Feasibility Conditions of Interference Alignment via Two Orthogonal Subcarriers

Transcription

1 Feasblty Condtons of Interference Algnment va Two Ortogonal Subcarrers Stefan Ders and Gerard Kramer Insttute for Communcatons Engneerng Tecnsce Unverstät Müncen, Munc, Germany Emal: {stefan.ders, Wolfgang Zrwas Noa Semens Networs Munc, Germany Emal: arxv:0.5v [cs.it] Mar 0 Abstract Condtons are derved on e-of-sgt cannels to ensure te feasblty of nterference algnment. Te condtons nvolve coosng only te spacng between two subcarrers of an ortogonal frequency dvson multplexng OFDM sceme. Te maxmal degrees-of-freedom are aceved and even an upper bound on te sum-rate of nterference algnment s approaced arbtrarly closely. I. INTRODUCTION Interference algnment IA s a promsng metod because t aceves ger trougput n nterference lmted scenaros tan conventonal metods suc as tme- or frequency-dvson multplexng or treatng nterference as nose []. Te man dea of IA s to use precodng at te transmtters to algn nterference at eac recever n one subspace. Te ortogonal subspace s used for nterference-free communcaton. One commonly measures performance by te sum of te rates tat te users can transmt relably. Te degrees-offreedom DoF are defned as C sum SNR d = lm SNR logsnr, were C sum SNR s te sum-rate capacty at te sgnal to nose rato SNR. Te DoF represent te number of nonnterferng data streams tat can be smultaneously transmtted over te networ. For sngle antennas IA aceves te maxmal DoF asymptotcally wt an nfnte number of subcarrers or tme-slots []. We derve condtons for wc two subcarrers mae IA feasble for general cannels n Secton III and for e-ofsgt cannels n Secton IV. For e-of-sgt.e. sngletap cannels tese condtons are fulflled by coosng te subcarrer spacng carefully, wle n pror art te subcarrers are assumed to be fxed wen IA s appled. Hence for e-ofsgt cannels we aceve te maxmal DoF and even aceve an upper bound on te sum-rate of IA arbtrarly closely. II. SYSTEM MODEL Consder an nterference cannel wt K user pars, were eac transmtter sends eter one or two streams to ts recever. Eac node s equpped wt a sngle antenna and uses te same two ortogonal subcarrers. Te receved sgnal n te frequency doman at recever s K y = U H,V s + U H,V s +U z =, were s s te vector of symbols at transmtter wt lengt D {,}, V s a D precodng matrx, H, s a cannel matrx n te frequency doman between transmtter and recever, U s a D receve flter matrx. U s te complex conjugate transpose of matrx U, wle u s te complex conjugate of scalar u. Te precodng and te receve flter matrces are cosen to satsfy V F and U F, were F denotes te Frobenus norm. z s a proper complex AWGN vector of lengt and varance σ. Te frst term on te rgt and sde of carres te data of recever, wle te sum represents te nterference, and te last term s fltered nose. Cannels connectng te transmtter and recever of te same user par are called drect cannels; te oter cannels.e. H, are called cross cannels. For ortogonal subcarrers te cannel matrces H are dagonal. Te dagonal entres are denoted by l, C, were l ndcates te subcarrer ndex. We wrte [ ] H, =, 0 0, =,e j, 0 0,e j,, were x denotes te ampltude ofxand x denotes te pase of x n radans. For e-of-sgt cannels te ampltudes are equal for all subcarrers, wle te pase rotatons depend on te delay τ, and te subcarrer frequences f and f : [ ] e H LoS jπf τ, 0, =,. 4 0 e jπf τ, Te ampltudes are bounded as 0 < l, to avod degenerate cannel condtons. We assume perfect cannel nowledge of all cannel parameters at all nodes. III. FEASIBILITY OF INTERFERENCE ALIGNMENT VIA TWO SUBCARRIERS For sngle antenna nodes, te DoF are upper-bounded by / per user par []. Te precoder and receve flters reduce

2 to vectors v and u and nterference s algned f [] u H,v = 0, 5 u H,v > 0,. 6 Te equatons 5 mean tat te nterference les n te nullspace of te receve flter, wle te equatons 6 ensure tat te effectve cannel = u H,v wc s nterferencefree f te frst set of equatons s fulflled as unt ran. Te queston of IA feasblty ass f tere s a soluton for u and v suc tat 5 and 6 are fulflled. Suppose tat all cannel coeffcents are cosen ndependently wt a contnuous dstrbuton. Te condtons 6 are fulflled wt probablty f te precoder and receve flters satsfy 5. Hence we need to examne te feasblty of 5 to sow tat te maxmal DoF are acevable. Te queston of feasblty s tacled, e.g., n []-[5]. In [] t s sown tat te maxmal DoF are asymptotcally acevable wt IA for tme-varyng cannels by ncreasng te number of symbol extensons.e. te number of subcarrers or tme slots. We sow tat for nterestng cannel condtons IA s feasble wt two subcarrers. For ts we use Lemma to wrte te IA equaton set 5 as te sum of logartms of te cannel, precoder, and receve flter coeffcent fractons. Wt Lemma we prove Teorem wc states feasblty condtons on te cannel coeffcents for te user pars case. In te followng subsecton we examne K user pars. Lemma. For sngle antenna nodes and two ortogonal subcarrers te IA condtons 5 are u v, + = jπ+n u v, 7, for all, were n, Z can be any nteger. Proof: We wrte 5 as te equaton set u, v +u, v = 0,. 8 Tere exst trval solutons of 8: u = 0 or v = 0, wc bot volate 6; u =v =0 or v =u =0, wc, wen examnng te equaton set, lead to te nvald solutons u = 0 or v = 0. Oter trval solutons wt u l =0 or v l =0 do not exst, snce we ave, 0 and, 0 recall tat l, > 0. Hence all u and v are non-zero for nontrval solutons. Manpulatng 8 we obtan and terefore were n, Z. u, v u, v u, v u, v = 9 = jπ+n, 0 Note tat, = 0 or, = 0 ave zero probablty for contnuous dstrbutons. A. User Pars Teorem. Tree DoF over two subcarrers are feasble for tree user pars wt sngle antennas f te followng condton olds,,,,,,,,,,, =., Proof: For tree users tere are sx cross-cannels. Accordng to Lemma sx equatons of type 7 must be satsfed. We wrte tese equatons n te form Ax = b as follows: u u jπ+n,,, u u jπ+n,,, u u jπ+n,, v =, v jπ+n,.,, v v }{{} jπ+n,,, RanA=5 jπ+n, v v,, Snce te ran of A s 5, wc s less tan te number of equatons, a soluton exsts f and only f te ran of te augmented matrx A b s equal to te ran of A or b s n te column space or mage of A. Ts condton s fulflled for,,,, +,, + = jπn,,,,,,, were n = n, n, +n, n, +n, n, Z. Teorem can be expressed as two equatons: One for te subcarrer ampltudes,,,,,,,,,,,, = 4 Te proof can also be obtaned by examnng te subspaces spanned by te cannel matrx and te precodng vector as s done n Secton IV-D of []. For nterference to algn one must ave spanh, v = spanh, v and spanh, v = spanh, v and spanh, v = spanh, v. From ts one obtans spanv = spantv, were T = H, H, H, H, H, H,. Due to te dagonal structure of te cannel matrces, T s also dagonal. Unless T s a scaled dentty matrx te precoder v must be an egenvector of all cannel matrces, leadng to nterference not beng algned. Settng T as a scaled dentty matrx leads to.

3 and one for te subcarrer pase rotatons, +, +,,, +, +,,, +, +,, = πn. 5 B. K User Pars For K user pars tere are KK cross-cannels and we ence ave KK equatons of type 7. We collect tem nto an equaton system Ax = b, were A s of dmenson KK K, but as ran K. Te augmented matrx A b agan must ave te same ran as A for a soluton to exst. Transformng A to rowecelon form by usng Gaussan elmnaton results n a new matrx A were te last KK K = K K + 6 rows are zero. We apply te same transformatons to b to obtan b. Te last K K + entres of b must be zero, and are of te form, α [w]. jπ+n,,, 7 were α [w]. {,0,} are te wegts of te w-t row. Hence we obtan K K + equatons of type smlar to wc must be fulflled for feasblty of IA. IV. SPECIAL CASE: USER PAIRS AND LINE-OF-SIGHT CHANNELS We examne IA for te specal case of e-of-sgt cannels and K =. We sow tat te feasblty condton of te cannel can be fulflled by coosng te subcarrer spacng carefully. We also derve te ampltudes of te effectve cannels and sow tat for ncreasng bandwdt an upper bound on te sum-rate of te presented sceme can be reaced arbtrary closely. Corollary. For e-of-sgt cannels te condton of Teorem smplfes to f f τ, τ, +τ, τ, +τ, τ, = n 8 were n Z\{0}. Proof: For sngle tap cannels te subcarrer ampltudes satsfy = and ence only te pase rotaton,, dfference remans. Insertng l, = πfl τ, gves π f f τ, +π f f τ, π f f τ, +π f f τ, 9 π f f τ, +π f f τ, = πn. After some manpulatons one obtans 8. Coosng n = 0 volates te assumpton of ortogonal sub-carrers, snce ts means f = f. Accordng to 8 e-of-sgt cannels may create condtons were IA s feasble by coosng te sub-carrer spacng f = f f carefully. Ts means tat te precodng and receve flter vectors can be cosen suc tat 5 olds. Te requred spacng depends only on te delays of te cross cannels and te non-zero nteger n wc can be cosen freely. Hence we can dentfy a mnmal sub-carrer spacng f mn = /τ, τ, +τ, τ, +τ, τ, 0 for wc IA s feasble. Any multple of f mn, except 0, creates feasblty agan. For te specal case τ, τ, +τ, τ, +τ, τ, = 0 IA s drectly feasble and te subcarrer spacng can be cosen arbtrarly. For contnuously and ndependently dstrbuted delays te probablty of ts event s zero and s not treated furter. Note tat we are not lmted to usng two subcarrers. Snce te feasblty depends solely on te spacng, subcarrer par f +f offset and f +f offset s feasble f par f and f s. Even dfferent user pars, wc requre dfferent f mn, can be sceduled n one OFDM frame. It mgt not be possble to use all subcarrers wt IA n wc case te remanng subcarrers are used as usual. A. Effectve Cannel Ampltudes If 8 s fulflled, te ratos of te precodng and receve flter coeffcents are obtaned from te system of ear equatons. Snce A s ran-defcent tere s one ndependent varable n x, wc we coose wtout loss of generalty to be u /u. Te remanng varables are determned as v v v v u u u u v v = jπ+n, + fτ, = jπ+n, + fτ, = jπ+n, + fτ, = jπ+n, + fτ, = jπ+n, + fτ, u u u u v u v v u u From -6 one obtans, for {,,}, v = u v. 7 u Togeter wt v F and u F one obtans u v /. 8 v v

4 For all else eld fxed te -t ampltude s largest f = v = u = u = / 9 v wc we use wen obtanng te ampltudes. Te ampltude of te frst drect cannel s = u H,v =, u e jπf τ, v +u e πf τ, v a =, b =, +e jπ fτ,+ u /u + v /v e jπ f τ,+τ, τ,+τ,+n, n,+n, c =, snπn f mn τ 0 were τ = τ, +τ, τ, +τ,. For a we used 9. For b we nserted 6, nto wc we nserted 4 and. For c we used f = n f mn, e jθ = snθ/ and snθ+πl = snθ for l Z. Te ampltudes of te second and trd drect cannels follow smlarly and are =, snπn f mn τ =, snπn f mn τ were τ = τ, +τ, τ, +τ, and τ = τ, + τ, τ, +τ,. Examnng te effectve cannel ampltudes, we observe tat te ampltude of te -t cannel s bounded by 0,. For a gven cannel one can nfluence only te nteger n of te argument of te sne functon, as te τ and te f mn are fxed. B. Upper Bound Te sum-rate of te proposed sceme for a tree user pars system wt e-of-sgt cannels s upper bounded by R sum log + σ. Snce te sum-rate s dfferent for dfferent coces of n, one can optmze te coce of f = n f mn wtn te avalable bandwdt to obtan te optmal sum-rate. Lemma. For contnuously and ndependently dstrbuted delays te upper bound on te sum-rate of te presented sceme s aceved arbtrarly closely for ncreasng bandwdt. Proof: Te mnmal sub-carrer spacng depends only on te delays and te delays are contnuously and ndependently dstrbuted. Hence also te products λ = f mn τ are contnuously dstrbuted. Tey are even ndependently dstrbuted, snce τ, appears only n τ. We can wrte λ mod wt ts nfntely long decmal expanson as λ mod = 0.λ [] λ [] λ []..., 4 were eac element λ [l] of te sequence s..d. and taes on te values {0,,,...9} wt equal probablty. We ws to sow tat n {Z : 0 < n < N} wt N suc tat nλ mod s arbtrarly close to some number µ 0,. We do ts by loong for strngs of decmal places of λ wc are equal for all and wc are, wen sfted to te frst decmal places, close enoug to te desred number µ. We ten coose n to sft te resultng sequence to te frst decmal places. We coose R Z suc tat 0 R < ǫ, were 0 < ǫ <. { Our goal s to fnd an r suc, tat te random } varables M r = λ [w] :,w = r,r +,...,r+r λ [r] λ [r+] fulfll te condton...λ [r+r ] = µ [] µ []...µ [R],, 5 were µ [w] s te w-t poston of te decmal expanson of µ. Te probablty tat te varables M r fulfll te condtons 5 for a gven r s postve. Tere are nfnte ndependent realzatons of te set M r, ence r suc tat te set M r fulflls condtons 5. We complete te proof by coosng n = 0 r and µ = /. Lemma ensures tat by ncreasng te bandwdt and optmzng te coce of f = n f mn we can get arbtrarly close to te upper bound of te presented sceme. C. Connecton to Tme Based Interference Algnment Tme based IA algns nterference by transmttng only n every oter tme slot and by possbly usng dfferent offsets. Interference s algned wen at te recevers te nterference arrves n te same tme slot, wle te useful sgnals arrve n a dfferent tme slot. Analyses of tme based IA can be found n [4], [6] or [7] for example. We sow tat tme based IA s a specal case of subcarrer IA. Coosng a precoderv n te frequency doman translates to te tme doman sgnal [ X [t] X [t+] ] = [ ] }{{} F [ ] v v s = v +v v v s s 6 were F s te IDFT matrx. Snce for tme based IA notng s transmtted n te second tme slot, we ave v v = 0. Tus v /v = 0, follows. In a smlar way we obtan u /u = 0,. Ts means tat te rgtand sde n must be b = 0, wc automatcally fulflls and ence and 8. From b = 0 t follows tat jπ+n, = = jπ fτ,, 7,, from were we obtan te condtons on te subcarrer spacng f = +n, τ,,. 8 For K = tere are sx fractons tat must be equal to eac oter and wc determne f. Te denomnators

5 of te fractons are real numbers wle te numerators are ntegers. Snce te delays are..d., equalty of tese fractons s approaced only by coosng larger nteger numerators. Ts means tat feasblty s aceved only asymptotcally for ncreasng f, wc translates to decreasng slot lengts n te tme doman. Ts s precsely wat Teorem n [4] states. But we are able to determne subcarrer spacngs wc aceve feasblty exactly for K =. Ts sows tat restrctng te coce of te precoder, as tme based IA does, probts acevng te DoF exactly. V. SIMULATION RESULTS Consder a user par e-of-sgt cannel, were te transmtter-recever dstances d, are contnuously and ndependently dstrbuted. Te delays are related to te dstances by τ, = c d, 9 were c s te speed of wave propagaton, wc we set to te speed of lgt c = 0 8 m/s. Te cannel ampltudes are obtaned from te dstances as γ m, = 40 d, were we coose te pat-loss exponent γ =.76. Te dstances of te drect cannels are dstrbuted as d, [50m,50m], and te dstances of te cross cannels asd, [50m, 50m],. Te drect cannels tus ave te largest ampltudes and we do not ave too small dstances for wc treatng nterference as nose wors best. We average over 0 4 cannel realzatons. As bencmar scemes we consder I treatng Interference as Nose and II an ortogonal access sceme, were we use TDMA. For treatng Interference as Nose, eac transmtter transmts two streams for every cannel use and at te recevers te nterference s treated as nose. For te TDMA sceme, eac transmtter transmts only n everyk-t slot, but wt K tmes te power. Snce only one par communcates per slot, te recever can receve two streams wtout nterference. To obtan te precoder and receve flter for IA, we use te pseudo-nverse of A to obtan a soluton or a least-squares soluton, f IA s nfeasble for te system of ear equatons. Snce we are nterested manly n te DoF, we consder only nterference-zero-forcng approaces. Oter approaces, e.g. MaxSINR or MMSE, wll be examned n future wor. Te values of f mn seem to be Rayleg-dstrbuted, were more tan95% of te values are between0 6 Hz and0 8 Hz for te consdered scenaro. Tese values depend strongly on te dstances and te speed of wave propagaton. For ncreasng dstances or decreasng c e.g. under-water communcaton te dstrbuton of f mn s sfted to lower frequences. Fgure sows te average sum-rate of te bencmar scemes and of IA for an average receved SNR from te drect cannels of 0dB. Te x-axs s normalzed to / f mn, were f mn s dfferent for every cannel realzaton. As expected, te bencmar scemes perform ndependent of te subcarrer spacng. For IA we plot tree curves. Te curve labeled IA ZF s te average sum-rate wt te current subcarrer spacng. As expected, we observe peas at multples of f mn. Note tat for small devatons from te optmal f mn tere are small reductons n sum-rate. A subcarrer spacng between multples of f mn leads to leaage nterference, wc prevents acevng te maxmal DoF. But for fnte SNR we aceve a good performance wen te drect cannel s ampltude s large. Te curve labeled Max IA ZF s obtaned n two steps: For eac cannel realzaton te maxmal sum-rate wtn te bandwdt equal to te x-axs value s determned. In te next step we tae te average and obtan te curve labeled Max IA ZF. A steep ncrease of ts curve can be observed around f mn due to te feasblty of IA. Wt ncreasng bandwdt te curve labeled Max IA ZF approaces te curve labeled IA Upper Bound, wc s te average of te upper bounds. Average sum rate n bts per use TDMA Interference as Nose IA ZF Max IA ZF IA Upper Bound f/ f mn Fg.. Average sum-rate for randomly dstrbuted dstances, were d, [50m,50m] and d, [50m,50m], and γ =.76 and te average receved SNR from te drect cannels s 0dB. VI. CONCLUSIONS We derved condtons for feasblty of IA va two ortogonal subcarrers. For e-of-sgt cannels tese condtons can be fulflled by carefully coosng te subcarrer spacng. ACKNOWLEDGMENT S. Ders and G. Kramer were supported by te German Mnstry of Educaton and Researc n te framewor of an Alexander von Humboldt Professorsp. REFERENCES [] S. A. Jafar, Interference Algnment A New Loo at Sgnal Dmensons n a Communcaton Networ. Foundatons and Trends n Communcatons and Informaton Teory, 00, vol. 7, no.. [One]. Avalable: [] V. R. Cadambe and S. A. Jafar, Interference Algnment and Degrees of Freedom of te K-User Interference Cannel, IEEE Trans. Inf. Teory, vol. 54, no. 8, pp , Aug. 008.

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