On the Degrees of Freedom of Wide-Band Multi-Cell Multiple Access Channels With No CSIT

Transcription

1 On the Degrees of Freedom of Wide-Band Multi-Cell Multiple Access Channels With No CSIT Yo-Seb Jeon, Namyoon Lee, and Ravi Tandon Future IT Innovation Laboratory, POSTECH, Pohang, Gyeongbu, Korea Department of Electrical Engineering, POSTECH, Pohang, Gyeongbu, Korea Department of Electrical and Computer Engineering, The niversity of Arizona, Tucson, Arizona, SA Abstract This paper considers a K-cell multiple access channel with inter-symbol interference (ISI) The primary finding of this paper is that, without instantaneous channel state information at a transmitter, the interference-free sum degrees of freedom of K is asymptotically achievable when the number of users per cell is sufficiently large, and also when the number of channel-impulse-response taps of desired lins is greater than that of interfering lins This achievability is shown by a blind interference management method that exploits the relativity in delay spreads between desired and interfering lins I INTRODCTION A multi-cell multiple access channel (MAC) with intersymbol interference captures the communication scenario in which multiple uplin users per cell communicate with their associated base stations (BSs) by utilizing the same time and frequency resources across both the users and the BSs The spectral efficiency of this channel is fundamentally limited by three different types of interference: Inter-cell interference (ICI), which arises from simultaneous transmissions of users in neighboring cells; Inter-user-interference (II), which is caused by simultaneous transmissions of multiple users in the same cell; and Inter-symbol-interference (ISI), which occurs by the relativity between the transmit signal s bandwidth and the coherence bandwidth of a wireless channel Orthogonal frequency division multiple access (OFDMA) is the most well-nown approach to mitigate both II and ISI in the multi-cell systems 1, 2 The ey idea of OFDMA is to decompose a wideband (frequency-selective) channel into multiple orthogonal narrowband (frequency-flat) subchannels, each with no ISI By allocating non-overlapping sets of subchannels to the users in a cell, each user is able to send information data without suffering from ISI and II in the cell For instance, in a two-user MAC with ISI, which captures a single-cell uplin communication scenario, the capacity has been characterized by finding an optimal power allocation strategy across the subchannels 3, 4 These approaches, The wor of Y-S Jeon was supported by the ICT consilience creative program of MSIP/IITP, Korea IITP-R The wor of N Lee was supported by the ICT R&D program of MSIP/IITP, Korea 2017( ) The wor of R Tandon was supported by S NSF through the grant CCF however, still suffer from ICI, which is a major hindrance in attempting to increase spectral efficiency in multi-cell scenarios Multi-cell cooperation has been considered as an effective solution to manage ICI problems for future cellular networs where BSs are densely deployed and small cells overlap heavily with macrocells 5 7 The common idea of multicell cooperation is to form a BS cluster, which allows the information exchange among the BSs within the cluster, in order to jointly eliminate ICI Among multi-cell cooperation strategies, coordinated beamforming, which only requires channel state information (CSI) exchange among the BSs in the same cluster, provides a good tradeoff between the overheads for the information exchange and the gains on the spectral efficiency 6, 7 Interference alignment (IA) is a representative coordinated beamforming method, which aligns ICI in a subspace to confine the signal dimension occupied by interference 8 For example, in a single-input-single-output (SISO) interference channel, IA has shown to be an optimal strategy in the sense of sum degrees-of-freedom (DoF) that is the approximate sum spectral efficiency in a high signal-to-noise-ratio (SNR) regime 8 The concept of IA has also been extended to multi-cell MACs (or interfering MACs) in single antenna settings 9, 10 and multiple antenna settings One remarable result is that, by an uplin IA method, the sum-dof of K is asymptotically achievable in the K-cell SISO MAC as the number of uplin users per cell approaches infinity 9 11 The common requirement of the prior wor in 8 14 is global and perfect CSI at a transmitter (CSIT), which is a major obstacle in implementing these IA methods in practice All the aforementioned multi-cell cooperation strategies have focused on the mitigation of ICI under the premise of perfect II and ISI cancellation by OFDMA Recently, a blind interference management method has been proposed for the K-user SISO interference channel with ISI 15 The ey idea of this method is to exploit the relativity of multipath-channel lengths between desired and interfering lins This channel relativity allows the ICI alignment with discrete Fourier transform (DFT)-based precoding that needs no CSIT One remarable result in 15 is that, with completely no CSIT, the sum-dof of the K-user interference channel can be made to scale linearly with the number of communication pairs K,

2 Cell 1 Cell 2 BS 1 BS Fig 1 An illustration of the system model for a K-cell MAC with ISI, in which users are associated with the -th BS under some conditions on ISI channels In this paper, we consider the K-cell SISO MAC with ISI, as illustrated in Fig 1 Continuing in the same spirit with 15, we attempt to characterize the sum-dof of the multi-cell MAC with ISI in the absence of CSIT Our major contribution is to demonstrate that, without instantaneous CSIT, the sum-dof of the considered channel is ( 1 L ) I K, provided that the number of users per cell is lager than L I with > 2L I, where and L I are the maximum channelimpulse-response (CIR) lengths of desired and ICI lins in each cell, respectively Our result implies that interferencefree DoF per cell (ie, sum-dof of K) is achievable as LI approaches infinity with a sufficiently large number of users per cell This result extends the sum-dof result in 15, where the sum-dof of K 2 is shown to be achievable without CSIT when each BS communicates with a single user Therefore, our result also shows that communicating with multiple users in a cell provides a significant DoF gain compared to the singleuser case, even in the absence of CSIT To show our result, we modify the blind interference management method introduced in 15 The underlying idea of this method is to exploit the relativity in delay spreads between desired and ICI lins Specifically, by adding the cyclic prefix at transmitters with an appropriate length and by removing it at receivers, we create non-circulant matrices for the desired lin, while generating circulant channel matrices for the ICI lins in the absence of CSIT nowledge This relativity in the matrix structure allows us to align all the ICI signals to the same direction, while maing the desired signals spread over the entire signal dimensions As a result, all ICI can be simply canceled by using linear receive beamforming that does not depend on channel realizations After the ICI cancellation, each BS reliably decodes data symbols sent from the associated users by eliminating the remaining II and ISI perfectly, based on local CSI at a receiver (CSIR) II SYSTEM MODEL We consider a K-cell SISO MAC with ISI, where uplin users attempt to access the BS in the -th cell, K {1,2,, K}, by using the common time-frequency resources Let (,u) be the user index denoting the u-th user in cell We assume that all users and BSs are equipped with a single antenna The CIR between a user (a transmitter) and a BS (a receiver) is represented by a finite number of channel taps Thus, we denote the CIR between user (i,u) and the -th BS by {hi,u l}l,i 1, where L,i is the length of the CIR This length is typically defined as L,i T D, i,u W BW, where WBW and T D, i,u are the transmission bandwidth of the system and the delay spread of the wireless channel from user (i,u) to the -th BS, respectively We assume a bloc-fading channel model in which channel coefficients {hi,u l} are time-invariant during each bloc transmission We also assume that channel coefficients {h,i l} are drawn from a independent and continuous distribution For example, in a rich-scattering propagation environment, channel coefficients can be modeled as circularly symmetric complex Gaussian random variables Let x,u n be the transmitted signal of user (,u) at time slot n with the power constraint of E x,u n 2 P Then the received signal of the -th BS at time slot n is y n i i1 L,i 1 h i,u lx i,un l + z n, (1) where z n is noise at the -th BS in time slot n We assume that z n is independent and identically distributed (IID) circularly symmetric complex Gaussian random variable with zero mean and variance σ 2, ie, CN (0,σ 2 ) Throughout the paper, we assume no CSIT, implying that all users (transmitters) do not have any nowledge of channel realizations {h i,u l}l,i 1 for i and i, K Furthermore, it is assumed that each BS is available to access nowledge of CSI between itself and the associated users in the cell, ie, {h,u l}l, 1 for K This is referred to as local CSIR Note that local CSIR is necessary to perform coherent detection at the BSs Each user sends independent m,u data symbols to the associated BS during M time slots In this case, the rate of the log 2 m,u -th BS is given by R (P) M The rate R (P) is achievable if the -th BS is able to decode the transmitted data symbols with an arbitrarily-small error probability by choosing a sufficiently large M Then the sum-dof that characterizes an approximate sum of achievable rates of the system at high SNR is defined as K1 R (P) d Σ lim (2) P log (P) III MAIN RESLT The following theorem is the main result of this paper, which characterizes the sum-dof of a K-cell SISO MAC with ISI in the absence of CSIT

3 Theorem 1 Consider a K-cell MAC with ISI, each cell with uplin users Let L I max max i L,i and max L, The achievable sum-dof of this channel with completely no CSIT is d MAC Σ max where (x) + max{x,0} 1 Proof: See Section IV min {, (L, L I ) +} max {,2L I 1},1, (3) Theorem 1 shows the achievable sum-dof for a general ISI condition, yet it is unwieldy to provide a clear intuition in the result Considering a symmetric ISI scenario, we simplify Theorem 1 to the following Corollary: Corollary 1 (Symmetric ISI condition) Consider a K-cell MAC with symmetric ISI, ie, L, and L,i L I for i and i, K When the number of users per cell is larger than L I with > 2L I for K, one can achieve the sum-dof of d MAC Σ ( 1 L ) I K K, as L I (4) Proof: This result is directly obtained from (3) for the considered scenario Corollary 1 implies that interference-free DoF per cell is asymptotically achievable even without CSIT, as the number of users per cell approaches infinity under the derived ISI condition In other words, a total of K DoF is asymptotically achievable if there exists a sufficient number of users whose channel condition satisfies that L I IV PROOF OF THEOREM 1 We start by providing a lemma that is essential for our proof Lemma 1 A circulant matrix C C n n is decomposed as C FΛF H, (5) where F f 1,f 2,,f n C n n is the n-point IDFT matrix, f 1 n 1, ω 1, ω 2( 1),, ω (n 1)( 1) H, and ω exp ( ) j 2π n for {1,2,,n} Proof: See 17 In this proof, we focus on the case that 1 min {, (L, L I ) +} max {,2L I 1} > 1, (6) because otherwise, the trivial sum-dof of one is achievable by using time-division multiple access with OFDMA in each cell Furthermore, we assume that the number of active users in cell is given by min{, (L, L I ) + } among users This assumption implies that when L, L I, none of the users in cell transmit the desired signal The ey proof idea is similar to 15 using a bloc transmission method We consider a bloc transmission that consists of B subbloc transmissions each with N N + L I 1 time slots, where N max{ L I + 1, L I } At the end of each transmission bloc, we use 1 additional time slots to avoid inter-bloc interference between two subsequent bloc transmissions Therefore, the total number of time slots needed for a bloc transmission is M B N + 1 Let x b,u CN be the input data vector of user (,u) during the b-th subbloc transmission, defined as x b,u x,u (b 1) N + 1,, x,u (b 1) N + N, (7) where b {1,2,, B} In each bloc b, user (,u) sends the data symbol s b,u by using precoding vector f 1, ie, x b,u f 1 s b We generate the transmitted signal vector of the b-th,u subbloc with length N by adding a cyclic prefix with length L I 1 to x b,u as follows: xb b,cp, x,u,u,,u xb where x b,cp,u x,u (b 1) N + N L I + 2,, x,u (b 1) N + N (8) After the transmission of B subblocs, we append 1 zeros at the end of each bloc transmission, to avoid interbloc interference The transmitted signal vector from user (,u) during a bloc transmission is given by x,u ( x,u) 1 ( ), x 2 ( ),u,, x B,u,0,,0 (9) 1 From (1), the received signal of the -th BS at time slot n of the b-th subbloc transmission is y (b 1) N + n + L, 1 i L,i 1 i1,i h,u lx,u(b 1) N + n l h i,u lx i,u(b 1) N + n l + z (b 1) N + n, (10) where n {1,2,, N} and b {1,2,, B} Let ȳ b y (b 1) N + L I,, y (b 1) N + N C N and z b z (b 1) N + L I,, z (b 1) N + N C N be the received signal and noise vectors of the -th BS during the b-th subbloc transmission after discarding the cyclic prefix, respectively Then the received signal vector ȳ b is expressed in a vector form: ȳ b H,u f 1s b,u + i i1,i H i,u f 1s b i,u + zb, (11) where H i,u CN N is a matrix representation of the convolution involved with x b i,u For the ease of exposition, we define Circ(c) as an n by n circulant matrix with its first column is a vector c C n Then the effective channel matrices of all the interfering lins, H i,u for i, are represented as H i,u Circ( h i,u 0,, h i,u L,i 1,0,,0 N L,i ) (12)

4 Whereas, the effective channel matrix of the desired lin, H,u for K, is decomposed into two matrices: where H,u H C,u + H NC,u, (13) H C,u Circ( h,u 0,, h,u N 1, 0,,0 (N L, ) + with N min{l,, N}, and H NC,u H NC,u ), (14) has the form of (15) H upp,u 0 (N LI +1) (L I 1) 0 (LI 1) (N L I +1),u In (15), H upp C (N L I+1) (N L I +1) and,u,u C (L I 1) (L I 1) are upper and lower toeplitz matrices defined in (16) (see the top of the next page) Note that when N L,,,u 0 (L I 1) (L I 1) As seen in (12) and (13), H i,u for i is a circulant matrix, while H consists of both circulant,u and non-circulant matrices This difference is due to the fact that the length of cyclic prefix is selected as L I 1 such that L,i 1 L I 1 < L, 1 for i and i, K This creates the relativity between desired lins and ICI lins by a matrix structure Inter-cell-interference cancellation: We explain a ICI cancellation method that exploits the relativity in the matrix structure between desired and ICI lins Plugging (13) into (11), we have ȳ b ( H NC,u + ) H C,u f1 s b i,u + H i,u f 1s b i,u + zb i1,i (17) In (17), H C,u and H i,u for i and i, K are circulant matrices, so from Lemma 1, the column vectors of a N-point IDFT matrix F are the eigenvectors of these circulant matrices Since f 1 is the first eigenvector of these matrices, we can rewrite the received signal expression in (11) as: ȳ b H NC,u f 1s b,u + zb + λ C,u,1 sb,u + i i1,i λ i,u,1 sb i,u f 1, (18) where λ C,u,1 and λ i,u,1 are the first eigenvalues of H C,u and H i,u that correspond to the eigenvector f 1, respectively As can be seen in (18), all the ICI signals are aligned in the same direction of f 1 As a result, it is possible to eliminate all the ICI signals by multiplying a projection matrix W f 2,f 3,,f N H C (N 1) N to ȳ b in (18), which yields ỹ b Wȳb W H NC,u f 1s b,u + W zb, (19) where ỹ b C N 1 is the effective received signal vector of the -th BS from the b-th subbloc transmission Let s b s b,1, sb,2,, sb C, be the data symbol vector received at the -th BS, and H C (N 1) be an effective channel matrix defined as H W H NC,1 f 1, H NC,2 f 1,, H NC, f 1 (20) Then the received signal vector in (19) simplifies to ỹ b H s b + zb, (21) where z b W zb CN 1 is an effective noise vector Note that the distribution of z b is invariant with zb because W is a unitary transformation matrix The effective received signal vector ỹ b in (21) only contains the transmitted signals from own-cell users without ICI Decodability of subbloc data: To accomplish our proof, we need to show the decodability of the data symbols in each subbloc transmission The following lemma shows that the ran of the effective channel matrix H equals, which is the number of symbols sent by the users in cell Lemma 2 The ran of a matrix H defined in (20) is ran ( H ), for K (22) Proof: Due to the space limitation, here, we only present the ey steps of the proof First, motivated by the fact that H NC,u f 1 L, 1 ll I h,u le l for some basis vectors e LI,,e L, 1, we decompose H as follows: H W e LI,e LI +1,,e L, 1 h eff,1 heff, } {{ }, (23) } {{ } E C N (L, L I ) H eff C(L, L I ) where h eff,u h,u L I,, h,u L, 1 C L, L I Next, we apply Frobenius inequality to (23), which says that r A + r B + r C n AB n BC ran (ABC) min{r A,r B,r C }, for three matrices A C n A n AB, B C n AB n BC, and C C n BC n C, with the rans r A, r B, and r C, respectively Finally, by calculating the rans of three matrices W, E, and H eff, we obtain the ran of H as given in (22) Lemma 2 implies that for sufficiently large SNR, all data symbols in s b can be reliably decoded Inter-subbloc-interference cancellation: We have shown that data symbols are decodable for each subbloc transmission, by assuming that there is no inter-subbloc interference (ISBI) nfortunately, ISBI between two subsequent subblocs is unavoidable because the length of the cyclic prefix is shorter than the number of CIR taps for desired lin Thus, we need to cancel ISBI for multiple subbloc transmissions After discarding 1 zeros at the end of the transmission bloc, we concatenate the received vectors ȳ b for b {1,2,, B} Then, when ignoring noise, the total input-

5 H upp,u 0 0 h,u N 1 h,u L I 0 0 h,u N and,u h,u N 0 0 h,u N 0 h,u L, 1 0 (16) h,u L, 1 h,u N output relationship during an entire bloc transmission is H 0 sub 0 sub ỹ 1 ỹ 2 H sub H s 1 0 sub H sub s 2 ḳ, (24) ỹ B H 0 sub 0 sub 0 sub H sub H s B where H sub C (N 1) is the effective channel matrix for ISBI at the -th BS, and 0 sub 0 (N 1) By the definition, one can easily verify that H sub is given by H sub W H sub,1 f 1, H sub,2 f 1,, H sub, f 1 (25) where H sub,u CN N is H sub,u 0(N LI +1) (L I 1) H upp,u, (26) 0 (LI 1) (L I 1) 0 (LI 1) (N L I +1) by letting N L, Since there is no ISBI during the first subbloc transmission, ie, ỹ 1 H s 1 + z1, it is possible to reliably decode s 1 for sufficiently large SNR Once the data symbol vector for the previous subbloc transmission, saying s b 1, is reliably decoded, it is possible to decode s b by subtracting the effect of s b 1 from the received signal ỹ b as follows: ỹ b H sub sb 1 H s b + zb Achievable sum-dof calculation: Applying the above ISBI cancellation strategy over B subblocs recursively, each BS is capable of decoding B data symbol vectors s 1,s2,,sB, with M B N + 1 channel uses For sufficiently large coherence time, B can be taen to be infinity, so the achievable DoF of the -th BS is B d lim B B N + 1 N (27) By plugging N N + L I 1, N min{ L I + 1, L I }, and min{, (L, L I ) + } to (27), we arrive at the expression in Theorem 1 V CONCLSION In this wor, we showed that the interference-free sum- DoF of K is asymptotically achievable in a K-cell SISO MAC with ISI, even in the absence of CSIT This achievability was demonstrated by a blind intereference management method that exploits the relativity in delay spreads between desired and interfering lins We observed that a significant DoF gain compared to the result in 15 is obtained when multiple users exist in a cell by improving the utilization of signal dimensions REFERENCES 1 R W Chang, Synthesis of band-limited orthogonal signals for multichannel data transmission, Bell Syst Tech J, vol 45, pp Dec J A C Bingham, Multicarrier modulation for data transmission: An idea whose time has come, IEEE Commun Mag, vol 28, no 5, pp 17 25, Mar R S Cheng and S Verdú, Gaussian multiaccess channels with ISI: Capacity region and multiuser water-filling, IEEE Trans Inf Theory, vol 39, no 3, pp , May C Y Wong, R S Cheng, K B Letaief, and R D Murch, Multicarrier OFDM with adaptive subcarrier, bit, and power allocation, IEEE J Sel Areas Commun, vol 17, no 10, pp , Oct D Gesbert, S Hanly, H Huang, S S Shitz, O Simeone, and W Yu, Multi-cell MIMO cooperative networs: A new loo at interference, IEEE J Sel Areas Commun, vol 28, no 9, pp , Dec H Dahrouj and W Yu, Coordinated beamforming for the multicell multiantenna wireless system, IEEE Trans Wireless Commun, vol 9, no 5, pp , May N Lee, D Jimenez, A Lozano, and R W Heath, Jr, Spectral efficiency of dynamic coordinated beamforming: A stochastic geometry approach, IEEE Trans Wireless Commun, vol 14, no 1, pp , Jan V R Cadambe and S A Jafar, Interference alignment and the degrees of freedom of the K user interference channel, IEEE Trans Inf Theory, vol 54, no 8, pp , Aug V R Cadambe and S A Jafar, Interference alignment and the degrees of freedom of wireless X networs, IEEE Trans Inf Theory, vol 55, no 9, pp , Sep C Suh and D Tse, Interference alignment for cellular networs, in In proc 46th Annual Allerton Conf Commun Control Comput (Allerton), Sep A Chaaban, A Sezgin, B Bandemer, and A Paulraj, On Gaussian multiple access channels with interference: Achievable rates and upper bounds, in Proc 4th Int Worshop Multiple Access Commun (MACOM), Sep T Kim, D J Love, B Clercx, and D Hwang, Spatial degrees of freedom of the multi-cell MIMO multiple access channel, in Proc IEEE Global Commun Conf (GLOBECOM), Dec N Lee, W Shin, R W Heath, Jr, and B Clercx, Interference alignment with limited feedbac for two-cell interfering MIMO-MAC, in Proc IEEE Int Symp Wireless Commun Systems (ISWCS), pp , Aug H J Yang, W-Y Shin, B C Jung, and A Paulraj, Opportunistic interference alignment for MIMO interfering multiple-access channels, IEEE Trans Wireless Commun, vol 12, no 5, pp , May N Lee, Interference-free OFDM: Rethining OFDM for interference networs with inter-symbol interference, Sep 2016 Online Available: 16 S A Jafar, Blind interference alignment, IEEE J Sel Topics Signal Process, vol 6, no 3, pp , June G H Golub and C F V Loan, Matrix Computations, 3rd ed The Johns Hopins niversity Press, 1996