Abstract
Multicell cooperative processing (MCP) has the potential to boost spectral efficiency and improve fairness of cellular systems. However the typical centralised conception for MCP incurs significant infrastructural overheads which increase the system costs and hinder the practical implementation of MCP. In Frequency Division Duplexing systems each user feeds back its Channel State Information (CSI) only to one Base Station (BS). Therefore collaborating BSs need to be interconnected via low-latency backhaul links, and a Control Unit is necessary in order to gather user CSI, perform scheduling, and coordinate transmission. In this paper a new framework is proposed that allows MCP on the downlink while circumventing the aforementioned costly modifications on the existing infrastructure of cellular systems. Each MS feeds back its CSI to all collaborating BSs, and the needed operations of user scheduling and signal processing are performed in a distributed fashion by the involved BSs. Furthermore the proposed framework is shown to be robust against feedback errors when quantized CSI feedback and linear precoding are employed.
base stations and 
mobile stations overall is considered. We consider the case of single-antenna BSs and MSs for simplicity, although our results can be easily generalized to the multiple antenna case. Downlink communication is taken into account, and frequency flat fading is assumed. The received signal of the 
th MS can be described as
represents the channel vector of the 
th user, 
is the vector containing the transmit antenna symbols, and 
is the independent complex circularly symmetric additive Gaussian noise coefficient. An average per antenna power constraint has been considered, 
for 
. It is assumed that the system operates in FDD mode and that each MS 
obtains a perfect estimate of its own channel state 
. In addition, we consider delayless feedback links which are utilised by the MSs in order to feed back their CSI to the system infrastructure. The users feed back limited CSI (quantized CSI) which can be corrupted by errors introduced by the feedback channel.
single antenna BSs are assumed; hence 
MSs are scheduled for transmission in each time slot. The vector of transmit symbols 
is mapped directly to the transmit antennas 
. Therefore the 
th MS receives the following signal when 
is its associated BS:
corresponds to the channel coefficient related to the useful signal and 
corresponds to the detrimental ICI. Thus the Signal to Interference plus Noise Ratio (SINR) of the user 
is
and 
represent the respective power allocation levels. In this paper equal power allocation is considered across MSs for simplicity.
and 
denote the transpose and the Hermitian transpose, respectively. 
stands for the Euclidean norm of a vector, 
denotes the cardinality of a set, 
refers to the 
th element of a matrix diagonal, and 
represents the complex space of 
dimensions. 
denotes the expectation operator, and 
represents the angle between vectors 
and 
.
single-antenna BSs jointly perform linear precoding on the downlink, the BS antennas combine and serve at most 
single-antenna mobile stations simultaneously. The complete channel matrix of the system is
is the channel vector of the 
th MS. Let 
be the set of MSs scheduled to be served in a specific time slot, where 
. Therefore 
is the channel matrix related to these MSs. The vector of transmit symbols 
with power 
, where 
, is mapped to the transmit antennas as follows:
is the precoding matrix of size 
which is a function of the received CSI of the scheduled users, and 
is the beamforming vector corresponding to MS 
. Therefore with linear precoding, the 
th MS, where 
, receives
represents the detrimental ICI. In matrix notation the scheduled users receive
is the received signal vector and 
is a vector of independent complex circularly symmetric additive Gaussian noise components. The SINR 
of the 
th MS is
corresponds to the intercell interference power.
for 
. For equal power allocation 
, where 
is a column vector of 1s with dimension 
, the elements of the power allocation vector must meet the following constraint [
th MS is
for 
, the expression for the power allocation vector (10) reduces to 
.
of the matrix H, due to quantization error (limited digital feedback) and to errors introduced by the feedback channel. The chosen precoding scheme is zero-forcing, where the precoding matrix inverts the imperfect channel matrix describing the received CSI. Hence the precoding matrix is
is a diagonal matrix that normalizes the columns of 
to unit norm. Note that other choices of linear precoding apart from zero-forcing (e.g., MMSE) can be considered [
there is a quantization codebook 
consisting of 
vectors of unit norm, where 
is the number of feedback bits. This codebook is known both by the user and by the scheduling entity. Each MS after obtaining an estimate of its channel vector 
(in this paper we assume a perfect estimate) quantizes its direction 
to the vector from the codebook 
that best approaches it, which is the one leading to the smallest angle separation [
results from the inner product rule. The quantity determining the efficiency of quantization is the quantization error defined as 
. The codebook should be user specific in order to avoid multiple users quantizing their channel direction to the same vector.
as the fed back CQI which does not capture the interuser interference. This is because we are interested in investigating the precoding performance and not the efficiency of scheduling; therefore the consideration of more complex CQI metrics is unnecessary.
and 
.
. The binary indices corresponding to the CSI vectors (lines of 
) are fed back to the scheduling entity through one or several radio channels, depending on the MCP framework, that introduce errors. Therefore the concatenated channel possessed by the scheduling entity is 
, where 
is a function of the errors introduced by the feedback channel.
feeds back a sequence of 
bits, 
, 
. These bits are the index of the vector in the employed codebook 
that best describes the CDI of a user in binary form. It is assumed that each transmitted bit 
is received in error with probability 
. Therefore
estimates the channel vector 
.
quantizes its channel direction 
to 
.
affected by the errors on the feedback channel (Figure 
.
utilises only one radio link to transmit the bit sequence describing its quantized CSI 
, the link to the Master BS. Therefore if each bit faces an independent error probability per feedback link 
(15), the probability that the 
th user's bit sequence 
is received in error by the scheduling entity (CU) is
denotes the binomial coefficient. The bit sequence is in error if at least one of its bits is in error; thus for a specific bit error probability 
, the more the feedback bits are, the more likely an error occurs. From this perspective it is of interest that the codebook size is kept as small as possible. However the smaller the codebook, the less accurately it can approach the actual channel state of the user, which leads to an inferior performance. Therefore a trade-off exists between the needed codebook precision and its size with respect to feedback errors.
; they only possess local CSI which is comprised of submatrices of 
; MCP could be achieved in a decentralised fashion if each involved BS obtained global CSI. Taking this into account, we propose a framework for the downlink whose phases are as follows.
estimates the channel vector 
.
quantizes its channel direction 
to 
.
), and the schedulers end up selecting exactly the same MSs. If feedback links introduce errors, the fed back CSI (CDI and CQI) needs to be protected by the use of appropriate techniques.
utilises for transmission the 
th line of the precoding matrix 
.
of the global CSI 
.
, 
is received in error in at least one of the 
collaborating BSs. This probability is
be the set of the cooperating BSs, where 
. Independent bit error probabilities on each feedback link might result in the selection of different channel vectors by some of the 
base stations for a specific MS. For example it is possible for the received feedback of BS 
that 
, where 
, 
. A set of BSs 
might receive a different CSI index than BS 
. This potential CSI discrepancy can lead to performance degradation of the decentralised framework (degradation on scheduling and precoding design). Therefore a good index for the discrepancy of the possessed CSI between the 
collaborating BSs is the probability of this discrepancy to occur. Consequently this is also an index of the performance gap between the two MCP frameworks. This probability of CSI discrepancy 
is given by the following expression:
links if the same bits in sequence are either correct or in error. In this case there is no CSI discrepancy and the probability for this event to occur if 
base stations collaborate and 
bits are transmitted is 
. Consequently the probability of discrepancy is 
(18). The probability of CSI information discrepancy as a function of the number of feedback bits and the number of independent transmission links can be seen in Figures 
) and 
). It can be seen that the probability of CSI discrepancy heavily relies on the probability of bit error 
. Furthermore, the more the transmitted bits and the cooperating BSs, the more likely a CSI discrepancy occurs. In the physical layer, the bit error probability can be reduced by the use of advanced error correction techniques whereas the number of transmitted bits can be reduced by an intelligent codebook design. CSI discrepancy can also be prevented in the Medium Access Control (MAC) layer, when error detection is enabled, with the appropriate signaling techniques [
in the decentralised framework as a function of the number of independent feedback links (equal to the number of involved BSs) and the number of feedback bits (18). The assumed bit error probability is 
.
in the decentralised framework as a function of the number of independent feedback links (equal to the number of involved BSs) and the number of feedback bits (18). The assumed bit error probability is 
.
th MS and the 
th sector is
is the distance between the 
th MS and the 
th sector, 
is the path-loss exponent and 
the path-loss constant. For the pathloss, the 3GPP Long Term Evolution (LTE) pathloss model has been used:
is the corresponding log-normal coefficient which models large-scale fading (shadowing), 
, and 
is the complex Gaussian coefficient which models small-scale fading, 
. 
is the sector antenna power gain as a function of the angle 
in degrees, also following the LTE evaluation parameters:
obtains a perfect estimate of the channel vector associated to all cooperating BSs (
). This estimate is quantized and then fed back omnidirectionally (CDI and CQI feedback). In the centralised framework each MS's feedback is received only by its Master BS (Figure 
, whereas for the red curves 
. "Cent'' denotes the centralised framework and "Decent'' the decentralised one. It can be seen that with 16 bits we get relatively close to the performance of the perfect CSI case in the absence of feedback errors (black uppermost line).
, whereas for the red curves 
. "Cent'' denotes the centralised framework and "Decent'' the decentralised one. It can be noted that we need at least 20 bits in order to approach the performance of the perfect CSI case in the absence of feedback errors (black uppermost line).
when MSs are scheduled in a round-robin fashion for system SNR of 10 and 20 dB, respectively. When the system SNR is 10 dB, 16 bits are chosen for feedback whereas at 20 dB, 20 feedback bits are considered. This increase in bits is justified by the higher power regime of operation which demands larger codebooks for guaranteeing low interuser interference. It can be seen that feedback errors have a much greater impact in the case of 20 bits which is justified by (16) and (17). Furthermore in the case of 20 bits the performance gap between the decentralised and centralised frameworks is increased, an increase justified by the greater probability of CSI discrepancy (18). Therefore a more intelligent codebook design can lead to a reduced number of feedback bits for the same performance targets and also provide better robustness against feedback errors. Generally, the centralised framework is a little more robust to feedback errors than the decentralised one, although for a bit error probability less than 
the difference is negligible. Furthermore the presented results reflect the case where no error detection/correction schemes have been employed. These schemes can significantly reduce the bit error probability 
; thus, they have the potential to eliminate the impact of feedback errors.
for both the decentralised and the centralised frameworks, when System SNR = 10 dB and 16 bits are employed for quantization. In the decentralised case the CSI is fed back through 3 independent radio links.
for both the decentralised and the centralised frameworks, when System SNR = 20 dB and 20 bits are employed for quantization. In the decentralised case the CSI is fed back through 3 independent radio links.