1. Introduction
To achieve the required peak data rates up to 1 Gbit/s for low mobility and 100 Mbit/s for high mobility in the 4G standards, a straightforward method is to increase the transmission antennas, e.g., the long term evolution (LTE) system allows for up to 8 antenna ports at the base station (BS). However, it is challenging for antenna configurations, especially for the situation with large scale antenna arrays [1]. Another way to increase the transmission antennas is coordinated multi-point (CoMP) transmission, a kind of distributed antenna systems [2] or named as network multi-input multi-output (MIMO), which employs BS cooperation among the neighboring cells for joint signal transmission, taking advantage of the distributed multiple antennas to achieve spatial multiplexing gain or transmit diversity gain. It has been considered as one of the potential technologies for LTE-Advanced [3][4].
CoMP transmission is divided into coordinated scheduling/beamforming (CS/CB) and joint processing (JP). In CS/CB, data is only transmitted from a single BS [5][6], but coordination BSs exchange channel state information (CSI) with each other, so that scheduling can be performed to reduce inter-cell interference. In JP, data is simultaneously transmitted from coordination BSs, requring all coordination BSs to share data and CSI. This paper focuses on JP system, and CoMP refers in particular to JP below.
The gain of CoMP system largely depends on the availability of CSI at BSs. In practical, the CSI may be available due to the channel reciprocity for time division duplexing systems. But for frequency division duplexing systems, channel quantization at user equipment (UE) is needed to report CSI. Relative to MIMO systems, the acquirement of CSI at BS is more challenging, as UE should report CSI of all cooperation BSs [7]. It is a heavy burden for the low-capacity feedback links.
In general, a predefined codebook is designed for channel quantization by vector quantization theory, based on which, the UE quantizes CSI and feeds back the index of the codeword to the BS [8]. Since UE should feedback the CSI of all the cooperation BSs in CoMP system, the codebook for CSI quantization should be studied [9]. An intuitive method is to design a large size codebook by treating the cooperation BSs as a super BS, which is named as the joint-cell codebook approach [10]. The joint-cell codebook method is optimal. However, in the practical system, the method is infeasible because of the asymmetry feature of CoMP channels, high searching complexity and varying codebook size caused by the dynamic number of cooperation BSs [11]. Therefore, the per-cell codebook method for CSI quantization has drawn much attention, in which each cooperation BS has an independent codebook [11-17]. The codebook designed for single-cell MIMO system can be used for per-cell codebook for CoMP system as well.
When UE uses the per-cell codebook scheme, there are different codeword selection schemes. [9] proposed two codeword selection schemes, joint codeword selection (JCS) and independent codeword selection (ICS), and analyzed the complexity of these two schemes. For JCS, the codewords are selected jointly to minimize the quantization error of the cooperation cells channel direction information (CDI), and ICS is to minimize the quantization error of single cell CDI with the codewords selected independently for cooperation BSs. JCS is superior to ICS in system throughput. However, the complexity of the codeword selection is high. Since the preferred codeword for each BS is obtained by exhaustive search over all the codebooks of the multiple cells, JCS has exponential complexity with respect to (w.r.t.) the size of the codebook. [12] proposed a codebook compression scheme to reduce the complexity of JCS. By selecting suitable codewords from original codebooks, the sub-codebooks with smaller size are combined for CDI quantization.
Conversely, the selection complexity of ICS is low, but the phase ambiguity (PA) derived from the per-cell quantization degrades the system performance, including distributed array gain, macro diversity gain and normalized quantization gain [13-15]. The authors of [13][14] had proved that the received signal to noise ratio (SNR) approaches zero with probability 1 when the number of the coordinated BSs approximates infinity, even though the codebook size is asymptotically large. [15] pointed that only for cell edge users, the PA will lead to significant performance degradation of the joint CDI quantization. Therefore, [16] proposed a new quantization technique to solve the PA problem, where the quantization of each of the real and the imaginary parts is performed independently using the same real codebook. Additionally, feeding back a few phase information (PI) bits in each cell for PA is an effective way to compensate the performance loss [17][18], even 1-bit PI feedback can greatly increase the system performance [13]. [17] analyzed the average quantization performance of ICS with or without PI feedback, and showed the necessity of the PI quantization especially for cell-edge users. [18] quantified the specific required bits number for PI quantization to ensure an allowed quantization error loss. However, the introduction of the PI feedback will increase the control information feedback and overhead.
Under the total number of feedback bits constraint, rational allocating the number of feedback bits for CDI and PI quantization can improve the system performance without increasing the feedback overhead [18][19]. In [19], a bit allocation scheme, by maximizing quantization accuracy, was proposed and derived the solution by searching the possible bit combinations. [18] derived closed-form solutions of the allocated feedback bits for CDI and PI quantization. The scheme to allocate feedback bits between CDI and PI for relay system, aimed at maximizing UE’s rate, is given in [20]. Besides CDI and PI, feedback-bit allocation among users is also needed for multi-user (MU) systems [21]. [22] concluded that users with higher requested quality-of-service, i.e., lower outage probabilities and higher downlink rates, should use larger shares of the feedback rate. The feedback-bit allocation algorithm among users can offer a 20% performance gain over the equal bit allocation scheme [23].
The above researches proposed two codeword selection schemes, JCS and ICS, without analyzing the different effect on system throughput. For ICS, the necessity and effectiveness of PI feedback for improving quantization accuracy had been analyzed. However, the performance of ICS with PI feedback (called for ICS below) and JCS are not compared comprehensively. In this paper, we study JCS and ICS for channel quantization based on the per-cell codebook in CoMP system. Several respects of ICS and JCS, including feedback overhead, selection complexity and system throughput, are compared. For ICS, the scheme of optimal feedback-bit allocation by maximizing system throughput is given. The numerical and simulation results are given to verify the theoretical system throughput. With regard to the problems of each scheme found through theoretical analysis and simulation results, some recommendations, selection complexity reducing for JCS and feedback-bit allocation for ICS, are given at last in the paper.
The contributions of the paper are summarized as follows.
This paper gives the upper bounds of system throughput of JCS and ICS. However, for ICS, the solution of feedback-bit allocation by maximizing the system throughput is derived by exhaustively searching. The close-form expression of the number of feedback-bit will be solved in the future work.
As for notations, we use uppercase boldface letters to denote matrices and lowercase boldface to denote vectors. The operators (·)T ,(·)H ,(·)† stand for transpose, Hermitian and pseudo-inverse respectively. E(·) is the expectation operator. ║·║ represents norm operation.
2. System Description
2.1 System Model of CoMP Joint Processing
Consider a CoMP system with N BSs, each equipped with nt antennas, cooperatively serving K single-antenna UEs. The nearest BS to the kth UE is the serving BS, and the other N-1 BSs are the coordinated BSs for UE k. As shown in Fig.1, N BSs cooperatively serve the UE k.
Fig. 1.An example of CoMP system. The dotted lines with arrow denote wireless links.
As to the UE k, the N cooperation BSs seem as a virtual super BS with Nnt antennas. Similarly, the channel vector between the virtual super BS and UE k (called “global CSI”, denote hk in Fig.1) with 1×Nnt dimension is the combination of the single cell channel vectors (called “local CSI”, denote hk1,⋯hki⋯,hkn in Fig.1). Therefore, the global CSI for UE k can be expressed by
where αki (i=1,⋯,N) is the large scale fading gain including path loss and shadowing component, hki ∈ C1×nt is local CSI from the ith BS to UE k, whose entries are independent and identically distributed (i.i.d.) complex Gaussian variables with zero mean and unit variance.
We assume that the transmit power P is uniformly allocated to K UEs, that is, the transmit power of each BS for UE k is p = P/K. The received signal of the kth UE is
where is the precoding vector for UE k, is the precoding of the ith BS with power constraint , which is obtained according to the limited feedback information (details in Section 2.2). xk is the transmitted signal of UE k. The second term on the right-hand side of (2) is the MU interference, and nk is Gaussian noise variable with zero mean and σ2 variance.
2.2 Characteristics of Global CDI
From the structure of the global CSI shown in (1), we can conclude that the global CSI is no longer i.i.d. due to the existence of the heterogeneous large scale fading gains of multiple single cell channels.
The CDI of the global CSI is
We assume ║hki║ and are the channel magnitude information and CDI of hki respectively, that is, (3) can be rewritten as
where is an aggregation of the CDI of local CSI, is the ith channel gain of user k, and Gk = diag｛gk1,⋯,gkN｝is the channel gain matrix of user k.
Expression (4) implies that the CDI of global CSI has the following characteristics.
▪ It is not the simple combination of local CDI . It also depends on the channel gain matrix Gk, which results in the entries of being no longer i.i.d..
▪ The ratio between different BSs’ channel gain gki／gkj = αki／αkj (i ≠ j) varies frequently and is fluctuant in a large range when UEs move, especially with high-speed, as αkj is highly depends on UE’s location.
▪ Under the scenario of dynamic cooperation cells, the cooperation sets of cells for each UE is dynamically adjusted according to the predefined criterion, in other words, N is varies from UEs and with time, as which the dimension of is not fixed.
2.3 Per-cell Codebook based CDI Quantization
In the limited feedback system, CSI quantization is processed at UE before reported to BS. In order to highlight the impact of codebook on CDI quantization, we assume that the large scale fading gain αki (i = 1,⋯,N) and the small scale fading channel norm ║hki║,i = 1,⋯,N are perfectly obtained at BSs.
The kth UE is assumed to have perfect and instantaneous knowledge of hki, i = 1,2,⋯,N. Codebook is necessary for CDI quantization, which is fixed beforehand and is known to both the BSs and the UEs. With per-cell codebook, the CDI of each BS is quantized to one of the codewords in the codebook for each BS, and the index of each selected codeword is perfectly fed back from the UE to the serving BS. Then, each BS uses the codeword corresponding to the index as the CDI. Assume that ĥki is the quantized version of and = ［ĥk1,⋯,ĥkn］, the global CDI obtained at BSs can be reconstructed as
Assume that the per-cell quantization codebook for each BS is denoted as Ci (i = 1,⋯,N), which consists of unit norm vectors cij (j = 1,⋯,2Bc) in Cnt×1, and Bc is the number of feedback bits allocated to each BS.
We use the minimum chordal distance criterion to quantize the vectors. The criterion for ICS can be expressed as
while the JCS (ĥk1,ĥk2,⋯,ĥkN) = QJCS () follows the criterion of
In the following, we assume that the random vector quantization codebook is used. Therefore, each of the vectors within Ci is selected randomly and independently from the uniform distribution on the complex unit sphere. We analyze the performance averaged over all such choices of random codebooks.
Define as the average quantization accuracy, which is [24]
where β(·,·) is the Beta function. It is also shown in [24] that D (2Bc ,nt) is tightly bounded as
In order to facilitate the description, the codebook size for JCS and ICS is denoted as 2BJc and 2BIc, respectively.
3. Performance Analysis of CDI Quantization
In this section, we analyze the performance of CDI quantization with two codeword selection schemes, ICS and JCS. The theoretical upper bounds of system throughput are derived for ICS and JCS both in SU and MU situation. Then we compare the two schemes in the feedback overhead and selection complexity.
3.1 Throughput Analysis
3.1.1 SU scenario (K=1)
In this scenario, there is no MU interference, that is, the received signal of the UE comprises of the first and the third term in the right-hand side of (2). In this sector, all of the superscript k on variables in previous sectors is replaced by 1.
For per-cell codebook based limited feedback, the system throughput is given by
where ρ denotes the ratio between the transmit power of each BS for each UE and the noise variance, i.e. ρ = p／σ2. The superscript 1 on R is taken to state the expression denoting the throughput for the system with K=1 UE.
In this situation, maximum ratio transmission is adopted to boost the signal power. Therefore, is a quantized vector of that is, = . Then, the precoding vector of the UE can be denoted as = . As all of the quantized variables ĥ11,⋯,ĥ1N have unit norm, we can achieve ║║ = .
In order to analyze the system throughput of JCS with per-cell codebook, we first state a lemma demonstrated in [11], i.e., JCS with per-cell codebook scheme and joint-cell codebook scheme can achieve the same average quantization accuracy with sufficiently large nt and finite N. Assume that with unit norm is the combination of the quantized CDIs selected from the joint-cell codebook, which consists of 2NBJc entries in Nnt×1 [9]. So the average quantization accuracy of JCS can be given by
The system throughput of JCS can be obtained by
Here, (a) follows Jensen’s inequality and by substituting for h1. (b) is satisfied by using (11). Step (c) is arrived at by noting that is independent with and the use of (8). Finally, substituting (9) in (12), we get the upper bound of the system throughput of the JCS
For the special case where α11 = ⋯ = α1N = α1 , corresponding to some cell edge UE, (13) can be derived as
since is chi-square with nt degrees of freedom.
Different from JCS, the codewords selected by ICS aim to maximum the quantization accuracy for each local CSI. However, it cannot guarantee the maximum of the received signal power on account of the existence of PA, which is caused by the property of the selection criterion,
where θ is an arbitrary phase rotation. The received signal power can be written as
where θi (called PI in this paper) is the phase of It shows that the received signal not only depends on quantization accuracy , but related to the phase θi. The phase differences between θi (i = 1,⋯,N) lower the power of the received signal. Feedback quantization version of θi can reduce the effect of PI. Denote as the quantization of θi with BPI bits, then is the PI quantization error. The average received signal power is computed as follows.
where the approximate of (a) is derived by using that is demonstrated in [17]. As θi submits the uniform distribution in [0,2π]and is quantized uniformly as , Δθi is uniformly distributed between Then we have [17]
Based on the above analysis on average received signal power, the system throughput of the ICS with BPI bits of PI quantization feedback is
where (a) satisfies Jensen’s inequality and ϕ(BPI) is an increasing function of BPI when BPI ≥ 0. As ϕ(BPI = 0) = 0 and ϕ(BPI → ∞) = 1, the range of ϕ(BPI) is [0,1]. Substitute (9) in (19), the upper bound of the system throughput of the ICS with PI quantization feedback is
For some cell edge user whose single cell gains approximately satisfied α11 = ⋯ = α1N = α1, (20) can be simplified as
According to (20), the upper bound of R1ICS is an increasing function of ϕ(BPI), which illustrates that with more bits for PI feedback, the system throughput would be higher, and the maximum value of R1ICS corresponding to ϕ(BPI) =1 is denoted as
According to (13) and (22), we get It illustrates that the performance of ICS, benefiting from PI, exceeds JCS. Meanwhile, the feedback of PI increases the overhead of the feedback link. For the feedback link with low capacity, the system throughput shown in (22) is unattainable. But the reasonable feedback-bit allocation between BIc and BPI can improve the system throughput. The allocating bits between BIc and BPI can be formulated as
where B is the total feedback bits per UE for feeding back CDI and PI.
Notice: The R1ICS is an increasing function of BIc, and the feedback bits should be real integer, so the resulting BIc is given by ⎾BIc⏋ or ⎿BIc⏌ (⎾BIc⏋ or ⎿BIc⏌ denotes the ceiling or the floor of BIc). We can get the optimal solution by exhaustive search of all possible compositions of B since BIc and BPI are integer and BIc + BPI = B.
3.1.2 MU scenario (K＞1)
In this section, the performance of JCS and ICS for K＞1 (MU-CoMP) is compared. In this paper, we use ZF precoding to reduce the MU interference. Compiling αki║hki║ĥki(k = 1,⋯,K) into , the ZF precoding matrix is given by The precoding for kth UE is the normalized vector of the kth column of
In MU-CoMP with ZF, CDI feedback is used and the system throughput can be written as
where the approximation is achieved since we employ the Jensen’s inequality to both the numerator and the denominator [20][25]. is computed firstly, as it has no relationship with codeword selection scheme.
where (a) is derived as are independent with each other, hki and are also independent. (b) is obtained as follows a chi-square distribution with (nt - K + 1) degrees of freedom [26].
For JCS, (24) can be rewritten as
According to [27], the interference power satisfies
For JCS, . Substituting (25) and (27) into (26), we get
where step (a) is arrived at by using (9).
For MU-CoMP with ZF precoding, the signal is mainly determined by the degrees of freedom, while the interference is related to the quantization error sin2θ. Therefore, sin2θ in ICS is different from that in JCS, which is denoted as
Similar as (17), is computed as
The system throughput of the ICS with per-cell codebook is
where
For some cell edge user whose single cell gains approximately satisfied αk1 = ⋯ = αkN =αk , (28) and (31) can be simplified as
where
Similar as (23), the feedback-bit allocation between BIc and BPI, with the fixed total number of feedback bits, to maximize the RKICS is given by
3.2 Feedback Overhead and Selection Complexity Analysis
For convenient comparison, we assume Bc = BJc = BIc in this section.
For JCS, each BS need Bc for CDI feedback, besides that, each BS need extra BPI overhead for ICS with PI. Therefore, the feedback overhead for the two schemes are NBc and N(Bc + BPI) in the N cooperation BSs conditions.
According to the codeword selection criterions shown in (6), the quantized vector for each BS is selected from 2Bc codewords and the quantization for N cooperation BSs is independently, so the selection complexity of ICS with PI is N2Bc. In the case of JCS, it has exponential complexity w.r.t. the codebook size as shown in (7). Since there are codeword combinations, the selection complexity of JCS is 2NBc.
Feedback overhead and selection complexity of two codeword selection schemes are summarized in Table 1, and JCS is used as the baseline to calculate the gain of feedback overhead and selection complexity for ICS with PI.
Table 1.Feedback overhead and selection complexity of two codeword selection schemes
Table 2.Example of Feedback overhead and selection complexity of two codeword selection schemes
4. Simulation Results and Analysis
In this section, we compare the system throughput of ICS and JCS via numerical and simulation results. In the evaluation, In the evaluation, path loss is modeled as α = r-θ. r is the distance between the BS and user, and θ is the path loss coefficient (θ = 2 is assumed in this paper). we set nt = 4 and the impact of different value of SNR ρ, the number of cooperation BSs N, the number of bits for PI feedback BPI to the system throughput is considered.
4.1 System Throughput with K=1
Fig. 2 show the comparison between the practical throughput and the theoretical upper bound of system throughput of JCS and ICS with PI feedback. Both the figures are plotted with varying SNR ρ when N = 3 and BJc = BIc = 4. It is observed that the practical throughput is close to the theoretical throughput upper bound both for JCS and ICS. Fig. 3 verifies the comparison under different number of cooperation BSs with ρ = 5 dB and BJc = BIc = 4. They prove the availability of the throughput upper bound.
Fig. 2.Comparison between practical throughput and theoretical throughput upper bound versus SNR when N = 3, BJc = BIc = 4.
Fig. 3.Comparison between practical throughput and theoretical throughput upper bound versus number of cooperation BSs when ρ = 5, BJc = BIc = 4.
The performance of R1ICS with different number of BPI and R1JCS are compared in Fig. 4 when BJc = BIc = 4 and ρ = 5 dB. The results illustrate the benefit of PI feedback well, and shows that the ICS with two bits PI for coordinated BSs can achieve the approximate throughput with JCS.
Fig. 4.Throughput of R1ICS with different number of BPI and R1JCS when nt = 4, BJc = BIc = 4.
With the increase of the number of cooperation BSs, the gap between ICS without PI and JCS becomes larger. This result shows that, in the ICS without PI, the difference on PI of each cooperation BS impedes the throughput increasing. With PI feedback, the throughput of ICS increase greatly, which makes the throughput gain is more obvious when the cooperation BSs numbers increase.
With the fix feedback bits, Fig. 5 depicts the effect of feedback-bit allocation to system throughput when ρ = 5 dB, assuming the same amount of feedback bits for JCS and ICS, i.e. BJc = BIc + BPI = 4. The blue line is throughput of ICS with feedback-bit allocation, which follows the criterion of (23). Via optimal feedback-bit allocation, ICS can greatly enhance the system throughput with no additional feedback overhead, i.e. about 32.5% when N=3, and the improvement will be enhanced when the cooperation BSs’ number increases. However, with the same amount of feedback bits, ICS still cannot surpass JCS.
Fig. 5.Throughput of R1JCS and R1ICS with bits allocation between BIc and BPI when nt= 4, BJc = BIc + BPI = 4.
4.2 System Throughput with K＞1
Corresponding to the situation of K=1, the comparison between the practical throughput and the theoretical throughput upper bound for JCS and ICS with PI feedback are shown in Fig. 6. Both the figures are plotted with varying SNR ρ when N = 3, K=3 and BJc = BIc = 4. Fig. 7 verifies the comparison under different number of cooperation BSs with ρ = 5 dB, K=3 and BJc = BIc = 4. The simulation results demonstrate the validity of the theoretical analysis.
Fig. 6.Comparison between practical throughput and theoretical throughput upper bound versus SNR when N = 3, BJc = BIc = 4, K = 3.
Fig. 7.Comparison between practical throughput and theoretical throughput upper bound versus number of cooperation BSs when ρ = 5, BJc = BIc = 4, K = 3.
In Fig. 8, the performance comparison between RKICS with different number of BPI and RKJCS is given when ρ = 5 dB, BJc = BIc = 4 and K=3. It also shows that the number of the cooperation BSs has the effect on the difference between these schemes. Fig. 9 verifies the effectiveness of bits allocation to BIc and BPI with the criterion shown in (13). The results are derived under the same assumption as Fig.12 except for BJc = BIc + BPI = 4.
Fig. 8.Throughput of RKICS with different number of BPI and RKJCS when nt = 4, BJc = BIc = 4, K = 3 .
Fig. 9.Throughput of RKJCS and RKICS with bits allocation between BIc and BPI when nt = 4, BJc = BIc + BPI = 4, K = 3 .
Since Fig. 8 and Fig. 9 have the same tendency as Fig. 4 and Fig. 5, we can obtain the similar conclusions. JCS is superior in throughput. With the same amount of feedback bits, ICS, employing feedback-bit allocation strategy between CDI and PI, still cannot surpass JCS. Only with the additional PI feedback, ICS will outperform JCS when the number of bits for PI exceed two, as shown in Fig. 4 and Fig. 8. Therefore, the scheme of ICS with PI can be regarded as a tradeoff scheme between JCS and ICS in system throughput.
From theoretical analysis and the simulation results shown in the Fig. 4, Fig. 5, Fig. 8 and Fig. 9, we have the following conclusions.
5. Conclusion
In this paper, we study the codeword selection schemes for per-cell codebook in CoMP limited feedback system. The upper bounds of system throughput achieved by ICS and JCS are analyzed. Several respects of ICS and JCS, including feedback overhead, selection complexity and system throughput, are compared. The theoretical analysis and the simulation results show that JCS is a better choice for system performance and feedback overhead. The ICS has obvious advantage with lower complexity, but it needs additional PI feedback if it obtains the same system throughput of JCS. Under the same number of feedback bits constraint, allocating the number of feedback bits for CDI and PI quantization can increase the system throughput, but ICS is still inferior to JCS.
JCS is a better choice for system performance and feedback overhead but the exponential complexity of codeword selection w.r.t. the codebook size. This disadvantage makes JCS have limitation in the practical application. So we should give some methods to reduce the complexity of JCS. Reducing the size of the per-cell codebook before joint selection shown in (7) is a simple way to solve this problem. The specific criterion for the codebook size reduction can refer to [12], which decreases the selection complexity of JCS greatly with tiny loss on performance and no additional feedback overhead. As to the situation of allowing only low-complexity for codeword selection at UE, ICS with PI can be considered with bit allocation strategy [18].
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