End-to-End Rate Enhancement in C-RAN Using Multi-Pair Two-Way Computation
aa r X i v : . [ c s . I T ] A ug End-to-End Rate Enhancement in C-RAN UsingMulti-Pair Two-Way Computation
Mahmoud A. Hasabelnaby and Anas Chaaban
School of Engineering, University of British Columbia, Kelowna, V1V1V7 BC, Canada { mahmoud.hasabelnaby,anas.chaaban } @ubc.ca Abstract —Cloud radio-access networks (C-RAN) have beenproposed as an enabling technology for keeping up with therequirements of next-generation wireless networks. Most existingworks on C-RAN consider the uplink or the downlink sepa-rately. However, designing the uplink and the downlink jointlymay bring additional advantage, especially if message source-destination information is taken into account. In this paper, thisidea is demonstrated by considering pairwise message exchangebetween users in a C-RAN. A multi-pair two-way transmissionscheme is proposed which targets maximizing the end-to-end userdata rates. In the proposed scheme, a lattice-based computationstrategy is used, where the baseband processing unit (BBU) pooldecodes integer linear combinations of paired users’ codewordsinstead of decoding linear combinations of individual codewords.The BBU pool then compresses the computed signals and for-wards them to the remote radio heads (RRHs), which decompressthe signals and send them to the users. Finally, each user decodesits desired message using its own message as side information.The achievable rate of this scheme is derived, optimized, andevaluated numerically. Results reveal that significant end-to-endrate improvement can be achieved using the proposed schemecompared to existing schemes.
Index Terms —C-RAN; Compress-and-forward; Compute-and-forward; Lattice codes; Reverse quantized-compute-and-forward.
I. I
NTRODUCTION
The cloud radio-access network (C-RAN) architecture isone of the methods that enhance communication capabilitiestowards meeting the critical requirements of next-generationwireless networks. Centralized processing in the C-RANenables coordination between remote radio heads (RRHs)over a large geographical service area, thus allowing RRHsto perform joint transmission and reception. This providesan interference mitigation capability, thereby improving theoverall performance [1], [2].The performance of a C-RAN is constrained by the limitedcapacity of fronthaul links that connect RRHs with the base-band processing unit (BBU) pool. This puts a constraint on theamount of information that can be exchanged between RRHsand the BBU pool. Therefore, advanced signal processing andrelaying techniques are required in both uplink and downlinkin order to make efficient use of the fronthaul links [3]. Thishas been the topic of many studies recently as discussed next.In the uplink, multiple users transmit their codewords tothe RRHs. Different relaying strategies can be used to relayinformation from RRHs to the BBU pool, such as decode-and-forward (DF) [1], compute-and-forward (CoF) [4], andcompress-and-forward (CF) [5]. CoF is better than the CF under small fronthaul capacity values. However, the averageperformance of CF is better at moderate and high fronthaulcapacities [1], [6]. Several low complexity CF strategies havebeen proposed, which vary based on the amount of utilizedside information in the compression process, such as single-user (SU) compression [1] and Wyner-Ziv (WZ) compression[7], with the latter being superior, but requiring channelstate information at all RRHs and the BBU pool. Moreover,a lattice-based compression scheme called integer-forcingsource and channel coding (IFSC+IFCC) can achieve similarperformance as Wyner-Ziv compression [6]. In the downlink,transmission strategies that enable cooperation between RRHsinclude data-sharing strategies [8], [9], compression basedstrategies [1], [10], reverse compute-and-forward (RCoF)[11], and reverse quantized-compute-and-forward (RQCoF)[4]. Since the exact characterization of the downlink C-RANcapacity is still an open problem, most works optimize theschemes using uplink-downlink duality to achieve downlinkrates greater than or equal to the uplink rates [12], [13].Note that all aforementioned works study either the uplinkor the downlink, separately. This may incur performance loss,especially in scenarios where intra-cloud message exchangeis desired. This paper demonstrates this idea via studying aC-RAN with intra-cloud pairwise communication. Such a sce-nario can occur in video conferencing or gaming applicationsfor instance. A multi-pair two-way transmission scheme isproposed to maximize the end-to-end achievable rate. Usinga lattice-based compression strategy, RRHs compress theirobservations and forward them to the BBU pool, which in turncomputes integer linear combinations of codeword-pairs. Thisreduces the required number of computation steps at the BBUpool, thereby reducing the number of rate constraints. TheBBU pool compresses the linear combinations and forwardsthem to the RRHs, which decompress the signals and transmitthem to the users. Finally, users decode their desired messageusing their own messages as side information. The achievablerate of the scheme is derived, optimized, and evaluated nu-merically, showing superior performance to existing schemesin the literature.In the sequel, the following notations will be used. Columnvectors and matrices are donated by boldface lowercase anduppercase letters, such as x and X , respectively. The transposeof matrix X is donated by X ⊤ . All the logarithms are to thebase 2, and log + ( x ) = max(0 , log( x )) .I. S YSTEM M ODEL
We consider a C-RAN consisting of M single-antennauser pairs ( K = 2 M users), L single-antenna RRHs, anda central processor (BBU pool). Each RRH ℓ ∈ { , , . . . , L } ,is connected to the BBU pool via a digital noiseless fronthaullink with a limited capacity C ℓ .User pairs wish to communicate with each other usingthe C-RAN architecture. In other words, users k, k ′ ∈{ , , . . . , K } , k = k ′ , exchange messages with each other.As a result, a pairing matrix W with dimensions of M × K is defined, where w m,k ∈ { , } is a user-pair associationindicator, i.e., w m,k = 1 if user k belongs to pair m , and w m,k = 0 otherwise. Note that P Mm =1 w m,k = 1 for all k .The message exchange among the user-pairs is completed intwo stages over n channel uses each, an uplink phase and adownlink phase. In the uplink, user k ∈ { , . . . , K } encodes its message g k with rate R k into a codeword x ulk ∈ R n and sends it to theRRHs. This signal is subject to a power constraint p ulk . Notethat we consider a real-valued transmission model for the sakeof simplicity, bearing in mind that a complex-valued modelcan be addressed using the real-valued vector representation.The received signal at the RRHs is given by Y ul = H ul X ul + Z ul (1)where Y ul = [ y ul , y ul , . . . , y ulL ] ⊤ , y ulℓ ∈ R n is the receivedsignal at RRH ℓ , H ul ∈ R L × K is the uplink channel gain ma-trix between all users and RRHs, X ul = [ x ul , x ul , . . . , x ulK ] ⊤ ,and Z ul ∈ R L × n is additive white Gaussian noise withindependent and identically distributed (i.i.d.) componentswith zero mean and unit variance N (0 , . We assume thatchannels are Rayleigh fading and remain fixed across thetransmission of a complete codeword (block fading). RRH ℓ ∈ { , . . . , L } processes the signal y ulℓ into a message e ulℓ ,and sends this message to the BBU pool using the fronthaullink.The downlink is described as follows. First, the BBU pro-cesses the received messages from all RRHs, then constructsmessages e dlℓ , ℓ ∈ { , . . . , L } , and sends e dlℓ to RRH ℓ in thedownlink using the fronthaul links. The RRH then processes e dlℓ to construct a transmit signal x dlℓ ∈ R n with powerconstraint p dlℓ for transmission to the users. The receivedsignals at the users can be written in a matrix form as Y dl = H dl X dl + Z dl , (2)where Y dl = [ y dl , y dl , . . . , y dlK ] ⊤ , H dl ∈ R K × L is thedownlink channel gain matrix between all RRHs and users, X dl = [ x dl , x dl , . . . , x dlL ] ⊤ , and Z dl ∈ R K × n is additivewhite Gaussian noise with i.i.d N (0 , components. Notethat we assume channel reciprocity, i.e., H dl = H ul ⊤ . Finally,each user uses its received signal in combination with its ownmessage to decode the message of the paired user. This work can be extended to deal with MIMO systems. We assume that the uplink and downlink occur over the same frequencyband, which may be either achieved in a full-duplex or a half-duplex fashion.
The goal is to design an uplink/downlink transmissionscheme which takes this user pairing into account, and toderive its achievable rate. This is discussed in the followingsections which discuss the uplink phase and the downlinkphase, respectively.III. U
PLINK T RANSMISSION
We start by describing the encoding at the users, followedby the processing at the RRHs and the BBU pool.
A. Encoding at the users
Using nested lattice coding [4], the lattice chain Λ c,K ⊆ . . . ⊆ Λ c, ⊆ Λ f is generated, consisting of n -dimensional lat-tices. The coarse lattices Λ c,k , k ∈ { , , . . . , K } are chosen tobe good for channel coding and quantization simultaneously,whereas the fine lattice Λ f is good for quantization only. User k generates its nested lattice codebook as C ulk = { Λ f ∩ υ Λ c,k } ,where υ Λ c,k is the fundamental Voronoi region of the coarselattice. Using a one-to-one mapping, it maps its message g k into a lattice point s ulk ∈ C ulk . To make the transmittedsignal independent on the lattice point, a random dither t ulk ,uniformly distributed over υ Λ c,k and known to all nodes in thenetwork, is added. The result is then reduced using a modulo-lattice operation with respect to Λ c,k . This leads to the signal x ulk = ( s ulk + t ulk ) mod Λ c , k (3)which is then transmitted to the RRHs. The power constraint p ulk is met by selecting a coarse lattice Λ c,k with secondmoment σ (Λ c,k ) = n E [ k x ulk k ] ≤ p ulk . B. Compression at the RRHs
The received signal y ulℓ at RRH ℓ is processed as follows.Given a lattice chain ˜Λ c, ⊆ . . . ⊆ ˜Λ c,L ⊆ ˜Λ f,L ⊆ . . . ⊆ ˜Λ f, ,RRH ℓ generates its codebook as κ ℓ = { ˜Λ f,ℓ ∩ υ ˜Λ c,ℓ } withrate equal to the fronthaul capacity C ℓ . The coarse lattice ˜Λ c,ℓ is good for channel coding and quantization simultaneously,thereby the probability of error can be neglected. In addition,the fine lattice ˜Λ f,ℓ must be good for quantization in order tobe able to find a precise relationship between the quantizationrates and distortion levels [14]. Then, the ℓ th RRH adds arandom dither vector ˜ t ulℓ , uniformly distributed over υ ˜Λ f,ℓ ,to its observation to make the quantization error independentof the received signal y ulℓ . Moreover, using its generatedcodebook κ ℓ , the ℓ th RRH compresses its dithered observedsignal as follows ¨ y ulℓ = [ Q ˜Λ f,ℓ ( y ulℓ + ˜ t ulℓ )] mod ˜Λ c ,ℓ . (4)Then, RRH ℓ maps ¨ y ulℓ to an index e ulℓ ∈ { , , . . . , nC ℓ } ,and forwards it to the BBU pool via its fronthaul link. C. Computations at the BBU pool
Once the BBU pool receives the indices e ul , e ul , . . . , e ulL ,it recovers ¨ y ul , ¨ y ul , . . . , ¨ y ulL , then subtracts the dithers, andeduces the result using the modulo-lattice operation withrespect to ˜Λ c,ℓ as ˜ y ulℓ = [ ¨ y ulℓ − ˜ t ulℓ ] mod ˜Λ c ,ℓ ( a ) = [ y ulℓ + q ulℓ ] mod ˜Λ c ,ℓ = [ ˆ y ul ℓ ] mod ˜Λ c ,ℓ (5)where ( a ) is obtained from the distributive law of the modulo-lattice operation, q ulℓ = − [ y ulℓ + ˜ t ulℓ ] mod ˜Λ f ,ℓ is thequantization error which is independent of y ulℓ and uniformlydistributed over υ ˜Λ f,ℓ , q ulℓ ∼ N (0 , d ulℓ ) where d ulℓ is thedistortion power, and ˆ y ulℓ = y ulℓ + q ulℓ . After that, theBBU pool proceeds to decode L integer linear combinations v ulr, , v ulr, , . . . , v ulr,L as in [6], where v ulr,i = " L X ℓ =1 a ulr,i,ℓ ˆ y ulℓ mod ˜Λ c , i ( b ) = L X ℓ = a ulr , i ,ℓ ˆ y ul ℓ (6) i ∈ { , , . . . , L } , a ulr,i,ℓ ∈ Z is an integer coefficient, and ( b ) is obtained with high probability (w.h.p.) if ˜Λ c , i is good forchannel coding and n E [ k v ulr,i k ] < σ (˜Λ c,i ) [14]. We write(6) in a matrix form as V ulr = A ulr ˆY ul , where A ulr is a L × L full-rank integer coefficient matrix with full rank sub-matrices A ulr, [1: i ] for i ∈ { , , . . . , L } and ˆY ul = [ ˆ y ul , ˆ y ul , . . . , ˆ y ulL ] ⊤ .The integer coefficients can be selected to reduce the varianceof the compressed signals which in turn decrease the requiredcompression rates.The compression rate at the ℓ th RRH can be expressed as R ulr,ℓ = 12 log + a ul ⊤ r,ℓ ( H ul P ul H ul ⊤ + I + D ul ) a ulr,ℓ d ulℓ ! (7)where R ulr,ℓ ≤ C ℓ , a ul ⊤ r,ℓ is the ℓ th row of A ulr , P ul isa K × K diagonal matrix whose diagonal elements areequal to diag( p ul1 , p ul2 , . . . , p ulK ) , I is a L × L identity ma-trix, and D ul is a L × L uplink quantization error effec-tive covariance matrix whose diagonal elements is equal to d ul = [ d ul , d ul , . . . , d ulL ] ⊤ . The achievable uplink distortionlevels d ul can be written in terms of the achievable uplinkcompression rates C , . . . , C L as d ul = C ul ξ ul , where ξ ulℓ = a ul ⊤ r,ℓ ( H ul P ul H ul ⊤ + I ) a ulr,ℓ is the ℓ th element of ξ ul and C ul is a L × L matrix represented as follows C ul = C − ( a ulr, , ) . . . C − ( a ulr, ,L ) ... . . . ... C L − ( a ulr,L, ) . . . C L − ( a ulr,L,L ) − (8)By multiplying V ulr by the inverse of the integer coefficientmatrix A ulr , inv , the BBU pool can recover ˆY ul = H ul X ul + Z ul + Q ul (9)where Q ul = [ q ul , q ul , . . . , q ulL ] ⊤ .The BBU pool proceeds to decode M integer linear com-binations of user-pairs’ codewords (instead of decoding K integer linear equations of users’ individual codewords as in[6]), to obtain v ulψ,j = " M X m =1 a ulψ,j,m " K X k =1 w m,k s ulk mod Λ c , K , (10)where j ∈ { , , . . . , M } and a ulψ,j,m ∈ Z is an integer coef-ficient. Computing M combinations instead of K decreasesthe number of constraints on the computation rate, whichimproves performance. Note that (10) can be rewritten inmatrix form as V ul ψ = [ A ul ψ WS ul ] mod Λ c , K , where A ul ψ isan M × M integer coefficient matrix with a rank of M = K/ ,and S ul = [ s ul , s ul , . . . , s ulK ] ⊤ . This decoding can be done bylinearly processing ˆ Y ul with a scaling equalizer ρ ulj , removingthe dither t ulk , and reducing the result modulo Λ c,K as µ ulj = " ρ ul ⊤ j ˆ Y ul − K X k =1 t ulk mod Λ c , K = (cid:2) ρ ul ⊤ j H ul S ul + ρ ul ⊤ j ( Z ul + Q ul ) (cid:3) mod Λ c , K = [ a ul ⊤ ψ,j WS ul | {z } desired signal + z ul eff , j ] mod Λ c , K (11)from which the BBU pool computes [ a ul ⊤ ψ,j WS ul ] mod Λ c , K ,where z ul eff , j = ( ρ ul ⊤ j H ul − a ul ⊤ ψ,j W ) S ul ρ ul ⊤ j ( Z ul + Q ul ) isthe effective noise with power σ ulj = 1 n E [ k z ul eff , j k (12) = k ( ρ ul ⊤ j H ul − a ul ⊤ ψ,j W )( P ul ) k + ρ ul ⊤ j ( I + D ul ) ρ ulj . In order to minimize the effective variance in (12), ρ ulj ischosen as the MMSE scaling equalizer given by ρ ul ⊤ j = a ul ⊤ ψ,j WP ul H ul ⊤ ( H ul P ul H ul ⊤ + I + D ul ) − (13)By substituting the MMSE solution into (12) and applyingthe matrix inversion lemma, the effective noise power σ ulj can be rewritten as σ ulj = a ul ⊤ ψ,j [ W ( P ul − + H ul ⊤ ( I + D ul ) − H ul ) − W ⊤ ] a ulψ,j = a ul ⊤ ψ,j [ F ulψ F ul ⊤ ψ ] a ulψ,j = k F ulψ a ulψ,j k (14)where F ulψ is the Cholesky decomposition satisfying F ulψ F ul ⊤ ψ = W ( P ul − + H ul ⊤ ( I + D ul ) − H ul ) − W ⊤ .Finally, the uplink computation rate for the k th user in user-pair m can be expressed as R ulψ, k = 12 log + p ulk M X m =1 w m,k σ ulm ! − (15)where σ ulm is given by (14). Instead of recovering the originalmessages as in [6], the BBU compresses the previouslycomputed equations directly and forwards them to the RRHsthrough the fronthaul links as described next.IV. D OWNLINK T RANSMISSION
The basic idea of the downlink is to employ a reverse-quantized-compute-and-forward scheme [4]. . Compression at the BBU pool
At first, the BBU pool uses the beamforming matrix B dl with dimensions L × M to produce S dl = B dl V ulψ = B dl [ A ulψ WS ul ] mod Λ c , K (16)where S dl = [ s dl , s dl , . . . , s dlL ] ⊤ . In order to enable each RRHto extract its desired quantized signal, the BBU pool pre-inverts the S dl with A dlr,inv as follows V dlr = A dlr,inv S dl (17)where V dlr = [ v dl , v dl , . . . , v dlL ] ⊤ and A dlr,inv is the inverseof the L × L full rank integer coefficient matrix A dlr . Then,the BBU pool uses a lattice chain ˆΛ c, ⊆ ˆΛ c, ⊆ ... ⊆ ˆΛ c,L ⊆ ˆΛ f , where the coarse lattices and the fine latticehave the same properties as mentioned in the user encodingstep. Next, the BBU pool adds a random dither matrix ˆ T dl =[ˆ t dl , ˆ t dl , . . . , ˆ t dlL ] ⊤ to V dlr which is uniformly distributed over υ ˆΛ f . The dithered output is then quantized as ˆV dlr = Q ˆΛ f ( V dlr + ˆ T dl ) (18)where Q ˆΛ f is applied to each row of the dithered matrixseparately. The BBU pool proceeds to generate integer linearcombinations ˜V dlr = A dlr ˆV dlr and performs the modulo-latticeoperation with respect to ˆΛ c,i , i ∈ { , , . . . , L } to each i throw in ˜V dlr to obtain ˜ v dlr,i = h a dl ⊤ r,i ˆV dlr i mod ˆΛ c , i = h a dl ⊤ r,i Q ˆΛ f ( V dlr + ˆ T dl ) i mod ˆΛ c , i (19)Finally, the BBU pool maps its compressed linear equation ˜ v dlr,i to an index e dli ∈ { , , . . . , nC l } , and forwards it to the i th RRH. B. Decompression at the RRHs
Once the ℓ th RRH receives the index e dlℓ , it recovers ˜ v dlr,ℓ ,then subtracts the dither ˆ T dl , and reduces the result using themodulo-lattice operation with respect to ˆΛ c,ℓ to obtain x dlℓ = h ˜ v dlr,ℓ − a dl ⊤ r,ℓ ˆ T dl i mod ˆΛ c ,ℓ = h a dl ⊤ r,ℓ ( V dlr + ˆ T dl + Q dl ) − a dl ⊤ r,ℓ ˆ T dl i mod ˆΛ c ,ℓ ( c ) = s dl ⊤ ℓ + a dl ⊤ r,ℓ Q dl (20)where Q dl = [ q dl , q dl , . . . , q dlL ] ⊤ is the quantization errorwith a L × L downlink effective covariance matrix D dl whosediagonal elements is equal to diag( d dl1 , d dl2 , . . . , d dlL ) , and (c)is obtained w.h.p. if n E [ k x dlℓ k ] < σ (ˆΛ c,ℓ ) . The downlinkcompression rate at RRH ℓ is given by R dlr,ℓ = 12 log + b dl ⊤ ℓ P ulψ b dlℓ + a dl ⊤ r,ℓ D dl a dlr,ℓ d dlℓ ! (21)where R dlr,ℓ ≤ C ℓ and P ulψ = p ulK I m is M × M diagonal powermatrix whose diagonal elements are equal to p ulK . Finally, after the ℓ th RRH recovers its desired signal, it broadcasts x dlℓ ∈ R n to the users with power n E [ k x dlℓ k ] = b dl ⊤ ℓ P ulψ b dlℓ + a dl ⊤ r,ℓ D dl a dlr,ℓ , ˜ p dlℓ ≤ p dlℓ . (22) C. Decoding at the users
The received signals at all the users can be written in amatrix form as Y dl = H dl ( B dl V ulψ + A dlr Q dl ) + Z dl . (23)The k th user scales its received signal y dlk by a linear scaling ρ dlk and reduces the result modulo Λ c,k ′ as follows µ dlk = [ ρ dlk y ulk ] mod Λ c , k ′ = [ ρ dlk h dlk ( B dl V ulψ + A dlr Q dl ) + ρ dlk z dlk ] mod Λ c , k ′ = [ a dlψ,k V ulψ | {z } intended signal + z dl eff , k ] mod Λ c , k ′ (24)where z dl eff , k is the effective noise given by ( ρ dlk h dlk B dl − a dlψ,k ) V ulψ + ρ dlk ( h dlk A dlr Q dl + z dlk ) , a dlψ,k is the k th row of A dlψ , a matrix with dimensions of K × M and rank of M , h dlk is the k th row of H dl , A dlψ = W ⊤ A ulψ,inv , A ulψ,inv is theinverse of A ulψ matrix, v dlψ,k = [ a dlψ,k V ulψ ] mod Λ c , k ′ is the k thuser’s intended signal that includes the sum of the codewordsof the user-pair k, k ′ ∈ { , , . . . , K } , k = k ′ . The power ofthe effective noise z dl eff , k is given by σ dlk = 1 n E [ k z dl eff , k k ] (25) = k ( ρ dlk h dlk B dl − a dlψ,k )( P ulψ ) k (26) + ρ dlk ( h dlk A dlr D dl A dl ⊤ r h dl ⊤ k + 1) This effective variance can be minimized by obtaining theMMSE coefficient for the linear scaling equalizer ρ dlk as ρ dlk = a dlψ,k P ulψ B dl ⊤ h dl ⊤ k h dlk ( A dlr D dl A dl ⊤ r + B dl P ulψ B dl ⊤ ) h dl ⊤ k + 1 (27)Finally, user k decodes [ a dlψ,k V ulψ ] mod Λ c , k ′ , and uses itsown codeword [ s ulk ] mod Λ c , k as side information to recoverits desired codeword [ s ul k ′ ] mod Λ c , k ′ as follows (cid:2) [ a dlψ,k V ulψ ] mod Λ c , k ′ − s ulk (cid:3) mod Λ c , k ′ = [ a dlψ,k A ulψ WS ul − s ulk ] mod Λ c , k ′ = " M X m =1 w m,k K X u =1 w m,u s ulu − s ulk mod Λ c , k ′ = [ s ulk + s ulk ′ − s ulk ] mod Λ c , k ′ = [ s ulk ′ ] mod Λ c , k ′ (28)Using this procedure, user k downlink rate is given by R dlψ, k = 12 log + ( p ulk ( σ dlk ) − ) (29)At this point, we can summarize the end-to-end achievablerate of the proposed scheme as given next. heorem 1: The end-to-end data rate of user k achieved bythe proposed scheme is given by R k = min { R ulψ, k , R dlψ, k } , (30)where R ulψ, k and R dlψ, k are given in (15) and (29), respectively. Proof:
This statement follows since the achievable end-to-endrate is bound by the smallest between the uplink rate and thedownlink rate.V. E ND - TO -E ND U SER - RATE O PTIMIZATION
In this section, we propose an iterative multi-pair two-wayrate optimization (MPTWR) algorithm to optimize the end-to-end rate in (30). The algorithm is carried in two steps, wherethe uplink and downlink user-rates are optimized iteratively.
1) Uplink Rate Optimization:
Given H ul and P ul , theachievable uplink rate R ulψ, k can be optimized by selectingproper full rank integer coefficient matrices A ulr and A ulψ ,and selecting the uplink quantization error covariance matrix D ul to satisfy the fronthaul capacity constraint. The uplinkoptimization problem can be formulated as followsmax A ulr , A ulψ , D ul R ulψ, k subject to rank( A ulr ) = L , rank( A ul ψ ) = MR ulr,ℓ ≤ C ℓ ∀ ℓ ∈ { , , . . . , L } (31)Selecting the proper integer coefficient matrices A ulr and A ulψ is related to the Shortest Independent Vector Problem (SIVP)which is NP-hard [15]. However, sub-optimal solutions canbe obtained using the LLL algorithm [16]. For the sake ofsimplicity, it is assumed that all RRHs choose equal distortionlevels, i.e., d ulℓ = d ul , ℓ ∈ { , , . . . , L } . To maximizethe uplink rates, we start by initializing d ul to two extremevalues and then calculate the corresponding A ulr using theLLL algorithm on F ulr which is the Cholesky decompositionsatisfying F ulr F ul ⊤ r = d ul H ul P ul H ul ⊤ + I ( d ul +1) . Next, weupdate d ul using bisection until (31) is satisfied with equality.Finally, we use the obtained d ul to calculate A ulψ using theLLL algorithm on F ulψ defined after (14). This is explainedin detail in Algorithm 1. The results of this algorithm will beused as inputs to optimize the achievable downlink user-rate.
2) Downlink Rate Optimization:
The aim of this step is toobtain an achievable downlink user-rate greater or equal tothe achievable one in the uplink, i.e., R dlψ, k ≥ R ulψ, k . Given thedownlink channel matrix H dl = H ul ⊤ , the downlink powers p dlℓ , ℓ = 1 , . . . , L , and the outputs of the uplink optimizationalgorithm, the achievable downlink user-rate R dlψ, k shouldbe tuned by selecting proper integer coefficient matrix A dlr ,beamforming matrix B dl , and downlink quantization errorcovariance matrix D dl . One way to formulate this problemis as followsmin A dlr , B dl , D dl f un ( A dlr , B dl , D dl ) = K X k =1 ( R dlψ,k − R ulψ,k ) subject to ˜ p dlℓ ≤ p dlℓ , and R dlr,ℓ ≤ C ℓ ∀ ℓ (32) Algorithm 1
Iterative uplink optimization (IUO) Initialization:
Set d min = 0 and d max = d ul = δ (large)such that R ulr,ℓ < C ℓ ∀ ℓ . while max ℓ ( C ℓ − R ulr,ℓ ) > ǫ or max ℓ ( R ulr ,ℓ − C ℓ ) > do if max ℓ ( R ulr ,ℓ − C ℓ ) > then d min = d ul else d max = d ul end if d ul = ( d max + d min ) / F ulr = Chol( ul H ul P ul H ul ⊤ + I ( ul + 1)) A ulr = LLL( F ulr ) R ulr,ℓ = log + ( || F ulr a ulr,l || ) end while Calculate C ul using (8). F ulψ = Chol( W ( P ul − + H ul ⊤ ( I + D ul ) − H ul ) − W ⊤ ) A ulψ = LLL( F ul ψ ) Calculate σ ulm using (14) ∀ m ∈ { , , . . . , M } . Calculate r ulψ = [ R ulψ, , . . . , R ulψ,K ] ⊤ using (15). return ( A ulr , C ul , A ulψ , r ulψ ) Algorithm 2
Iterative downlink optimization (IDO) Initialization: set H dl = H ul ⊤ , A dlr = A ul ⊤ r , C dl = C ul ⊤ , and r dlψ = [ R dlψ, , . . . , R dlψ,K ] ⊤ = 0 K × . while K P k =1 | R dlψ,k − R ulψ,k | > ǫ or max ℓ (˜ p dlℓ − p dlℓ ) > do Set δ = 0 , initialize B dl = L × M matrix of i.i.d. N (0 , , and Γ = I . while termination condition for BFGS method do Compute line search Θ δ = − Γ δ ∇ f un ( B dlδ ) , andstep length γ δ > . Calculate B dlδ +1 = B dlδ + γ δ Θ δ and d dl = C dl ξ dl Calculate ˜ p dlℓ , ρ dlk , and r dlψ = [ R dlψ, , . . . , R dlψ,K ] ⊤ using (22), (27), and (29), respectively Calculate β δ = B dlδ +1 − B dlδ . Calculate Ω δ = ∇ f un ( B dlδ +1 ) − ∇ f un ( B dlδ ) . Update Γ δ using β δ and Ω δ as in [17]. Set δ = δ + 1 . end while end while return ( r ulψ , r dlψ )To simplify this problem we choose A dlr = A ul ⊤ r . Then, weobtain B dl that minimize (32) using the BFGS Quasi-Newtonalgorithm with a cubic line search procedure [17] and calculatethe downlink distortion levels d dl = [ d dl , d dl , . . . , d dlL ] ⊤ using d dl = C dl ξ dl , where C dl = C ul ⊤ , ξ dl = [ ξ dl , . . . , ξ dlL ] ⊤ ,and ξ dlℓ = b dl ⊤ ℓ P ulψ b dlℓ . The line search in the BFGS algorithmmust satisfy the Wolfe conditions in order to ensure sufficientstep length taken in each search direction. Finally, the optimal B dl matrix is obtained when the partial derivatives of B dl aresufficiently too small [17]. If the constraints in (32) are not
20 0 20 40 60 80
SNR [dB] E nd - t o - E nd S u m R a t e [ b i t s / s e c / H z ] MPTWR schemeIFSC+IFCCWZ+IFCCSC+IFCC
Fig. 1. End-to-End sum rate versus SNR in dB at C ℓ = 4 bits/sec/Hz. Fronthaul Capacity per RRH [bits/sec/Hz] E nd - t o - E nd S u m R a t e [ b i t s / s e c / H z ] MPTWR schemeIFSC+IFCCWZ+IFCCSC+IFCC
Fig. 2. End-to-End sum rate versus C ℓ in bits/sec/Hz at SNR = 30 dB. satisfied, we update the initial value of B dl , and repeat untilthe constraints are satisfied. The details of this procedure aregiven in Algorithm 2.VI. N UMERICAL R ESULTS
In this section, the proposed MPTWR optimization schemeis evaluated and compared to the conventional integer-forcingsource and channel coding (IFSC+IFCC) scheme. We alsocompare with optimized Wyner-Ziv (WZ+IFCC) and single-user (SC+IFCC) compression schemes with integer-forcingchannel coding. We use 5000 realizations of the L × K channelmatrix H = H ul = H dl ⊤ , where each element h ℓ,k is i.i.d N (0 , . We set L = 2 RRHs, K = 4 users ( M = 2 user-pairs). It is assumed that the transmitted power of all user-terminals are equal to p ulk = SNR / K , where SNR is thesignal-to-noise ratio in dB. Also, the capacities of the fronthaullinks are assumed to be equal, i.e., C = C .Fig. 1 and 2 show the total achievable end-to-end rate ofour proposed scheme and conventional ones in bits/sec/Hzversus SNR at C ℓ = 4 bits/sec/Hz and different fronthaullink capacity values at SNR of dB, respectively. Thesefigures demonstrate that our proposed scheme has a superiorperformance over other conventional approaches. This is dueto the exploitation of the multi-pair lattice-based computationstrategy that reduces the number of decoded linear combina-tions at the BBU pool to 2 equations instead of the 4 equations required by other IF schemes. In addition, the performanceof the IFSC+IFCC scheme is nearly the same as that in theoptimized WZ scheme. Further, the optimized SC has thepoorest performance as usual.VII. CONCLUSION
We proposed a multi-pair two-way user-rate optimizationscheme for a C-RAN network, where users are grouped intocommunicating pairs. We used a multi-pair lattice-based com-putation strategy, where the BBU pool decodes integer linearcombinations of paired users’ codewords instead of decodinglinear combinations of individual codewords. This reducesthe required number of computation steps at the BBU pool,thereby reducing the number of rate constraints. In addition,instead of recovering the original messages as common in theBBU pool, the previously computed equations are compresseddirectly and forwarded to the RRHs through the fronthaullinks. The scheme achieves significant improvement in theend-to-end rate compared to existing schemes.R
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