Design and measurement-based evaluations of coherent JT CoMP: a study of precoding, user grouping and resource allocation using predicted CSI

1.1 Contributions

We investigate and develop a transmit strategy for coherent JT CoMP by a step-by-step evaluation of its various components and interactions, leading to the following main conclusions and results.

First, one issue with CoMP is that significant coordination delays over backhaul links might eliminate the potential for CoMP gains. We show that channel prediction enables large average performance gains when using linear coherent joint transmission at pedestrian velocities for total delays of over 20 ms at 2.66 GHz. For lower delays, the same conclusion holds for higher-mobility users. CoMP would, e.g. remain possible at 500 MHz carrier frequencies for velocities up to 120 km/h, if the total delays are 5 ms.

Second, two parts of a JT CoMP design that are crucial for the average performance gains are the means for resource allocation over frequency-selective OFDM downlinks and the user grouping, i.e. the formation of groups of users who will share a particular time-frequency resource block.

We here introduce and evaluate a user grouping scheme with very low complexity, ‘User groups provided by cellular scheduling’. This user grouping strategy is based on local scheduling in the base stations, and it can (but does not have to) utilize already existing scheduling algorithms. In many papers with 2 to 3 base stations and single-carrier transmission, the authors have intuitively used a user grouping scheme similar to this, often with all users placed at the same distance to their nearest base station site. However, to the best of our knowledge, this has never been compared with other schemes nor is it usually motivated by the authors using it. At much lower complexity than, e.g. greedy user selection, this strategy provides spatially good (although not optimal) user groups that improve the sum rate performance when using linear precoding. It preserves multiuser diversity gains and also requires less feedback and less backhaul capacity than alternative strategies proposed previously. For systems with many users, the backhaul demand for transmission control can even be significantly lower than that for JS/JB CoMP. Using this scheme, JT CoMP can improve the sum capacity for essentially all investigated combinations of user positions. On average over random sets of user positions, it is increased by up to 54% as compared to cellular transmission, with imperfect CSIT at full system load.

Third, a main mechanism behind the sometimes disappointing performance of JT CoMP is highlighted: The different distances involved from sets of transmitters to the different receivers will often generate hard-to-invert joint channel matrices. This results in precoders with large differences in the scaling of their elements. A joint linear precoding design under a per-antenna power constraint is then forced to reduce the transmit powers of the closest base station to a user far below the allowed power to obtain a balanced solution. This effect reduces the total transmit power for a cluster of transmitters that participate in joint transmission, often with the result that out-of-cluster interference and noise reduce performance below that of single-cell transmission. The proposed user grouping strategy alleviates this problem.

Finally, since the CSIT is uncertain, robust techniques for joint precoder design are of interest. The robust linear precoder (RLP) design, introduced in [9], is here investigated further and is developed into a versatile tool for design of linear joint precoders. Robust design is most easily performed for mean square error (MSE) criteria. The RLP is here designed to optimize more general criteria by using a low-dimensional iteration over weighting matrices in a closed-form robust precoder design. We here provide sufficient conditions for the closed-form robust design to minimize a weighted sum of intracluster interference and transmit powers under imperfect CSIT accuracy for known second-order moments of the statistical uncertainties. We also show that imperfect CSIT due to quantization is straightforwardly included into the design. We investigate under what conditions a robust JT design provides benefits by comparing to a simple zero-forcing (ZF) design. Also, we observe that the interplay between channel prediction errors, opportunistic scheduling and precoder design increases the multiuser scheduling gain when using CoMP, relative to single-cell transmission.

These results, taken together, in our opinion indicate that large performance gains are indeed possible by using linear JT CoMP techniques that can be designed with reasonable computational complexity.

1.2 Assumptions, design choices and related work

The potential for coherent JT CoMP was shown in [10] to be highest for low-mobility users, as compared to joint scheduling and to the use of noncoherent JT CoMP. We therefore here focus on coherent JT CoMP, also referred to as network multiple-input multiple-output (MIMO) or multi-cell MIMO (see, e.g. [5, 6, 11, 12]), for low-mobility users.

Although, the largest gains are achieved with nonlinear precoding techniques such as dirty paper coding [6], complexity currently makes nonlinear precoding unfeasible for most realistic systems. We here focus on a low-complexity linear precoding solution. Zero-forcing linear precoders [13] are here a frequently studied alternative.

Coordination over a very wide area would provide the highest performance, but would be unrealistic due to computational complexity, delay constraints and capacity constraints in the fixed network. Therefore, we consider the use of CoMP within limited coordinated sets (clusters) of N transmitters distributed over N _B cells. In cellular transmission, the transmitters belonging to each cell are coordinated, but they are uncoordinated to the transmission in other cells. In CoMP that uses clustered joint transmission, the aim is to suppress the intracluster interference when jointly transmitting to M _g users. With perfect CSIT, the intracluster interference can then be eliminated by phase cancellation when N≥M _g .

The cluster size, i.e. the number of cooperating cells per cluster, involves a trade-off. A larger size ideally provides larger gains relative to cellular transmission, since a lower fraction of users are then located at cluster edges, but introduces a higher computational burden. Investigations in [11, 14] show that a cluster size above 7 to 9 cells will not provide large additional gains for systems with MIMO links. In [15], for few base station antennas, a cluster that used transmitters at three separate sites was adequate to attain most of the achievable CoMP gains (see also [16]). Our evaluations in Sections 6 and 7 focus on a cluster size of three sites, partially motivated by the results of [15] and partially due to the limitations of our measurements.

An important aspect is to limit the remaining intercluster interference. An interesting scheme proposed in [14] and further evaluated in [17] uses cluster-specific antenna tilting and power control for this purpose. We have in our investigations adjusted the interference statistics to approximate the one that would be generated by the scheme of [14].

Near accurate CSIT is important for multi-user MIMO [18] and for coherent JT CoMP [19]. We here evaluate schemes under the imperfect CSIT that would be due to the main unavoidable causes: noisy estimates and outdated CSIT due to signaling delays. Users are assumed to move at pedestrian velocities at 2.66 GHz. This setting results in large channel estimation errors due to outdating when channel prediction is not used. It has previously not been clear if the use of channel prediction helps CoMP performance in a significant way. Promising results based on simulations were reported in [19], using adaptive recursive least squares prediction. A preliminary simulation study in [20] investigated a two-user, two-cell scenario. The recent paper [21] investigated this question theoretically, in the limit of large numbers of antennas per base station, but did not use a per-base station transmit power constraint, so it is hard to draw conclusions from these results.

Channel predictors are here assumed to be located in the user terminals. They report the predictions to their strongest base station. The base stations then transmit the reports over a backhaul link to a central control unit (CU) for the cluster which jointly designs the beamformers.

Kalman prediction of MIMO OFDM channels, outlined in Section 3 and Appendix 1 has been investigated in, e.g. [22, 23]. We here investigate its use in a CoMP setting, focusing on two requirements that are peculiar to this setting: (1) Transmit antennas located at different sites will be at different distances while their channels, with differing signal-to-interference-and-noise ratio (SINR), have to be estimated jointly. The weakest signals will in general be estimated with the lowest accuracy. The effects of this on the choice of pilots, the resulting precoder matrices and capacity performance need to be understood. (2) Channels may need to be predicted over long prediction horizons, due to the coordination delays.

Since significant model errors will be present, the precoder (the set of joint beamformers) should furthermore be designed to be robust with respect to (w.r.t.) the expected errors. Implementation without unrealistic computational complexity is here in focus, so we will restrict attention to linear precoders. We mainly use a versatile scheme with reasonable design complexity, the iteratively adjusted RLP introduced in [9] and further developed in Section 5 and in Appendix 2. This averaged robust design is used since it is less conservative than the minimax schemes in, e.g. [24, 25]. A useful property of the RLP is that the channel uncertainty in the form of covariance matrices that are provided by Kalman predictors can be directly used in its adjustment.

In the optimization of a criterion such as the weighted sum capacity for the involved terminals, the RLP design utilizes the analytical solution to an MSE-optimal linear robust precoder and iteratively optimizes over criterion weights used by this design. This MSE-optimal analytical solution constitutes a special case of robust feedforward control filters for dynamic (frequency-selective) systems, previously developed in [26–28]. Robust linear precoders that minimize MSE by averaging over CSIT uncertainty have more recently been highlighted for multiple-input single-output (MISO) transmit schemes by [29, 30] and for multiuser and MIMO downlinks in [24, 31]. Very few solutions have been proposed for robust linear precoder design for more general performance criteria.

Many proposals form user groups for CoMP, as, e.g. [32, 33], by first forming the user group and then allocating it to a transmission resource. This can provide groups with spatially compatible users, but may sacrifice some of the potential multiuser scheduling gain, since the frequency-domain variability of channels to users is not taken into account. Another approach is to use a greedy algorithm as in, e.g. [34–36] that assigns one user at a time to a given resource, forming a near-optimal solution both in terms of spatially compatible users and exploiting multiuser diversity. This, however, requires repeated pre-evaluation of beamformers, resulting in a high complexity. Greedy user grouping will in Section 7 be compared to the user grouping scheme we propose, but due to high complexity, we use a block-fading model rather than the whole measured channel statistics for this particular comparison.

Notations

In the following, $\mathit{Ē}\left[·\right]$ averages over the distribution of channel model errors, E[·] averages over the statistics of noise and message symbols, ∥·∥ denotes the 2-norm of a vector, tr(·) is the trace of a matrix, Re(·), (·) ^T and (·)^∗ denote the real part, the transpose and the Hermitian transpose of a matrix, respectively. The unit matrix is denoted I. For simplicity, we shall enumerate the users such that users {1,…,M _g } are in the selected user group for the subcarriers considered. The Kronecker delta function is denoted δ _{i j} . Unless otherwise explicitly stated, (·) _{j n} denotes element (j,n) and (·) _j denotes column j of a matrix or the j th element of a vector. The indices i and m are user indices, j and n are base station indices, t and τ are time indices and k and q are subcarrier indices. We shall denote the base station that, on average over all subcarriers and over the small-scale fading, has the strongest channel gain to a user as that user’s master base station.

3.1 Short-term fading models

for Kalman predictor number $q=0,\dots ,⌊\frac{K_{\text{CRS}}-1}{w}⌋$ where K_CRS is the number of RS-bearing subcarriers. Note that the superscript index q w,q w + 1… in (4) represents a frequency spacing of Δ f, while k in (1) represents a frequency spacing of Δ f/n_CRS where n_CRS is the RS spacing in number of subcarriers. The prediction accuracy can be improved by increasing the number w of subcarriers that are predicted jointly, by averaging the noise. However, this comes at a cost of higher computational complexity which grows as $\mathcal{O}\left(w^3\right)$[22].

The matrices A, B, C and the covariance matrix Q can be updated based on past channel estimates at an interval that is related to the time constant of the shadow fading (see [23] and chapter 4 of [22]).

3.2 Kalman predictor

Please see Appendix 1 for further aspects on the filter design.

As mentioned above there is a trade-off in the choice of the number w of subcarriers estimated by each Kalman filter. We here keep this parameter low and, in a second step, reduce prediction errors further by Wiener smoothing over estimates for all subcarriers. The true prediction error covariances then differ from those of (7) due to two effects. First, the AR models (3) are imperfect which increases the errors. Second, Wiener smoothing over frequency decreases the errors. In our studies, these two effects leave the variance of the prediction error slightly less than that given by (7). The use of the accurate covariance instead of (7) would cause only minor noticeable difference in precoder performance and only for systems with very low noise power. We shall therefore use (7) in the precoder design in Section 5.

4.1 User groups provided by cellular scheduling (CUG)

This is our main proposed strategy to create diagonal-dominant channel matrices that then become relatively easy to invert in the CoMP precoder design. We first present this scheme, denoted as cellular user grouping (CUG), for single antenna base stations. All users with the same master base station are then locally scheduled on orthogonal subcarriers by a scheduler connected to their master base station, as shown in the example in Figure 2. This scheduling is based on a CQI metric. For the schedulers explored in this paper, the CQI for user i at resource block b, CQI_b,i, is given by the average estimated channel gains from all antennas at that user’s master base station.

On each resource block, the scheduled M _g ≤N users within the cooperation cluster (with equality if each base station is the master base station of at least one user) will then form a CoMP group. These users, which all belong to different cells, are to be served jointly by all base stations in the cluster, including base stations that are not the master base station of any of these users. The full CSIT used in the precoder design then only needs to be fed back and transmitted over backhaul by the users that have been scheduled and only for a scheduled resource. Two-step feedback approaches such as this have been investigated in [39] for multiuser MIMO and in [40] for CoMP.

Here > denotes a logical comparison resulting in 1 if true and 0 otherwise. The user with the highest score will be allocated to the resource block l. The use of score-based scheduling to create the user grouping will be denoted SB-CUG.

with ${\mathcal{P}}_{j_{\text{mast}:i},max}$ being the power constraint for the antennas of the master base station of user i. It is denoted best rate CUG (BR-CUG). The use of this metric to compare attainable rates presupposes that a well-functioning CoMP scheme will suppress intracluster interference.

For multiantenna base stations with N _A antennas, cellular scheduling proceeds similarly but may allocate up to N _A users per frequency resource and base station, using cell-specific beamforming.

4.2 Greedy user grouping (GUG)

Here, P_S,i, P_I,i and P_N,i are the powers of the signals, the interference and the scalar noise powers at the receiver antenna i = 1,..,M _g , respectively. Calculations of the expected values of the powers based on the prediction error statistics is discussed in Appendix 3. If α _i  = 1 for all i the sum rate is maximized. We shall denote this GUG with best rate (GUG-BR). If instead ${\alpha }_i=1/\overline{r_i}$ with $\overline{r_i}$ being the average throughput of user i over already scheduled resources, we get a proportional fair scheduler [42], which will be denoted GUG with proportional fair scheduling (GUG-PF).

5.1 Target system

The system model used for precoder design is shown in Figure 3. Here, $u\in {\mathbb{C}}^{N\times 1}$ is the transmit signal vector, and $z=\frac{1}{c}\mathit{\text{Ds}}\in {\mathbb{C}}^{M_g\times 1}$ is the desired received vector. Its desired properties are modeled by a target matrix D which is diagonal, representing the ideal of a complete interference suppression. In a generalization to multiple receiver antennas, D would be block-diagonal. The distances between terminals and transmitters will differ substantially in a CoMP setting. It would therefore be unrealistic to demand equal received power at all users by setting D = I. Instead, the targeted received signal magnitudes (the diagonal elements of D) should be set to realistically attainable levels. This can be done in different ways. We here adjust the targeted received signal magnitudes to the amplitude of the strongest channel for each user

$D_{\mathit{\text{ii}}}=\underset{j}{\max }\left|{\mathit{Ĥ}}_{\mathit{\text{ij}}}\right|,i=1,\dots ,M_g.$

(12)

This is a very simple way of choosing D. For channel matrices with a dominant diagonal, which often appear, e.g. if all users in a CoMP group have different master base stations, (12) provides a sum rate close to the sum rate that is obtained if D is optimized.

Alternatively, in [43] all users are given the same fraction of the transmit power in combination with zero-forcing precoding. This corresponds to an alternative strategy for adjusting the diagonal elements of D. We have investigated both that alternative and numerical optimization of D with respect to the sum rate. We then found little differences in the end result as compared with the use of (12). (However the use of D = I, which is commonly used in zero-forcing precoders for single-cell multiuser MIMO problems, would cause a large loss in system performance in CoMP settings).

5.2 Robust linear precoder (RLP)

Here, V is a diagonal positive definite matrix and S is a positive semidefinite matrix, both real-valued. The use of these weighting matrices in the design is discussed in Sections 5.2.1 to 5.2.3 below.

6.1 Channel measurements

6.2 Simulation method and assumptions

To simulate velocities of pedestrian users, and to make the model more 3GPP-LTE like, the data has been upsampled 25 times in time resulting in the parameters presented in the right-hand column of Table 1. The upsampling is done using the fast Fourier transform to ensure that no extra frequency components are added.

In the present investigation, we have focused only on the prediction error part in the error model (2).

6.3 Prediction performance

For JT CoMP, assume that an interfering scalar complex-valued channel is given by $g=ĝ+\mathrm{\Delta g}$, with $ĝ$ known, $\mathit{Ē}\left[\mathrm{\Delta g}\right]=0$, $\mathit{Ē}\left[\mathrm{ĝ\Delta }{g}^{\ast }\right]=0$ and an NMSE $\mathit{Ē}\left[{\left|\mathrm{\Delta g}\right|}^2\right]/\mathit{Ē}\left[{\left|g\right|}^2\right]$. If this interference is to be canceled by receiving another channel component h, from another base station, then the resulting interference power $\mathit{Ē}\left[{\left|g+h\right|}^2\right]$ is minimized by setting $h=-ĝ$ resulting in $\mathit{Ē}\left[{\left|g+h\right|}^2\right]=\mathit{Ē}\left[{\left|\mathrm{\Delta g}\right|}^2\right]$. Therefore, the maximum attainable relative dampening factor would become $\mathit{Ē}\left[{\left|g\right|}^2\right]/\mathit{Ē}\left[{\left|\mathrm{\Delta g}\right|}^2\right]$. Hence, a channel error with an NMSE of -x dB indicates that we can reduce the interference from that base station by at most x dB. For example, at a prediction horizon of ϑ=18, the interference from the weakest base station at a given user can on average only be suppressed by 3 to 5 dB. The prediction performance of the weakest base station is far below that of the average performance over all base stations. These poor predictions might become ‘bad apples’ that infect the quality of the total precoding solutions.

A closer study of the effect of using different noise floors and RS SNRs is shown in Figures 4 and 5. As expected, a low noise floor increases the prediction performance. The impact of the RS SNR is largest at short prediction horizons. This is because at long prediction horizons the fading statistics, rather than the noise, is the main limiting factor of the prediction performance, as also discussed in [22].

6.4 Precoding performance

With RUG-RR, SC transmission outperforms RLP for 34% of the groups. For 17% of the groups, the per-cell sum rate is more than 1 bps/Hz/cell higher for SC transmission. With cellular user grouping combined with score-based scheduling (SB-CUG), these numbers decrease to 7% and 0.6%, respectively. The improvement is due to better conditioned 3×3 channel matrices H resulting in the need for on average smaller power scaling factors c in (10). These results indicate that even with few users to choose from in the system, local scheduling will provide good user groups for CoMP. This phenomenon will be further validated in Section 7.

With SB-CUG for M = 9 users, CoMP improves the average sum rate by 54% as compared to SC transmission. For the worst combinations of positions of scheduled users (the 5% percentile), the sum rate improves by 47%.

It is seen from Figure 6 that the highest sum rate gains from using CoMP are achieved when the noise floor is low. The system is then intracluster interference limited. The performance for ZF with perfect CSIT has been added for comparison. As the noise floor decreases, the gap between ZF with perfect CSIT and ZF with predicted CSIT increases. For low noise floors, RLP does not outperform ZF since RLP can only compensate for inaccurate CSIT by allocating transmit power over the more reliable channels, but it cannot compensate for the actual phase errors in the CSIT. As the noise floor decreases, and the channels become more accurate as a result (see Table 2), it therefore cannot perform better than ZF, even for the tough user groups.We now in Figure 6 compare ZF, RLP and ZF with perfect CSIT in the case with a noise floor of -110 dBm using RUG-RR. ZF with perfect CSIT then performs worse than RLP with predicted channels, which may seem surprising. However, as mentioned, the regularizing third term in the inverse in (14) affects the power allocation such that more power is transmitted over accurate channels than over very inaccurate channels. Since generally the most accurate channels are also the strongest channels, the power allocation is automatically better than that of the ZF solution, even when ZF uses perfect CSIT.

All investigated scenarios above suggest that using SB-CUG instead of RUG-RR is especially important for ZF precoding. User grouping based on cellular scheduling increases the average sum rate performance of ZF precoding so that it becomes equal to that of RLP. The 5% percentile sum rate is increased by up to a factor 6.7. This is because SB-CUG generates well-conditioned matrices. The channel errors from the weak base stations will then have less effect on the final solution. This is most evident in the lowest percentiles, since these include the user groups with the largest channel errors.

It is noticeable, from Table 3 and Figure 6, that with SB-CUG, ZF sometimes outperforms RLP. In our studies, we have seen that this is due to the approximations made when calculating $\mathit{Ē}\left[\Delta H^{\ast }{V}^{\ast }\mathrm{V\Delta H}\right]$ in (14) by using (7), (15) and (16). This overestimates the variance of the prediction error as discussed in Section 3.2. RLP then becomes overly cautious, yielding a slightly worse solution. However, these effects are small and only noticeable at the lowest noise floor.

In all the above, we have assumed that the quantization error is small compared to the prediction error and therefore negligible. As the prediction errors are mostly in the regions of over -20 dB, a feedback cost of 8 to 10 bits per complex-valued scalar channel would ensure this. With an adaptive quantization scheme, the poor channels might only need 4 to 6 bits per complex-valued scalar channel for the quantization error to be negligible compared with the prediction error, so the feedback cost can then be lowered. The overhead required to notify the base station on how many bits each channel require is low, as this relates to the shadow fading and only needs to be fed back on a slow varying time scale, related to the shadow fading.

An idea of how a nonnegligible adaptive quantization error would affect the results can be gained by studying the performance differences between different noise floors. The higher noise floors lead to less accurate predictions, and quantization errors would amplify this effect. However, with a fixed quantization granularity, the size of the quantization error would be independent of the channel prediction quality. Then, in the presence of nonnegligible quantization errors, other effects might occur, which are not present in the results presented her. This is a topic of importance, which will be left for future studies.

7.1 Results

Comparisons between all the user grouping and scheduling schemes described in Section 4, as well as RUG-RR are presented in terms of sum rate (Figure 7) and average user rate (Figure 8) for M = 9 users. Note that the CUG scheme performs close to the much more complex GUG algorithm both for the near sum rate optimal groups, comparing GUG-BR with BR-CUG and for the ‘fair’ user groups, comparing GUG-PF with SB-CUG. Both GUG-BR and BR-CUG also perform close to the sum rate optimal user grouping obtained by exhaustive search. In terms of the lowest percentiles of the average user rates for the fair algorithms, GUG-PF is more fair than SB-CUG. This can be explained by the SB-CUG being restricted to allocating resources fair amongst users in the same cell. Therefore, when the users are unevenly distributed, e.g. when 80% of the users belong to the same master base station, then these users will be allocated to less resources than the other 20% of the users. The low percentiles of SB-CUG are still much better than those obtained with RUG-RR and with the sum rate optimal user grouping algorithms. In Figure 9, we see that the multiuser scheduling gain for the BR-CUG algorithm is on level with that of the sum rate optimal algorithm. For the more fair SB-CUG, the gain in terms of sum rate is less.

Kalman filter

Here, the measurement noise $\aleph \left(t\right)={\left[n_i^{\mathit{\text{qw}}}\left(t\right),\cdots ,n_i^{\left(q+1\right)w-1}\left(t\right)\right]}^T$ is assumed zero mean with known covariance matrix R ^ℵ and the matrix $\Phi \in {\mathbb{C}}^{w\times w·N}$ contains only known reference symbols and zeros. Reference symbols may be transmitted at orthogonal time-frequency resources by different transmitters. Alternatively, to reduce the RS overhead or to increase the RS pattern density in time and/or frequency, we may use quasi-orthogonal RS. Such ‘overlapping’ or ‘code orthogonal’ pilots, have, e.g. been proposed in [23] and evaluated in [9, 17]. One benefit from using the later technique is that the addition of transmit antennas would not necessary cause an increase in overhead. However, whenever jointly estimated subchannels are not perfectly flat fading, code orthogonality is lost in the receiver. In [17], it was shown that this leads to a large degradation of prediction quality for the weakest channels when these are much weaker (e.g. by more than 10 dB) than the strongest channels. The energy leaking from the RS transmitted over strong channels will then be large in comparison to the energy of the received RS transmitted over weak channels. This leaked energy can be regarded as an extra noise term in the measurement, thus causing a noticeably lower experienced SNR for the weak channels. Since channels from antennas located at different base stations will generally have large gain differences while those located at the same base station will not, we here assume the use of orthogonal RS for antennas located at different base stations while quasi-orthogonal RS may be used for those located at the same base station. As an example, with two transmitters that use overlapping RS with BPSK symbols {1,-1}, two users and four jointly estimated subcarriers, w = 4, the matrices Φ in (21) could be given by Φ = [ I, I ] for user i = 1 and Φ = [ diag{1,-1,1,-1}, diag{1,-1,1,-1} ] for user i = 2, while for K_CRS = K, $h\left(t\right)={\left[H_{i1}^0,H_{i1}^1,H_{i1}^2,H_{i1}^3,H_{i2}^0,H_{i2}^1,H_{i2}^2,H_{i2}^3\right]}^T$, i = 1,2.

where $\widehat{x}\left(t_1\right|t_2)$ is an estimate of the state space vector in (3) at time t₁ based on measurements up to time t₂, $P\left(t_1\right|t_2)=E\left[\left(x\left(t_1\right)-\widehat{x}\left(t_1\right|t_2)\right){\left(x\left(t_1\right)-\widehat{x}\left(t_1\right|t_2)\right)}^{\ast }\right]$ and $\mathcal{K}\left(t\right)$ is the Kalman filter gain. These recursively computed estimates are based on a set of past measurements Υ(t),Υ(t-1),… that grows in time, without requiring an increasing memory size to store the measurements.

Iterative adjustment of the penalty matrix S

The criterion (18) can be optimized by adjusting the transmit powers with a step-by-step Greedy algorithm to reduce power normalization loss problems due to the scaling of (1/c) in (10). We outline this procedure below for single-antenna base stations.

First, calculate the optimal precoder from (14) with V = I and S = ϵ I. Here, ϵ≪1 is a small real-valued number ensuring that S is positive definite. The resulting precoder minimizes the intracluster interference which might not be optimal w.r.t. (18). This precoder is then used as the initial value for a sequence of iterative, one-dimensional searches where we sequentially adjust the penalties on the transmit powers used by each base station.

For clusters with few base stations, it is often sufficient to adjust only one scalar parameter in S related to the strongest base station as for (23). For clusters with many base stations, further improvements are obtained by adjusting additional diagonal elements in S starting with that associated with the second strongest base station.

For multiantenna base stations, all the co-located transmit antennas of one cell have average channel gains of the same order of magnitude. They should therefore be penalized using the same order of magnitude. Then, one penalty parameter value ρ _j can be adjusted simultaneously for all antennas at one base station j at a time as for the single-antenna base station example above.

Proof of Theorem 1

Assuming that $\mathit{Ē}\left[\mathrm{\Delta H}\right]=\mathit{Ē}\left[\Delta H_{\text{quant}}\right]=0$ and $\mathit{Ē}=\left[\Delta H^{\ast }{V}^{\ast }\mathrm{V\Delta }{H}_{\text{quant}}\right]=0$ and inserting (2) and (14) into (31) we get J₁ = 0 for all T. Hence, we cannot choose a matrix T that will decrease the cost function, J = J₀ + ϵ²J₂. The minimum J = J₀ is attained only by setting ϵ = 0, so R = R_RLP minimizes the cost function.