1466 IEEE TANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 8, AUGUST 2007 Interference Avoidance and Multiaccess Vector Channels Diitrie C. Popescu, Senior Meber, IEEE, Otilia Popescu, Meber, IEEE, and Christopher ose, Fellow, IEEE Abstract In this paper we present application of interference avoidance in the context of a general ultiple access vector channel odel. We show that this onotonically increases su capacity, and discuss algoriths for code division ultiple access (CDMA) codeword optiization based on this procedure. A greedy interference avoidance algorith for ultiaccess vector channels is presented in the paper, for which we discuss convergence to a class of codeword ensebles that satisfy a siultaneous water filling solution and axiize su capacity. Nuerical results obtained fro siulations that corroborate our analytical results are also presented. Index Ters CDMA, distributed codeword adaptation, interference avoidance, vector channels. I. INTODUCTION INTEFEENCE avoidance has eerged in the literature as a ethod for transitter optiization that provides distributed algoriths for codeword adaptation in CDMA systes [1] [5]. Using interference avoidance, CDMA codewords are independently adapted by users in response to changing patterns of interference through an iterative process which requires only that each user has access to the syste covariance inforation and now its channel to the base station (which ay be obtained through a feedbac channel [6], [7] fro the base station). The distributed nature of interference avoidance algoriths also aes the applicable to dynaic systes with changing nubers of active users and/or quality of service (QoS) requireents through adaptive/increental ipleentations [8] [10]. This is different fro centralized CDMA codeword optiization algoriths [11] [18] that are perfored at the base station without considering the channels between the users and the base, which are static in the nuber of users and assign optial codewords to users upon coputing the. Moreover, centralized algoriths do not allow variable QoS, and are not aenable to dynaic systes [19] and/or adaptive ipleentations. Using CDMA codewords that are optiized for particular counication channels can also iprove perforance. Wireless channels are in general dispersive due to the existence of ultiple paths between the transitter and receiver, and algoriths for optiizing codewords in ultipath CDMA channels have been proposed in [20] [22]. We note that interference Paper approved by. Schober, the Editor for Wireless Counication of the IEEE Counications Society. Manuscript received October 17, 2005; revised May 17, 2006 and Noveber 4, 2006. This wor was presented in part at the 2002 IEEE International Syposiu on Inforation Theory. D. C. Popescu is with the Departent of Electrical and Coputer Engineering, Old Doinion University, Norfol, VA 23529 USA (e-ail: dpopescu@odu.edu). O. Popescu and C. ose are with the Wireless Inforation Networ Laboratory (WINLAB), utgers The State University of New Jersey, Technology Center of New Jersey, North Brunswic, NJ 08902-3390 USA (e-ail: otilia@winlab.rutgers.edu; crose@winlab.rutgers.edu). Digital Object Identifier 10.1109/TCOMM.2007.902528 avoidance algoriths can also be used to obtain optial CDMA codewords for ultipath and dispersive channels [5], [23]. In this paper we present application of interference avoidance in the context of a general ultiple access vector channel odel. This wor extends the use of interference avoidance fro the dispersive channel odel in [5] in which user channel atrices were restricted to be diagonal, to a vector channel with no restrictions on user channel atrices. Such a channel odel provides a theoretical fraewor for the analysis of a wide variety of counication scenarios, which include CDMA systes with synchronous or asynchronous users that transit over channels with eory or channels with ultiple antennas in the transitter and/or receiver. Exaples of ultiaccess vector channels in the wireless literature are available in [22], [24] [26]. We note that coplex-valued ultiacces vector channels that ay occur in the context of ultiple antenna and/or OFDM-based systes are always representable in ters of a real-valued ultiaccess vector channel of double diension by using the isoorphis between the field of coplex nubers and the field of 2 2 real and sew-syetric atrices which is discussed in ore details in [5]. For siplicity of the atheatical presentation and with no loss of generality we choose to wor with real-valued ultiacces vector channels in our paper. The paper is organized as follows: in Section II, we present the syste odel under consideration, and state our proble. In Section III, we show how interference avoidance applies in the context of ultiaccess vector channels for a given user through whitening of the interference-plus-noise and projection of the received signal onto its corresponding signal space. In Section IV, we present a greedy interference avoidance algorith for ultiaccess vector channels, for which we discuss fixed point properties and convergence to optial codeword ensebles which axiize su capacity and present nuerical results obtained fro siulations. Final rears and conclusions are presented in Section V. II. POBLEM STATEMENT We consider a syste with L users that transit to a coon receiver as it is the case in the uplin of a wireless counication syste where several obile terinals counicate with the sae base station. We assue that distinct users transit signals residing in different signal spaces, with possibly different diensions and potential overlap aong the, and all being subspaces of the receiver signal space. Each user s signal space as well as the receiver signal space are of finite diension as iplied by a finite signaling interval T and finite bandwidths W l for each user l, and W (which includes all W l s corresponding to all users) for the receiver [27]. Matheatically this syste is 0090-6778/$25.00 2007 IEEE
POPESCU et al.: INTEFEENCE AVOIDANCE AND MULTIACCESS VECTO CHANNELS 1467 described by a ultiaccess vector channel equation r = H l x l + n (1) l=1 where x l of diension N l is the input vector corresponding to user l, l =1,...,L,r of diension N N l is the received vector at the coon receiver corrupted by the additive Gaussian noise vector n of the sae diension N with covariance atrix W = E[nn ].TheN N l channel atrix H l corresponding to user l defines a linear transforation between user l s signal space and the receiver signal space, and is the atheatical representation of the physical channel between user l and the base station that ebeds characteristics lie attenuation, ultipath, or ultiple antennas [5], [22] [26]. In this signal space setting, a given user l (l =1,...,L) transits a frae of M l inforation sybols b l = [b (l) 1...b(l) M l ] using a ulticode CDMA approach: each sybol b (l) in user l s frae is assigned a distinct codeword s (l), and the transitted signal by user l is a superposition of all codewords scaled by their corresponding inforation sybols. Thus, user l transits the signal vector x l = M l =1 s (l) b (l) = S l b l (2) where S l =[s (l) 1...s(l)...s (l) M l ] is the N l M l codeword atrix corresponding to user l whose coluns s (l), = 1,...,M l, are the codewords assigned to each of the M l sybols in user l s frae. We assue that all codewords have the sae nor equal to one so that user l transitted power is P l = Trace [S l S l ]=M l, l =1,...,L. (3) The received signal at the coon receiver is rewritten as r = H l S l b l + n. (4) l=1 In this paper, we extend application of interference avoidance fro the dispersive CDMA channel odel in [5] in which user channel atrices are restricted to be diagonal, to the general ultiaccess vector channel odel in (4) where no restrictions are placed on user channel atrices. We note that this channel odel allows also that distinct users transit in distinct subspaces of the receiver signal space, with different diensions and potential overlap aong the, which is different fro the initial fraewor for interference avoidance [3], [4], which assues the sae N-diensional signal space for the receiver and all users. III. GEEDY INTEFEENCE AVOIDANCE FO VECTO CHANNELS Interference avoidance allows users in a CDMA syste to adapt their signatures to achieve better perforance by axiizing signal-to-interference-plus-noise-ratio (SIN) at the receiver. In order to see how interference avoidance can be applied to ultiaccess vector channels we rewrite the received signal in (4) fro the perspective of particular user r = H S b + l=1,l H l S l b l + n } {{ } z where z represents the interference-plus-noise seen by user, and has covariance atrix Z = E[z z ]= H l S l S l H l + W. (6) l=1,l Since Z is syetric it can be diagonalized (5) Z = E E. (7) Furtherore, because Z is a positive definite covariance atrix we define the whitening transforation T = 1/2 E (8) which applied to (5) yields r = T r = T H S b + T z = H S b + w (9) where H = T H is the effective channel atrix seen by user in the new coordinates and w = T z is the equivalent white noise with covariance atrix E[w w ]= T Z T = I equal to the identity atrix. We now apply the singular value decoposition (SVD) [28, p. 442] to the transfored channel atrix corresponding to user H = U D V (10) where atrix U of diension N N has as coluns the eigenvectors of H H,atrixV of diension N N has as coluns the eigenvectors of H H, and atrix D of diension N N contains the singular values of H on the ain diagonal and zero elsewhere. We note that because T is invertible the ran of H will be equal to that of H. Without loss of generality we assue that H has full ran 1 N. Thus, the singular value atrix D can be partitioned as [ ] D D = (11) 0 with D an N N diagonal atrix containing the non-zero singular values along the diagonal and zeros in rest. The left inverse of D is D = [ D 1 0 ] = D D = I N. (12) eturning to equation (9) in which the SVD for transfored channel atrix H has been applied, we have r = U D V S b + w. (13) 1 This is not a restriction, since if H is not full ran then soe diensions of user signal space will have zero projection on the output space. Therefore, we can redefine a reduced codeword atrix S, which uses only diensions with nonzero projections on the output space.
1468 IEEE TANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 8, AUGUST 2007 We pre-ultiply by U r = U r = D V S b + U w (14) and define S = V S and w = U w. Note that because both U and V are unitary atrices they preserve nors of vectors. Thus, coluns of S are also unit nor as were the coluns of S. Also, because the equivalent noise ter w is white, then w will reain white with the sae covariance atrix equal to the identity atrix r = D S b + w. (15) At this point, we pre-ultiply with the left inverse of D to obtain the equivalent received signal fro user s perspective r = D r = S b + z (16) with E[ z z 2 ]= D. The received signal in (16) is processed by a ban of linear filters atched to the transfored codewords S to yield the decision variables for user. Thus, the SIN corresponding to a given sybol of user is expressed in ters of the transfored codeword s () (the th colun of S ) as where γ () = () 1 s () () s() (17) = () s () s() (18) is the autocorrelation atrix of the interference-plus-noise affecting the decision variable for sybol of user, and () = E[ r r ]= S S + D 2 (19) is the autocorrelation atrix of the equivalent received signal in equation (16). Greedy interference avoidance consists of changing s () the transfored codeword of user such that the corresponding SIN in equation (17) is axiized, and is accoplished [3] by replacing s () with the iniu eigenvector of atrix () in (18). In addition to greedy axiization of the SIN corresponding to codeword of user l, this procedure iplies also a onotonic increase in su capacity which for the vector channel in equation (4) is given by C s = 1 2 log 1 log W (20) 2 where is the covariance atrix of the received signal in (4), and is given by = H l S l S l H l + W. (21) l=1 We note that the expression of su capacity in (20) is not achievable with linear receivers, and we use it erely to argue that codeword replaceent based on greedy interference avoidance converges to a fixed point. In order to see this, we note that is directly related to D 2 () = S S + through the whitening transforation T and the unitary transforations iplied by the SVD. Thus,. After codeword re- + x (), where x () + s () placeent this becoes () = () is the iniu eigenvector of () a onotonic increase in () iplies a onotonic increase in, and iplicitly in su capacity. We now loo at () before and after codeword replaceent. Before codeword replaceent we have () = () () + x () x() s() x(), and we need to show that () + s () s(). (22) Using the approach outlined by [29, p. 46, eqs. (I), (II)] for coputing the deterinant of a bloc partitioned atrix, we can write for N N atrix A and N 1 vector y A y y 1 = A + yy =1+y A 1 y (23) and applying the bloc deterinant forula (23) to (22), we can rewrite it as [ 1+ x () ( () ) 1 x () which further reduces to ( () x () ) 1 x () ] [ 1+ s () s () ( () ) 1 s () ] (24) ( ) () 1 s (). (25) When x () is the iniu eigenvector of (), then it is also the axiu eigenvector of ( () ) 1, and properties of the ayleigh quotient [28, pp. 348 349] iply that (25) is true. Thus, (22) is true, and as a consequence, we have that greedy interference avoidance onotonically increases su capacity. We note that, as it is the case with all interference avoidance procedures, no siultaneous updates are allowed and only one codeword update can occur at a given tie. IV. INTEFEENCE AVOIDANCE ALGOITHMS FO MULTIACCESS VECTO CHANNELS Nuerous interference avoidance algoriths can be forulated based on repeated application of the greedy interference avoidance procedure presented in the Section III. These are defined by the various ways in which user codewords are selected for update. For exaple, one algorith could be defined by the update at a given step of one codeword of a given user, followed by update of a randoly selected codeword of a randoly selected user. Alternatively, at a given step of the algorith, one could update the codeword with the lowest SIN over all codewords and users. Or, one could update the codeword, which will yield the axiu increase in su capacity. We note that, regardless of the order in which codewords are selected for update, convergence of all algoriths based on greedy interference avoidance to a fixed point is guaranteed by the onotonic increase in su capacity iplied by greedy interference avoidance, along with the fact that su capacity is an upper bounded easure. We also note that this does not iply that codewords converge to a particular codeword enseble, but rather that repeated application of the greedy interference avoidance procedure converges to a class of codewords, which iply the sae value for su capacity, the so-called convergence in class [4]. To investigate fixed-point properties and illustrate the with
POPESCU et al.: INTEFEENCE AVOIDANCE AND MULTIACCESS VECTO CHANNELS 1469 nuerical exaples, we consider the interference avoidance algorith forally stated here: Greedy Interference Avoidance Algorith for Multiaccess Vector Channels 1) Initialize: user codeword atrices {S l } L l=1, channel atrices {H l } L l=1, and noise covariance atrix W. 2) For each user =1 L a) Apply the whitening transforation in (8) followed by the SVD in (10) to obtain the equivalent proble in (16). b) For each codeword =1,...,M of user. eplace s () by the iniu eigenvector of the corresponding () in (18). c) Update user codeword atrix S = V S. 3) epeat Step 2 until a fixed point is reached. This algorith is different than the one presented in [30] and analyzed in ore detail in [31], which consists of updating all codewords of a given user sequentially until convergence to a fixed point for the given user is reached and then iterating for all users in the syste, as opposed to the greedy interference avoidance algorith in which Step 2 defines an enseble iteration that updates all codewords of all users once. Nuerically, a fixed point of the algorith is defined with respect to a stopping criterion. That is, we say that a fixed point is reached when the difference between two consecutive values of the stopping criterion is within a specified tolerance ɛ. The stopping criterion can be an individual one, lie the codeword SIN or the Euclidean distance between codewords and their corresponding replaceents, or a global one lie su capacity. We note that in the case of individual stopping criteria, all values corresponding to all codewords ust be within the specified tolerance for the algorith to stop. As we have already entioned, convergence of the algorith to a fixed point is guaranteed by the fact that each codeword update onotonically increases su capacity, and that su capacity is upper bounded. However, this does not necessarily iply that the fixed point is unique, and theoretically any fixed points of the algorith are possible. A fixed point is characterized by the fact that any codeword of any user is iniu eigenvector of its corresponding () atrix in (18) as () s() = 1 γ () s (), (26) with γ () being the equilibriu SIN corresponding to codeword of any user. In addition, all codewords of any given user are eigenvectors of (), the autocorrelation atrix of the equivalent received signal in (16), i.e., ( ) () () s = 1 γ () +1 s () =1,...,M. (27) With respect to su capacity, the optial point with axiu su capacity corresponds to siultaneous water filling by all users in their respective signal spaces, provided that users transit covariance atrices are allowed to span the entire transit signal space [26]. In our fraewor, the transit covariance atrix of a given user expressed in ters of its codewords is S S, and siultaneous water filling requires that the nuber of codewords assigned for transission to user l be equal to or larger than the diension of its signal space, i.e., M N. According to [26], siultaneous water filling iplies that the following conditions are satisfied for any user. 1) The transit directions of user align with the rightsingular vectors of its effective channel. 2) The eigenvalues of user transit covariance atrix, and of its corresponding interference-plus-noise covariance atrix satisfy a water filling condition [32, p. 253]. The first condition is always satisfied at a fixed point of the greedy interference avoidance, algorith as user codeword atrix is expressed as S = V S with V containing the right-singular vectors of H. In addition, transfored codewords are eigenvectors of the autocorrelation atrix of user equivalent received signal [3] in (16), and the water-filling condition 2) iplies that they satisfy () () s = c s () =1,...,M (28) where c is the corresponding waterar. Thus, the optial point that corresponds to a su capacity axiizing codeword enseble is also a fixed point of the greedy interference avoidance algorith, and corresponds to the case when all codewords of user have unifor SIN equal to 1/(c 1). We note that the axiu su capacity point can be reached by applying the iterative water-filling algorith [26]. In this case, the distribution of user power over signal space diensions is explicitly optiized, and optial transit covariance atrices are obtained by applying iterative water filling. These can be used to obtain optial codeword ensebles with axiu su capacity by applying codeword construction algoriths at the transitter [13] [15]. However, greedy interference avoidance can yield an optial codeword enseble that axiizes su capacity directly, by distributed codeword adaptation, and in this case, the distribution of user power over signal-space diensions is optiized iplicitly by changing codeword coponents. More precisely, even though all user codewords are constrained to have unit nor, the power that a given user places in signal diension n is given by the su of squares of the nth eleents of all its codewords, i.e. p () n = M =1 s ()2 n n =1,...,N (29) and changes when the user codeword atrix is updated, such that the total user power N P = n=1 p () n = N M n=1 =1 s ()2 n, =1,...,L (30) reains constant. While we have not been able to prove that, in general, the greedy interference avoidance algorith always reaches the optial fixed point that corresponds to an enseble of codewords, which axiizes su capacity, we have perfored extensive siulations that corroborate this fact. To eep results general, we have not restricted siulations to particular odulation and
1470 IEEE TANSACTIONS ON COMMUNICATIONS, VOL. 55, NO. 8, AUGUST 2007 Fig. 1. Convergence of su capacity for 100 trials of the greedy interference avoidance algorith for a syste with L =10 users, in a signal space of diension N =15, white noise with unit variance at the base station, and randoly generated channel atrices. Each user has M =15codewords, and one enseble iteration consists of 150 codeword updates. propagation odels, but rather perfored the in the arbitrary signal space with channel atrices generated randoly. We have perfored siulations for both white noise with covariance atrix equal to scaled identity atrix, as well as colored noise with covariance atrix a randoly generated positive definite atrix. Siulations of the greedy interference avoidance algorith for signal-space diensions ranging fro 2 to 100, with different nubers of users and at least as any codewords per user as signal-space diensions (i.e., M l N l, l =1,...,L), indicate that the proposed algorith reaches the optial point corresponding to axiu su capacity fro rando initial codeword ensebles without special assistance, and no suboptial fixed points were observed. That is, the su capacity value at these points is identical to that obtained by applying iterative water-filling [26], and for the resulting codeword ensebles, all the codewords of any given user are eigenvectors of their corresponding () with identical eigenvalues γ /(1 + γ ), where γ is the corresponding SIN. Furtherore, at such fixed points, the reaining eigenvalues of () are saller than γ /(1 + γ ), which shows that a siultaneously water filling solution is satisfied. The siulations have also shown rapid convergence of the algorith when su capacity was used as a global convergence etric, usually within 3 4 enseble iterations, which is siilar to convergence of the iterative water-filling algorith to the axiu su capacity value [26]. This is illustrated in Fig. 1, which is typical for all the siulations. In addition, codeword convergence was observed. That is, siulations have shown that codewords converged to within tight nor difference tolerances Fig. 2. Codeword convergence for 100 trials of the greedy interference avoidance algorith for the sae syste with L =10users, in a signal space of diension N =15, white noise with unit variance at the base station, and randoly generated channel atrices. ( s () (i +1) s () (i) ε) when starting fro different rando initializations. However, codeword convergence was uch slower than convergence in su capacity as can be seen fro Fig. 2 (>40 enseble iterations for ε =10 3 ). We conclude this section by noting that, when users have less codewords than available signal-space diensions (i.e., M l <N l ), the greedy interference avoidance algorith still converges to a fixed point, but whether this fixed point corresponds to axiu su capacity is an open question whose answer is contingent upon solving the ore general proble of axiizing su capacity subject to power and ran constraints iposed on users. We note that when M l <N l, then user l s transission is restricted to a subspace of diension at ost M l, and axiization of su capacity subject to power and ran constraints iposed on users is still an open proble [31], [33]. V. CONCLUSION In this paper we considered a ultiuser counication syste in a general signal-space fraewor with distinct users residing in different possibly overlapping signal spaces, all being subspaces of the coon receiver signal space in which inforation is transitted using ulticode CDMA. The syste is odeled as a ultiaccess vector channel for which interference avoidance is applied to codeword adaptation in a given user s signal space after whitening of the interference-plus-noise seen by the user and projection using the SVD. We showed that application of greedy interference avoidance for any given user/codeword onotonically increases su capacity, and presented an algorith for codeword adaptation based on greedy interference avoidance. We note that while
POPESCU et al.: INTEFEENCE AVOIDANCE AND MULTIACCESS VECTO CHANNELS 1471 suboptial ensebles of codewords are theoretically possible, nuerical experients have shown that these were never obtained when starting fro randoly chosen initial codewords, and the algorith has consistently yielded su capacity axiizing codeword ensebles. This is consistent with siilar observations on related interference avoidance algoriths [1], [3] [5], which converge to codeword ensebles with axiu su capacity fro rando initializations. Convergence of the greedy interference avoidance algorith to axiu su capacity occurs within 3 4 enseble iterations, which is siilar to the iterative water-filling procedure described in [26]. However, interference avoidance algoriths are not substitutes for the iterative water-filling procedure. We note that the iterative water filling algorith perfors explicit optiization of user power over signal-space diensions and yields optial transit covariance atrices, which axiize su capacity, while the greedy interference avoidance algorith optiizes the distribution of user power over signal-space diensions iplicitly by adaptation of user codewords, and yields optial CDMA codeword ensebles that axiize su capacity. We also note that when optial transit covariance atrices are sought, then iterative water filling is the way to go, although when optial enseble of codewords to be used for transission are desired, as it is the case in CDMA systes, application of greedy interference avoidance offers a viable alternative to iterative water filling: convergence to axiu su capacity occurs within the sae nuber of enseble iterations and the greedy interference avoidance algorith yields directly an optial CDMA codeword enseble, while additional procedures ust be applied to obtain such an enseble fro the optial transit covariance atrices yielded by the iterative water filling algorith. ACKNOWLEDGMENT The authors are grateful to Dr.. Schober and the anonyous reviewers for their constructive coents on the paper. EFEENCES [1] S. Uluus and. Yates, Iterative construction of optiu signature sequence sets in synchronous CDMA systes, IEEE Trans. Inf. Theory, vol. 47, no. 5, pp. 1989 1998, Jul. 2001. [2] P. 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