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Design of double codebook based on 3D dualpolarized channel for multiuser MIMO system
EURASIP Journal on Advances in Signal Processing volume 2014, Article number: 111 (2014)
Abstract
This paper proposes a precoding method based on double codebook for multiuser threedimensional multipleinput multipleoutput (3D MIMO) system deployed with dualpolarized antenna array. Considering the block diagonal feature of the transmit correlation matrix of 3D dualpolarized channel, we adopt the double codebook structure, where the precoding matrix is composed of an inner longterm matrix and an outer shortterm matrix. Then, we introduce a set of 3D beams which possess good discrimination in horizontal and vertical domains to be the basic beam set and adopt a topology describing the 3D beam set to illustrate the angle information in an intuitive way. For the purpose of multiuser transmission, we also propose a corresponding limited channel information feedback scheme and a user scheduling criteria based on this double codebook structure. Simulation results demonstrate that the precoding method based on double codebook for the dualpolarized 3D MIMO system effectively improves system throughput compared with 2D transmission scheme.
1 Introduction
By applying multiple antennas at the transmitter and receiver, multipleinput multipleoutput (MIMO) techniques can exploit spatial degrees of freedom and improve the spectral efficiency[1]. Although initial work on multipleantenna design was primarily motivated by the singleuser paradigm, more recent attention has been on the theory and practice of multiuser (MU) multipleantenna communications[2–5]. A transmission strategy using dirty paper coding (DPC) was proposed in[6, 7] and shown to achieve the system capacity. However, it is extremely computationally intensive, especially in the multiuser context. This makes it less attractive in practical systems, and low complexity sumoptimal schemes are desirable. Therefore, many linear precoding methods appear, such as zeroforcing precoding and block diagonalized precoding. In 3GPP Release 10, a double codebookbased precoding method was proposed[8]. It adopts discrete Fourier transform (DFT) beams as the basic beam set, which behaves well under uniform linear array (ULA) antenna array deployment. The scheme takes advantage of the block diagonal feature of the channel’s transmit correlation matrix and selects beams according to both longterm and shortterm channel state information (CSI). However, most traditional MUMIMO systems are designed with respect to a twodimensional (2D) world. By choosing beams with different azimuths, the base station (BS) can simultaneously provide service to several users located in different horizontal directions. All the beams are with the same downtilt; therefore, users in the same horizontal direction referring to BS cannot be scheduled at the same time, as shown in Figure1a. If several users are distributed in different floors of a tall building, only one of them will be served.
With the rapid development of highrise buildings in modern cities, there has been a growing range of floors users may stay in. This challenges the traditional channel model and transmission techniques. In recent years, threedimensional (3D) MIMO techniques, which fully exploit the degrees of freedom in 3D space, have attracted more and more attention[9–20]. Compared with 2D MUMIMO, the 3D MUMIMO system is able to simultaneously serve several users who have the same azimuth but are located at different heights and therefore improves transmission efficiency greatly, as shown in Figure1b. To realize 3D MIMO, changes are required for both channel model and transmission scheme. A new 3D channel model has been set up by 3GPP[9], where the concept of downtilt or elevation angle is added as an important channel parameter, BS is deployed with uniform planar antenna array (UPA), and users are distributed in every floor instead of just moving on the ground. Based on the 3D channel model, some 3D beamforming methods that adjust the beam downtilt adaptively according to elevation sectorization or peruser elevation were proposed[10–15]. A 3D beamforming method for hexagonal multicell network was proposed in[10], which implements an implicit coordination scheme to control multicell interference by vertically and horizontally assigning different resources to adjacent sectors. The methods in[11] and[12] are based on cellular segmentation, the whole cell is split into several concentric annular zones, and each of them is allocated with a fixed beam downtilt. Two userspecific elevation beamforming methods based on adaptive codebooks derived from channel correlation matrix are proposed in[13, 14]. However, they are designed for singleuser transmission systems. Another peruser adaptive beamforming method raised in[15] calculates the downtilt by comprehensive channel estimation and then applies the estimated downtilt to traditional 2D LTE precoding. Existing techniques generally consider horizontal domain and vertical domain independently; therefore, relevant beamforming is implemented in horizontal and vertical domains separately.
In this paper, we investigate the 3D MUMIMO downlink transmission system and propose a jointdomain precoding method which considers both horizontal and vertical domains simultaneously. This method is based on double codebook and designed for MUMIMO systems under the deployment of dualpolarized UPA antenna array at BS. The double codebook structure is adopted here under the consideration of the block diagonal feature for the channel’s transmit correlation matrix. The precoding matrix is composed of two components: inner precoding matrix and outer precoding matrix. The inner precoding matrix roughly orients the direction of the channel’s main lobe and is decided by longterm CSI, i.e., transmit correlation matrix. The outer precoding matrix is used for precise positioning and is decided by instantaneous CSI. A set of 3D beams which has good discrimination in horizontal and vertical domains is selected to be the basic beam set, and a UPA topology is introduced to demonstrate the angle relations among these 3D beams. This paper also proposed a best companion cluster (BCC) limited feedback scheme and a user pairing criteria in multiuser transmission mode. Users calculate the optimal inner precoding matrix index (PMI1), the optimal outer precoding matrix index (PMI2), the optimal pairing cluster index (CI), and channel quality indicator (CQI) according to downlink channel estimation results and feed these values back to the BS, and then the BS uses feedback values to implement user pairing and scheduling. Moreover, a fast codebook searching method is presented for the purpose of saving calculation time when the codebook size is huge. Simulation results show that the proposed 3D transmission method can effectively improve system throughput compared to the 2D transmission method.
The rest of this paper is organized as follows. Section 2 presents the antenna array model and 3D MUMIMO downlink transmission model. Basic beam set and precoding method based on double codebook are described in Section 3. In Section 4, the CSI feedback scheme, user scheduling criteria, and fast codebook searching method are illustrated. Simulation results and analysis are given in Section 5. Finally, Section 6 concludes the paper.
Notation: In this paper, matrices and vectors are denoted by uppercase and lowercase boldface letters, respectively. We use I to denote the identity matrix and 0 to denote the allzero matrix. The superscripts (·)^{H}, (·)^{T}, and (·)^{∗} represent the conjugatetranspose, transpose, and conjugate operations, respectively. Symbol\mathbb{E}\{\xb7\} represents the expectation with respect to all random variables within the brackets. A ⊗ B is the Kronecker product of matrices A and B. We use · and ∥·∥ to denote taking absolute value and modulus operations, respectively.
2 System model
In this section, we provide detailed introduction to the system model, focusing on the antenna array topology and the downlink transmission scheme. We consider the singlecell 3D MUMIMO downlink transmission system. The BS is located at the cell center. A total number of K users are distributed randomly in the cell. According to the typical scenario in the 3GPP 3D channel model, some of the users are located in high rises, while others are moving on the ground at a low speed.
2.1 Antenna array model
It has been proved that largescale multiple antenna array deployment can significantly improve spatial resolution and diminish interuser interference (IUI) due to its extremely narrow transmit beams[21], along with the challenges of terminal size and price. Dualpolarized antennas become widely employed for saving half of the space that the antenna array covers while maintaining the same data transmission rate.
Based on the above considerations, in this paper, a dualpolarized UPA antenna array is adopted at BS as the 3GPP 3D channel model recommends, and each pair of dualpolarized antennas is positioned equidistantly on a wall, as shown in Figure2. In this UPA array, there are{N}_{\mathit{\text{tx}}}^{(h)} and{N}_{\mathit{\text{tx}}}^{(v)} antenna pairs in each row and column, and the distances between any two adjacent rows and columns are{d}_{\mathit{\text{tx}}}^{(h)} and{d}_{\mathit{\text{tx}}}^{(v)}, respectively. Thus, the total number of dual polarized antenna pairs is{N}_{\mathit{\text{tx}}}={N}_{\mathit{\text{tx}}}^{(h)}{N}_{\mathit{\text{tx}}}^{(v)}, and the total number of transmit antenna elements is 2 × N_{ tx }. Now, for simplicity, we consider a single polarized direction and suppose there are only verticalpolarized antenna elements, as shown in Figure3, where γ_{ t } and θ_{ t } are the beam downtilt and azimuth, respectively, and each antenna port serves one antenna element[19]. The antenna array response can be expressed in the following way
where{\mathrm{\Psi}}_{t}=\frac{{d}_{\mathit{\text{tx}}}^{(v)}}{\lambda}sin{\gamma}_{t},{\mathrm{\Omega}}_{t}=\frac{{d}_{\mathit{\text{tx}}}^{(h)}}{\lambda}cos{\gamma}_{t}cos{\theta}_{t}, and λ is the carrier wavelength. The derivation of (1) can be found in the Appendix. At the user side, N_{ r x } pairs of dualpolarized antennas are assigned into the ULA array. A numbering method for UPA is also introduced in Figure3 for convenience.
2.2 Downlink transmission
In this paper, we consider a twouser spatial division multiple access (SDMA) system. In the 3D MUMIMO system, users firstly estimate CSI using downlink pilot signals and feed the estimated CSI back to BS, and then BS uses the information to conduct downlink multiuser transmission. The estimated CSI includes information about the selected precoding matrix, i.e., PMI1 and PMI2, and user pairing and scheduling accordance, i.e., CQI and CI. Figure4 shows the block diagram of the downlink transmission system.
Assume that the two scheduled users are u and s. The received signal at user u can be written as
where{\mathbf{x}}_{u}\in {\mathbb{C}}^{{N}_{l}\times 1} and{\mathbf{x}}_{s}\in {\mathbb{C}}^{{N}_{l}\times 1} are signals for users u and s with power constraint\mathbb{E}\left\{{\mathbf{x}}_{u}^{H}{\mathbf{x}}_{u}\right\}=\mathbb{E}\left\{{\mathbf{x}}_{s}^{H}{\mathbf{x}}_{s}\right\}=P/2, N_{ l } is the transmission rank, P is the total transmit power,{\mathbf{n}}_{u}\in {\mathbb{C}}^{2{N}_{\mathit{\text{rx}}}\times 1} is zeromean complex additive Gaussian noise satisfying\mathbb{E}\left\{{\mathbf{n}}_{u}{\mathbf{n}}_{u}^{H}\right\}={\sigma}_{n}^{2}{\mathbf{I}}_{2{N}_{\mathit{\text{rx}}}},{\mathbf{H}}_{u}\in {\mathbb{C}}^{2{N}_{\mathit{\text{rx}}}\times 2{N}_{\mathit{\text{tx}}}} is the channel matrix of user u, and{\mathbf{W}}_{u}\in {\mathbb{C}}^{2{N}_{\mathit{\text{tx}}}\times {N}_{l}} and{\mathbf{W}}_{s}\in {\mathbb{C}}^{2{N}_{\mathit{\text{tx}}}\times {N}_{l}} are the precoding matrices for users u and s, respectively. Minimum mean square error (MMSE) receivers are applied at the user side[22], and the signaltointerferenceplusnoise ratio (SINR) matrix of user u can be expressed as
where{\mathbf{F}}_{u}^{(s)}={\mathbf{H}}_{u}{\mathbf{W}}_{s}. Based on (2) and (3), the system throughput can be calculated as follows:
The target of the proposed scheme is to maximize the system throughput with respect to W_{ u } and W_{ s }. In the rest of this paper, we will consider the rank 1 transmission mode, i.e., N_{ l } = 1. In this case, vectors x_{ u } and x_{ s } are reduced to scalars x_{ u } and x_{ s }.
3 Precoding based on double codebook
In this section, we discuss the design of the precoding matrix. As is mentioned above, the double codebook structure is adopted in our precoding matrix design, where the precoding matrix W is the product of two matrices W = W_{1}W_{2}, and they are selected from a longterm codebook and a shortterm codebook, respectively. Inner precoding matrix W_{1} roughly orients the direction of the channel main lobe and is decided by longterm CSI, i.e., channel’s transmit correlation matrix. In the case of flat fading channel whose statistical information is regarded to be unchanged for a long period, W_{1} remains constant during this period. Outer precoding matrix W_{2} is used for precise positioning and is decided by instantaneous CSI. Since such information changes over time, W_{2} should be synchronously updated with the instantaneous channel. Due to the block diagonal feature of the transmit correlation matrix of the 3D dualpolarized channel, the length of each codeword is only half of the total number of transmit antennas, i.e., N_{ tx }, which contributes to dimensionality reduction. Comprehensive utilization of both longterm and shortterm CSI also results in feedback period extension. To discuss the design of inner precoding matrix W_{1}, we first introduce the selection of basic beam set and then explain the design of inner precoding and outer precoding matrix in detail.
3.1 Basic beam set
Theoretical research has shown that the deployment of UPA topology used in BS antenna array can greatly exploit the vertical domain and adjust the beam direction in 3D space feasibly[20]. The orientation of a beam is determined by downtilt γ_{ t } and azimuth θ_{ t }, as shown in Figure3. We can split the whole 3D space using γ_{ t } and θ_{ t } and select N_{ v } downtilt values from [π/2,π/2] and N_{ h } azimuth values from [0,2π]. If we combine these values together, the complete set of basic beams will be defined, which covers N_{ v } × N_{ h } directions totally. These basic beams can be regarded as a set of basic vectors in 3D space, and any spatial vector can be represented using this set. Let c_{ i } represent the i th beam. The basic beam set can be displayed in the form of UPA topology shown in Figure3, where the downtilts of the beams in a row are the same, so do the azimuths of the beams in a column, as shown in Figure5.
In practical situations, users are located in a small range in the vertical domain. Downtilt values of the basic beams are not uniformly distributed in [π/2,π/2] but centralized in a small range. In this case, as presented in Figure5, the right semicircle is nonuniformly segmented. However, the UPA topology for the basic beam set is still available. Existing 3D beamforming techniques satisfying this UPA topology includes the Kronecker productbased codebook (KPC) precoding scheme[23]. KPC is derived from the traditional DFT codebook by making the Kronecker product of horizontal DFT codewords and vertical DFT codewords as follows:
where β is a parameter related to the abovementioned regional centralization of user distribution in the vertical domain. Obviously, KPC codewords are designed according to the transmit antenna array response (1); therefore, they can match the 3D spatial channel model well. Figure6 gives an example of the projection of KPC beams onto the horizontal plane when{N}_{\mathit{\text{tx}}}^{(h)}={N}_{\mathit{\text{tx}}}^{(v)}=8,{N}_{h}=32,{N}_{v}=16,\beta =4. Beams c_{1} and c_{9}, c_{225} and c_{233} lie in the same row of the UPA array, respectively; meanwhile, c_{1} and c_{225}, c_{9} and c_{233} are in the same column. Energy distribution of these beams illustrates that the downtilts of c_{1} and c_{9} are the same, as well as the azimuths of c_{1} and c_{225}, demonstrating that the KPC beams satisfy the previous angle requirement for the basic beam set.
According to the 3GPP 3D channel model[9],{\mathbf{c}}_{m}^{(v)} in (5) is no longer able to cover the highrise situation. Knowing that the value set of parameter m corresponds to a N_{ v }size subset taken out from a β N_{ v }size downtilt value set, it can be changed according to the practical situation. For example, for the case in Figure7, the value set of m can be selected as {3,2,⋯,6}, covering all the potential situation including both the high rise and the ground.
3.2 Precoding matrix design
Inner precoding roughly orients the channel and captures the channel’s main lobe by choosing a few basic beams that are adjacent to each other. Under the deployment of dualpolarized antenna array, transmit correlation matrix R of the channel holds the following block diagonal feature:
where A and B are the transmit correlation submatrices of two polarization directions and are approximately equal. According to this longterm character, the k th inner precoding matrix in the longterm codebook is defined as
where X_{ k } is a N_{ tx } × N_{ b } matrix representing the k th subset of the basic beam set which contains N_{ b } basic beams, N_{ b } = N_{ bv }N_{ bh }, N_{ bv } and N_{ bh } are the row and column number of each basic beam subset. Figure8 gives an example of X_{ k } and illustrates its position in the complete basic beam set as the dotted line surrounds.{\mathbf{X}}_{k}=\left[\begin{array}{cccc}{\mathbf{c}}_{i}& {\mathbf{c}}_{i+1}& \cdots & {\mathbf{c}}_{i+{N}_{b}1}\end{array}\right], i = 1,⋯,N_{ h }N_{ v }, and the relationship between i and k is decided by the definition of basic beam subset.
If the complete set is uniformly segmented and nonoverlapping exists among subsets, the numerical relation between the complete set and the beam subsets satisfies
where m,n are integers. Referring to the Release 10 scheme[8], to avoid border vagueness, we can consider the method of overlapping among subsets, and the overlapping ratio between adjacent subsets is 1/2. In such case, the selected subsets have higher possibility to cover the channel main lobe, but the amount of subsets doubles afterwards. Here we use three overlapping definitions, that is, overlapping in only the horizontal domain, in only the vertical domain, and in both domains. Take N_{ v } = 4,N_{ h } = 32,N_{ bv } = 2,N_{ bh } = 4 case as an example, and four different segment methods are shown in Figure9.
Outer precoding is used for beam matching. The beams best matching the current instantaneous channel will be picked out from the previous selected subset. Considering rank 1 transmission, only one beam will be selected, and therefore, the l th outer precoding matrix in the shortterm codebook is expressed as follows:
where
i = 1,⋯,N_{ b } and α is a phase parameter from the QPSK signal set {1,j,1,j}. There are altogether 4N_{ b } alternatives for W_{2}.
4 Multiuser transmission based on double codebook precoding scheme
Double codebook structure can greatly reduce the amount of CSI feedback, shorten the feedback period, and improve transmission efficiency of the FDD system. In this paper, we choose BCC as the corresponding limited feedback scheme. There are four values to be calculated, that is, inner precoding matrix index PMI1 which is related to parameter k, outer precoding matrix index PMI2 which is related to parameter l, the cluster indices CIs that a user himself and his best companion are included in, and the channel quality indicator CQI reflecting channel condition when interference exists. According to Figure4, users estimate CSI and use these estimation results to match beams step by step until PMI1 and PMI2 are settled. CI and CQI will also be calculated and sent back to the BS along with PMI1 and PMI2. In downlink, the BS uses these feedback values to conduct user scheduling and beam selecting, then implements multiuser transmission. Since a large size basic beam set is usually adopted, in this section, we propose a fast codebook searching method for the purpose of saving PMI1 calculation time.
4.1 Calculating the feedback values
(1) PMI1 calculation: When selecting PMI1, statistic channel information is used and assumed to be constant during T sampling intervals. For user u, we firstly calculate its transmit correlation matrix
Under the deployment of dualpolarized antennas, R_{ u } has block diagonal structure[24]
where{\mathbf{R}}_{u}^{(1)},{\mathbf{R}}_{u}^{(2)}\in {\mathbb{C}}^{{N}_{\mathit{\text{tx}}}\times {N}_{\mathit{\text{tx}}}}, and{\mathbf{R}}_{u}^{(1)}\approx {\mathbf{R}}_{u}^{(2)}. Therefore, we use the following equivalent transmit correlation matrix{\hat{\mathbf{R}}}_{u} to represent the transmit correlation matrix
Implement singular value decomposition (SVD) to{\hat{\mathbf{R}}}_{u} and choose its principle eigenvector v_{1} as the longterm channel main lobe direction. Then, calculate the distances between v_{1} and all the basic beams, and select the nearest one c_{ i } and the farthest one c_{ j },
If nonoverlapping method is used, we can calculate PMI1 directly from c_{ i }. When utilizing overlapping methods, there are more than one subset that contains c_{ i }, and further processing is required to choose the best subset. In this case, the second primary eigenvector v_{2} of{\hat{\mathbf{R}}}_{u} can be taken out to match with the beams included in the several selected subsets, until the final PMI1 is selected.
(2) PMI2 calculation: PMI2 is decided according to the instantaneous channel information H_{ u }. Outer precoding W_{2} should maximize the power of the received signal, and therefore, PMI2 for user u is determined as follows:
(3) CI calculation: A simple way of defining CI proposed in Release 10 feedback scheme[24] is to directly regard the subset index PMI1_{ u } wherein the fittest beam c_{ i } is referred to as the belonged cluster index CIi_{ u }, and regard the subset index PMI1_{ s } wherein the farthest beam c_{ j } is referred to as the optimal paired companion cluster index CIj_{ u }, that is, CIi_{ u } = PMI1_{ u }, CIj_{ u } = PMI1_{ s }. However, in order to reduce the size of the shortterm codebook, only a few beams are included in X_{ k }, resulting in complex subset segmentation and large amount of subsets. If we apply the above method and only use PMI1 to implement user scheduling and pairing, successful paired rate will be low unless the number of users is large. In this paper, we revise the definition of CI, cluster several adjacent PMI1 into a CI, and increase the paired rate by means of decreasing the number of companion clusters. PMI1 array in the form of UPA is shown in Figure9. Considering the concentrated user distribution in the vertical domain and the uneven segmentation that KPC acts on 3D space[23], CI calculation is illustrated in Figure10, where the gray blocks represent PMI1 values. While uniformly clustered in the horizontal direction, PMI1 values are vertically clustered according to the sparsity of beam downtilts in the ratio of 1:3. This method will maximize user paired rate and guarantee IUI as slight as possible.
(4) CQI calculation: CQI of user u is conservatively calculated as follows:
4.2 User scheduling
Having received all the information that users feed back, BS uses these feedback values to implement user pairing and scheduling as follows:

Step 1 Primary user selection. BS finds out the maximum CQI and chooses the corresponding user for primary consideration.

Step 2 Paired cluster selection. According to the longterm information (CIi_{ u },CIj_{ u }) of user u, BS finds out its paired cluster whose CI values are exactly (CIj_{ u },CIi_{ u }).

Step 3 User scheduling. Choose user s whose CQI is the maximum one of successful paired cluster as the best pairing user and simultaneously scheduled with primary user u.
4.3 Fast codebook searching method
The deployment of a largescale antenna array can exploit enough spatial degrees of freedom; nevertheless, it results in largescale codebook, i.e., with 10^{2} or more codewords. Generally, when using CSI to do beam matching, we calculate the distances individually and take the best one. It is a traversal method (TM) and is highly timeconsuming when conducted for searching a 3D codebook. Here, we propose a new fast codebook searching method (FCSM), utilizing the UPA topology of the complete basic beam set shown in Figure5. Generally, the value of N_{ v } or N_{ h } is a power of 2, so here we consider a quartering method.

Quartering step 1 We divide the complete beam set arranged in UPA form into four subsets A_{1},A_{2},A_{3}, and A_{4}, signed as the firstorder subset. Figure11 shows quartering step 1, where small circles are codewords, and the four red circles at 1/4 and 3/4 positions of the top and bottom rows represent reference codewords{\stackrel{~}{\mathbf{c}}}_{1},{\stackrel{~}{\mathbf{c}}}_{2},{\stackrel{~}{\mathbf{c}}}_{3}, and{\stackrel{~}{\mathbf{c}}}_{4}. When calculating PMI1, firstly compare CSI with these four codewords and pick out the optimum one, for example,{\stackrel{~}{\mathbf{c}}}_{1}, then search subset A_{1} and no longer consider A_{2},A_{3}, and A_{4}.

Quartering step 2 If the size of subset A_{1} is still too huge, we further conduct quartering and get the secondorder subsets B_{1},B_{2},B_{3}, and B_{4}. Now we use codewords in four vertices instead of the 1/4 and 3/4 positions of the top and bottom rows as new reference codewords, as shown in Figure12. Continue to compare CSI with the new reference codewords and select the best secondorder subset, such as B_{2}, without further consideration about the remaining three subsets.

Quartering step 3 Repeat quartering step 2 until there is no need for a higherorder subset. Then, we simply compare the CSI with beams in the smallest subset individually and find the best one.
Notation: Since the first and the last columns of the complete beam set are actually adjacent with each other, we select the center, not the vertex of each firstorder subset, to be the reference codewords in quartering step 1. However, for each secondorder subset, its two sidecolumns lose adjacency, becoming suitable for reference in quartering step 2.
The idea of quartering is applicable for not only codebook searching but also other utilities. If only the samples to be searched can be arranged uniformly in UPA form, quartering method will be applicable.
5 Numerical results
The performance of the proposed precoding method based on double codebook for the 3D dualpolarized MUMIMO system is investigated through simulation. Numerical results about the design of precoding matrix including four different methods to segment the complete set of basic beams and the multiuser transmission scheme are presented in this section. The performance of 2D precoding under the same deployment is also shown for comparison. Simulation parameters are listed in Table1.
Consider a 120° sector of a 500mradius cell. Users are uniformly distributed in the sector, with a minimum 2D distance of 10 m from BS. To better fit modern urban condition, we assume that all users stay in an eightfloor building whose storey height is 3 m, and the bottom floor is equivalent to the ground; therefore, the effect of user height is significant. Energy loss caused by signals traveling through walls from outdoor to indoor and other similar factors are ignored here. WINNER 3D channel model is adopted, and C2 urban macro in NLOS condition is selected[25]. When implementing 2D scheme, the downtilt of each transmit beam is fixed to 12°, and the value set of CI is {1,2,…,8}.
Figure13 compares the throughput of the 3D multiuser transmission scheme based on dualpolarized channel and that of the 2D scheme when the size of BS antenna array is{N}_{\mathit{\text{tx}}}^{(v)}\times {N}_{\mathit{\text{tx}}}^{(h)}=8\times 8, i.e., the total number of transmit antenna elements is 2 × (8 × 8). From the figure, it can be seen that the throughput of the MUMIMO system using the proposed precoding method based on 3D beams is higher than the traditional precoding method which uses 2D beams, since 3D beams perform better than 2D beams in describing and matching 3D channel. Precoding based on 3D beams can adaptively adjust the beam downtilt according to the vertical channel information, while precoding using 2D beams simply unifies all the beam downtilt and only makes use of the horizontal channel information. Moreover, if we increase the number of users, successful paired rate grows, and system throughput increases. Performance comparison of the four different methods to segment the basic beam set is also shown in Figure13. When there are only a few users, the performance of nonoverlapping method is not better than that of overlapping method. While with the growth of user number and successful paired rate, the advantage of overlapping method becomes gradually significant. Successful paired rate of these above methods is compared in Table2. It is obvious that the overlapping method in both domains maximizes system throughput; thus, it is more suitable for our multiuser feedback scheme in single sector scenario. The achieved throughput when using the vertically overlapping method is higher than when using the horizontally overlapping method. If more comprehensive division in the vertical domain is employed for calculating feedback value CI, the advantage of vertically overlapping method will be further exploited.
Since beams in each row of KPC cover the whole 360° plane, performance of the proposed scheme is also tested in the whole cell scenario. The previous directive antennas are replaced by omnidirectional antennas, i.e., A_{ H }(ϕ) = 0, and users are distributed uniformly in the cell. Figure14 and Table3 give the simulation results. In this scenario, the overlapping method in both domains loses the excellence achieved in the 120° sector scenario, although surpasses nonoverlapping method when user number is large, and it does not perform better than the two singledomain overlapping methods. In conclusion, overlapping method in both domains is recommended in the 120° sector scenario and vertically overlapping method is suggested in the whole cell scenario.
To investigate the performance of fast codebook searching method, the principle eigenvector v_{1} of the transmit correlation matrix of the channel between user 1 and BS is taken out and matched with beams using the proposed method and traditional traversal method, respectively. A total of four KPC codebooks with different sizes are tested here, and the numerical results are given in Table4. In order to diminish iterations in fast searching method and maximize its calculating efficiency, the quartering is stopped when the thirdorder subset is segmented. According to the data in Table4, if a smallsize codebook is used, for example, N_{ v } × N_{ h } = 16 × 32, these two methods will cost the same amount of time, and thus fast searching method cannot save time. With the enlargement of codebook size, time consumption of traversal searching grows linearly, while the increased time for fast searching is small, especially in N_{ v } × N_{ h } = 64 × 64 condition, and fast searching costs only 1/6 of the traversal searching time. Searching results of the two methods show that the correct ratio of the proposed fast searching is high. Therefore, our proposed fast searching codebook method can effectively reduce the calculation time of the feedback value PMI1 and at the same time guarantee high accuracy when the size of codebook is large.
6 Conclusion
In this paper, we considered 3D dualpolarized MUMIMO system and proposed a precoding method based on double codebook. Considering the block diagonal feature of the transmit correlation matrix of 3D dualpolarized channel, we adopted the double codebook structure for precoding design, conducted beam matching, and calculated feedback values according to both longterm and shortterm CSI. We also proposed a multiuser transmission scheme based on BCC limited feedback scheme and designed a fast codebook searching method for largescale codebooks. Simulation results showed that when compared with 2D precoding method, the proposed precoding method based on 3D beams can effectively improve system throughput, and the fast codebook searching method can reduce the calculation time of the feedback value and at the same time guarantee high accuracy.
Appendix
Derivation of Equation 1
In this appendix, we give the specific derivation process of the UPA antenna array response in (1). According to Figures2 and3, when a signal gets through the UPA array, we can analyze its phase shift in horizontal and vertical domains, respectively. If we take out all the antenna elements in the n th column, then a uniform linear array (ULA) is formed, as shown in Figure15. The phase shift between the m th antenna element and its samecolumn adjacent element is[26, 27]
where c represents the propagation rate of the carrier satisfying c = λ f. From Figure15, we can easily find that
and therefore, φ^{(v)} can be expressed as
Similarly, the phase shift between UPA element (m,n) and its samerow adjacent element is
When compared with the first antenna array element, the total phase shift of (m,n) is the sum of the values in horizontal and vertical domains
and the corresponding array element response is
According to the numbering method in Figure3, the UPA array response is expressed as
Then, it can be reduced to Kronecker product expression
and (23) is equivalent to (1).
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Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grant 61222102, the National Basic Research Program of China (973 Program 2012CB316004), National Natural Science Foundation of China under Grants 61101089 and 61271018, Research Project of Jiangsu Province under Grants BK2012021 and BK20130019, and NEC Research Fund.
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Han, Y., Jin, S., Li, X. et al. Design of double codebook based on 3D dualpolarized channel for multiuser MIMO system. EURASIP J. Adv. Signal Process. 2014, 111 (2014). https://doi.org/10.1186/168761802014111
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DOI: https://doi.org/10.1186/168761802014111