Signaling over MIMO Multi-Base Systems: Combination of Multi-Access and Broadcast Schemes

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1 Signaling over MIMO Multi-Bae Sytem: Combination of Multi-Acce and Broadcat Scheme Mohammad Ali Maddah-Ali Abolfazl S. Motahari and Amir K. Khandani Coding & Signal Tranmiion Laboratory ( Dept. of Elec. and Comp. Eng. Univerity of Waterloo Waterloo ON Canada N2L 3G Tel: Fax: mohammad abolfazl Abtract A new tructure for multi-bae ytem i tudied in which each uer receive data from two nearby bae tation rather than only from the tronget one. Thi ytem can be conidered a a combination of broadcat and multi-acce channel. By taking advantage of both perpective an achievable rate region for a dicrete memoryle channel modeled by r(y y 2 x x 2) i derived. In thi model x and x 2 repreent the tranmitted ignal by the tranmitter one and two repectively and y and y 2 denote the received ignal by the receiver one and two repectively. In thi derivation it i aumed that each tranmitter i unaware of the data of the other tranmitter and therefore x and x 2 are independent. To invetigate the advantage of thi cheme an efficient ignaling method which work at a corner point of the achievable region for multiple-antenna cenario i developed. In the propoed cheme each bae tation only require the tate information of the channel between the other bae tation and each uer. In thi paper the ignaling cheme i elaborated for the cae that each tranmitter/reciever i equipped with three antenna. It i proven that in uch a cenario the multiplexing gain of four i achievable which outperform any other conventional cheme. I. Introduction In conventional wirele ytem each uer receive information from one bae tation which i generally the tronget one. In thi cae the performance of the ytem can be dramatically deteriorated by the interference from the other pair of tranmitter/reciver known a cochannel interference. A number of reearch work have invetigated the effect of co-channel interference in multi-input multi-output (MIMO) multi-uer ytem. In [] the capacity of ytem for a group of interfering uer employing ingle-uer detection i tudied. In [2] [3] multi-uer detection and turbo decoding are exploited to improve the performance of the ytem where it i aumed that all uer have complete information of their channel with the bae tation. A well-known metric to evaluate the performance of the ignaling cheme i the o-called multiplexing gain which i defined a the ratio of the um-rate over log(snr) for large ignal-to-noie-ratio (SNR) value. Simulation reult indicate that the multiplexing gain of the ignaling cheme propoed in [2] i zero. In other word the umrate converge to a fixed value a SNR increae. In fact the un-canceled term of co-channel interference in the denominator of the ignal-to-interference-plu-noie-ratio (SINR) do not allow further increae in the rate. In [4] the multiplexing gain of MIMO multiuer cheme are invetigated. Specially it i proven that in an interference channel with two tranmitter and two receiver where each of them i equipped with η antenna the conventional ignaling cheme can achieve the maximum multiplexing gain i η. To mitigate the co-channel interference the cooperation among bae tation i propoed [5]. In [5] the infinitecapacity link among bae tation i aumed which reduce the ytem to a ingle broadcat channel. The QR decompoition cheme a a ignaling method over MIMO broadcat channel i applied which completely cancel the interference. The interference cancelation i baed on a reult known a dirty paper coding (DC) due to Cota 983 [6]. In [7] the performance of the method propoed in [5] i evaluated for a more practical channel model. In [8] the idea of [5] i explored taking into account the individual power contraint per bae tation and making ue of uplink downlink duality. Cooperative bae tation in uplink i conidered in [9]. By auming a full cooperation among bae tation the ytem i implified to a ingle MIMO multi-acce ytem. In thi paper a new ignaling cheme for multi-bae ytem in downlink i propoed. In thi cheme each uer receive data from two nearby bae tation rather than only from the cloet one. In thi cae we can conider the ytem a a et of broadcat channel (from bae tation point of view) or a et of multi-acce channel (from uer point of view). We benefit from both perpective to derive an achievable rate region for a dicrete memoryle channel modeled by r(y y 2 x x 2 ). In thi derivation it i aumed that each tranmitter i unaware of the data of the other tranmitter and therefore x and x 2 are independent. Thi derivation i baed on a combination of two achievable region: (i) Marton rate region for the memoryle broadcat channel [0] [] and (ii) rate region for the memoryle multi-acce channel [2]. By focuing on a corner point of the derived achievable region an efficient ignalling cheme for uch ytem in

2 propoed. In the propoed cheme each bae tation only require the tate information of the channel between the other bae tation and each uer. In thi paper the propoed ignalling cheme i elaborated for the cae that each of the tranmitter and receiver i equipped with three antenna. It i proven that in uch a cenario the multiplexing gain of four i achievable which outperform any other conventional cheme. II. Achievable Region In the following an achievable rate region for a general dicrete memoryle channel modeled by r(y y 2 x x 2 ) i derived. In the uggeted rate region the auxiliary random variable W and Z contain information from the tranmitter one to the receiver one and two repectively. Similarly the auxiliary random variable W 2 and Z 2 contain information from the tranmitter two to the receiver two and one repectively (ee Fig. ). W Z Z 2 W 2 Fig.. X X 2 r(y y 2 x x 2 ) Y Y 2 A General Dicrete Memoryle Channel Theorem Conider a dicrete memoryle channel modelled by r(y y 2 x x 2 ). Aume that the tranmitter t t = 2 tranmit data to the receiver r r = 2 with the rate R rt. Then an achievable region i given by the et of all rate in the convex cloure of the quadruple (R R 2 R 2 R 22 ) atifying R q I(Y ; W Z 2 ) () R 2 q 2 I(Y 2 ; Z W 2 ) (2) R 22 q 22 I(Y 2 ; W 2 Z ) (3) R 2 q 2 I(Y ; Z 2 W ) (4) R + R 2 q + q 2 I(Z ; W ) (5) R 22 + R 2 q 22 + q 2 I(Z 2 ; W 2 ) (6) q + q 2 I(Y ; W Z 2 ) (7) q 22 + q 2 I(Y 2 ; W 2 Z ) (8) for ome joint ditribution of r(w z x w 2 z 2 x 2 ) = r(w z x ) r(w 2 z 2 x 2 ). In addition by defining R r = R r + R r2 r = 2 a the total rate of the receiver r an achievable rate region for R and R 2 i given by convex cloure of all rate (R R 2 ) atifying R I(Y ; Z 2 W ) R 2 I(Y 2 ; Z W 2 ) R + R 2 I(Y ; Z 2 W ) + I(Y 2 ; Z W 2 ) I(Z ; W ) I(Z 2 ; W 2 ) for ome joint ditribution a in the firt part. roof: Fix r(w z x w 2 z 2 x 2 ) = r(w z x ) r(w 2 z 2 x 2 ). Random Encoding for the Tranmitter One: Generate 2 nq i.i.d. equence w n A n ɛ according to the uniform ditribution over A n ɛ (W ) where A n ɛ (W ) denote the typical et for the random variable Z n. Label the elected i.i.d. equence a w n (k ) k =... 2 nq. Similarly generate 2 nq2 i.i.d. equence z n A n ɛ according to the uniform ditribution over A n ɛ (Z ) where A n ɛ (Z ) denote the typical et for the random variable Z n. Label the elected i.i.d. equence a z n (l ) l =... 2 nq2. For i [ 2 nr ] and j [ 2 nr2 ] define the cell B () i = w n (k ) : k [(i )2 n(q R) + i 2 n(q R)+ ] C () j = z n (l ) : l [(j )2 n(q2 R2) + j 2 n(q2 R2)+ ] D () i j = (w n (k ) z n (l )) : w n (k ) B i z n (l ) C j (w n (k ) z n (l )) A n ɛ To end a meage pair (i j ) chooe one pair (w n (k ) z n (l )) from D () i j and find an x n (i j ) that i jointly typical with that pair. Random Encoding for the Tranmitter Two: Similarly for the uer two generate w 2 (k 2 ) for k 2 2 nq22 z 2 (l 2 ) for l 2 2 nq2 and cell B (2) i 2 C (2) j 2 D (2) i 2j 2 for i 2 2 nr22 and j 2 2 nr2. For meage pair (i 2 j 2 ) chooe one pair (w n 2 (k 2 ) z n 2 (l 2 )) from D (2) i 2j 2 and find an x n 2 (i 2 j 2 ) that i jointly typical with that pair. Decoding: Receiver one find the unique indice pair (k l 2 ) uch that (w n (k ) z n 2 (l 2 ) y n ) A n ɛ. Similarly receiver two find the unique indice pair (k 2 l ) uch that (w n 2 (k 2 ) z n (l ) y n 2 ) A n ɛ. Uing the above random coding cheme we can prove that the average probability of error converge to zero a n if inequalitie () to (8) are atified. The econd part of the theorem i derived directly from the firt part. A. A Corner oint To how the advantage of thi cheme we focu on one of the corner point of the achievable region for the rate vector (R R 2 R 2 R 22 ) for a fixed joint probability on input and auxiliary random variable. To thi end we chooe for R and R 22 the maximum poible value i.e.. R = I(Y ; W Z 2 ) (9) R 22 = I(Y 2 ; W 2 Z ). (0) With thee choice of R and R 22 we can how that the maximum poible value for R 2 and R 2 are equal to R 2 = I(Y ; Z 2 ) I(Z 2 ; W 2 ) () R 2 = I(Y 2 ; Z ) I(Z ; W ). (2)

3 Here we invetigate the rate of the data received by the uer i.e. R and R 2. Equation (9) implie that to achieve the highet rate for W receiver one firt decode Z 2 and then W. The formula obtained for R 2 i baically the ame a that of the rate of the channel with noncaually known i.i.d. tate at the tranmitter derived by Gelfand-inker [3]. In fact if the tranmitter two firt chooe a codeword for W 2 it interference over Z 2 at the receiver one terminal i non-caually known by the tranmitter and therefore rate of () i achievable. For the pecial cae of additive white Gauian noie with Gauian ditribution for auxiliary random variable W 2 and Z equation () implie that the interference of W 2 over Z at the receiver one terminal can be effectively canceled out [6]. Thi reult i known a the dirty paper coding due to Cota [6]. The above obervation lead u to a ignaling cheme which i elaborated for MIMO cenario in the next ection. III. Signaling Method for MIMO Sytem Conider a MIMO multi-bae ytem with the bae tation t t = 2 a the tranmitter and the uer r r = 2 a the receiver. A an example here we focu on a cae where each bae tation t i equipped with M t = 3 antenna and imilarly each uer r r = 2 i equipped with N r = 3 antenna. However the propoed cenario can be generalized to the cae of different number of antenna. Auming flat fading environment the channel between the bae tation t and the uer r i repreented by the channel matrix H rt where H rt C 3 3. The received vector y r C 3 by uer r r = 2 i given by y = H x + H 2 x 2 + n (3) y 2 = H 2 x + H 22 x 2 + n 2 (4) where x t C 3 repreent the tranmitted vector by the bae tation t. The vector n r C 3 i a white Gauian noie with zero mean and identity covariance matrix. It i aumed that E(x t x t) for t = 2. In the propoed cenario each bae tation tranmit two data tream. The bae tation t end the data tream d t to the uer and the data tream d 2t to the uer 2. The tranmitted vector are equal to the linear uperpoition of the modulation vector with d rt t r = 2 a the coefficient i.e. x = d v + d 2 v 2 (5) x 2 = d 2 v 2 + d 22 v 22 (6) where the unit vector v rt C 3 r t = 2 denote the modulation vector. The power of p rt i allocated to the data tream d rt. A mentioned in the previou ection the interference of d over d 2 and the interference of d 22 over d 2 are canceled out baed on the dirty-paper-coding theorem. Motivated by the proof of the dirty-paper-coding theorem in [6] we embed data in ˆd 2 and ˆd 2 where d 2 = ˆd 2 α 2 (v 2 H 2H 2 v 2 ) v 2 H 2H 2 v d d 2 = ˆd 2 α (v 2 H 2H 2 v 2 ) v 2 H 2H 2 v 22 d 22 α = p 2 (p 2 + (v 2 H 2H 2 v 2 ) ) α 2 = p 2 (p 2 + (v 2 H 2H 2 v 2 ) ) where (.) denote tranpoe conjugate operation. H 2 and H 2 are defined later in (20) and (26). A mentioned at the receiver ide the ucceive decoding (SD) cheme i employed. The tructure of the receiver i a follow: firt uer decode ˆd 2 and ubtract it effect from the received vector y. Then d i decoded. Similarly uer 2 firt decode ˆd 2 and ubtract it effect from y 2 then decode d 22. The detail of the detection are depicted in Fig. 2. To decode ˆd 2 at the uer terminal the ignal received from bae tation i.e. d and d 2 are treated a interference. The propoed precoding cheme i uch that the data tream d 22 ha no interference on the data tream ˆd 2. The filter Ψ 2 = R 2 2 i ued to whiten the interference plu noie H (v d + v 2 d 2 ) + n with the variance matrix R 2 R 2 = [ p 0 H [v v 2 ] 0 p 2 ] [v v 2 ] H + I.(7) Thi formula i baed on the reult in [6] which implie d and d 2 are independent. The output of Ψ 2 i paed through the filter u 2 which maximize the effective SINR. The deign of the precoding and the filter u 2 will be explained later. Here the uer one decode ˆd 2 and then ubtract it effect from the received ignal y i.e. where ỹ = y H 2 v 2 ˆd2 = Q H 2 v 22 d 22 + H v d + H 2 v 2 d 2 Q = I H 2 v 2 α (v 2 H 2H 2 v 2 ) v 2 H 2Ψ 2 In the next tep the uer one decode d from ỹ. Firt the filter Ψ i ued to whiten the interference of d 22 over d. Note that the data tream d 2 ha no interference over d due to the precoding at the tranmitter. The interference plu noie i equal to Q H 2 v 22 d 22 + n with the covariance matrix R = Q H 2 v 22 p 22 v 22 H 2 Q + I. Then the whitening filter i equal to Ψ = R 2. The output of the whitening filter Ψ i paed through the filter u which maximize the SNR of the data tream d. Similarly for the uer 2 there are two whitening filter

4 Ψ 2 = R 2 2 and Ψ 22 = R 2 2 where [ ] p2 0 R 2 = H 22 [v 2 v 22 ] [v 0 p 2 v 22 ] H 22 + I 22 R 22 = Q 2 H 2 v p v H 2 Q 2 + I Q 2 = I H 2 v 2 α 2 (v 2 H 2H 2 v 2 ) v 2 H 2Ψ 2 Similarly at the uer two terminal u 2 and u 22 are ued to detect d 2 and d 22 repectively. Figure 2 how ome more detail. In what follow we explain the derivation of the modulation vector v rt and the demodulation vector u rt r t = 2. To thi end we conider the econd perpective of the ytem a a et of two broadcat channel. A depicted in Fig. 2 the following MIMO broadcat channel i viewed from the bae tation one ŷ = H x + n (8) ˇy 2 = H 2 x + ň 2 (9) where n and ň 2 are whitened noie term and H = Ψ H (20) H 2 = Ψ 2 H 2. (2) For ignaling we apply the cheme propoed in [4] for the MIMO broadcat ytem with multiple receive antenna. According to [4] the modulation vector v i equal to the optimizing vector of the following maximization problem σ 2 = max H H (22).t. = where σ i the gain of the equivalent ingle-antenna channel on which the data tream d i ent. The demodulation vector u i given by u = Hv σ. v 2 i the optimizing vector of the following maximization problem σ2 2 = max H 2H 2 (23).t. = v = 0 where σ 2 i the gain of the equivalent channel on which the data tream d 2 i ent. The demodulation vector u 2 i given by u 2 = H2v2 σ 2. A hown in [4] by uing thi cheme the data tream d 2 ha no interference over the data tream d. A mentioned knowing the elected codeword for data tream d the bae tation one can effectively cancel out the interference of the data tream d over d 2 baed on the dirty-paper coding theorem. Conequently the broadcat channel i reduced to a et of two parallel channel with gain σ and σ 2. Waterfilling i applied to optimally allocate the power p and p 2 to the data tream d and d 2 repectively where p + p 2. Similarly from the bae tation 2 we have a MIMO broadcat channel modeled by ˇy = H 2 x 2 + ň (24) ŷ 2 = H 22 x 2 + n 2 (25) where ň and n 2 are whitened noie and H 2 = Ψ 2 H 2 (26) H 22 = Ψ 22 H 22. (27) Here we apply the ame algorithm to derive the modulation and demodulation vector for the bae tation 2. v 22 i equal to the optimizing vector of the following maximization problem σ22 2 = max H 22H 22.t. = and u 22 = H22v22 σ 22. In addition v 2 i equal to the optimizing vector in the following problem σ2 2 = max H 2H 2.t. = v 22 = 0 and u 2 = H2v2 σ 2. Similar to the firt bae tation d 2 ha no interference over d 22. Selecting the codeword for d 22 the tranmitter two can effectively cancel it interference over d 2 uing the dirty paper coding. Water-filling i ued to optimally divide the total power between p 2 and p 22. At the end thi method reduce the ytem to four parallel channel with the channel gain σ rt r t = 2. Therefore the um-rate of the propoed cheme i obtained by 2 2 R Sum Rate = log 2 ( + σrtp 2 rt ). (28) r= t= Note that to compute v and v 2 in (22) and (23) v 2 and v 22 are needed (Ψ and Ψ 2 are function of v 2 and v 22 ) and vie vera. To derive the modulation vector we can initialize v rt r t = 2 randomly and iteratively follow (22) to (28) until the reulting vector converge. Simulation reult how that the algorithm converge very fat. IV. erformance Analyi Although finding the optimal power allocation i traight-forward to implify the analyi we aume that each bae tation divide the total power equally between data tream i.e. p rt = /2 t r = 2. In thi cae we have R = 2 Q H 2 v 22 v 22 H 2 Q + I. Let δ = Q H 2 v 22 and v 22 = QH2v22 Q H 2v 22. Conider the unit vector ν and ν 2 uch that [v 22 ν ν 2 ] form a unitary matrix. Then we can how that H = δ v 22 ν ν 2 H. (29)

5 ŷ u Decoding d H 2v 2 Ψ ỹ d v H Ψ DC 2 ˆd 2 ˇy H 2 v n d 2 2 u 2 Decoding ˆd 2 d 22 n 2 ˆd 2 v 22 DC H 2 H 22 Ψ 2 ˇy 2 u 2 Decoding ˆd 2 H 2v 2 d 2 v 2 ỹ 2 Ψ 22 ŷ 2 u 22 Decoding d 22 Fig. 2. Block Diagram of the ropoed recoding and Detection Scheme In high SNR 0. Thu we have H = 2 δ2 + [0 ν ν 2 ] H. Regarding (22) σ i equal to the maximum ingular value of H which i a rank 2 matrix for large SNR. Therefore σ converge to a nonvanihing poitive contant. Similar tatement are valid for σ 22. Conequently each of the data tream d and d 22 achieve multiplexing gain of one. Now we invetigate the multiplexing gain of the data tream d 2 and d 2. Let p = p 2 = 2. From (7) we have R 2 = 2 H [v v 2 ][v v 2 ] H + I. Applying the SVD decompoition we have H [v v 2 ][v v 2 ] H = λ 2 ϖ ϖ + λ2 2ϖ 2 ϖ 2 where λ λ 2 0 and ϖ and ϖ 2 are two unit orthogonal vector. Conider ϖ 3 uch that the matrix [ϖ ϖ 2 ϖ 3 ] form a unitary matrix then we can how that H 2 = λ λ ϖ ϖ 2 ϖ 3 H 2. (30) A it i hown in [4] σ 2 in (23) i equal to the maximum ingular value of H 2 where H 2 = H 2 [ϕ ϕ 2 ] and ϕ and ϕ 2 are two unit vector uch that [v ϕ ϕ 2 ] form a unitary matrix. In high SNR 0 and + 2 λ λ2 0. Conequently H 2 converge to a matrix with rank one. Therefore σ 2 defined in (23) converge to non-vanihing poitive number. Thu the data tream d 2 archive multiplexing gain of one. Similar tatement are valid for d 2. Theorem 2 In a MIMO ytem with two tranmitter and two receiver each of them equipped with three antenna the propoed cheme achieve multiplexing gain of four. A mentioned if we apply conventional cheme for thi ytem the maximum achievable multiplexing gain i three [4]. The above reult clearly how the advantage of the propoed cheme. Note that in thi cheme the ignal of the tranmitter one and two are uncorrelated. In fact the only information which ha to be hared between the bae tation are all the channel matrice. Reference [] B. S. Blum MIMO capacity with interference IEEE Journal on Selected Area in Communication vol. 2 pp June [2] H. Dai A. F. Molich and H. V. oor Downlink capacity of interference-limited MIMO ytem with joint detection IEEE Tranaction on Wirele Communication vol. 3 pp March [3] H. Dai and H. V. oor Aymptotic pectral efficiency of multicell MIMO ytem with frequency-flat fading IEEE Tranaction on Signal roceing vol. 5 pp Nov [4] S. A. Jafar Degree of freedom in ditributed mimo communication in IEEE Communication Theory Workhop [5] S. Shamai (Shitz) and B.M. Zaidel Enhancing the cellular downlink capacity via co-proceing at the tranmitting end in IEEE Vehicular Technology Conference May 200 vol. 3 pp [6] M. Cota Writing on dirty paper IEEE Tran. Inform. Theory vol. 29 pp May 983. [7] G. J. Fochini H. Huang K. Karakayali R. A. Valenzuela and S. Venkatean The value of coherent bae tation coordination in Conference on Information Science and Sytem (CISS) March [8] S. A. Jafar and A. J. Goldmith Tranmitter optimization for multiple antenna cellular ytem in IEEE International Sympoium on Information Theory 2002 p. 50. [9] C. Roe O. opecu Sum capacity and TSC bound in collaborative multi-bae wirele ytem IEEE Tranaction on Information Theory vol. 0 pp October [0] K. Marton A coding theorem for the dicrete memoryle broadcat channel IEEE Tranaction on Information Theory vol. 25 pp May 979. [] A. El Gamal and E. van der Meulen A proof of marton coding theorem for the dicrete memoryle broadcat channel (correp.) IEEE Tranaction on Information Theory vol. 27 pp Jan. 98. [2] H. Liao Multiple acce channel h.d. thei Dep. Elec. Eng. Univ. of Hawaii [3] S. Gel fand and M. inker Coding for channel with random parameter roblem of Control and Information Theory vol. 9() pp [4] M. A. Maddah-Ali M. Anari and Amir K. Khandani An efficient ignaling cheme for MIMO broadcat ytem: Deign and performance evaluation IEEE Tranaction on Information Theory July 2005 Submitted for ublication.