An Uplink-Downlink Duality for Cloud Radio Access Network

Transcription

1 An Uplin-Downlin Duality for Cloud Radio Access Networ Liang Liu, Prati Patil, and Wei Yu Department of Electrical and Computer Engineering University of Toronto, Toronto, ON, 5S 3G4, Canada s: Abstract Uplin-downlin duality refers to the fact that the Gaussian broadcast channel has the same capacity region as the dual Gaussian mtiple-access channel under the same sumpower constraint This paper investigates a similar duality relationship between the uplin and downlin of a cloud radio access networ C-RAN), where a central processor CP) cooperatively serves mtiple mobile users through mtiple remote radio heads RRHs) connected to the CP with finite-capacity frontha lins The uplin of such a C-RAN model corresponds to a mtipleaccess relay channel; the downlin corresponds to a broadcast relay channel This paper considers compression-based relay strategies in both uplin and downlin C-RAN, where the quantization noise levels are functions of the frontha lin capacities If the frontha capacities are infinite, the conventional uplindownlin duality applies The main rest of this paper is that even when the frontha capacities are finite, duality continues to hold for the case where independent compression is applied across each RRH in the sense that when the transmission and compression designs are jointly optimized, the achievable rate regions of the uplin and downlin remain identical under the same sum-power and individual frontha capacity constraints As an application of the duality rest, the power minimization problem in downlin C-RAN can be efficiently solved based on its uplin counterpart I INTRODUCTION As a promising candidate for the future 5G standard, cloud radio access networ C-RAN) enables a centralized processing architecture, using mtiple relay-lie base stations BSs), named remote radio heads RRHs), to serve mobile users cooperatively under the coordination of a central processor CP) The practically achievable throughput of C-RAN is largely constrained by the finite-capacity frontha lins between the RRHs and the CP In the literature, a considerable amount of effort has been dedicated to the study of efficient frontha techniques [] In the uplin, the compression-based strategy can be used, where each RRH samples, quantizes and forwards its received signals to the CP over its frontha lin such that each user s messages can be jointly decoded [2] Liewise, in the downlin, the CP can pre-form the beamforming vectors, then compress and transmit the beamformed signals via the frontha lins to RRHs for coherent transmission [3] This paper focuses on these compression-based strategies and reveals an uplin-downlin duality for C-RAN Information theoretically, the uplin of C-RAN model corresponds to a mtiple-access relay channel, while the downlin corresponds to a broadcast relay channel When the frontha lins have infinite capacities, the C-RAN model reduces to a mtiple-access channel AC) in the uplin, and a broadcast channel BC) in the downlin, for which there is a well-nown uplin-downlin duality, ie, the achievable rate regions with linear beamforming or the entire capacity regions of the AC and the dual BC are identical under the same sum-power constraint [4] [7] In this paper, we extend such an uplindownlin duality relationship to the C-RAN model where the frontha capacities are finite We show that if independent compression is applied across each RRH, the achievable rate regions of the uplin and downlin C-RAN are identical under the same sum-power and the same individual frontha capacity constraints when all the terminals are equipped with a single antenna Specifically, given any downlin uplin) power assignment, transmit receive) beamforming and quantization noise levels, we construct the corresponding respective uplin downlin) power assignment, receive transmit) beamforming and quantization noise levels such that each user s access rate, each RRH s frontha rate, as well as the sum-power are all preserved Based on this duality rest, we further show that the downlin sum-power minimization problem can be efficiently solved based on its uplin counterpart Although this paper focuses on the compression-based s- trategies for C-RAN, we remar that other coding strategies can potentially outperform the compression strategy for certain channel parameters For example, the so-called compute-andforward and reverse compute-and-forward strategies have been applied for the uplin and downlin C-RAN, respectively, where linear functions of transmitted codewords are computed and then sent between the RRHs and CP [8] oreover, in the downlin, the CP may opt to share user data directly with the RRHs [9] However, the capacity of the uplin and downlin C-RAN model is still an open problem Notation: Scalars are denoted by lower-case letters, vectors denoted by bold-face lower-case letters, and matrices denoted by bold-face upper-case letters I and denote an identity matrix and an all-one vector, respectively, with appropriate dimensions For a square matrix S, S denotes its inverse if S is fl-ran) For a matrix of arbitrary size, [] i,j denotes the entry on the ith row and jth column of Diagx,, x ) denotes a diagonal matrix with the diagonal elements given by x,, x C x y denotes the space of x y complex matrices II SYSTE ODEL The uplin and downlin models for a C-RAN consist of one CP, RRHs, and users It is assumed that all the RRHs

2 where w [w,,, w, ] T with w 2 denotes the decoding beamforming vector for user s message, and ỹ [ỹ,, ỹ ]T denotes the collective compressed signals from all RRHs The total transmit power of all the users is expressed as P { i }) E[ x i 2 ] i 4) oreover, in this paper we assume that the compression process is done independently across RRHs Based on ratedistortion theory, the frontha rate for transmitting ỹm is expressed as Fig System model of the uplin and downlin C-RAN and users are equipped with a single antenna In the uplin, the channel from user to RRH m is denoted by h m, ; in the dual downlin, the channel from RRH m to user is then given by h H m, The sum-power constraint is denoted by P for both the uplin and downlin It is assumed that RRH m is connected to the CP via a noiseless digital frontha lin with capacity C m In this paper, we focus on compression-based strategies to relay the information between the CP and RRHs via frontha lins A Uplin C-RAN The uplin C-RAN model is as shown in Fig a) The discrete-time baseband channel between the users and RRHs can be modelled as y y h, h, h, h, x x + z z, ) where x denotes the transmit signal of user, and z m CN 0, σ 2 ) denotes the additive white Gaussian noise AWGN) at RRH m In this paper, we assume that each of the users transmits using a Gaussian codeboo, ie, x s,, where s CN 0, ) denotes the message of user, and p denotes the transmit power of user After receiving the wireless signals from users, RRH m compresses ym and sends the compressed signals to the CP, m The quantization noise can be modeled as an independent Gaussian random variable, ie, ỹ m y m + e m h m, x + z m + e m, m, 2) where e m CN 0, qm), and qm denotes the variance of the compression noise at RRH m After receiving the compressed signals, the CP first decodes the compression codewords and then applies linear beamforming to decode each user s message, ie, s w H ỹ,, 3) Cm{ i }, qm) Iym; ỹm) i h m,i 2 + qm + σ 2 q m, m 5) Finally, with interference treated as noise, the achievable rate of user is expressed as R { i, w i }, {qm}) Is ; s ) i wh h i 2 + qm w,m 2 + σ 2 j j wh h j 2 + q m w,m 2 + σ 2,, 6) where h [h,,, h, ] T denotes the collective channel from user to all RRHs Given the individual frontha capacity constraints C m s and sum-power constraint P, define the set of feasible transmit power, compression noise levels, and receive beamforming vectors as T {C m }, P ) { { i, w i }, {qm}) : P { i }) P, Cm{ i }, qm) C m, m, w i 2, i } 7) With independent compression and linear decoding, the achievable rate region in the uplin C-RAN is thus given by R {C m }, P ) { r,, r) : r B Downlin C-RAN { i,wi},{q m }) T {C m},p ) R { i, w i }, {q m}), } 8) The downlin C-RAN model is as shown in Fig b) The discrete-time baseband channel model between the RRHs and the users is the dual of the uplin channel given by y y h H, h H, h H, hh, x x + z z, 9) where x m denotes the transmit signal of RRH m, and z CN 0, σ 2 ) denotes the AWGN at receiver

3 The transmit signals x m s are compressed versions of the beamformed signals x m s Similar to 2), the compression noise is modelled as a Gaussian random variable, ie, x m x m + e m, m, 0) where e m CN 0, qm), and qm denotes the variance of the compression noise at RRH m Similar to the uplin, a Gaussian codeboo for each user is used for downlin transmission Define the beamformed signal intended for user to be transmitted across all the RRHs as v s,, where s CN 0, ) denotes the message for user, denotes the transmit power, and v [v,,, v, ] T with v 2 denotes the transmit beamforming vector The aggregate signal intended for all the RRHs is thus given by v i p i s i, which is compressed and then sent to the RRHs via frontha lins The transmit signal across the RRHs is therefore expressed as x v i, p i s i e + x v ) i, p i s e i Then, the transmit power of all the RRHs is expressed as P { i }, {q m}) E[ x m 2 ] q m 2) Again, the compression is done independently across RRHs As a rest, the frontha rate for transmitting x m is expressed as Cm{ i, v i }, qm) I x m; x m) i v i,m 2 + qm q m, m 3) Finally, with linear encoding, the achievable rate of user can be expressed as R { i, v i }, {qm}) Is ; y ) i vh i h 2 + qm h m, 2 + σ 2 j j vh j h 2 + q m h m, 2 + σ 2, 4) Given the individual frontha capacity constraints C m s and sum-power constraint P, define the set of feasible transmit power and beamforming vectors as well as compression noise levels as T {C m }, P ) { { i, v i }, {qm}) : P { i }, {qm}) P, Cm{ i, v i }, qm) C m, m, v i 2, i } 5) With independent compression and linear encoding, the achievable rate region in the downlin C-RAN is thus given by R {C m }, P ) { r,, r) : r { i,vi},{q m }) T {C m},p ) R { i, v i }, {q m}), } 6) III UPLIN-DOWNLIN DUALITY FOR C-RAN In this section, we establish a duality relationship for C- RAN under the compression strategy by comparing the rate regions R {C m }, P ) and R {C m }, P ) As mentioned before, if the frontha lins between the RRHs and the CP have infinite capacities, ie, C m, m, then the uplin and downlin C-RAN models reduce to the AC and BC, respectively, thus the conventional uplin-downlin duality applies, ie, R {C m }, P ) R {C m }, P ) With finite values of C m s, however, it is not obvious if duality still holds On one hand, the compression noise contributes to the transmit power in the downlin, but has no contribution to the transmit power in the uplin On the other hand, bacground Gaussian noise contributes to the frontha rate in the uplin, but it does not affect the frontha transmission in the downlin Interestingly, the following theorem shows that these two effects counteract with each other and the uplindownlin duality continues to exist in C-RAN even with finite frontha capacities Theorem : Consider a C-RAN model implementing compression-based strategies in both the uplin and the downlin, where all the users and RRHs are equipped with one single antenna Then, any rate tuple that is achievable in the uplin is also achievable with the same sum-power and individual frontha capacity constraints in the downlin, and vice versa, ie, R {C m }, P ) R {C m }, P ) Proof: First, we show that given any feasible uplin solution { i, w i}, { q m}) T {C m }, P ), the following downlin problem is always feasible find { i, v i }, {qm} 7) st R { i, v i }, {qm}) R { i, w i }, { q m}),, 8) Cm{ i, v i }, qm) Cm{ i }, q m), m, 9) P { i }, {qm}) P { i }) 20) According to 5), we have q m i h m,i 2 + σ 2, m, 2) where 2 C m { p i }, q m ) By substituting the quantization noise level from 2) in 6), it follows that γ j j wh h j 2 + wh h 2 where γ 2 R { p i, w i},{ q m }) i hm,i 2 +σ 2 ) w,m 2 + σ 2,, 22)

4 Define, τ C with the th elements denoted as and w,m 2 /, respectively, D diagγ / w H h 2,, γ / w H h 2 ), and Ā C with [Ā ] h m,i 2 w i,m 2, if i j, i,j w H i h j 2 + h m,j 2 w i,m 2, otherwise 23) Then, we can rewrite 22) in the following matrix form: I D Ā ) σ 2 D + σ 2 D τ 24) Since > 0 is a feasible solution to 24) by choice, according to [0, Theorem 2], it follows that ρ D Ā ) <, where ρ ) denotes the spectral radius of the argument matrix Next, consider problem 7) It follows from constraint 9) that q m i v i,m 2, m 25) By substituting the quantization noise level from 25) in 8), we have γ j j vh j h 2 + vh h 2 i vi,m 2 h m, 2 + σ 2, 26) We can rewrite 26) in the following matrix form: I D A ) σ 2 D, 27) where C with the th element denoted as, D diagγ / v H h 2,, γ / v H h 2 ), and A C with [ A ] i,j h m,i 2 v i,m 2, if i j, v H j h i 2 + h m,i 2 v j,m 2, otherwise Next, we choose the downlin beamforming solution as 28) v w, 29) Then we have D D and A Ā ) T As a rest, 27) reduces to I D Ā ) T ) σ 2 D 30) Since D Ā and D Ā ) T possess the same eigenvalues, we have ρ D Ā ) T ) ρ D Ā ) < According to [0, Theorem 2], this implies the existence of a feasible power solution > 0 to equation 27), which is given by I D ) Ā ) T σ 2 D 3) By substituting the above downlin power solution into 25), the corresponding quantization noise power levels can be obtained Given any feasible uplin solution { i, w i}, { q m}) T {C m }, P ), we have constructed a downlin solution as given by 29), 3) and 25) which satisfies constraints 8) and 9) in problem 7) Next, we show that this solution also satisfies the sum-power constraint 20): P { i }, {q m}) q m i v i,m 2 T + τ ) T σ 2 + τ ) T I D Ā ) T ) D σ 2 T I D Ā ) D + τ ) T i P { i }) 32) As rest, given any feasible uplin solution { i, w i}, { q m}) T {C m }, P ), the constructed downlin solution given in 29), 3) and 25) satisfies all the constraints in problem 7) In other words, any rate tuple that is achievable in the uplin is also achievable in the downlin Similarly, we can show that given any feasible downlin solution { i, v i}, { q m}) T {C m }, P ), the following uplin problem is always feasible find { i, w i }, {qm} 33) st R { i, w i }, {qm}) R { i, v i }, { q m}),, 34) Cm{ i }, qm) Cm{ i, v i }, q m), m, 35) P { i }) P { i }, { q m}) 36) In other words, any rate tuple that is achievable in the downlin is also achievable in the uplin Theorem is thus proved Remar : Besides linear encoding and decoding, to achieve larger rate regions, dirty-paper coding can be used in the downlin and successive interference cancellation can be used in the uplin at the CP Consider again the compressionbased strategy C-RAN with single-antenna terminals Using similar proof technique as for Theorem, it can be shown that by reversing the encoding and decoding order, any rate tuple that is achievable in the uplin is achievable with the same sum-power and individual frontha capacity constraints in the downlin, and vice versa As a rest, uplin-downlin duality also applies with non-linear encoding and decoding techniques Remar 2: In [7], it is shown based on the minimax optimization technique that the capacity region of the BC with perantenna power constraints is the same as that of its dual AC with an uncertain noise at the receiver constrained by a certain positive semidefinite covariance matrix Following a similar

5 approach, it can be shown that the achievable rate region of the downlin C-RAN with per-rrh power constraints is the same as that of its dual uplin C-RAN with uncertain noises across all the RRHs IV APPLICATION OF UPLIN-DOWNLIN DUALITY This section illustrates an application of uplin-downlin d- uality We first show that the sum-power minimization problem in the uplin C-RAN can be globally solved by the celebrated fixed-point method [] Then, the optimal solution to the downlin sum-power minimization problem is obtained based on the uplin solution Specifically, in the uplin, the sumpower minimization problem is formated as { i,wi},{q m } subject to P { i }) 37) R { i, w i }, {qm}) R, Cm{ i }, qm) C m, m,, where R denotes the rate requirement of user It can be shown that with the optimal solution, the frontha capacities shod be fly used, ie, 2) holds with 2 Cm, m oreover, the optimal beamforming solution is the wellnown minimum-mean-square-error SE) based receiver, ie, w j j h j h H j + Q ) + σ 2 I h,, 38) where C with the th element denoted by, and Q ) diagq,, q ) is a diagonal matrix with q m s given by 2) By substituting the quantization noise levels and receive beamforming vectors with 2) and 38), problem 37) reduces to the following power control problem P { i }) 39) subject to Γ ), where Γ ) C with the th element as [Γ )] 2 R ), 40) h H j h jh H j + Q ) + σ 2 I h j It can be shown that Γ ) is a standard interference function [] According to [, Theorem 2], the following fixed-point algorithm can converge to the globally optimal power solution to problem 39) given any initial point:,n+) Γ,n) ), 4) where,n) denotes the power allocation solution obtained in the nth iteration of the above method After the optimal power allocation is obtained, the optimal quantization noise levels and receive beamforming vectors to problem 37) can be obtained according to 2) and 38), respectively Next, consider the sum-power minimization problem in the downlin { i,vi},{q m } subject to P { i }, {q m}) 42) R { i, v i }, {qm}) R,, Cm{ i, v i }, qm) C m, m In general, the downlin sum-power minimization problem is more involved since unlie the uplin, the optimal transmit beamforming vectors are not easy to obtain However, Theorem indicates that the optimal values of problems 37) and 42) are identical As a rest, we can find the optimal solution to problem 37) first, based on which we can then construct the optimal downlin solution according to 29), 3), and 25) Remar 3: Similar to the power minimization problem, uplin-downlin duality can be applied to solve the downlin weighted sum-rate maximization problem with per-rrh power constraints based on the dual uplin, which is typically easier to deal with see Remar 2) V CONCLUSION An uplin-downlin duality relationship is established for C-RAN Specifically, we show that if transmission and compression designs are jointly optimized, the achievable rate regions of the uplin and downlin C-RAN are identical under the same sum-power and individual frontha capacity constraints when independent compression is performed across RRHs Furthermore, this duality rest proves usef for solving the downlin power minimization problem based on its dual problem in the uplin REFERENCES [] O Simeone, A aeder, Peng, O Sahin, and W Yu, Cloud radio access networ: virtualizing wireless access for dense heterogeneous systems, to appear in J Commun Networs, Apr 206 [Online] Available: [2] Y Zhou and W Yu, Optimized bacha compression for uplin cloud radio access networ, IEEE J Sel Areas Commun, vol 32, no 6, pp , Jun 204 [3] S H Par, O Simeone, O Sahin and S Shamai, Joint precoding and mtivariate bacha compression for the downlin of cloud radio access networs, IEEE Trans Signal Process, vol 6, no 22, pp , Nov 203 [4] F Rashid-Farrohi, J R Liu, and L Tassias, Transmit beamforming and power control for cellar wireless systems, IEEE J Sel Areas Commun, vol 6, no 8, pp , Oct 998 [5] P Viswanath and D N C Tse, Sum capacity of the mtiple antenna Gaussian broadcast channel and uplin-downlin duality, IEEE Trans Inf Theory, vol 49, no 8, pp 92-92, J 2003 [6] S Vishwanath, N Jindal, and A Goldsmith, Duality, achievable rates and sum rate capacity of Gaussian IO broadcast channels, IEEE Trans Inf Theory, vol 49, pp , Oct 2003 [7] W Yu, T Lan, Transmitter optimization for the mti-antenna downlin with per-antenna power constraints, IEEE Trans Signal Process, vol 55, no 6, pp , Jun 2007 [8] S N Hong and G Caire, Compute-and-forward strategies for cooperative distributed antenna systems, IEEE Trans Inf Theory, vol 59, no 9, pp , Sep 203 [9] B Dai and W Yu, Sparse beamforming and user-centric clustering for downlin cloud radio access networ, IEEE Access, vol 2, pp , 204 [0] E Seneta, Non-Negative atrices and arov Chains: Springer, 98 [] R D Yates, A framewor for uplin power control in cellar radio systems, IEEE J Select Areas Commun, vol 3, pp , Sep 995