Coordinated multipoint (CoMP) transmission has been regarded as a potential technology for LTEAdvanced. In frequency division duplexing systems, channel quantization is applied for reporting channel state information (CSI). Considering the dynamic number of cooperation base stations (BSs), asymmetry feature of CoMP channels and high searching complexity, simply increasing the size of the codebook used in traditional multiple antenna systems to quantize the global CSI of CoMP systems directly is infeasible. Percell codebook based channel quantization to quantize local CSI for each BS separately is an effective method. In this paper, the theoretical upper bounds of system throughput are derived for two codeword selection schemes, independent codeword selection (ICS) and joint codeword selection (JCS), respectively. The feedback overhead and selection complexity of these two schemes are analyzed. In the simulation, the system throughput of ICS and JCS is compared. Both analysis and simulation results show that JCS has a better tradeoff between system throughput and feedback overhead. The ICS has obvious advantage in complexity, but it needs additional phase information (PI) feedback for obtaining the approximate system throughput with JCS. Under the same number of feedback bits constraint, allocating the number of bits for channel direction information (CDI) and PI quantization can increase the system throughput, but ICS is still inferior to JCS. Based on theoretical analysis and simulation results, some recommendations are given with regard to the application of each scheme respectively.
1. Introduction
T
o 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 multipoint (CoMP) transmission, a kind of distributed antenna systems
[2]
or named as network multiinput multioutput (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 LTEAdvanced
[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 intercell 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 lowcapacity 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 jointcell codebook approach
[10]
. The jointcell 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 percell codebook method for CSI quantization has drawn much attention, in which each cooperation BS has an independent codebook
[11

17]
. The codebook designed for singlecell MIMO system can be used for percell codebook for CoMP system as well.
When UE uses the percell 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 subcodebooks with smaller size are combined for CDI quantization.
Conversely, the selection complexity of ICS is low, but the phase ambiguity (PA) derived from the percell 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 1bit 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 celledge 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 closedform 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, feedbackbit allocation among users is also needed for multiuser (MU) systems
[21]
.
[22]
concluded that users with higher requested qualityofservice, i.e., lower outage probabilities and higher downlink rates, should use larger shares of the feedback rate. The feedbackbit 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 percell 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 feedbackbit 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 feedbackbit allocation for ICS, are given at last in the paper.
The contributions of the paper are summarized as follows.

▪ We derive the upper bounds of system throughput of two codeword selection schemes, JCS and ICS, both in SU and MU scenarios for CoMP system. The system throughput upper bound is a function of the number of feedback bits for CDI quantization. When PA is considered for ICS, the system throughput upper bound of ICS is also a function of the number of feedback bits for PI quantization.

▪ We compare the throughput of JCS and ICS by Monte Carlo simulation. We also compare JCS and ICS in feedback overhead and selection complexity. While JCS has higher selection complexity, ICS has higher feedback overhead.

▪ According to the feature of JCS and ICS analyzed in this paper, we give some recommendations with regard to the application of each scheme respectively.
This paper gives the upper bounds of system throughput of JCS and ICS. However, for ICS, the solution of feedbackbit allocation by maximizing the system throughput is derived by exhaustively searching. The closeform expression of the number of feedbackbit 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 pseudoinverse 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
n_{t}
antennas, cooperatively serving
K
singleantenna UEs. The nearest BS to the
k
th 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
.
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
Nn_{t}
antennas. Similarly, the channel vector between the virtual super BS and UE
k
(called “global CSI”, denote
h^{k}
in
Fig.1
) with
1×Nn_{t}
dimension is the combination of the single cell channel vectors (called “local CSI”, denote
h^{k}_{1},⋯h^{k}_{i}⋯,h^{k}_{n}
in
Fig.1
). Therefore, the global CSI for UE
k
can be expressed by
where
α^{k}_{i}
(i=1,⋯,N)
is the large scale fading gain including path loss and shadowing component,
h
^{k}_{i}
∈
C
^{1×nt}
is local CSI from the
i
th 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
k
th UE is
where
is the precoding vector for UE
k
,
is the precoding of the
i
th BS with power constraint
, which is obtained according to the limited feedback information (details in Section 2.2).
x_{k}
is the transmitted signal of UE
k
. The second term on the righthand side of (2) is the MU interference, and
n_{k}
is Gaussian noise variable with zero mean and
σ
^{2}
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 ║
h
^{k}_{i}
║ and
are the channel magnitude information and CDI of
h
^{k}_{i}
respectively, that is,
(3) can be rewritten as
where
is an aggregation of the CDI of local CSI,
is the
i
th channel gain of user
k
, and
G
^{k}
=
diag
｛
g^{k}_{1}
,⋯,
g^{k}_{N}
｝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
G
^{k}
, which results in the entries of
being no longer i.i.d..
▪ The ratio between different BSs’ channel gain
g^{k}_{i}／g^{k}_{j}
=
α^{k}_{i}／α^{k}_{j}
(
i
≠
j
) varies frequently and is fluctuant in a large range when UEs move, especially with highspeed, as
α^{k}_{j}
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 Percell 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
α^{k}_{i}
(
i
= 1,⋯,
N
) and the small scale fading channel norm ║
h
^{k}_{i}
║,
i
= 1,⋯,
N
are perfectly obtained at BSs.
The
k
th UE is assumed to have perfect and instantaneous knowledge of
h
^{k}_{i}
,
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 percell 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
ĥ
^{k}_{i}
is the quantized version of
and
= ［
ĥ
^{k}_{1}
,⋯,
ĥ
^{k}_{n}
］, the global CDI obtained at BSs can be reconstructed as
Assume that the percell quantization codebook for each BS is denoted as
C_{i}
(
i
= 1,⋯,
N
), which consists of unit norm vectors
c
_{ij}
(
j
= 1,⋯,2
^{Bc}
) in
C
^{nt×1}
, and
B_{c}
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 (
ĥ
^{k}
_{1}
,
ĥ
^{k}
_{2}
,⋯,
ĥ
^{k}_{N}
) =
Q_{JCS}
(
) follows the criterion of
In the following, we assume that the random vector quantization codebook is used. Therefore, each of the vectors within
C_{i}
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
(2
^{Bc}
,
n_{t}
) is tightly bounded as
In order to facilitate the description, the codebook size for JCS and ICS is denoted as 2
^{BJc}
and 2
^{BIc}
, 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 righthand side of (2). In this sector, all of the superscript
k
on variables in previous sectors is replaced by 1.
For percell 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／σ
^{2}
. 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
ĥ
^{1}
_{1}
,⋯,
ĥ
^{1}
_{N}
have unit norm, we can achieve ║
║ =
.
In order to analyze the system throughput of JCS with percell codebook, we first state a lemma demonstrated in
[11]
, i.e., JCS with percell codebook scheme and jointcell codebook scheme can achieve the same average quantization accuracy with sufficiently large
n_{t}
and finite
N
. Assume that
with unit norm is the combination of the quantized CDIs selected from the jointcell codebook, which consists of 2
^{NBJc}
entries in
Nn_{t}
×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
h
^{1}
. (
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
α
^{1}
_{1}
= ⋯ =
α
^{1}
_{N}
=
α
^{1}
, corresponding to some cell edge UE, (13) can be derived as
since
is chisquare with
n_{t}
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
θ_{i}
(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
θ_{i}
. The phase differences between
θ_{i}
(
i
= 1,⋯,
N
) lower the power of the received signal. Feedback quantization version of
θ_{i}
can reduce the effect of PI. Denote
as the quantization of
θ_{i}
with
B_{PI}
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
θ_{i}
submits the uniform distribution in [0,2
π
]and is quantized uniformly as
, Δ
θ_{i}
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
B_{PI}
bits of PI quantization feedback is
where (
a
) satisfies Jensen’s inequality and
ϕ
(
B_{PI}
) is an increasing function of
B_{PI}
when
B_{PI}
≥ 0. As
ϕ
(
B_{PI}
= 0) = 0 and
ϕ
(
B_{PI}
→ ∞) = 1, the range of
ϕ
(
B_{PI}
) 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
α
^{1}
_{1}
= ⋯ =
α
^{1}
_{N}
=
α
^{1}
, (20) can be simplified as
According to (20), the upper bound of
R
^{1}
_{ICS}
is an increasing function of
ϕ
(
B_{PI}
), which illustrates that with more bits for PI feedback, the system throughput would be higher, and the maximum value of
R
^{1}
_{ICS}
corresponding to
ϕ
(
B_{PI}
) =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 feedbackbit allocation between
B^{I}_{c}
and
B_{PI}
can improve the system throughput. The allocating bits between
B^{I}_{c}
and
B_{PI}
can be formulated as
where
B
is the total feedback bits per UE for feeding back CDI and PI.
Notice:
The
R
^{1}
_{ICS}
is an increasing function of
B^{I}_{c}
, and the feedback bits should be real integer, so the resulting
B^{I}_{c}
is given by ⎾
B^{I}_{c}
⏋ or ⎿
B^{I}_{c}
⏌ (⎾
B^{I}_{c}
⏋ or ⎿
B^{I}_{c}
⏌ denotes the ceiling or the floor of
B^{I}_{c}
). We can get the optimal solution by exhaustive search of all possible compositions of
B
since
B^{I}_{c}
and
B_{PI}
are integer and
B^{I}_{c}
+
B_{PI}
=
B
.
 3.1.2 MU scenario (K＞1)
In this section, the performance of JCS and ICS for
K
＞1 (MUCoMP) is compared. In this paper, we use ZF precoding to reduce the MU interference. Compiling
α^{k}_{i}
║
h
^{k}_{i}
║
ĥ
^{k}_{i}
(
k
= 1,⋯,
K
) into
, the ZF precoding matrix is given by
The precoding for
k
th UE
is the normalized vector of the
k
th column of
In MUCoMP 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,
h
^{k}_{i}
and
are also independent. (
b
) is obtained as
follows a chisquare distribution with (
n_{t}

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 MUCoMP with ZF precoding, the signal is mainly determined by the degrees of freedom, while the interference is related to the quantization error sin
^{2}
θ
. Therefore, sin
^{2}
θ
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 percell codebook is
where
For some cell edge user whose single cell gains approximately satisfied
α^{k}
_{1}
= ⋯ =
α^{k}_{N}
=
α^{k}
, (28) and (31) can be simplified as
where
Similar as (23), the feedbackbit allocation between
B^{I}_{c}
and
B_{PI}
, with the fixed total number of feedback bits, to maximize the
R^{K}_{ICS}
is given by
 3.2 Feedback Overhead and Selection Complexity Analysis
For convenient comparison, we assume
B_{c}
=
B^{J}_{c}
=
B^{I}_{c}
in this section.
For JCS, each BS need
B_{c}
for CDI feedback, besides that, each BS need extra
B_{PI}
overhead for ICS with PI. Therefore, the feedback overhead for the two schemes are
NB_{c}
and
N
(
B_{c}
+
B_{PI}
) in the
N
cooperation BSs conditions.
According to the codeword selection criterions shown in (6), the quantized vector for each BS is selected from 2
^{Bc}
codewords and the quantization for
N
cooperation BSs is independently, so the selection complexity of ICS with PI is
N
2
^{Bc}
. 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 2
^{NBc}
.
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.
Feedback overhead and selection complexity of two codeword selection schemes
Feedback overhead and selection complexity of two codeword selection schemes
Example of Feedback overhead and selection complexity of two codeword selection schemes
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
n_{t}
= 4 and the impact of different value of SNR
ρ
, the number of cooperation BSs
N
, the number of bits for PI feedback
B_{PI}
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
B^{J}_{c}
=
B^{I}_{c}
= 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
B^{J}_{c}
=
B^{I}_{c}
= 4. They prove the availability of the throughput upper bound.
Comparison between practical throughput and theoretical throughput upper bound versus SNR when N = 3, B^{J}_{c} = B^{I}_{c} = 4.
Comparison between practical throughput and theoretical throughput upper bound versus number of cooperation BSs when ρ = 5, B^{J}_{c} = B^{I}_{c} = 4.
The performance of
R
^{1}
_{ICS}
with different number of
B_{PI}
and
R
^{1}
_{JCS}
are compared in
Fig. 4
when
B^{J}_{c}
=
B^{I}_{c}
= 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.
Throughput of R^{1}_{ICS} with different number of B_{PI} and R^{1}_{JCS} when n_{t} = 4, B^{J}_{c} = B^{I}_{c} = 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 feedbackbit allocation to system throughput when
ρ
= 5 dB, assuming the same amount of feedback bits for JCS and ICS, i.e.
B^{J}_{c}
=
B^{I}_{c}
+
B_{PI}
= 4. The blue line is throughput of ICS with feedbackbit allocation, which follows the criterion of (23). Via optimal feedbackbit 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.
Throughput of R^{1}_{JCS} and R^{1}_{ICS} with bits allocation between B^{I}_{c} and B_{PI} when n_{t}= 4, B^{J}_{c} = B^{I}_{c} + B_{PI} = 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
B^{J}_{c}
=
B^{I}_{c}
= 4.
Fig. 7
verifies the comparison under different number of cooperation BSs with
ρ
= 5 dB,
K
=3 and
B^{J}_{c}
=
B^{I}_{c}
= 4. The simulation results demonstrate the validity of the theoretical analysis.
Comparison between practical throughput and theoretical throughput upper bound versus SNR when N = 3, B^{J}_{c} = B^{I}_{c} = 4, K = 3.
Comparison between practical throughput and theoretical throughput upper bound versus number of cooperation BSs when ρ = 5, B^{J}_{c} = B^{I}_{c} = 4, K = 3.
In
Fig. 8
, the performance comparison between
R^{K}_{ICS}
with different number of
B_{PI}
and
R^{K}_{JCS}
is given when
ρ
= 5 dB,
B^{J}_{c}
=
B^{I}_{c}
= 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
B^{I}_{c}
and
B_{PI}
with the criterion shown in (13). The results are derived under the same assumption as Fig.12 except for
B^{J}_{c}
=
B^{I}_{c}
+
B_{PI}
= 4.
Throughput of R^{K}_{ICS} with different number of B_{PI} and R^{K}_{JCS} when n_{t} = 4, B^{J}_{c} = B^{I}_{c} = 4, K = 3 .
Throughput of R^{K}_{JCS} and R^{K}_{ICS} with bits allocation between B^{I}_{c} and B_{PI} when n_{t} = 4, B^{J}_{c} = B^{I}_{c} + B_{PI} = 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 feedbackbit 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.

▪ By comparing the criterion of (6) and (7), the ICS obviously has lower complexity of codeword selection, but its system throughput is poor without PI feedback. Therefore, ICS with PI can be seen as a tradeoff scheme between ICS and JCS in system throughput.

▪ With the same amount of feedback bits, the system throughput obtained by JCS is much higher. The system throughput of ICS will be increased by allocating feedback bits betweenBIcandBPI, but ICS is still inferior to JCS, as shown inFig. 5andFig. 9.

▪ With additional feedback bits of PI feedback, ICS can significantly improve the system throughput. With the same codebook size for CDI quantization, the system throughput of ICS with PI feedback will outperform JCS when the feedback bits of PI exceed two bits for coordinated BSs, as shown inFig. 4andFig. 8. However, the PI feedback will increase feedback overhead with the number of coordinated BSs linearly.

▪ Without considering of the selection complexity, JCS has a better tradeoff between system throughput and feedback overhead.
5. Conclusion
In this paper, we study the codeword selection schemes for percell 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 percell 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 lowcomplexity for codeword selection at UE, ICS with PI can be considered with bit allocation strategy
[18]
.
BIO
Zhirui Hu is currently pursuing the Ph.D. degree in Communication and Information Systems at Beijing University of Posts and Telecommunications (BUPT). Her current research field is in the areas of wireless communications, multipleantenna technology, cooperative communications and signal processing technology.
Chunyan Feng received the B.S. degree in Communications Engineering, the M.S. and Ph.D. degrees in Communication and Information Systems, all from Beijing University of Posts and Telecommunications (BUPT), Beijing, China. She is currently a Professor with the School of Information and Communication Engineering, BUPT. Her research interests are in the areas of broadband networks and wireless communication systems. Current research focuses on cognitive radio, key technology of B3G/4G systems, and green wireless communications.
Tiankui Zhang BSc(Eng), PhD is a associate professor at the School of Informationand Communication Engineering of Beijing University of Posts and Telecoms(BUPT). He received his BEng and PhD degrees from BUPT, in the areas of Wireless Telecommunications in the years 2003 and 2008 respectively. His research field is in the areas of next generation wireless networks with particular focus on radio resource management and green radio.
Qiubin Gao received his B.S. and Ph.D. in control science and engineering from Tsinghua University. He is currently a senior research engineer at Datang Wireless Mobile Innovation Center of China Academy of Telecommunication Technology (CATT). His current research interests include physical layer design for mobile communication, multipleantenna technology, CoMP, and system performance evaluation. He is inventor /coinventor of more than 100 patents in wireless communications, and author/coauthor of a number journal and conference papers.
Shaohui Sun received a B.S. with auto control engineering and M.S. with computer engineering from Xidian University in 1994 and 1999, respectively, and a Ph.D. in communication and information system from Xidian University in 2003. He has been deeply involved in the development and standardization of LTE/LTEAdvanced since 2005. His research area of interest includes multipleantenna technology, heterogeneous wireless network and relay.
Rusek F.
,
Persson D.
,
Lau B. K.
,
Larsson E. G.
,
Marzetta T. L.
,
Edfors O.
,
Tufvesson F.
2013
“Scaling up MIMO: opportunities and challenges with very large arrays”
IEEE Signal Processing Magazine
Article (CrossRef Link).
30
(1)
40 
46
DOI : 10.1109/MSP.2011.2178495
You X. H.
,
Wang D. M.
,
Sheng B.
,
Gao X. Q.
,
Zhao X. S.
,
Chen M.
2010
“Cooperative distributed antenna systems for mobile communications [Coordinated and Distributed MIMO]”
IEEE on Wireless Communications
Article (CrossRef Link).
17
(3)
35 
43
DOI : 10.1109/MWC.2010.5490977
Sawahashi M.
,
Kishiyama Y.
,
Morimoto A.
,
Nishikawa D.
,
Tano M.
2010
“Coordinated multipoint transmission/reception techniques for LTEadvanced [Coordinated and Distributed MIMO]”
IEEE on Wireless Communications
Article (CrossRef Link).
17
(3)
26 
34
DOI : 10.1109/MWC.2010.5490976
Lee D.
,
Seo H.
,
Clerckx B.
,
Hardouin E.
,
Mazzarese D.
,
Nagata S.
,
Sayana K.
2012
“Coordinated multipoint transmission and reception in LTEadvanced: deployment scenarios and operational challenges”
IEEE Commu. Mag.
Article (CrossRef Link).
50
(2)
148 
155
DOI : 10.1109/MCOM.2012.6146494
Kim Tae Min
,
Sun Fan
,
Paulraj A. J.
2013
“LowComplexity MMSE precoding for coordinated multipoint with perantenna power constraint”
IEEE Signal Processing Letters
Article (CrossRef Link).
20
(4)
395 
398
DOI : 10.1109/LSP.2013.2246152
Sun Fan
,
De Carvalho. E
2012
“A leakagebased MMSE beamforming design for a MIMO interference channel”
IEEE Signal Processing Letters
Article (CrossRef Link).
19
(6)
368 
371
DOI : 10.1109/LSP.2012.2196040
Rantelobo K.
,
Hendrantoro G.
,
Affandi A.
,
Zhao H. A.
2013
“Adaptive combined scalable video coding over MIMOOFDM systems using partial channel state information”
KSII Transactions on Internet and Information Systems
Article (CrossRef Link).
7
(12)
3200 
3219
Love D. J.
,
Heath R. W.
,
Lau V. K. N.
,
Gesbert D.
,
Rao B. D.
,
Andrew M.
2008
“An overview of limited feedback in wireless communication systems”
IEEE Journal Select Areas Communications
Article (CrossRef Link).
26
1341 
1365
DOI : 10.1109/JSAC.2008.081002
Su D.
,
Hou X. Y.
,
Yang C. Y.
2011
“Quantization based on percell codebook in cooperative multicell systems”
in Proc. of IEEE Wireless Communications and Networking Conference (WCNC)
March
Article (CrossRef Link).
1753 
1758
Kim J. H.
,
Zirwas W.
,
Haardt M.
2008
“Efficient feedback via subspacebased channel quantization for distributed cooperative antenna systems with temporally correlated channels”
EURASIP J. Adv. Signal Process
Article (CrossRef Link).
2008
(2)
1 
13
Cheng Y.
,
Lau V. K. N.
,
Long Y.
2010
“A scalable limited feedback design for network MIMO using percell product codebook”
IEEE Transactions on Wireless Communications
Article (CrossRef Link).
9
(10)
3093 
3099
DOI : 10.1109/TWC.2010.082110.091189
Hu Z. R.
,
Zhang T. K.
,
Feng C. Y.
2013
“Study on codeword selection for percell codebook with limited feedback in CoMP systems”
in Proc. of IEEE Wireless Communications and Networking Conference (WCNC)
April
Article (CrossRef Link).
3140 
3145
Zeng E.
,
Zhu S.
,
Xu M.
2008
“Impact of limited feedback on multiple relay zeroforcing precoding systems”
IEEE International Conference on Communications
May
Article (CrossRef Link).
4992 
4997
Zeng E.
,
Zhu S.
,
Liao X.
,
Zhong Z.
2008
“Impact of limited feedback on the performance of MIMO macrodiversity transmission”
in Proc. of IEEE Wireless Communications and Networking Conference (WCNC)
March
Article (CrossRef Link).
672 
677
Yuan F.
,
Yang C. Y.
2011
“Phase ambiguity quantization for percell codebook based limited feedback coordinated multipoint transmission systems”
in Proc. of IEEE Vehicular Technology Conference (VTC)
Article (CrossRef Link).
Hassan M. H.
,
Fahmy Y. A.
,
Khairy M. M.
2012
“Phase ambiguity mitigation for percell codebook based limited feedback coordinated multipoint transmission systems”
IET Communications
Article (CrossRef Link).
6
(15)
2378 
2386
DOI : 10.1049/ietcom.2012.0132
Su D.
,
Yang C. Y.
2011
“Necessity of phase ambiguity quantization for limited feedback coordinated multipoint transmission”
IEEE Vehicular Technology Conference (VTC)
September
Article (CrossRef Link).
Yuan F.
,
Yang C. Y.
2012
“Bit allocation between perCell codebook and phase ambiguity quantization for limited feedback coordinated multipoint transmission systems”
IEEE Transactions on Communications
Article (CrossRef Link).
60
(9)
2546 
2559
DOI : 10.1109/TCOMM.2012.071312.110510
Yu S.
,
Kong H. B.
,
Kim Y. T.
,
Park S. H.
,
Lee I.
2012
“Novel feedback bit allocation methods for multicell joint processing systems”
IEEE Transactions on Wireless Communications
Article (CrossRef Link).
11
(9)
3030 
3036
DOI : 10.1109/TWC.2012.062012.111169
Wu Y. L.
,
Ding M.
,
Zou J.
,
Li X. N.
2011
“Efficient limited feedback for MIMORelay systems”
IEEE Communication Letters
Article (CrossRef Link).
15
(2)
Xu X. R.
,
Yao Y. D.
,
Hu S. Q.
,
Yao Y. B.
2013
“Joint subcarrier and bit allocation for secondary user with primary users’ cooperation”
KSII Transactions on Internet and Information Systems
Article (CrossRef Link).
7
(12)
3037 
3054
Khoshnevis B.
,
Wei Y.
2012
“Bit allocation laws for multiantenna channel feedback quantization: multiuser case”
IEEE Transactions on Signal Processing
Article (CrossRef Link).
60
(1)
367 
382
DOI : 10.1109/TSP.2011.2169250
Park E.
,
Kim H.
,
Park H.
,
Lee I.
2013
“Feedback bit allocation schemes for multiuser distributed antenna systems”
IEEE Communication Letters
Article (CrossRef Link).
17
(1)
DOI : 10.1109/LCOMM.2012.120612.122138
AuYeung C.
,
Love D. J.
2007
“On the performance of random vector quantization limited feedback beamforming in a MISO system”
IEEE Transactions on Wireless Communications
Article (CrossRef Link).
6
(2)
458 
462
DOI : 10.1109/TWC.2007.05351
Han S. Q.
,
Yang C. Y.
2011
“Downlink multicell cooperative transmission with imperfect CSI sharing”
in Proc. of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
Article (CrossRef Link).
3024 
3027
Zhang J.
,
Heath R. W.
,
Kountourix M.
,
Andrews J. G.
2009
“Mode switching for the multiantenna broadcast channel based on delay and channel quantization”
EURASIP Journal on Advances in Signal Processing
Article (CrossRef Link).
2009
(1)
1 
15
DOI : 10.1155/2009/802548
Yoo T.
,
Jindal N.
,
Goldsmith A.
2007
“Multiantenna downlink channels with limited feedback and user selection”
IEEE Journal Select Areas Communications
Article (CrossRef Link).
25
(8)
1478 
1491
DOI : 10.1109/JSAC.2007.070920