatve Scenaros: Nash Equlbrums [5][9] Table I summarzes the varous orms o the multuser ntererence channel co-exstence problems, the pror work, and n whch sectons we present solutons that mprove upon these pror results. We suggest uture research drectons n the concluson. Due to space lmtatons, proo detals whch can be ound n [] are omtted here n avor o explanng the sequence o results and ther sgncance. II. CHANNEL MODEL AND TWO BASIC CO-EXISTENCE STRATEGIES An n-user ntererence channel s modeled by y = Hx + xjh + n, =,,..., n, where x j s the transmtted sgnal o user, and y s the receved sgnal o user ncludng addtve Gaussan nose n (a user corresponds to a par o transmtter and recever). H are the drect channel gans, whereas H are the cross couplng gans. We assume that the channel s requency selectve over the band (, ), where W s the total bandwdth. The channel gans are denoted by H ( ) and H ( ). The transmt power spectrum densty (PSD) o user s denoted by P ( ), and the nose PSD at recever by σ ( ). We assume that ntererence s treated as nose and random Gaussan codebooks are used. The achevable rate or user P( ) H( ) s R = log ( + ) d. σ ( ) + P ( ) H ( ) j j Normalzng the channel gans and nose power by the drect channel gans, we have P ( ) R = log ( + ) d, N ( ) + P ( ) α ( ) j j σ ( ) H ( ) where N ( ) and α ( ). H ( ) H ( ) To acltate analyzng the optmal spectrum management scheme, we ntroduce two basc co-exstence strateges: Flat Frequency Sharng and Flat FDMA, both dened n lat channels. These two strateges are the buldng blocks o all non-lat co-exstence strateges n requency selectve channels, and wll be used to establsh general condtons or FDMA to be optmal. Consder a two-user lat channel: (, ), N( ) = n, N ( ) = n, α( ) = α, α ( ) = α, () a lat requency sharng scheme o two users s dened as any power allocaton n the orm o P( ) = p, P( ) = p, (, ); () a lat FDMA scheme o two users s dened as any power allocaton n the orm o P( ) P( ) = 0 and P( ) + P( ) = p, (, ). Next, we dene the lat FDMA reallocaton to be the ollowng power nvarant transorm that reallocates a lat requency sharng scheme to be a lat FDMA scheme: user reallocates ts power wthn a sub-band W = ( p/ p + p) W wth a lat PSD p = p + p ; user reallocates ts power wthn another dsjont sub-band W = ( p / p + p ) W wth the same lat PSD p = p+ p. Illustratons o the power allocatons o the two basc co-exstence strateges beore and ater a lat FDMA reallocaton are depcted n Fg.. Smlarly, lat requency sharng schemes, lat FDMA schemes, and lat FDMA reallocaton n n-user lat channel cases can be dened. III. STRONG INTERFERENCE SCENARIO: THE CONDITIONS FOR THE OPTIMALITY OF FDMA In ths secton, we nvestgate the condtons under whch PSD p p W Flat Frequency Sharng Flat FDMA (Beore lat FDMA reallocaton) (Ater lat FDMA reallocaton) Fg.. Power allocatons o lat requency sharng and lat FDMA, also an llustraton o lat FDMA reallocaton.
the optmal spectrum and power allocaton s FDMA, and our objectve s to encompass all Pareto optmal ponts. Frstly, we show a couplng condton under whch FDMA s optmal wthn a group o strongly coupled users. We then show that ths couplng condton also works when there are other users that are not strongly enough coupled. The two basc co-exstence strateges serve as a powerul tool n provng the general condton or the optmalty o FDMA. We begn wth two-user lat channels. Theorem : Consder a two-user lat ntererence channel (). Suppose the two users co-exst n a lat requency sharng manner (). I α / and α /, then wth a lat FDMA power reallocaton, both users rates wll be hgher (or unchanged). Proo : See [], secton III.A. Theorem can be generalzed to n-user cases n requency selectve channels []. We summarze these results as ollows: pck any sub-band (, ), as long as all the users havng power wthn ths sub-band are strongly coupled wth α ( ) /, j, (, ), then or any power allocaton scheme havng requency sharng happenng anywhere wthn ths sub-band, there always exsts an FDMA power reallocaton scheme (wth the total power unchanged or each user) that leads to a rate hgher than or equal to the orgnal sharng scheme or every exstng user. We have shown the condton or FDMA schemes to be optmal wthn strongly coupled users. In real communcaton networks, however, there are usually users not strongly enough (maybe just moderately) coupled wth some other users. For these users outsde the strongly coupled group, we show that they always benet rom an FDMA wthn the strongly coupled group. We begn wth two-ntererer lat channels. Theorem : Consder a three-user (one user + two ntererers) lat channel: N( ) = n, α ( ) = α. Suppose the three users co-exst n a lat requency sharng manner: P( ) = p, (, ), = 0,,. From user 0 s perspectve, a lat FDMA power reallocaton o ts two ntererers user and user always leads to a hgher (or equal) rate or user 0. Proo: See [], secton III.B. Theorem can be generalzed to an arbtrary number o users n requency selectve channels, provng that an FDMA wthn a subset o users s always preerred by every user who s not n ths subset, and ths s true or all couplng condtons []. In the case that, j, α ( ) / and α ( ) /, combnng Theorem and Theorem gves us a very strong nsght nto the condtons under whch the optmal co-exstence strateges must be FDMA: Suppose there are n ( ) users, or any two users and j among them, or any requency band (, ), the normalzed cross couplng gans α ( ) / and α ( ) /, (, ), then j j no matter rom whch o the n users pont o vew, an FDMA o user and user j wthn ths band s always preerred. Ths parwse condton s very convenent to use because t makes determnng whether any two users should be orthogonally channelzed depend only on the couplng condtons between the two o them. On the other hand, snce ths condton guarantees that an FDMA between user and user j benets every exstng user, we conclude (wth an mmedate proo by contradcton) that under ths condton, all the Pareto optmal ponts o the n-user achevable rate regon can be acheved wth these two users beng orthogonalzed (FDMA). In ths secton, we have shown that ths condton s sucent. In Secton IV, we show that t s also necessary,.e. t cannot be urther weakened. IV. GENERAL INTERFERENCE SCENARIO: OPTIMAL SPECTRUM MANAGEMENT IN FREQUENCY SELECTIVE CHANNELS In ths secton, we contnue to analyze the optmal spectrum management n the cases n whch α ( ) can be less than /. We rst analyze two-user symmetrc lat channels, and then extend our results to the general cases. (Smlar results or the symmetrc lat channel case have also been ndependently developed n [].) A. Soluton o Sum-rate Maxmzaton n Two-user Symmetrc Flat Channels wth Equal Power Constrants Consder the sum-rate maxmzaton problem n a two-user symmetrc lat Gaussan ntererence channel: α ( ) = α( ) = α, N( ) = N ( ) = n, (, ) (3) Frst, we have the ollowng theorem on the sucent and necessary condton or a lat FDMA scheme to be better than a lat requency sharng scheme. Theorem 3: For any lat requency sharng power allocaton, a lat FDMA power reallocaton (Fg. ) leads to a hgher or unchanged sum-rate and only ( p + p )/ n /α / α. ( ) Proo: See [], secton VI.A. Dene the crtcal pont p ( 0 = /α / α). Clearly, whenα < /, p0 > 0, and wthn the non-empty trangular power regon 0 < p+ p < np0 lat requency sharng s better than lat FDMA (whch s the optmal FDMA scheme n lat channels). It thus shows the necessty o the condton α / or FDMA to be always optmal. Next, we mpose an equal power constrant P/ or both users. The optmzaton problem becomes: P max R+ R, st.. P( ) d, P( ) 0, =,. (4) We normalze the sgnal power by the nose power and let n =. Dene an average densty p = P/ W. Then the maxmum achevable sum-rate wth lat requency sharng s
6 5 4 3 50 60 70 80 90 00 0 0 0 0 50 00 50 00 50 Fg.. The maxmum achevable sum-rate as the convex hull o the rates o lat FDMA and lat requency sharng, 0. α =. * p / ( p) = W log + []. The maxmum + α p / achevable sum-rate wth FDMA s h * ( p) = W log( + p). Dene r( p) = max( ( p), h ( p)). It can be vered that * ( p ) and h * ( p ) ntersects at p = p0 (drectly mpled by Theorem 3), and r( p ) s not concave n [0, ). Next, dene r * ( p ) to be the unque convex hull o r( p ). A typcal plot o * ( p ), h * ( p ), and the convex hull * ) Dene the set o unctons r ( p ) when α < / s S = { r ( P( ); ) r ( P( ); ) concave n P( ); gven n Fg.. When the power constrant p alls between r ( P( ); ) r( P( ); ), P( ) 0} the two ponts o tangency p and p h on the convex hull, ) r * ( P( ); ) s the unque uncton satsyng r * ( p ) can be acheved by applyng lat FDMA and lat * requency sharng n dsjont sub-bands respectvely []. r ( P( ); ) S Next, we show n the ollowng theorem that r ( p ) s not r ( P( ); ) r ( P( ); ), P( ) 0, r ( P( ); ) S only achevable, but also an upper bound (Proo by Jensen s Replacng the orgnal non-concave rate densty uncton nequalty n contnuous requency []), and hence optmal. r( P ( ); ) n (6) by ts convex hull r * ( P( ); ) at every Theorem 4: In a two-user symmetrc lat Gaussan requency pont, we obtan the ollowng convex ntererence channel (3), the maxmum achevable sum-rate optmzaton: wth power constrant P/ (4) or both users s r * ( p ). * max r ( ( ); ) d Proo: See [], secton VI.A. P ( ), =,,..., n P (7) The major mplcatons o Theorem 4 are as ollows []: ) The maxmum achevable sum-rate n ths case s s.. t P( ) d P, P( ) 0, (, ) computable. In act, ater solvng the two ponts o tangency Clearly, (7) has an optmal value that upper bounds that o as n Fg., we have an analytc expresson or the maxmum the orgnal problem (6), because the convex hull sum-rate as a concave uncton o the power constrant. r * ) The optmal (potentally requency selectve) spectrum ( P ( ); ) upper bounds r( P ( ); ) tsel at every and power allocaton s a combnaton o lat requency sharng and lat FDMA n two dsjont bands, combned accordng to where the power constrant p les on the curve o * ( ) r p. ) In lat channels, the convex hull o any achevable sum-rate uncton (as a uncton o power constrants) s also 4.8 4.7 4.6 4.5 4.4 4.3 4. achevable. B. Prmal Doman Convex Optmzaton Formulaton or General Frequency Selectve Intererence Channels We now consder the general weghted sum-rate maxmzaton n n-user non-symmetrc requency selectve channels wth arbtrary ndvdual power constrants: max wr, P ( ), =,,..., n n =.. P( ) P, P( ) 0, (, ) s t d where P ( ) = ( P ( ),..., Pn ( ) ) are P = ( P P ) (5), and the power constrants,..., n. Dene the rate densty uncton as n P ( ) r( P( ); ) w log( + ). = N ( ) + P ( ) α ( ) Problem (5) can then be rewrtten as max r( P( ); ) d P ( ), =,,..., n j j s.. t P( ) d P, P( ) 0, (, ) At every requency pont, r( P ( ); ) s a non-concave uncton o P ( ), makng optmzaton non-convex and hard. Now, we dene r * ( P ( ); ) as the convex hull o r( P ( ); ) along the n dmensons o users power: requency pont. On the other hand, by treatng every requency pont as an nntesmal lat channel, any achevable objectve value or (7) s also achevable or the orgnal non-convex one (6). We then have the ollowng theorem: Theorem 5: The convex optmzaton (7) has the same optmal value as the orgnal non-convex optmzaton (6). (6)
Proo: See [], secton V. From Theorem 5, we see that the orgnal non-convex optmzaton (6) can be transormed n the prmal doman to convex optmzaton (7) wthout loss o optmalty. The optmal spectrum and power allocaton o (7) can be transormed to that o (6) accordng to a weghtng uncton wth whch the ponts on r( P ( ); ) are weghted averaged (convexly combned) to be those on r * ( P ( ); ) []. Furthermore, we now show that Theorem 5 drectly leads to the zero dualty gap result n the lterature [0]. For the prmal problem (6), the Lagrange dual s L ( P ( ), λ T ) = r ( P ( ); ) d λ ( P ( ) d P ). The dual objectve s g( λ) = sup L( P( ), λ ). P( ) 0 For the prmal problem (7), the Lagrange dual s L * ˆ( P ( ), λ ) ( ( ); ) T = r P d λ ( ( ) d ) P P. The dual objectve s gˆ( λ) = sup Lˆ ( P( ), λ ). P( ) 0 Snce r * ( P( ); ) r( P( ); ), P ( ),, we have L ˆ( P( ), λ) L( P( ), λ ) and gˆ( λ) g( λ) always. Denote the prmal optmal values or (6) and (7) by p and ˆp, and the dual optmal values by d * = mn g( λ ) st.. λ 0 and ˆ* d = mn gˆ ( λ) st.. λ 0. From gˆ( λ) g( λ ), λ 0, we get ˆd d. From weak dualty or (6), p * d *. Snce problem (7) s a convex optmzaton, t has strong dualty * ˆ* ˆp = d []. Thus ˆp = dˆ d p. From Theorem 5, (6) and (7) are equvalent, and thus ˆp * = p *. Thereore, the orgnal non-convex optmzaton (6) also have strong dualty (zero dualty gap),.e. p = d. V. CONCLUDING REMARKS In ths paper, we have analyzed the optmal spectrum and power allocaton n all couplng condtons. We have shown that or any two users, as long as the two normalzed cross couplngs between them are both larger than or equal to /, an FDMA between these two users benets every exstng user, and hence can be used to acheve any Pareto optmal pont o the n-user achevable rate regon. Because ths ntererence condton has a parwse nature, t leads oreseeably to dstrbuted mplementaton. Ths condton cannot be urther lowered as shown n two user symmetrc lat channels. For the sum-rate maxmzaton problem n ths case wth equal power constrants, we analytcally obtaned the optmal spectrum and power allocaton whch has a clear ntuton o combnng lat FDMA and lat requency sharng n an optmal way. For the general n-user weghted sum-rate maxmzaton problems n requency selectve channels, we generalzed our nsght rom the lat channel case and ormulated the orgnally non-convex optmzaton nto an equvalent prmal doman convex optmzaton by replacng the non-concave objectve uncton at every requency pont wth ts convex hull. Ths result provdes the perormance lmt and a new perspectve nto optmal algorthm desgns n spectrum management. Ths paper has worked on the contnuous requency doman problems, and hence has nnte-dmenson varables. The deas can be appled to dscrete requency spectrum management va approxmaton. Wth the new nsghts we obtaned or ths optmzaton problem, the desgn o novel practcal algorthms to approach the optmal spectrum management s an nterestng uture research drecton. We note n partcular that whle t s n general dcult to compute hgh dmensonal convex hull unctons, prelmnary nvestgatons reveal that ether pure FDMA or pure sharng strateges suer lttle loss compared to the optmal scheme over a consderable range o the couplng parameter α. Consequently, relatvely smple strateges and algorthms may suce n practce. REFERENCES [] Bhaskaran, S.R.; Hanly, S.V.; Badruddn, N.; Evans, J.S., "Maxmzng the sum rate n symmetrc networks o ntererng lnks", Inormaton Theory and Applcatons Workshop, 009, Feb. 8 009 - Feb. 3 009 [] Boyd, S.P., Vandenberghe, L., "Convex Optmzaton," Cambrdge Unversty Press, 004. [3] Cendrllon, R.; Janwe Huang; Mung Chang; Moonen, M., "Autonomous Spectrum Balancng or Dgtal Subscrber Lnes," Sgnal Processng, IEEE Transactons on, vol.55, no.8, pp.44-457, Aug. 007. 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