Fronthaul Compression and Passive Beamforming Design for Intelligent Reflecting Surface-aided Cloud Radio Access Networks
aa r X i v : . [ ee ss . SP ] F e b Fronthaul Compression and Passive BeamformingDesign for Intelligent Reflecting Surface-aidedCloud Radio Access Networks
Yu Zhang, Xuelu Wu, Hong Peng, Caijun Zhong and Xiaoming Chen
Abstract —This letter studies a cloud radio access network (C-RAN) with multiple intelligent reflecting surfaces (IRS) deployedbetween users and remote radio heads (RRH). Specifically, weconsider the uplink transmission where each RRH quantizesthe received signals from the users by either point-to-pointcompression or Wyner-Ziv compression and then transmits thequantization bits to the BBU pool through capacity limitedfronthhual links. To maximize the uplink sum rate, we jointlyoptimize the passive beamformers of IRSs and the quantiza-tion noise covariance matrices of fronthoul compression. Anjoint fronthaul compression and passive beamforming designis proposed by exploiting the Arimoto-Blahut algorithm andsemidefinte relaxation (SDR). Numerical results show the per-formance gain achieved by the proposed algorithm.
Index Terms —C-RAN, IRS, fronthaul compression, Arimoto-Blahut algorithm.
I. I
NTRODUCTION
Cloud radio access network (C-RAN) is a prospectivemobile network architecture, which provides an efficient wayfor multi-cell interference management [1]. In a C-RAN, thebaseband processing function of conventional base stations isbackward migrated into a baseband unit (BBU) pool and radioremote heads (RRH) are deployed close to users. Nevertheless,high-speed fronthaul links are required to connect the RRHsand BBU pool [2], which leads to high implementation costand complexity for dense deployment of RRHs.To tackle this issue, in this letter we propose the use ofthe recently-emerging intelligent reflecting surface (IRS) toenhance the access link between users and RRHs in the C-RAN. IRS consists of a large number of reflecting elementswith which controllable phase shifts can impose on the impingwaves [3]. Benefited from this, IRS can generate desiredreflection beams and create favorable propagation conditions
This work was supported partially by Zhejiang Provincial Natural ScienceFoundation of China under Grant LY21F010008 and LD21F010001, by theopen research fund of National Mobile Communications Research Laboratory,Southeast University (No. 2020D10), and the National Natural ScienceFoundation of China (No. 61871349, No. 61803336).Y. Zhang, X. Wu and H. Peng are with the College of In-formation Engineering, Zhejiang University of Technology, China. (e-mail: [email protected] , [email protected] ). Y. Zhang is also withthe National Mobile Communications Research Laboratory, Southeast Uni-versity, China.C. Zhong ang X. Chen are with the College of Information Sci-ence and Electronic Engineering, Zhejiang University, China (e-mail: [email protected] , [email protected] ) [4]. Since IRS is basically a passive device and solely requiresa low-rate control link, it provides an energy-efficient andcost-effective way to enhance the C-RAN. Most recently, aplethora of works have studied the design of IRS-assistedwireless communication systems [5]- [7]. In particular, IRSwas considered in multi-cell systems to assist coordinatedmultiple point transmission (CoMP) [8] and enhance the celledge performance [9]. The IRS-aided cell-free systems wereinvestigated in [10] and [11], where the weighted system sumrate and network energy efficiency were optimized, respec-tively. Moreover, the authors in [12] exploited the advantageof deploying IRS to improve the accuracy of the over-the-aircomputation in the C-RAN.In this letter, we focus on the uplink transmission designfor a multi-IRS-assisted C-RAN. Due to the limited fronthaulcapacity, the received signals at each RRH are compressed firstbefore being conveyed to the BBU pool. There are mainlytwo approaches, i.e, point-to-point compression and Wyner-Ziv compression, wherein the latter has better performancebut higher signal processing complexity [2]. Therefore, tofully exploit the advantage of IRS, the passive beamform-ing of each IRS should be jointly designed together withfronthaul compression, which has not been considered in theaforementioned works for multi-cell systems and cell-freenetworks. Specifically, with the goal of maximizing the uplinksum rate, we jointly optimize the IRS beamformers and thequantization noise covariance matrices under Wyner-Ziv-basedfronthaul compression, which leads to a non-convex problem.By exploiting the Arimoto-Blahut scheme and semi-definiterelaxation (SDR), we proposed a joint fronthaul compressionand passive beamforming design algorithm. Note that the pro-posed algorithm can be simply extended to the case of point-to-point compression. Finally, numerical results are providedto show the performance gain from deploying IRS in the C-RAN under both point-to-point and Wyner-Ziv-based fronthaulcompression. Notation : For a matrix A , | A | , Tr ( A ) , A T and A H denotethe determinant, trace, transpose and conjugate transpose of A . diag ( A ) denotes a column vector formed with the diagonalsof A . For an index set S , unless otherwise specified, A S denotes the matrix with elements A i whose indices i ∈ S and diag (cid:0) { A i } i ∈S (cid:1) denotes the block diagonal matrix formedwith A i on the diagonal where i ∈ S . A ⊙ B denotesthe Hadamard product of A and B . I denotes the identitymatrix. E [ . ] stands for the expectation. Let K = { , · · · , K } , L = { , · · · , L } and M = { , · · · , M } . Fig. 1: An uplink C-RAN system aided by multiple IRSs.II.
PRELIMILARY
A. System Model
As depicted in Fig. 1, we consider the uplink transmission ofa C-RAN, where K single-antenna users communicate with theBBU pool through L RRHs, each equipped with N R antennas. M IRSs are deployed to aid the communication between usersand RRHs, each of which has N I reflecting elements. Forsimplicity, we assume global channel state information (CSI)at the BBU pool. Note that the CSI acquisition for the IRSlink has been discussed in [13].On the access link, user k ∈ K transmits the signal x k toRRHs. Let x = [ x , ..., x k ] T with x ∼ CN (0 , P I ) , where P denotes user transmit power. Then the signals received byRRH l ∈ L can be expressed as y l = H l x + M X m =1 G l,m Θ m H R,m x + n l = ( H l + G l, M ΘH R, M ) x + n l , (1)where H l ∈ C N R × K , G l,m ∈ C N R × N I and H R,m ∈ C N I × K represent the channel matrix between users andRRH l , between IRS m and RRH l , and between users andIRS m , respectively, G l, M = [ G l, , ..., G l,M ] , H R, M =[ H TR, , ..., H TR,M ] T , Θ m = diag( θ m, , ..., θ m,N I ) representsthe passive beamformer of IRS m (we assume that the IRScan only adjust the phase shift, i.e., | θ m,n | = 1 ), Θ =diag( { Θ m } m ∈M ) , and n l ∼ CN (0 , σ I ) is the additive whiteGaussian noise.RRH l compresses its received signals and then transmits thequantization bits to the BBU pool through a wired fronthaullink with limited capacity. By adopting the Gaussian testchannel model, the compressed signal recovered by the BBUpool can be expressed as [2] ˆy l = y l + q l , (2)where q l ∼ CN (0 , Ω l ) represents the quantization noisefor RRH l and Ω l denotes its covariance matrix which isdetermined by the corresponding quantization codebook. B. Uplink Sum Rate and Fronthaul Constraints
From (1) and (2) , the achievable uplink sum rate of theconsidered C-RAN is given by R sum = I ( x ; ˆy L )= log (cid:12)(cid:12) I + P V L V H L (cid:12)(cid:12) − log (cid:12)(cid:12) σ I + Ω L (cid:12)(cid:12) , (3)where ˆy L = [ ˆy T , ..., ˆy TL ] T , V L = H L + G L ΘH R, M , H L = [ H T , ..., H TL ] T , G L = [ G T , M , ..., G TL, M ] T and Ω L =diag( { Ω l } l ∈L ) .The compression rates at each RRH should not exceed thefronthaul link capacity. With point-to-point compression, thecorresponding fronthaul constraints are given by [2] I ( y l ; ˆy l ) ≤ C l , ∀ l ∈ L , (4)where C l represents the fronthaul capacity from RRH l tothe BBU pool. The mutual information term can be evaluatedaccording to (1) and (2). Let V l = H l + G l M ΘH R, M .Constraints (4) can be written as log (cid:12)(cid:12) P V l V Hl + σ I + Ω l (cid:12)(cid:12) − log | Ω l | ≤ C l . (5)For the case where Wyner-Ziv compression is applied, weconsider sequential decompression at the BBU pool, whichis easy for practical implementation [2]. Explicitly, the BBUpool sequentially recovers the compressed signal from eachRRH and the recovered signals are utilized as side informationfor the decompression of the signals from the remainingRRHs. Denoting the decompression order as π ( · ) , then thecorresponding fronthaul constraints are given by [2] I ( y π ( l ) ; ˆy π ( l ) | ˆy ˜ L ( l − ) ≤ C π ( l ) , ∀ l ∈ L , (6)where ˜ L ( l ) = { π (1) , ..., π ( l ) } . By evaluating the mutualinformation term on the left, the above constraints can berewritten as log (cid:12)(cid:12)(cid:12) P V ˜ L ( l ) V H ˜ L ( l ) + σ I + Ω ˜ L ( l ) (cid:12)(cid:12)(cid:12) − log (cid:12)(cid:12) Ω π ( l ) (cid:12)(cid:12) − log (cid:12)(cid:12)(cid:12) P V ˜ L ( l − V H ˜ L ( l − + σ I + Ω ˜ L ( l − (cid:12)(cid:12)(cid:12) ≤ C π ( l ) , (7)where V ˜ L ( l ) = H ˜ L ( l ) + G ˜ L ( l ) ΘH R, M and Ω ˜ L ( l ) =diag( { Ω l } l ∈ ˜ L ( l ) ) .III. J OINT D ESIGN OF
IRS B
EAMFORMING AND F RONTHUAL C OMPRESSION
Due to the fact that the received signals are correlated acrossdifferent RRHs, Wyner-Ziv compresssion, with which jointdecompression is performed at the BBU pool, is expected tobe superior to point-to-point compression [2]. In this section,we first consider the joint optimization of IRS beamformingand fronthaul compression under Wyner-Ziv compression. Theoptimization for the case of point-to-point compression will bediscussed later.Recalling (6), the decompression order affects the fronthaulconstraints as well as the overall system performance. How-ever, it is prohibitive to find the optimal decompression order π ( · ) , since there are L ! possible order in total. Therefore, weadopt an efficient heuristic order selection scheme proposed in [2] and [14]. Specifically, the BBU decompresses firstly forthe RRH with a larger value of: C l − log det( P V l V Hl + σ I ) , (8)where Θ = I in V l . The rationale is to decompress first thesignals from the RRH with either larger fronthual capacity orlower received signal power, which will suffer lower quanti-zation noise.By fixing π ( l ) , with the goal of maximizing the uplink sumrate (3) under the fronthaul constraints (6), the problem forjoint optimizing IRS beamforming and fronthual compressioncan be formulated as follows: max Θ , Ω l log (cid:12)(cid:12) P V H L V L + σ I + Ω L (cid:12)(cid:12) − log (cid:12)(cid:12) σ I + Ω L (cid:12)(cid:12) s.t. log (cid:12)(cid:12)(cid:12) P V ˜ L ( l ) V H ˜ L ( l ) + σ I + Ω L ( l ) (cid:12)(cid:12)(cid:12) − log (cid:12)(cid:12) Ω π ( l ) (cid:12)(cid:12) − log (cid:12)(cid:12)(cid:12) P V ˜ L ( l − V H ˜ L ( l − + σ I + Ω ˜ L ( l − (cid:12)(cid:12)(cid:12) ≤ C π ( l ) , ∀ l, | θ m,n | = 1 , ∀ m, n, Ω l (cid:23) , ∀ l. (9)The above problem is non-convex, which makes it non-trivialto find the optimal solution. In the following, we reformulatethe non-convex objective and constraints to make the problemtractable.Firstly, consider the objective in problem (9). By exploitingthe Arimoto-Blabut algorithm [15] [16, Lemma 10.8.1, p. 33],we rewrite the objective (3) as follows: R sum = max q ( x | ˆy ) E (cid:20) log( q ( x | ˆy ) p ( x ) ) (cid:21) , (10)where the optimal q ∗ ( x | ˆy ) for (10) is the posterior probability p ( x | ˆy ) . According to [17, Theorem 10.3, p. 32], p ( x | ˆy ) follows the complex Gaussian distribution CN ( Wˆy , Σ) with W ∗ = √ P V H L (cid:0) P V L V H L + σ I + Ω L (cid:1) − (11) and Σ ∗ = I − √ P W ∗ V L . (12)Then we tackle constraints (7) in problem (9). Firstly weequivalently transform (7) into the following form: log | Γ l | − P l ∈ ˜ L ( l ) log | Ω l | ≤ P l ∈ ˜ L ( l ) C l , ∀ l ∈ L (13)where Γ l = P V ˜ L ( l ) V H ˜ L ( l ) + σ I + Ω ˜ L ( l ) . According to [14,Lemma 1], the first term on the left is upper bounded by log | Γ l | ≤ log | E l | + Tr( E − l Γ l ) − lN R , (14)for E l (cid:23) . The equality is achieved by: E ∗ l = Γ l . (15)With this, we approximate constraints (13) by the followingconstraints: log | E l | + Tr( E − l Γ l ) − log (cid:12)(cid:12)(cid:12) Ω ˜ L ( l ) (cid:12)(cid:12)(cid:12) ≤ ˜ C l , ∀ l ∈ L , (16) where ˜ C l = P l ∈ ˜ L ( l ) C l + lN R . With (10) and (16), we reformu-late the original problem (9) as follows: max W , Σ , E l , Θ , Ω l E (cid:20) log (cid:18) CN ( Wˆy , Σ ) CN (0 , P I ) (cid:19)(cid:21) s.t. log( E l ) + Tr( E − l Γ l ) − log (cid:12)(cid:12)(cid:12) Ω ˜ L ( l ) (cid:12)(cid:12)(cid:12) ≤ ˜ C l , ∀ l | θ m,n | = 1 , ∀ m, n, Ω l (cid:23) , ∀ l. (17) Remark 1 : According to (14), any feasible solution toproblem (17) is also feasible to original problem (9), whichindicates that we can solve problem (17) to obtain a sub-optimal solution to the original problem.Then we tackle problem (17) using an efficient alternatingoptimization approach. In each iteration, we first update theauxiliary variables W , Σ and E l , while fixing all the othervariables. Obviously, the optimal W and Σ is given by (11)and (12), respectively. E l is updated with (15), which followsthe successive convex optimization approach [18].By fixing the auxiliary variables, we optimize Θ and Ω l .Firstly, we evaluate the expectation term in the objective asfollows: − E (cid:20) log (cid:18) CN ( Wˆy , Σ ) CN (0 , P I ) (cid:19)(cid:21) = E [( x − Wˆy ) H Σ − ( x − Wˆy )]+log | Σ | − K ( a ) = ˆ θ H ( A ⊙ B T )ˆ θ + 2Re(Tr(ˆ θ H z ))+ Tr( W H Σ − WΩ L ) + J b ) = Tr( Ψ ¯Θ ) + Tr( W H Σ − WΩ ) + J , (18)where in (a), we have the following notations: ˆ θ = diag( Θ ) , A = G H L W H Σ − WG L , B = P H R, M H HR, M , z = diag( P G H L W H Σ − WH L H Hr −√ P G H L W H Σ − H Hr ) ,J = Tr( P W H Σ − WH L H H L ) − √ P Σ − WH L ))+Tr( σ W H Σ − W ) + Tr( Σ − ) + log | Σ | − K, and in (b), we have ¯ θ = h ˆ θ T , i T , ¯Θ = ¯ θ ¯ θ H and Ψ = A ⊙ B T z H z ! . Similarly we can rewrite constraints (16) in problem (17)as follows:
Tr( Υ l ¯Θ ) + Tr( E − l Ω ˜ L ( l ) ) − log (cid:12)(cid:12)(cid:12) Ω ˜ L ( l ) (cid:12)(cid:12)(cid:12) ≤ ˜ C l − J ,l , (19)where A l = G H ˜ L ( l ) E − L ( l ) G ˜ L ( l ) , z ,l = diag( G H ˜ L ( l ) E − L ( l ) H ˜ L ( l ) H Hr ) ,J ,l = log | E l | + Tr( E − l ( σ I + P H ˜ L ( l ) H H ˜ L ( l ) )) , Υ l = A l ⊙ B T z H ,l z ,l ! . Now the optimization problem (17) can be rewritten asfollows: min ¯Θ , Ω l Tr(
Ψ ¯Θ ) + Tr( W H Σ − WΩ L ) s.t. Tr( Υ l ¯Θ ) + Tr( E − l Ω ˜ L ( l ) ) − log (cid:12)(cid:12)(cid:12) Ω ˜ L ( l ) (cid:12)(cid:12)(cid:12) ≤ ˜ C l − J ,l , ∀ l rank( ¯Θ ) = 1 , ¯Θ (cid:23) , (cid:12)(cid:12) ¯Θ i,i (cid:12)(cid:12) = 1 , ∀ i, Ω l (cid:23) , ∀ l. (20)We apply SDR by relaxing the rank-one constraint and theresulted problem becomes convex. Thus it can be effectivelysolved by standard convex optimization tools like CVX [19].Note that the obtained ¯Θ may not be exactly rank-one ingeneral. We apply the efficient randomization techniques givenin [20] to generate suboptimal candidates and choose the oneachieving the minimal objective function.To this end, we summarize the proposed joint fronthaulcompression and IRS beamforming design algorithm: Algorithm 1 Fix π ( l ) according to (8). Initialize Θ , Ω l feasible for problem (9). Update Σ , W , E l using (11), (12), (15), respectively. Solve problem (20), Update ( Θ , Ω l ) if the objectivedecreases. Repeat Step 3-4, until convergence.
Remark 2 : Note that in Step 4, since SDR is applied, wesolely obtain a suboptimal solution to problem (20). Therefore,we check whether the obtained Θ and Ω l decrease theobjective value compared with that achieved by the solution inthe last iteration. With this, the convergence of the proposedalgorithm can be guaranteed since the objective function inproblem (17) is monotonically increasing for both Step 3 andStep 4. The computational complexity of Algorithm 1 is dom-inated by Step 3 which involves solving problem (20). Sinceproblem (20) is convex after SDR, it can be efficiently solvedby the interior-point method with computational complexity inthe problem size given by ( M N I ) + LN R [14]. The overallcomplexity of Algorithm 1 is given as the product of thenumber of iterations and the above complexity. Remark 3 : The extension to the case of point-to-pointcompression is straightforward, by replacing constraints (7)in problem (9) with fronthaul constraints (5) for point-to-pointcompression. We can similarly tackle the non-convex objectiveand fronthaul constraints as (10) and (14). Then the jointoptimization algorithm can be obtained by simply removingStep 1 and modifying Steps 3 and 4 in Algorithm 1. Due tothe limited space, the detailed algorithm is not given here.IV. N
UMERICAL R ESULTS
In this section we present numerical results to validate theeffectiveness of the proposed algorithm. In the simulation, fourusers are uniformly distributed within a circle centered at theorigin with radius of m, two RRHs each equipped with fourantennas are located at (-20m,80m) and (20m,80m), respec-tively. Two IRSs are deployed at (-25m,80m) and (25m,80m). U p li nk Su m R a t e [ b i t \ s \ H z ] Subopt , continuous, N I =15Subopt , continuous, N I =25P2P, continuous, N I =15P2P, continuous, N I =25 Fig. 2: Average sum rate versus number of iterations.The path loss is modeled as k = ξd − α , where d is the linkdistance, α is the path loss exponent, and ξ is set to -30dB.We model both the user-IRS link and IRS-RRH link as LoSchannel. The path loss exponent α for user-RRH link, user-IRS link, IRS-RRH link is set to 3.6, 2.2 and 2.2, respectively.As for small-scale fading, we assume that the user-RRH linkfollows Rayleigh fading, and the user-IRS link and the IRS-RRH link follow Rician fading with a Rician factor of 10dB.The relative reflection gain of the IRS element over the user-RRH link is set to 5dB [21]. The Gaussian noise variance isset to -80dBm.Before the performance comparison, we first numericallyverify the convergence of Algorithm (denoted as “Subopt π , continuous”) and the joint optimization modified fromAlgorithm 1 for the case of point-to-point compression (de-noted as “P2P, continuous”). Fig. plots the average sumrate versus the iteration number under different number ofreflecting elements of each IRS, where the user transmit poweris set as P = 10 dBm and the fronthaul capacity for each RRHequals 5bps/Hz. It can be observed that both algorithms canconverge in a few rounds of iterations, which validates theconvergence analysis.Besides the proposed algorithms, we also simulate thefollowing cases for comparisons. For Wyner-Ziv compression,the benchmark schemes are: 1) Opt π , continuous: Step 1 inAlgorithm 1 is replaced by exhaustively searching the optimaldecompression order; 2) Subopt π , b-bit: the IRS phase shifttakes b discrete value, i.e., θ m,n = { , e j π b , ..., e j (2 b − π b } .In this case the optimized IRS phase shifts obtained by Algo-rithm 1 are projected to the nearest discrete values and Ω l isscaled to meet the fronthaul constraints. 3) Subopt π , random:IRS phase shifts are uniformly distributed within [0 , π ) whileonly Ω l is optimized; 4) Subopt π , no IRS: the IRS is removedwhile Ω l is optimized. For point-to-point compression, thefollowing benchmark schemes are considered: 1) P2P, b-bit;2) P2P, random; 3) P2P, no IRS, which are defined accordingly.Fig. 3 plots the average achieved uplink sum rate by allthe aforementioned schemes versus the fronthual capacity.Firstly, it can be found that for both point-point compressionand Wyner-Ziv compression, deploying IRSs can enhance the Fronthaul Capacity [bit/s/Hz] U p li nk Su m R a t e [ b i t / s / H z ] Opt π , continuousSubopt π , continuousSubopt π , 2bitSubopt π , 1bitSubopt π , randomSubopt π , no IRSP2P, continuousP2P, 2bitP2P, 1bitP2P, randomP2P, no IRS 3.8 4 4.25.65.866.2 Fig. 3: Average sum rate versus fronthaul capacity, where P =15dBm and N I = 35.
15 25 35 45 55 65 75
Number of Reflection Elements U p li nk Su m R a t e [ b i t / s / H z ] π , continuousSubopt π , continuousSubopt π , 2bitSubopt π , 1bitSubopt π , randomSubopt π , no IRSP2P, continuousP2P, 2bitP2P, 1bitP2P, randomP2P, no IRS Fig. 4: Average sum rate versus number of IRS elements,where P = 10dBm, C l = 5 bps/Hz, ∀ l .system performance, especially with the proposed joint opti-mization algorithm. Furthermore, the restriction for discreteIRS phase shift solely brings limited rate loss. It can also beobserved that Wyner-Ziv compression generally outperformspoint-to-point compression, in accordance with the existingliterature. Finally, it is shown in Fig. 3 that the heuristicdecompression order selection is quite efficient, i.e., the rateloss is negligible. Fig. 4 plots the average achieved sum rateversus the number of reflecting elements for each IRS, whichalso validates the performance gain of the proposed algorithm.It can be observed that the uplink sum rate increases alongwith the increase of N I . Nevertheless, under fixed fronthaulcapacity, the growth of sum rate becomes slower when N I islarge. V. C ONCLUSIONS
We have studied a joint design of passive beamforming andfronthual compression for multi-IRS-aided C-RAN uplink. Weproposed an alternating approach which efficiently optimizethe IRS beamformers and the quantization noise covariancematrices to maximize the uplink sum rate under point-to-point compression and Wyner-Ziv compression. Numericalresults verified that deploying IRS can significantly improve the system rate with the proposed optimization algorithm.Finally, it is noted that this work can be extended to the uplinkscenario with multi-antenna users. We leave it as the futurework. R
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