Interference Alignment and the Generalized Degrees of Freedom of the X Channel

Transcription

1 Iterferece Aligmet ad the Geeralized Degrees of Freedom of the X Chael Chiachi Huag, Viveck R. Cadambe, Syed A. Jafar Electrical Egieerig ad Computer Sciece Uiversity of Califoria Irvie Irvie, Califoria, USA {chiachih, vcadambe, syed}@uci.edu Abstract We study the sum capacity of the X chael geeralizatio of the symmetric -user iterferece chael. I this X chael, there are idepedet messages, oe from each trasmitter to each receiver. We characterize the sum capacity of a determiistic versio of this chael, ad obtai the geeralized degrees of freedom characterizatio for the Gaussia versio. The regime where the X chael outperforms the uderlyig iterferece chael is explicitly idetified, ad a iterestig iterferece aligmet scheme based o a cyclic decompositio of the sigal space is show to be optimal i this regime. I. INTRODUCTION The X chael is a geeralizatio of the -user iterferece chael with idepedet messages, oe from each trasmitter to each receiver. It holds special sigificace for a umber of reasos: ) Iterferece Aligmet: It is the smallest (i terms of the umber of odes) etwork where the ewly discovered cocept of iterferece aligmet [], [] becomes relevat. ) Structured Codes: The X chael is related to the double dirty multiple access chael [] where liear codes have bee show to outperform radom codes. This is because, like the double dirty multiple access chael, each trasmitter i a X chael has side-iformatio about its ow potetially iterferig trasmissios to the other receiver. I this paper, we explore the capacity of the X chael as a steppig stoe to a improved uderstadig of the role of iterferece aligmet ad structured codig i wireless etworks. For simplicity of expositio, we choose as a baselie the symmetric iterferece chael model ad study its X chael extesio. We idetify the regime where iterferece aligmet is helpful so that the X chael has a higher capacity tha the uderlyig symmetric iterferece chael. The iterferece aligmet is accomplished usig a liear codig scheme based o a cyclic decompositio of the sigal space which is of iterest i ad of itself. Etki, Tse, ad Wag [] itroduced the otio of geeralized degrees of freedom (GDOF) to study the performace of various iterferece maagemet schemes. As its ame suggests, the idea of GDOF is a geeralizatio of the cocept of degrees of freedom. Ulike the covetioal degrees of freedom perspective where all sigals are approximately equally strog i the db scale, the GDOF perspective provides a richer characterizatio by allowig the full rage of relative sigal stregths i the db scale. A useful techique i the characterizatio of the GDOF of a wireless etwork is the determiistic approach [5]. The determiistic approach essetially maps a Gaussia etwork to a determiistic chael, i.e, a chael whose outputs are determiistic fuctios of its iputs. The determiistic chael captures the essetial structure of the Gaussia chael, but is sigificatly simpler to aalyze. Referece [6] showed that the determiistic approach leads to a GDOF characterizatio of the -user iterferece etwork, which leads to a costat bit approximatio of its capacity. I this paper, usig a determiistic approach, we characterize the sum capacity of a determiistic X chael, ad obtai the GDOF characterizatio for the Gaussia versio. Although the models are simplified to symmetric settig, our results provide a iterestig view of the structural differece betwee the X chael ad the iterferece chael. I terms of GDOF, both chaels perform equally well whe iterferece aligmet is ot applicable. But whe iterferece aligmet is applicable, the X chael has larger GDOF ad, therefore, higher capacity tha the iterferece chael. The rest of the paper is orgaized as follows. Sectio II describes the models. Sectio III summarizes our mai results. I Sectio IV, we explore the sum capacity of the determiistic X chael. Sectio V provides the GDOF characterizatio of the symmetric Gaussia X chael. II. SYSTEM MODEL A. The Symmetric Determiistic X Chael The symmetric determiistic X chael is described by the iput-output equatios Y (t) S q X (t)+s q c X (t) () Y (t) S q c X (t)+s q X (t) () where, c Z +,q max(, c ), X i (t), Y i (t) F q for i,, ad S is a q q shift matrix. The determiistic X chael is physically the same chael as the determiistic iterferece chael itroduced i [6], except that the X chael has idepedet messages. Please refer to [6] for the illustratio of the determiistic chael. The message set

2 ad stadard defiitios ad otatios of the achievable rates are similar to those i the Gaussia settig give i the ext subsectio. To avoid cofusio, sometimes we add the subscript det to distiguish the otatios of the determiistic chael from those for the Gaussia chael. W,W X " Z Y W, W B. The Symmetric Gaussia X Chael The symmetric -user Gaussia X chael is described by the iput-output equatios Y (t) H d X (t)+h c X (t)+z (t) () Y (t) H c X (t)+h d X (t)+z (t) () where at symbol idex t, Y j (t) ad Z j (t) are the chael output symbol ad additive white Gaussia oise (AWGN) respectively at receiver j. H c ad H d are the chael gai coefficiets, ad X i (t) is the chael iput symbol at trasmitter i. All symbols are real ad the chael coefficiets do ot vary with time. I the remaider of this paper, we suppress time idex t if o cofusio would be caused. The AWGN is ormalized to have zero mea ad uit variace ad the iput power costrait is give by E [ Xi ], i,. (5) There are idepedet messages i the X chael: W,W,W,W where W ji represets the message from trasmitter i to receiver j. We idicate the size of the message by W ji. For codewords spaig T symbols, rates log Wji R ji T are achievable if the probability of error for all messages ca be simultaeously made arbitrarily small by choosig a appropriate large T. The capacity regio C of the X chael is the set of all achievable rate tuples R (R,R,R,R ). We idicate the sum rate ad the sum capacity of the X chael by R Σ ad C Σ respectively. ) Geeralized Degrees of Freedom (GDOF): The same problem formulatio has bee give i [7], ad we iclude it for the sake of completeess. To motivate our problem formulatio, we briefly revisit the framework for the GDOF characterizatio of the symmetric iterferece chael. The iterferece chael is defied as: Y (t) SNRX (t)+ INRX (t)+z (t) (6) Y (t) INRX (t)+ SNRX (t)+z (t) (7) ad with the parameter α defied as follows α log(inr) log(snr) the GDOF metric is defied as [], (8) C Σ (SNR,α) d(α) lim sup SNR log(snr) (9) where C Σ (SNR,α) is the sum capacity of the iterferece chael. Sice our goal is to compare GDOF of the X chael with the iterferece chael, we use the same symmetric iterferece chael model described above as the physical W,W X Fig.. " Z -user Gaussia X chael. Y W, W chael model for the X chael. There is however, oe otatioal differece. Sice the termiology SNR, INR is ot as appropriate for the X chael, we istead use the parameter ρ to substitute for these otios, resultig i the followig system model for the X chael GDOF characterizatio: Y (t) ρx (t)+ ρ α X (t)+z (t) (0) Y (t) ρ α X (t)+ ρx (t)+z (t) () I other words, we have set H d ρ, H c ρ α. Note that (0), () represet the same physical chael as (6), (7). However, as metioed earlier, ulike the iterferece chael the X chael has idepedet messages: oe from each trasmitter to each receiver. The GDOF characterizatio for the X chael is defied as: d(α) lim sup ρ C Σ (ρ, α) log(ρ) () where C Σ (ρ, α) is the sum capacity of the X chael. Note that we use lim sup to esure that d(α) always exits. The half i the deomiator is because all sigals ad chael gais are real. III. MAIN RESULTS The first mai result of the paper is the characterizatio of the sum capacity of the symmetric determiistic X chael. The result is give i the followig theorem. Theorem : The sum capacity C Σ ( c, ) of the symmetric determiistic X chael is give as follows. c, 0 c < / c, / c < / ( c), / c < C Σ ( c, ), c () ( c ), < c /, / < c c, c > The key features of the sum-capacity-achievig scheme for the regimes of / c < ad < c / are show i Fig.. It ca be easily see that all iterferece is aliged ad that all iteded messages ca be recovered after cacelig both the decoded messages ad the decoded sums of the aliged iterferece.

3 d ( ) GDOF of X chael GDOF of iterferece chael obtaied by W W " GDOF of iterferece chael obtaied by W W " 5 Fig.. Geeralized degrees of freedom of the -user Gaussia X chael. Fig.. Sum-capacity-achievig scheme for the symmetric determiistic X chael with ( c, ) (, 5) The secod mai result of the paper builds upo the result of Theorem to fid the GDOF characterizatio of the Gaussia X chael. Theorem : The geeralized degrees of freedom d(α) of the symmetric Gaussia X chael is show i Fig. ad give as follows. α, 0 α< / α, / α< / α, / α< d(α), α () α, <α /, / <α α, α > It ca be easily see that the X chael outperforms the uderlyig iterferece chael i the regimes of / α< ad <α /. IV. SUM CAPACITY OF THE SYMMETRIC DETERMINISTIC CHANNEL The goal of this sectio is to prove Theorem. We start from the followig lemma, which follows trivially from the symmetry i the X chael. Lemma : C Σ ( c, )C Σ (, c ). A. Upperboud The followig lemma provides a set of outer bouds for all achievable rate tuples R of the symmetric determiistic X chael. Theorem : All achievable rate tuples (R,R,R,R ) satisfy R + R + R max(, c )+( c ) + (5) R + R + R max( c, )+( c ) + (6) R + R + R max(, c )+( c ) + (7) R + R + R max( c, )+( c ) + (8) R Σ max( c, ( c ) + ) (9) R Σ max(, ( c ) + ) (0) Proof: Iequality (5) to (8) are the upperbouds of the sum capacity for the four Z chaels cotaied i the X chael respectively. Iequality (9) ad (0) ca be see as the X chael extesios of the boud give i (9.c) of [8]. Please see the full paper [9] for the detailed proof i a more geeral asymmetric settig. It is easy to verify that (9) ad (0) lead to the tight upperbouds of the sum capacity i the regimes of 0 c < / ad c > / respectively. Addig (5) to (8) ad dividig both sides by, we have R Σ max ( c, )+ ( c ) + + ( c ) +. () The tight upperboud of the sum capacity for c follows from () ad the multiple access boud that if c, we have R Σ. B. Achievable Scheme The achievable scheme for c < c ad > follows trivially from [6]. We provide a outlie of the achievability proof for c. We start from the followig lemma. Lemma 5: Let ( c, ) Z + such that c <. The ) If c is divisible by, the there exists a V F c such that rak ([ V S c V S c V V ull ]) d where V ull is a ( c ) matrix whose colum vectors form a basis for the ullspace of S c ) There exists a V F c such that where rak ([ V H V H V V ull ]) d H S c 0 d 0 d 0 d S c 0 d 0 d 0 d S c

4 Rx Rx e e e He He He d c x [] x [] H e H e H e d c x [] x [] x [] x [] H e H e H e H e ker # H$ d c c d x [] x [] x [] x [] spa e, H e, " spa e,h e, " spa e, H e, " Fig. 5. Sigal levels at receivers for c <. Fig.. A pictorial represetatio of the cyclic decompositio of F with ( c, ) (0, ). H S c ad V ull represets the ( c ) matrix whose colum vectors form a basis for the ullspace of H. Proof: To prove the lemma, we make use of cyclic decompositio, the idea of decomposig F ito several disjoit ivariat subspaces. Fig. gives a illustratio of the cyclic decompositio. Pleas see [9] for the detailed proof. By Lemma, we oly eed to cosider the achievable scheme for c. The achievable scheme is split ito three differet regimes viz. c <, c <, ad c. Achievability for c is trivial, sice a optimal achievable scheme sets W W W φ ad uses all the levels for W at trasmitter. We will treat the other cases below. Case : c < The achievable scheme for this regime is illustrated i Fig. 5. Case : c < We first cosider the case where c is a multiple of. ) Trasmit Scheme: We use liear precodig at the trasmitters. Let V ull F ( c) satisfy S c V ull 0 d ( c). At trasmitter i, we use, as precodig vectors for W ii, colum vectors of the matrix [V V ull ] where V F c. We will shortly explai how V is chose, but here we metio that the colums of V are liearly idepedet of V ull. Note that this implies that S c V has a full rak of c /. For W ji, we use S c V as the precodig matrix so that the trasmitted codeword X i ca be represeted as X i V ˆX ii () + V ull ˆX ii () + S c V ˆX ji () for (i, j) {(, ), (, )}, where ˆX ii () F c ad ˆX ii () F c are colum vectors represetig the bits ecodig W ii. ˆX ji F c is the colum vector of the bits ecodig W ji. ) Receive Scheme: Cosider receiver. The received sigal Y ca be expressed as follows. ( ) V ˆX () + V ull ˆX () + S c V ˆX + ˆX () +S c V ˆX () Now, receiver wishes to decode ˆX (), ˆX (), ˆX usig liear decodig. Notice that the iterferece from ˆX, ˆX () aligs alog S c V. Suppose we choose V such that the colums of the matrix G [ V S c V S c V V ull ] are liearly idepedet, the clearly receiver caecode W,W usig liear decodig. Therefore, i order to show achievability, we eed to show that there exists V so that the matrix G has a full rak of. This is show i Lemma 5. Similar aalysis applies to receiver. Now, we cosider the case where c / is ot a iteger. I this case, we use a symbol extesio of the chael represeted below Y i(t) Y i (t + ) Y i (t + ) } {{ } Ȳ i F X i(t) X i (t + ) } X i (t + ) {{ } X i F + H X j(t) X j (t + ) } X j (t + ) {{ } X j F Like the case where c was a multiple of, a liear precodig ad decodig techique is applicable over this exteded chael. The oly differece i this case is that, we eed to show that there exists a c matrix V such that the matrix Ḡ [ V H V H V V ull ] has a full rak of, where V ull represets the ( c ) basis elemets of the ull space of H. This is show i Lemma 5 as well. This completes the proof of the achievability. Remark: I the regime of c <, the sum-capacityachievig scheme eeds to satisfy the challegig coditio of achievig (5) to (8) simultaeously. Ad the scheme solves the problem by efficietly aligig the iterferece: for the c levels that receiver j ca see the sigals from both trasmitters, c of them are used for W jj, c of them are used for W ji, ad c of them are used for aligig W ii ad W ij, i j. V. GENERALIZED DEGREES OF FREEDOM OF THE SYMMETRIC GAUSSIAN X CHANNEL To prove Theorem, we start from the followig lemma. Lemma 6: d(α) αd( α ). Proof: The lemma is proved usig the symmetry of the X chael. Please see [9] for the details.

5 By Lemma 6, we oly eed to fid d(α) for α. Sice d(α) for α has bee established i Theorem of [7], we oly cosider the remaiig case of <α. The upperbouds are derived usig the isights obtaied from the determiistic settig. Please see [9] for the detailed derivatio ad the coectio betwee the determiistic ad Gaussia cases. The achievable scheme is similar to those used i [0], [], ad we iclude a outlie of the proof for the sake of the completeess. For a give α [, ], we ca fid a (, c ) Z + ad a very small oegative value ɛ such that α ( c + ɛ( c )). () Note that whe α is a ratioal umber, ɛ is chose to be zero. But whe α is ot ratioal, ɛ( c )/ is used to compesate the differece betwee α ad a ratioal umber c that is very close to α. We choose ( c, ) such that () ca be achieved without symbol extesio for the symmetric determiistic chael with parameter ( c, ). Cosider the sequece of chaels, i.e. ρ idexed by N, such that ρ Q N ɛ (5) where Q is a very large but fixed positive iteger ad N is a positive iteger whose value grows to ifiity. Note that ρ grows to ifiity as N grows to ifiity. ) Trasmit Scheme: We impose the followig structure o the Q-ary represetatio of the trasmit sigal X i at trasmitter i for i {, }. X i N d x i,k Q k (6) ρ k0 The values of x i,k are restricted to the set {,..., Q } to esure that the additio of iterferece does ot produce carry over. It s easy to see that the power costrait is satisfied. Sice the sigal desig process developed i the Sectio IV-B also works i F N Q, we ca use it to fid the trasmit sigals X, X F N for the symmetric Q determiistic chael with parameter (N c, N ) ad the obtai the correspodig X,X R + by X i ρ [ Q N Q N Q Q ] X i. ) Receive Scheme : Each receiver takes the magitude of the received sigal, reduces to modulo Q N, discards the value below the decimal poit, ad expresses the result i Q- ary represetatio as Y i Y i mod Q N (7) N k0 y i,k Q k, y i,k {0,,..., Q } (8) Substitutig () ad (5) ito (0) ad (), we ca rewrite the iput output equatio as Y i X i + Q N(c ) X j + Z i, (i, j) {(, ), (, )} where X i ρxi Note that multiplicatio by Q N(c d) shifts the decimal poit i the Q-ary represetatio of X j by N( c ) places to the left. Thus, i the absece of oise, the N digits of X, X, Y, ad Y behave exactly like the symmetric determiistic chael with parameter (N c, N ). Followig the similar argumets i [0], [], we have R Σ R Σ,det (N c, N ) log Q ( Q )+o(n) (9) NR Σ,det ( c, ) log Q ( Q )+o(n) (0) Combiig (5), (0), ad (9), we ca show that d(α) is ot less tha NR Σ,det ( ( α ɛ lim sup N ɛ ), ) log Q ( Q )+o(n) N ɛ ɛ R Σ,det ( ( α ɛ ɛ ), ) log Q ( Q ) d Carryig out the substitutio of R Σ,det (, ), choosig Q ad ɛ to be arbitrarily large ad small respectively, ad comparig with the outerboud, we fiish the proof of Theorem. ACKNOWLEDGMENT This work was supported by NSF uder Grat CCF ad by ONR YIP uder Grat N REFERENCES [] S. Jafar ad S. Shamai, Degrees of freedom regio for the MIMO X chael, IEEE Tras. Iform. Theory, vol. 5, pp. 5 70, Ja [] M. Maddah-Ali, A. Motahari, ad A. Khadai, Commuicatio over MIMO X chaels: iterferece aligmet, decompositio, ad performace aalysis, IEEE Tras. Iform. Theory, vol. 5, pp , Aug [] T. Philosof ad R. Zamir, The rate loss of sigle-letter characterizatio: the dirty multiple access chael, arxiv:080.0v [cs.it], Mar [] R. Etki, D. Tse, ad H. Wag, Gaussia iterferece chael capacity to withi oe bit, IEEE Tras. Iform. Theory, vol. 5, pp , Dec [5] A. Avestimehr, S. Diggavi, ad D. Tse, A determiistic approach to wireless relay etworks, arxiv: [cs.it], Oct [6] G. Bresler ad D. Tse, The two-user Gaussia iterferece chael: a determiistic view, Europea Tras. i Telecommuicatios, vol. 9, pp. 5, Jue 008. [7] C. Huag, V. Cadambe, ad S. Jafar, Geeralized degrees of freedom of the (oisy) X chael, i Proc. Asilomar Coferece o Sigals, Systems, ad Cotrol, Oct [8] A. E. Gamal ad M. Costa, The capacity regio of a class of determiistic iterferece chaels, IEEE Tras. Iform. Theory, vol., pp. 6, March 98. [9] C. Huag, V. Cadambe, ad S. Jafar, O the capacity ad the geeralized degrees of freedom of the X chael, arxiv:080.7 [cs.it], Oct [0] S. Jafar ad S. Vishwaath, Geeralized degrees of freedom of the symmetric Gaussia k user iterferece chael. arxiv: [cs.it], Apr [] V. Cadambe, S. Jafar, ad S. Shamai, Iterferece aligmet o the determiistic chael ad applicatio to gaussia etworks, IEEE Tras. Iform. Theory, vol. 55, pp. 69 7, Ja. 009.