Downlink Channel Reconstruction for Spatial Multiplexing in Massive MIMO Systems
Hyeongtaek Lee, Hyuckjin Choi, Hwanjin Kim, Sucheol Kim, Chulhee Jang, Yongyun Choi, Junil Choi
aa r X i v : . [ c s . I T ] F e b Downlink Channel Reconstructionfor Spatial Multiplexing in Massive MIMO Systems
Hyeongtaek Lee, Hyuckjin Choi, Hwanjin Kim, Sucheol Kim, Chulhee Jang, Yongyun Choi, and Junil Choi
Abstract —To get channel state information (CSI) at a base sta-tion (BS), most of researches on massive multiple-input multiple-output (MIMO) systems consider time division duplexing (TDD)to get benefit from the uplink and downlink channel reciprocity.Even in TDD, however, the BS still needs to transmit down-link training signals, which are referred to as channel stateinformation reference signals (CSI-RSs) in the 3GPP standard,to support spatial multiplexing in practice. This is becausethere are many cases that the number of transmit antennas isless than the number of receive antennas at a user equipment(UE) due to power consumption and circuit complexity issues.Because of this mismatch, uplink sounding reference signals(SRSs) from the UE are not enough for the BS to obtain fulldownlink MIMO CSI. Therefore, after receiving the downlinkCSI-RSs, the UE needs to feed back quantized CSI to the BSusing a pre-defined codebook to support spatial multiplexing.In this paper, possible approaches to reconstruct full downlinkMIMO CSI at the BS are proposed by exploiting both theSRS and quantized downlink CSI considering practical antennastructures with reduced downlink CSI-RS overhead. Numericalresults show that the spectral efficiencies by spatial multiplexingbased on the proposed downlink MIMO CSI reconstructiontechniques outperform the conventional methods solely based onthe quantized CSI.
Index Terms —Massive MIMO systems, spatial multiplexing,downlink MIMO channel reconstruction, CSI-RS, SRS, TDD
I. I
NTRODUCTION M ASSIVE multiple-input multiple-output (MIMO) sys-tems, which deploy tens or hundreds of antennas ata base station (BS), have become one of the key featuresof future wireless communication systems including the up-coming fifth generation (5G) cellular networks [1]–[5]. It isnow well known that massive MIMO systems can effectivelymitigate inter-user interference with simple linear precoders(for downlink) and receive combiners (for uplink) and achievehigh spectral efficiency by supporting a large number of userssimultaneously [4], [5].All the benefits mentioned above are possible only when theBS has accurate channel state information (CSI). Although fre-quency division duplexing (FDD) dominates current wirelesscommunication systems, FDD massive MIMO suffers fromexcessive downlink training and uplink feedback overheads[3]–[7]. There has been much work on resolving these issues[8]–[15]. Especially, [13]–[15] exploited spatial reciprocity
H. Lee, H. Choi, H. Kim, S. Kim and J. Choi are with the School ofElectrical Engineering, Korea Advanced Institute of Science and Technology(e-mail: [email protected]; [email protected]; [email protected];[email protected]; [email protected]).C. Jang and Y. Choi are with the Network Business, SamsungElectronics Co., LTD (e-mail: [email protected];[email protected]). between the downlink and uplink channels in FDD. Since bothchannels experience the same environment and share somedominant channel parameters, e.g., path delays and directions,even in FDD, the proposed approaches could remove mostof the overheads in FDD massive MIMO. These works are,however, restricted to a single antenna user equipment (UE)case, and it is not straightforward to extend the techniques tomultiple antennas at the UE side. When the UE has multipleantennas, [12], [16], [17] proposed to exploit the sparse natureof channels and compressive sensing techniques to mitigatethe channel training overhead. Since not all channels wouldexperience the sparsity, however, it is difficult to extend theseapproaches into more general environments.The most common and direct approach to get rid of alldownlink training and uplink feedback issues is to adopttime division duplexing (TDD) to exploit the downlink anduplink channel reciprocity [1], [4], [5], [18]–[25]. In TDD,exploiting the channel reciprocity is useful especially whenthe BS supports multiple users simultaneously with a singledata stream per UE through multi-user MIMO, which havebeen the main focus of most of massive MIMO researches.However, single-user (SU) MIMO with spatial multiplexing,which has been neglected from massive MIMO researches sofar, is still important in practice and must be optimized formassive MIMO as well.For spatial multiplexing at the BS, however, exploiting thedownlink and uplink channel reciprocity may be insufficienteven in TDD. It is common in practice, but has not been takeninto consideration in most of MIMO researches before, thatthe UE is deployed with the number of transmit antennasless than the number of receive antennas because of manypractical constraints including power consumption and circuitcomplexity issues at the UE side [26], [27]. Therefore, theuplink sounding reference signals (SRSs), which are onlytransmitted from the transmit antennas at the UE, are notenough for the BS to obtain full downlink MIMO CSI byexploiting the channel reciprocity. This is why the 3GPPstandard defines downlink channel training using channel stateinformation reference signals (CSI-RSs) and CSI quantizationcodebooks (or precoding matrix indicator (PMI) codebooks)even for TDD [28].It is critical to reduce the downlink CSI-RS overhead formassive MIMO, and most effective way to reduce the overheadis by grouping multiple antennas at the BS as a single antennaport [29], [30]. Each antenna port transmits the same CSI- This problem also hinders from exploiting the proposed techniques in[13]–[15] when the UE has multiple antennas since the number of transmitantennas is less than that of receive antennas.
RS while the antennas in a port can have different weightsto beamform the CSI-RS [31], [32]. Since each antenna porttransmits the same CSI-RS, the UE is not able to distinguishthe different antennas in one port and only sees a lower di-mensional effective channel through CSI-RSs from the antennaports. The UE then quantizes this lower dimensional downlinkchannel, which we will refer to as
CSI-RS channel throughoutthe paper, with a pre-defined PMI codebook and feeds backthe index of selected codeword to the BS.The 3GPP standard has defined two kinds of codebooksfor efficient limited feedback, i.e., the Type 1 and Type 2codebooks [28]. The Type 1 codebook is a standard PMIcodebook consists of precoding matrices while the Type 2codebook is to quantize the CSI-RS channel itself or itssubspace. Although the Type 2 codebook gives better quan-tization performance than the Type 1 codebook, its feedbackoverhead increases significantly for higher layer transmission,making it unsuitable to spatial multiplexing [34], [35].In addition to PMI quantization error, there are two possiblefactors that could result in performance degradation for spatialmultiplexing: 1) usually the same beamforming weights forthe CSI-RS transmissions are used for spatial multiplexingwithout adapting to channel conditions, and 2) the dimensionof fed back PMI is usually much smaller than the dimensionof original MIMO channel between the BS and the UE,which makes the BS only have very limited knowledge ofthe downlink MIMO channel. These problems exist regardlessof the codebook types [28], [34], [35]. Therefore, there isa demand on finding the full dimensional downlink MIMOchannel from the low dimensional effective CSI-RS channelat the BS to maximize the performance of spatial multiplexing.To the best of our knowledge, there has been no priorwork on downlink MIMO reconstruction to support spatialmultiplexing from uplink channel information. Since there isno relevant prior work, in this paper, we propose and compareseveral possible approaches for the BS to reconstruct the fulldownlink MIMO channel when the number of transmit anten-nas is less than the number of receive antennas at the UE. Theproposed techniques exploit both the downlink CSI-RS anduplink SRS considering practical antenna structures to mitigatethe downlink CSI-RS overhead in massive MIMO systems.The proposed techniques range from a very simple approach tocomplex techniques based on convex optimization problems.Among many possible approaches, we verify using the realisticspatial channel model (SCM) [36], which is adopted in the3GPP standard, that it is possible to reconstruct the downlinkMIMO channel quite well using only basic matrix operations.The proposed techniques can be used for both uniform lineararrays (ULAs) and uniform planar arrays (UPAs) in a unifiedway. Numerical results show that the spectral efficiencies byspatial multiplexing based on the proposed downlink MIMOCSI reconstruction techniques outperform that of conventionalapproach, which only exploits the fed back PMI from the UE. Uplink feedback using the Type 2 codebook is often referred to as explicitfeedback [33]. Although [13]–[15] tackled similar problems, these works are limited tosingle antenna UEs, and it is difficult to extend the techniques in [13]–[15]to multiple antenna UEs as discussed before.
The remainder of this paper is organized as follows. Systemmodel and key assumptions are discussed in Section II. InSection III, the proposed downlink MIMO CSI reconstruc-tion techniques using the low dimensional effective CSI-RSchannel and the uplink SRS are presented. Numerical resultsthat verify the performance of the proposed techniques arepresented in Section IV, and conclusion follows in Section V.
Notations:
Lower and upper boldface letters represent col-umn vectors and matrices. A T , A H , and A † denote the trans-pose, conjugate transpose and pseudo-inverse of the matrix A . A (: , m : n ) denotes the submatrix consists of the m -th columnto the n -th column of the matrix A , A ( m : n, :) denotes thesubmatrix consists of the m -th row to the n -th row of thematrix A , and a ( m : n ) denotes the vector consists of the m -th element to the n -th element of the vector a . A H (: , k ) denotes the k -th column of A H , and A H ( k, :) denotes the k -th row of A H . |·| is used to denote the absolute value of acomplex number, k·k denotes the ℓ -norm of a vector, and k·k F denotes the Frobenius-norm of a matrix. m is used forthe m × all zero vector, and I m denotes the m × m identitymatrix. CN ( m, σ ) denotes the complex normal distributionwith mean m and variance σ . O ( · ) denotes Big-O notation.II. S YSTEM M ODEL
We consider a TDD massive MIMO system, especiallySU-MIMO with spatial multiplexing. We further considera standard PMI codebook, e.g., the Type 1 codebook, forCSI quantization. We assume the BS is equipped with N BS antennas, and the UE is equipped with M UE antennas asshown in Fig. 1. At the UE side, all M UE antennas are used forreception while only one of them, indexed as m Tx , is used fortransmission. The BS is deployed with either a ULA or a UPAwhile the UE is deployed with a ULA. The overall procedureof our full downlink MIMO CSI reconstruction framework inFig. 1 is first summarized as follows.Step 1: The BS transmits beamformed CSI-RSs to the UE.Since the BS groups multiple antennas as a singleport, the UE only sees low dimensional effectiveCSI-RS channel H CSI − RS − uq .Step 2: The UE quantizes H CSI − RS − uq using a pre-definedPMI codebook and feeds back the index of selectedPMI, H CSI − RS , to the BS.Step 3: The UE transmits uplink SRS, and the BS obtains h SRS relying on the downlink and uplink channelreciprocity in TDD.Step 4: Using both H CSI − RS and h SRS , the BS recon-structs full downlink MIMO CSI.Step 5: Based on the reconstructed MIMO CSI, the BSsupports the UE through spatial multiplexing.As a way of mitigating the downlink CSI-RS overhead,several physical antenna elements can form a single antennaport at the BS [37]. The same CSI-RS is transmitted fromeach antenna port so that the UE considers one antennaport as a single transmit antenna. To improve the quality ofCSI-RS at the UE, however, different antenna elements in It is possible to have more than one transmit antennas at the UE, and weleave this extension as a possible future work.
Fig. 1: Massive MIMO with (i) downlink channel training through CSI-RS and uplink limited feedback using a PMI codebook,(ii) uplink channel training using SRS, (iii) downlink channel reconstruction. The uplink SRS channel is denoted by h SRS ,and the index of UE antenna, which is used for transmission, is denoted by m Tx .Fig. 2: An example of practical UPA structure with N ver vertical antennas and N hor horizontal antennas. Each antennaport has J = N ver antenna elements in this example.one antenna port can have different beamforming weights.When each antenna port consists of J antenna elements, K = N BS /J antenna ports are deployed as in Fig. 2. Ifthe BS has prior knowledge of channel, for example throughuplink SRS exploiting the channel reciprocity, the BS candynamically select appropriate CSI-RS beamforming weights,otherwise, fixed CSI-RS beamforming weights can be applied[29]. After constructing the CSI-RS beamforming weights, theBS transmits known CSI-RS sequences successively to the UEthrough the antenna ports as in Fig. 3. In general, the CSI-RSsequences are based on pseudo-random sequences [38]. Sincethe proposed downlink channel reconstruction techniques workfor arbitrary CSI-RS sequences, as long as the BS and theUE share the same ones, we assume the CSI-RS transmitted Fig. 3: An example of downlink CSI-RS procedure with theantenna structure in Fig. 2. The CSI-RSs are transmittedsuccessively from Port 1 to Port K to the UE.from the k -th antenna port is a scalar x k , not a sequence, forsimplicity in this paper.Since the antenna ports transmit the CSI-RS successively,the received signals at the UE from the k -th antenna port y k is given as y k = H H p k x k + n k , (1)where H is the N BS × M UE downlink MIMO channelmatrix, x k is the CSI-RS satisfying | x k | = 1 , n k ∼CN ( M UE , σ I M UE ) is the M UE × noise vector, σ =1 /ρ DL , and ρ DL denotes the downlink signal-to-noise ratio(SNR). The N BS × CSI-RS beamforming vector p k isgiven by p k = h T( k − J , w T k , T N BS − kJ i T , (2) where w k is the J × CSI-RS beamforming weight vectorapplied to the k -th antenna port such that k w k k = 1 . Notethat all other antenna ports except the k -th port are silentduring the k -th CSI-RS transmission. As discussed above, theBS can dynamically adjust w k if it has prior knowledge ofchannel, e.g., through the uplink SRS. If the BS has no priorchannel knowledge, it needs to fix w k to have a widebeamshape to guarantee that the UE can experience a certain level ofquality of service regardless of its location on its serving cell.The performance according to different CSI-RS beamformingweight vectors will be compared in Section IV.After receiving all K CSI-RS transmissions, the UE canconstruct an M UE × K unquantized low dimensional effectiveCSI-RS channel matrix H CSI − RS − uq . We set x k = 1 forsimplicity throughout the paper since the proposed techniquesdo not rely on any CSI-RS structure. Then H CSI − RS − uq isrepresented by H CSI − RS − uq = [ y , y , · · · , y K ] , (3) = H H PX + N , (4) = H H P + N , (5)where P = [ p , p , · · · , p K ] , X = diag [ x , x , · · · , x K ] = I K , and N = [ n , n , · · · , n K ] . The UE then quantizes H CSI − RS − uq using a pre-defined PMI codebook and feedsback the index of selected PMI to the BS through limitedfeedback. Assuming the selected PMI is for layer L transmis-sion via spatial multiplexing, the selected PMI H CSI − RS = Q ( H CSI − RS − uq ) , (6)is a K × L matrix where Q ( · ) is a CSI quantization function.We consider Q ( · ) as a function of maximizing the spectralefficiency [6], which is given by H CSI − RS =argmax ˜ H CSI − RS ∈C log (cid:16) det (cid:16) I L + ρ DL L ˜ H HCSI − RS H HCSI − RS − uq H CSI − RS − uq ˜ H CSI − RS (cid:17)(cid:17) , (7)where C is the pre-defined PMI codebook. Since the UEonly sees the low dimensional effective channel H CSI − RS − uq ,the UE can find the codeword that maximizes (7) throughexhaustive search over C . Note that L is smaller than or equalsto M UE while the proposed downlink channel reconstructiontechniques can be applied to any values of L . We will shownumerical results with different values of L in Section IV.In TDD, relying on the downlink and uplink channel reci-procity, the BS can estimate the downlink channel from theuplink SRSs transmitted by the UE. Since we assume the UEhas only one transmit antenna, the BS can estimate an N BS × uplink SRS channel vector h SRS that corresponds to one ofthe columns of the downlink channel matrix H corrupted withnoise as h SRS = H (: , m Tx ) + v , (8)where v ∼ CN ( N BS , σ I N BS ) is the N BS × noise vector, σ = 1 /ρ UL , and ρ UL denotes the uplink SNR. Since the BS would not be able to know in advance whichantenna is used for transmission among M UE antennas at theUE, the BS does not know which column of H correspondsto the uplink SRS channel vector. In Section III, we firstassume m Tx is known to the BS to develop downlink channelreconstruction techniques using H CSI − RS and h SRS . Then, weshow that the effect of imperfect knowledge of m Tx would benegligible in terms of spectral efficiencies. We further verifythe effect of m Tx numerically in Section IV.III. P ROPOSED D OWNLINK
MIMO C
HANNEL R ECONSTRUCTION T ECHNIQUES
In this section, we propose and compare possible downlinkMIMO channel reconstruction techniques using the low di-mensional effective CSI-RS channel and the uplink SRS. Itturns out that some of proposed approaches, even complex,do not work well, i.e., even worse than the conventionalspatial multiplexing only using the quantized PMI. We willstill explain these approaches since there is no prior workon this problem, and readers may not know whether theseapproaches perform well or not.We assume perfect knowledge of m Tx at the BS. The UEfeeds back the quantized H CSI − RS , which is selected forlayer L spatial multiplexing, to the BS. Since the UE hasalready decided that the layer L spatial multiplexing wouldbe the best for the current channel after receiving the CSI-RSs, it is reasonable to assume that the BS would reconstructan N BS × L , not N BS × M UE , downlink channel if the BSobtains the layer L H CSI − RS from the UE. In this case, weassume m Tx is in between and L . Note that we considerthe conjugate transpose on H CSI − RS , i.e., H HCSI − RS , for thedownlink channel reconstruction since the original purpose ofthe PMI codebook is to inform the BS of beamformer forspatial multiplexing. Then, the BS needs to take the conjugatetranspose on the fed back PMI to consider it as a downlinkchannel. A. Ratio technique
Considering H = [ h , h , · · · , h M UE ] , the unquantizedeffective CSI-RS channel matrix H CSI − RS − uq in (5) can befurther represented by H CSI − RS − uq = h H1 p h H1 p · · · h H1 p K h H2 p h H2 p · · · h H2 p K ... ... . . . ... h H M UE p h H M UE p · · · h H M UE p K + (cid:2) n , n , · · · , n K (cid:3) , (9)where h m is the N BS × channel vector between the transmitantennas at the BS and the m -th receive antenna at the UEfor m = 1 , , · · · , M UE . Without the noise in (9), the ( m, k ) -th component of H CSI − RS − uq is the inner product between h m and the CSI-RS beamforming vector p k , and the onlydifference among components of H CSI − RS − uq in the samecolumn is h m with fixed p k . Using the knowledge of h SRS and m Tx , the BS can simply reconstruct the downlink channel in ablock-wise manner with the ratio of components of H CSI − RS .The reconstructed m ′ -th column of downlink channel basedon the ratio technique can be expressed as ˆ H ratio (: ,m ′ ) = " h TSRS (1 : J ) H HCSI − RS ( m ′ , H HCSI − RS ( m Tx , , h TSRS ( J + 1 : 2 J ) H HCSI − RS ( m ′ , H HCSI − RS ( m Tx , , · · · , h TSRS (( K − J + 1 : KJ ) H HCSI − RS ( m ′ , K ) H HCSI − RS ( m Tx , K ) T , (10)for m ′ = 1 , , · · · , L . Although this technique is quite simple,it works well when the BS is deployed with the ULA as shownin Section IV. B. Inner product (IP) maximization technique
For the
IP maximization technique , we set an optimizationproblem to reconstruct the downlink channel based on theknowledge of H CSI − RS and the CSI-RS beamforming matrix P . This technique tries to maximize the IP between each rowof H CSI − RS and ˆ H H P , which is the beamformed version ofestimated downlink channel ˆ H , as the main objective, i.e., ˆ H IP (: , m ′ ) = argmax ˆ H (: ,m ′ ) ∈ C N BS × (cid:12)(cid:12)(cid:12) ( ˆ H H ( m ′ , :) P ) H HCSI − RS (: , m ′ ) (cid:12)(cid:12)(cid:12) , (11)where ˆ H IP represents the reconstructed channel based on the IP maximization technique .Since the optimization in (11) is non-convex, we consideran equivalent convex problem as in [39], which is given by ˆ H IP (: , m ′ ) =argmin ˆ H (: ,m ′ ) ∈ C N BS × min α ∈ R + ω ∈ [0 , π ) (cid:13)(cid:13)(cid:13) ˆ H H ( m ′ , :) P − αe jω H CSI − RS ( m ′ , :) (cid:13)(cid:13)(cid:13) . (12)After the optimization, the m Tx -th column of ˆ H IP is replacedby h SRS , i.e., ˆ H IP (: , m Tx ) = h SRS , (13)since h SRS is, with sufficiently high uplink SNR, close to thetrue channel relying on the downlink and uplink channel reci-procity while the m Tx -th column of ˆ H IP is the estimated one. C. Element-wise technique
In the element-wise technique , we set a convex optimizationproblem to minimize the error between ˆ H H P and H CSI − RS as ˆ H ele = argmin ˆ H ∈ C N BS × L (cid:13)(cid:13)(cid:13) ˆ H H P − H HCSI − RS (cid:13)(cid:13)(cid:13) F + λ (cid:13)(cid:13)(cid:13) ˆ H (: , m Tx ) − h SRS (cid:13)(cid:13)(cid:13) , (14)where λ ∈ R + denotes the regularization factor, and ˆ H ele represents the reconstructed downlink channel based on the element-wise technique . Large λ implies large emphasis onminimizing the difference between the m Tx -th column of thereconstructed downlink MIMO channel and the known h SRS .It is not always better to have large λ , instead, it should beproperly adjusted to balance the two differences. Similar to(13), the m Tx -th column of ˆ H ele is replaced by h SRS afterthe optimization.Although the
IP maximization technique and element-wisetechnique exploit given information of H CSI − RS , h SRS , and P , they do not exploit any physical structure, e.g., angle-of-arrival (AoA) and angle-of-departure (AoD) of the channel H or antenna array at the BS and UE, making them have poorperformance as shown in Section IV. In addition, the dimen-sion of H is quite large in massive MIMO with large N BS ,resulting in high degree-of-freedom with only a few knownvariables for the optimization problems. In what follows, weimpose physical structures on the optimization problems toimprove reconstruction performance and mitigate optimizationcomplexity. D. Structure technique
In massive MIMO with a large number of antennas, thedownlink channel H is usually modeled as virtual channelrepresentation [40], [41], which is the weighted sum of theouter products of AoA array response vectors at the UE sideand the AoD array response vectors at the BS side. Themodeled channel is given by H Hv = P X p =1 Q X q =1 c p,q a r ( ψ p ) a Ht ( µ q ) , (15)where c p,q ∈ C is the complex gain of the path with theAoD µ q and AoA ψ p . Since the UE deploys the ULA, thearray response vector for AoA ψ p , assuming half wavelengthantenna spacing, is represented by a r ( ψ p ) = 1 √ L h , e jπ sin( ψ p ) , · · · , e j ( L − π sin( ψ p ) i T . (16)The UE deploys M UE physical receive antennas; however, wemodel the antenna array of the UE as deploying L antennas toreconstruct the N BS × L downlink channel at the BS. Similarly,the array response vector for AoD µ q , assuming the ULA atthe BS with half wavelength antenna spacing, is given as a t ( µ q ) = 1 √ N BS h , e jπ sin( µ q ) , · · · , e j ( N BS − π sin( µ q ) i T . (17)Although the UPA is also possible, we only consider the ULAat the BS for the structure technique , the reason will becomeclear at the end of this subsection.In a matrix form, the modeled channel in (15) can berewritten as H Hstr = A r CA Ht , (18)where C is the P × Q matrix with c p,q as the ( p, q ) -thelement, A r = [ a r ( ψ ) , a r ( ψ ) , · · · , a r ( ψ P )] , and A t =[ a t ( µ ) , a t ( µ ) , · · · , a t ( µ Q )] . Without any prior knowledgeof AoAs and AoDs of the channel, ψ p and µ q can be chosen randomly from [ − π/ , π/ considering practical cellstructures.To reconstruct the downlink channel assuming the channelstructure expressed in (18) and randomly chosen AoAs andAoDs, the convex optimization problem in (14) now becomes ˆ C str = argmin ˆ C ∈ C P × Q (cid:13)(cid:13)(cid:13) A r ˆ CA Ht P − H HCSI − RS (cid:13)(cid:13)(cid:13) F + λ (cid:13)(cid:13)(cid:13)h A t ˆ C H A Hr i (: , m Tx ) − h SRS (cid:13)(cid:13)(cid:13) , (19)where λ ∈ R + denotes the regularization factor as in (14). Thereconstructed channel based on the structure technique then isgiven as ˆ H Hstr = A r ˆ C str A Ht . (20)Similar to (13), the m Tx -th column of ˆ H str is replaced by h SRS after the optimization.The structure technique may suffer from randomly chosen ψ ’s and µ ’s, which could be misaligned with the true AoAs andAoDs. It is possible to resolve this problem by increasing thesize of P and Q but this would impose huge complexity on theoptimization in (19). With the UPA at the BS, the complexityissue becomes even worse since the BS needs to take boththe horizontal and vertical angles into account. Therefore, weonly consider the ULA at the BS for the structure technique . E. Pre-search technique
As a way of resolving the complexity issue in the structuretechnique , we first estimate the dominant AoDs and AoAsas preliminary information relying on the channel model (15)in the pre-search technique . Since the BS has H CSI − RS bylimited feedback from the UE, the dominant AoAs can beextracted by comparing the strengths χ r ( ˜ ψ i ) of the arrivalangle ˜ ψ i as χ r ( ˜ ψ i ) = (cid:13)(cid:13)(cid:13) a HAoA ( ˜ ψ i ) H HCSI − RS (cid:13)(cid:13)(cid:13) , (21) ˜ ψ i = − π πR ULA ( i − , (22)where i = 1 , , · · · , R ULA + 1 . Here, π/R
ULA represents theresolution of ˜ ψ i , and a AoA ( · ) is the same as a r ( · ) in (16)since the UE is assumed to deploy the ULA. To extract T AoA dominant angles for AoAs, we need to find T AoA local maximaof χ r ( ˜ ψ i ) where a local maximum is defined as χ r ( ˜ ψ i ) ≥ χ r ( ˜ ψ i +1 ) , χ r ( ˜ ψ i ) ≥ χ r ( ˜ ψ i − ) . (23)We denote an angle that gives a local maximum of χ r ( ˜ ψ i ) as ˆ ψ u for u = 1 , , · · · , T AoA . Note that the PMI codebookis pre-defined; therefore, it is possible to construct a lookuptable that defines T AoA dominant AoAs for each H CSI − RS in advance.Estimating T AoD dominant AoDs can be conducted simi-larly using h SRS . Different from the structure technique , nowit is possible to consider both the ULA and UPA for the BSantenna structure. Assuming the ULA at the BS, the strength χ t (˜ µ i ) of the departure angle ˜ µ i is given by χ t (˜ µ i ) = (cid:12)(cid:12) a HAoD (˜ µ i ) h SRS (cid:12)(cid:12) , (24) ˜ µ i = − π πR ULA ( i − , (25) where a AoD ( · ) is the same as a t ( · ) in (17). If the UPA isassumed at the BS, the strength χ t (˜ µ ℓ ver , ˜ µ ℓ hor ) of the verticaldeparture angle ˜ µ ℓ ver and horizontal departure angle ˜ µ ℓ hor arewritten as χ t (˜ µ ℓ ver , ˜ µ ℓ hor ) = (cid:12)(cid:12) a HAoD (˜ µ ℓ ver , ˜ µ ℓ hor ) h SRS (cid:12)(cid:12) , (26)where the array response vector for the UPA, assuming halfwavelength spacing, is given as a AoD (˜ µ ℓ ver , ˜ µ ℓ hor ) =1 √ N BS h , e jπ sin(˜ µ ℓ ver ) , · · · , e j ( N ver − π sin(˜ µ ℓ ver ) i T ⊗ h , e jπ sin(˜ µ ℓ hor ) cos(˜ µ ℓ ver ) , · · · ,e j ( N hor − π sin(˜ µ ℓ hor ) cos(˜ µ ℓ ver ) i T . (27)In (27), N BS = N ver N hor , and ⊗ denotes the Kroneckerproduct. Further, ˜ µ ℓ ver and ˜ µ ℓ hor are conditioned by ˜ µ ℓ ver = − π πR UPA , ver ( ℓ ver − , (28) ℓ ver = 1 , , · · · , R UPA , ver + 1 , (29) ˜ µ ℓ hor = − π πR UPA , hor ( ℓ hor − , (30) ℓ hor = 1 , , · · · , R UPA , hor + 1 , (31)where π/R UPA , ver and π/R UPA , hor represent the resolutionof ˜ µ ℓ ver and ˜ µ ℓ hor .Rather than just choosing the dominant T AoD departureangles, it is possible to increase the AoD estimation accuracyby the null space projection technique as in [42]. Once adominant AoD is found, we can exclude the componentcorresponding to that angle from h SRS through the null spaceprojection technique before searching another dominant AoD.The details are summarized in Algorithms 1 and 2 for theULA and UPA cases. We denote the sets of dominant AoDsfor the ULA and UPA cases as O ULA and O UPA in those twoalgorithms.After obtaining the dominant AoAs and AoDs, we have the L × T AoA matrix A AoA given as A AoA = h a AoA ( ˆ ψ ) , a AoA ( ˆ ψ ) , · · · , a AoA ( ˆ ψ T AoA ) i , (32)and the N BS × T AoD matrix A AoD expressed as A AoD = h a AoD (ˆ µ ) , a AoD (ˆ µ ) , · · · , a AoD (ˆ µ T AoD ) i , (33)assuming the ULA at the BS or A AoD = h a AoD (ˆ µ , ver , ˆ µ , hor ) , a AoD (ˆ µ , ver , ˆ µ , hor ) , · · · , a AoD (ˆ µ T AoD , ver , ˆ µ T AoD , hor ) i , (34)assuming the UPA at the BS. To reconstruct the downlinkchannel, we set a convex optimization problem to find thepath gain matrix ˆ C pre with the given A AoA and A AoD as ˆ C pre = argmin ˜ C ∈ C T AoA × T AoD (cid:13)(cid:13)(cid:13) A AoA ˜ CA HAoD P − H HCSI − RS (cid:13)(cid:13)(cid:13) F + λ (cid:13)(cid:13)(cid:13)h A AoD ˜ C H A HAoA i (: , m Tx ) − h SRS (cid:13)(cid:13)(cid:13) , (35) Algorithm 1
Estimation of the dominant AoDs for the ULAInitialize O ULA as an empty set h ← h SRS for v = 1 , , · · · , T AoD do Initialize i max for i = 1 , , · · · , R ULA + 1 do Calculate χ t (˜ µ i ) in (24) end for Calculate i max = argmax i χ t (˜ µ i )ˆ µ v ← ˜ µ i max h ← h − ( h H a AoD (ˆ µ v )) a AoD (ˆ µ v ) O ULA ← { O ULA , ˆ µ v } end for where λ ∈ R + denotes the regularization factor as in (14). Thereconstructed channel ˆ H pre based on the pre-search technique is given as ˆ H Hpre = A AoA ˆ C pre A HAoD . (36)Similar to (13), the m Tx -th column of ˆ H pre is replaced by h SRS after the optimization. Note that the pre-search technique has lower complexity for optimization than the structuretechnique since T AoA and T AoD would be smaller than P and Q to have the same performance in general. Even thoughthe size of optimization problem has become smaller, still itmight take much time to perform the pre-search technique in practice, which could prevent its use when the channelcoherence time is insufficient. F. Pseudo-inverse technique
All the above techniques except the ratio technique con-sider certain convex optimization problems for which theconvergence is guaranteed. However, the overall complexityto reconstruct the downlink channel through an optimizationproblem can be quite severe especially in massive MIMO. Wepropose another channel reconstruction technique that onlyexploits basic matrix operations and does not rely on anyoptimization to combat the complexity problem. This wouldbe especially beneficial when the channel coherence time isnot long enough to perform any complex optimization process.Adopting the channel model as in (36), H CSI − RS − uq in (5)can be represented by H CSI − RS − uq = H H P + N , (37) ≈ A AoA ˜ CA HAoD P + N , (38)where A AoA and A AoD are obtained by the same way as inthe pre-search technique . Note that T AoA and T AoD for findingthe dominant AoAs and AoDs are design variables that theBS can choose. By setting T AoA ≤ L and T AoD ≤ K , theleft pseudo-inverse of A AoA and the right pseudo-inverse of A HAoD P always exist. Then, the estimated ˆ C pinv is given by ˆ C pinv = A † AoA H HCSI − RS ( A HAoD P ) † , (39)and the reconstructed channel ˆ H pinv based on the pseudo-inverse technique is given as ˆ H pinv = A AoA ˆ C pinv A HAoD . (40) Algorithm 2
Estimation of the dominant AoDs for the UPAInitialize O UPA as an empty set h ← h SRS for v = 1 , , · · · , T AoD do Initialize ℓ ver , max , ℓ hor , max for ℓ ver = 1 , , · · · , R UPA , ver + 1 dofor ℓ hor = 1 , , · · · , R UPA , hor + 1 do Calculate χ t (˜ µ ℓ ver , ˜ µ ℓ hor ) in (26) end forend for Calculate ( ℓ ver , max , ℓ hor , max ) = argmax ℓ ver ,ℓ hor χ t (˜ µ ℓ ver , ˜ µ ℓ hor )ˆ µ v, ver ← ˜ µ ℓ ver , max ˆ µ v, hor ← ˜ µ ℓ hor , max h ← h − ( h H a AoD (ˆ µ v, ver , ˆ µ v, hor )) a AoD (ˆ µ v, ver , ˆ µ v, hor ) O UPA ← { O UPA , (ˆ µ v, ver , ˆ µ v, hor ) } end for Similar to (13), the m Tx -th column of ˆ H pinv is replacedby h SRS after the reconstruction.
G. Complexity analysis
Among the proposed techniques, the
IP maximization,element-wise, structure and pre-search techniques need tosolve the convex optimization problems. Although these prob-lems can be efficiently solved using the interior-point method,its complexity is incomparable to the complexity of ba-sic matrix-vector operations. On the contrary, the ratio and pseudo-inverse techniques solely rely on the basic matrix-vector operations. Specifically, the complexity of ratio tech-nique is O ( N BS L ) since it needs to obtain the inner product oftwo vectors L times. The pseudo-inverse technique requires tohave the dominant AoA/AoD information where the complex-ity of AoA estimation based on H CSI − RS is O ( LKR
ULA ) ,and that of AoD estimation using h SRS is O ( N BS R ULA ) forthe ULA and O ( N BS R UPA , hor R UPA , ver ) for the UPA at theBS. The complexity of channel gain matrix estimation in (39)is O ( LT + K + T AoD N BS K + LKT
AoA ) . Although thecomplexity of pseudo-inverse technique is higher than that of ratio technique , it is only proportional to N BS and much lowerthan the complexity of interior-point method. H. Effect of imperfect knowledge of transmit antenna index ofUE at BS
Until now, we assumed the BS has the perfect knowledge of m Tx , i.e., the transmit antenna index of the UE, to reconstructthe downlink channel. The BS, however, may have imperfectknowledge about m Tx in practice. To see the effect of imper-fect knowledge of m Tx on the spectral efficiency performance,we first assume L is the same as M UE for simplicity. Then,the spectral efficiency of channel is defined as [6] R = log (cid:18) det (cid:18) I M UE + ρ DL M UE F H HH H F (cid:19)(cid:19) , (41) F = V (: , M UE ) , (42) H H = U Σ V H , (43) where (43) is the singular value decomposition (SVD) of thetrue downlink channel H H , and F is the optimal data trans-mission beamformer. Let T be an arbitrary row permutationmatrix. Then the SVD on the row permuted downlink channel TH H is given as TH H = ( TU ) Σ V H , (44)where TU is still a unitary matrix. Since the right singularmatrix V is not altered by T , the spectral efficiency becomesthe same regardless of T .In our downlink channel reconstruction problem, the im-perfect knowledge of m Tx works as the row permutationmatrix T . Of course incorrect knowledge of m Tx would resultin a different reconstruction result in addition to the row per-mutation effect. It is difficult, however, to analytically derivethe impact of imperfect knowledge of m Tx on the downlinkMIMO CSI reconstruction. Therefore, we numerically studythis impact in Section IV where the result shows that theimperfect knowledge of m Tx has negligible impact on thespectral efficiency performance. This information could beimportant for practical implementation, e.g., symbol detectionat the UE, which could be an interesting future research topic.IV. N UMERICAL R ESULTS
In this section, we evaluate the performance of the proposeddownlink channel reconstruction techniques. The downlinkchannel H is generated based on the SCM channel that isextensively used in the 3GPP standard [36]. Unless explicitlystated, we adopt the scenario of urban micro (UMi) single cellwith carrier frequency 2.3 GHz for the SCM channel. Since theSCM channel takes cell structures with path loss into account,channel gains are usually very small. For numerical studies ofpoint-to-point communication using spatial multiplexing, wenormalize the average gain of all channel elements to one,i.e., E h | h n,m | i = 1 where h n,m is the ( n, m ) -th componentof H .We set the number of transmit antennas at the BS N BS = 32 (for the UPA, N ver = 8 , N hor = 4 ), the number of receiveantennas at the UE M UE = 4 , the number of antenna elementsfor an antenna port J = 8 , which gives the number ofantenna ports K = 4 , and the number of PMI feedbacklayer L = 2 or L = M UE = 4 . The regularization factor isnumerically optimized and set as λ = 0 . for all optimizationproblems. We use CVX [43], a well established optimizationsolver, for some of proposed approaches that need to solveconvex optimization problems. We also set the number ofrandomly selected angles P = Q = 20 for the structuretechnique , the number of dominant AoAs or AoDs T AoA = L, T
AoD = K and the resolution for finding dominant AoAsor AoDs R ULA = 3600 , R
UPA , ver = R UPA , hor = 200 forthe pre-search technique and pseudo-inverse technique . Thedownlink SNR ρ DL is assumed to be 20 dB since the spatialmultiplexing is intended to increase the spectral efficiency inhigh SNR regimes.For the CSI-RS beamforming weight vector w k in (2), weconsider a widebeam or dynamically selected beam based on h SRS . Specially, for the case of dynamically selected beam, we assume w j max is used where j max is the column index of J × J discrete Fourier transform (DFT) matrix D selected as j max = argmax j (cid:12)(cid:12)(cid:2) D H ( j, :) , T N BS − J (cid:3) h SRS (cid:12)(cid:12) . (45)Then, w j max is defined by the j max -th column of D . Notethat w k may vary depending on k in general; however, weassume those are the same for all k . For the PMI codebook C ,the Type 1 Single-Panel Codebook in [28] is adopted. As aperformance metric, we consider the spectral efficiency of thechannel with reconstructed downlink channel replacing M UE with L in (41). The data transmission beamformer F is set as F = ˆ V (: , L ) , (46) ˆ H H = ˆ U ˆΣ ˆ V H , (47)where (47) is the SVD of the reconstructed channel ˆ H H bythe downlink channel reconstruction techniques proposed inSection III.In the following figures, the term Pre-ULA (Pre-UPA) refersto the pre-search technique explained in Section III-E withthe ULA (UPA) assumption at the BS, and Pinv-ULA (Pinv-UPA) refers to the pseudo-inverse technique in Section III-Fwith the ULA (UPA) assumption at the BS. The Random isthe case when all the channel elements, except the m Tx -thcolumn replaced with h SRS , are randomly distributed follow-ing CN (0 , . As a baseline, we compare the conventionalscenario with F = PH CSI − RS . This baseline is denoted as“Type 1” in the following figures. We also compare the upperbound of conventional method without quantization loss as F = PV CSI − RS − uq where V CSI − RS − uq is the right singularmatrix of H CSI − RS − uq . Since this is the upper bound of theType 2 codebook, we denote this as “Type 2” in the figures.We also have the ideal case with F = V Ideal where V Ideal isthe right singular matrix of the true downlink channel H H .In Figs. 4 and 5, we consider the case when the BS adoptsa fixed widebeam for w k without any prior information ofchannel. We design the widebeam with boresight ◦ andbeamwidth about ◦ as in [44]. We consider L = M UE = 4 for the feedback layer and the N BS × M UE full MIMOdownlink channel reconstruction. Fig. 4 shows the averagespectral efficiency of proposed downlink channel reconstruc-tion techniques according to the uplink SNR ρ UL assumingthe ULA at the BS. It can be observed that the pre-searchtechnique and pseudo-inverse technique outperform the otherchannel reconstruction techniques. It is better for these twotechniques to assume the ULA for the reconstruction sincethe BS is deployed with the ULA in this scenario. Despite itssimplicity, the ratio technique shows quite good performancebecause of the simple structure of the ULA. As we discussedin Section III, the IP maximization , element-wise , and structuretechniques show poor performance, comparable to the Randomcase, because of not considering any channel structure or toomuch degree-of-freedom in the optimizations. Especially, theperformance difference between the structure and pre-searchtechniques clearly shows that it is essential to have judiciouspreprocessing before the optimization for downlink channelreconstruction. Note that the Type 1 and Type 2 cases show Fig. 4: Average spectral efficiency of the different channelreconstruction techniques according to ρ UL with the ULA andfixed widebeam at the BS. The N BS × M UE downlink channelis reconstructed through L = M UE = 4 layer feedback.Fig. 5: Average spectral efficiency of the different channelreconstruction techniques according to ρ UL with the UPA andfixed widebeam at the BS. The N BS × M UE downlink channelis reconstructed through L = M UE = 4 layer feedback.the same performance since the codewords of PMI codebookare unitary matrices when L = M UE .In Fig. 5, we considered the UPA at the BS. The pre-search technique and pseudo-inverse technique still outperformthe other channel reconstruction techniques, and the figureshows that it now becomes better for these techniques toassume the UPA for the reconstruction. Note that the pseudo-inverse technique is much more practical than the pre-searchtechnique since it only requires basic matrix operations. Unlikein Fig. 4, the ratio technique does not perform well since thereconstruction procedure of the ratio technique is not suitableto the UPA case. Through Figs. 4 and 5, it can be observedthat the structure technique has poor performance in spite ofits high complexity since it has no prior information about thedominant AoAs/AoDs and just set those randomly.In Figs. 6 and 7, we consider the case when the BS dynam- Fig. 6: Average spectral efficiency of the different channelreconstruction techniques according to ρ UL with the ULAand dynamically selected beam at the BS. The N BS × M UE downlink channel is reconstructed through L = M UE = 4 layer feedback.Fig. 7: Average spectral efficiency of the different channelreconstruction techniques according to ρ UL with the UPAand dynamically selected beam at the BS. The N BS × M UE downlink channel is reconstructed through L = M UE = 4 layer feedback.ically selects w k as in (45). We set L = M UE = 4 feedbacklayer for the N BS × M UE full MIMO downlink channelreconstruction. Fig. 6 shows the average spectral efficiencyof proposed downlink channel reconstruction techniques ac-cording to ρ UL assuming the ULA at the BS. Although theCSI-RS beamforming matrix P , which is a function of w k ,is now dynamically selected, the figure shows that there isno noticeable difference on the performance of the proposedtechniques compared to Fig. 4 since the data transmissionbeamformers of the proposed techniques are already adjustedwith the reconstructed channel and independent of P . TheType 1 and Type 2 cases, however, become better than Fig. 4 as ρ UL increases. This is because the BS exploits the prior chan-nel knowledge h SRS not only for the CSI-RS beamforming but Fig. 8: Average spectral efficiency of the different channelreconstruction techniques according to ρ UL with the ULA anddynamically selected beam at the BS. The N BS × L downlinkchannel is reconstructed through L = 2 layer feedback.Fig. 9: Average spectral efficiency of the different channelreconstruction techniques according to ρ UL with the UPA anddynamically selected beam at the BS. The N BS × L downlinkchannel is reconstructed through L = 2 layer feedback.also for the data transmission. Still, the proposed ratio , pre-search and pseudo-inverse techniques outperform the Type 1and Type 2 cases.In Fig. 7, we plot the average spectral efficiency of proposeddownlink channel reconstruction techniques according to ρ UL assuming the UPA at the BS. Similar to Fig. 6, there isspectral efficiency improvement of the Type 1 and Type 2cases compared to that in Fig. 5 as ρ UL increases; however, theproposed techniques still experience no noticeable differencein terms of their spectral efficiencies.Figs. 8 and 9 consider the same scenario as in Figs. 6 and 7except L = 2 for the PMI feedback layer. The BS then triesto reconstruct the N BS × L MIMO downlink channel. Thefigures show the average spectral efficiency of the proposeddownlink channel reconstruction techniques according to ρ UL assuming the ULA/UPA at the BS. Since the BS transmits Fig. 10: Average spectral efficiency of the different channelreconstruction techniques according to ρ UL with the UPA anddynamically selected beam at the BS. The UMa scenario wasconsidered for the SCM channel. The N BS × L downlinkchannel is reconstructed through L = 2 layer feedback.Fig. 11: Average spectral efficiency of the different channelreconstruction techniques according to ρ DL with the UPA anddynamically selected beam at the BS. The UMa scenario wasconsidered for the SCM channel with the fixed uplink SNR ρ UL = 0 dB. The N BS × L downlink channel is reconstructedthrough L = 2 layer feedback.data only through L = 2 layer spatial multiplexing, it canbe observed that the spectral efficiencies are lower than theprevious cases of L = M UE = 4 . The overall trends amongthe proposed downlink channel reconstruction techniques,however, are similar to those of Figs. 6 and 7. Since the pre-search technique and pseudo-inverse technique performquite well with L = 2 , we can conclude that the proposedtechniques are able to reconstruct the downlink channel evenwhen L is less than M UE . Note that, although the Type 2 casedoes not assume any CSI quantization error, the pre-searchtechnique and pseudo-inverse technique outperform the Type 2case when the BS is equipped with the UPA. This clearlyshows the loss of conventional methods by only using the low Fig. 12: Spectral efficiency CDF of the pseudo-inverse tech-nique according to differently assumed m Tx while the truevalue is m Tx = 1 . The BS is deployed with the UPAusing the fixed widebeam for the CSI-RS beamforming, and ρ UL = 0 dB is assumed. The N BS × M UE downlink channelis reconstructed through L = M UE = 4 layer feedback.dimensional effective CSI-RS channel for spatial multiplexing.In Figs. 10 and 11, we adopt a different scenario of urbanmacro (UMa) with the same carrier frequency for the SCMchannel. In Fig. 10, we plot the average spectral efficiency withthe same assumptions as in Fig. 9. It is clear from the figurethat the overall trends among the proposed downlink channelreconstruction techniques are the same with the UMi scenario.In Fig. 11, we plot the average spectral efficiency with thedownlink SNR ρ DL with the fixed uplink SNR ρ UL = 0 dBwhile other assumptions are the same as in Fig. 10. The figureshows the proposed techniques, especially the pre-search and pseudo-inverse techniques , work well for all range of ρ DL .In Fig. 12, we plot the spectral efficiency cumulativedistribution function (CDF) of the pseudo-inverse technique assuming the UMi scenario and the UPA at the BS to seethe effect of imperfect knowledge of m Tx . We consider thewidebeam weight for w k and L = M UE = 4 layer feedbackas in Figs. 4 and 5. We set the true transmit antenna indexof the UE m Tx as 1 while the BS assumes different valuesof m Tx for the downlink channel reconstruction. It is clearfrom the figure that the knowledge of m Tx does not affectmuch on the spectral efficiency performance as discussed inSection III-H. V. C ONCLUSION
In this paper, we proposed possible downlink massiveMIMO channel reconstruction techniques at the BS. Con-sidering practical antenna structures to reduce the downlinkCSI-RS overhead, the proposed techniques work in TDD byexploiting both the downlink CSI-RS and the uplink SRS.The numerical results showed that the spectral efficiencies byspatial multiplexing based on the proposed downlink chan-nel reconstruction techniques outperformed the conventionalmethods of using the fed back PMI directly in most cases. Among the proposed techniques, the pre-search technique and pseudo-inverse technique outperformed the other techniquesin terms of the spectral efficiency while the pseudo-inversetechnique is much more practical due to its low complexity. Inaddition, we showed that the proposed channel reconstructiontechniques are not affected by the imperfect knowledge of thetransmit antenna index of the UE at the BS.Possible future research directions would include practicalsymbol detection techniques at the UE assuming the BSmay not have perfect knowledge of the transmit antennaindex of the UE, and downlink channel reconstruction for thecase when the UE has multiple transmit antennas. It is alsoworth investigating the performance limit of downlink channelreconstruction using the CSI-RS and SRS to analyze how closethe proposed techniques to the limit.R
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