The

Joint transceiver designs with criteria such as minimum mean square error (MMSE), maximum sum capacity, and minimum bit error rate (BER), and so forth, for multiple-input-multiple-output (MIMO) systems, with both uplink and downlink configurations, have been studied intensively in recent literature (e.g., see [

Assuming perfect channel state information (CSI), joint MMSE transceiver design has been studied by many researchers. A closed form design subject to the total power constraint for a single-user MIMO system is derived in [

For the downlink configuration, iterative approaches such as in [

All of the above mentioned MMSE transceiver designs are based on perfect CSI. However, the CSI is usually estimated in practice and is therefore subject to CSI estimation errors and possibly quantized CSI feedback errors. Hence, in practice, joint transceiver design has to be based on imperfect CSI. One option is to ignore that the CSI is imperfect. This type of approach is herein called non-robust. Unfortunately, the system performances derived from the non-robust approaches depend strongly on the quality of the available CSI (performances get worse quickly if the CSI quality deteriorates). Moreover, an optimum design based on poor CSI could be worse than suboptimum designs using the same CSI. Therefore, a more appealing option is to model the CSI error and to incorporate the error model into the transceiver design. This type of approach is herein called robust. The robust approaches can better mitigate the degradation of system performances due to imperfect CSI than the non-robust approaches if the CSI error is modeled correctly. Two classes of imperfect CSI models are usually employed: the stochastic model for the CSI estimation errors and the deterministic model for the CSI feedback errors. If a stochastic model is used, a statistically robust design is usually performed to optimize some system performance functions. If a deterministic model is used, a minimax or maximin design aiming at optimizing the worst-case system performance is usually carried out.

To cope with CSI estimation errors, closed form solutions for the robust joint MMSE transceiver design subject to the total power constraint are developed for single-user MIMO systems in [

So far, no statistically robust approach has been shown optimum in the MMSE sense for the downlink MIMO systems (either single-cell or multicell) under the per-antenna power constraint. Proposed in this paper is the robust MMSE transceiver design with respect to CSI estimation errors for downlink multicell MIMO systems subject to arbitrary linear power constraints. Specifically, the per-antenna and per-cell power constraints are considered. The work is relevant to frequency division duplex (FDD) systems where channel estimation is done at each user equipment (UE) and then fed back to the base station(s) (denoted as evolved Node B or eNB) via a zero-delay and error free communication link. Note that CSI feedback errors are not considered in this paper. The work may possibly also be extended to time division duplex (TDD) systems where channel estimation is done at the eNBs.

We first extend the statistical model of imperfect CSI in [

The

The relationships between the

MMSE transceiver designs using the proposed robust approaches are performed for various single-cell and multicell examples with different system configurations, power constraints, channel correlations, and cooperation scenarios. System performances in terms of MSE and BER of various numerical examples are compared. Computational efficiency for various approaches is studied. Sensitivity studies with respect to channel statistics (channel correlations and path loss, estimated independently from channel estimation) are also investigated. The numerical results show that the proposed robust approaches are indeed superior to the non-robust approaches. Moreover, accurate channel correlations and path loss are not required in the robust approaches. With cell cooperation, the cell edge UEs perform as well as those UEs without inter-cell interferences.

Notations are as follows. All boldface letters indicate vectors (lower case) or matrices (upper case).

Consider the downlink of a multicell multiuser MIMO system with

In this system, there may be multiple groups where each group jointly designs its precoders and decoders but does so independently of the other groups. In the with-cooperation scenario (there is full cooperation among all eNBs), system-wide design is performed and there is only one group. In the without-cooperation scenario (there is no cooperation among eNBs), the eNB and UEs in a cell are one group. Let

At the

In the without-cooperation scenario, let the eNB serving the

Since there are multiuser precodings at eNB

In order to unify (

In order to account for path loss and spatial correlation, the channel

In practice, the CSI is estimated, resulting in estimation error. Thus,

For a given group and thus a given

To solve (

The central processing unit is assumed to have knowledge about the channel estimate(s),

Substituting (

On the other hand, for a given set of decoders

By setting the gradients of (

Given

Given

Given

Note that the

The

Similarly, substituting (

With (

Given

Given

Note that the number of data streams intended for the UE's

Note that when the CSI is perfectly known, the

When the source covariance matrices are all identity matrices multiplied by the same constant, that is,

We propose a novel

Given

Given

In practice, the

Similar to the

We observe poor convergence behavior of the SDP procedure for the

When the

Note that the

For the

The task of showing the equivalence of the KKT conditions of the two approaches which boils down to showing the above KKT condition of the

Without loss of generality, let

Four examples (two single-cell and two 3-cell examples) will be considered. Their system parameters are shown in Table

System parameters.

Example | 1 | 2 | 3 | 4 |
---|---|---|---|---|

No. of cells (eNBs): | 1 | 1 | 3 | 3 |

No. of Tx Antennas per eNB: | 4 | 4 | 2 | 4 |

Total No. of Tx Antennas of the System: | 4 | 4 | 6 | 12 |

Number of UEs associated per eNB | 2 | 2 | 1 | 2 |

Total No. of UEs of the System: | 2 | 2 | 3 | 6 |

No. of Rx Antennas per UE: | 2 | 2 | 2 | 2 |

No. of data streams per UE: | 2 | 1 | 2 | 2 |

System configurations of two multicell examples: (a) 3 eNBs and 3 UEs. (b) 3 eNBs and 6 UEs. The coordinates of relevant eNBs and UEs shown here are employed for simulations. Note that UE_{1} is right on the edge of three cells in both systems. In example 3, UE

Example

Example

In the simulation, no CSI feedback error is assumed. The only CSI error is the CSI estimation error.

Without loss of generality, we will numerically show the equivalence of the

Figure

MSE and BER as functions of

For the 1-data-stream scenario (i.e., example 2 in Table

Without loss of generality, we will compare the computational efficiency of the various proposed approaches with perfect CSI. Consider example 1 in Table

Note that the

Convergence of the

In Figure

Using the same single-cell example, the convergent properties of the SDP Procedure and the Numerically Efficient Procedure of the

Convergence of the

Comparing the

In the following sections, we will consider the situation where

Using the 3-cell configuration in Figure

MSE and BER as functions of

It is not surprising to see that the BER and the MSE of the without-cooperation scenario are much larger (worse) than the BER and the MSE of the with-cooperation scenario, respectively. Even with perfect CSI, the without-cooperation BER is larger than 10% even at high power. It is obvious that some kinds of time/frequency scheduling or code spreading are needed in order to reduce the cell edge interferences if no cooperation among eNBs is available. On the other hand, in the with-cooperation scenario, the BER of the

We now compare the results of example 3 with the results of example 4 in Table _{1} is right on the 3-cell edge and each of the other UEs is near at least one of the 2-cell edges.

Figure

MSE and BER as functions of

We make four main observations. First, the results for the per-cell and per-antenna power constraints are more or less the same for all of the approaches (the

Secondly, as expected, the

Thirdly, also as expected, the

Lastly, compared to the results in example 3, the MSE results for all the approaches are noticeably higher in example 4, but the degradation of BER results in example 4 compared to example 3 is not significant if the per-antenna power

Using the example 1 in Table

Figures

MSE’s as functions of

MSE as functions of

Using the 3-cell configuration in Figure

MSE and BER as functions of estimated

Using the 3-cell configuration in Figure

MSE and BER as functions of the estimated-to-actual-path loss ratio (EAPLR) under the per-antenna and per-cell power constraints (

Three robust approaches, the

The

MMSE transceiver designs using the three proposed approaches are performed for various single-cell and multicell examples with different system configurations, power constraints, channel spatial correlations, and cooperation scenarios. System performances in terms of MSE and BER are investigated. Important concluding remarks made from these numerical examples are list below. First of all, the robust approaches outperform their non-robust counterparts in most of the numerical simulations (even when the channel is highly correlated, when the CSI estimation errors are large, and when there exist estimation errors in statistics of channel parameters). Secondly, the performance of the with-cooperation scenario is much better than that of the without-cooperation scenario. With cell cooperation, the cell edge UEs perform as well as those UEs without inter-cell interferences and therefore the cell edge difficulties can be remedied. Thus, with full cell cooperation, the system throughput can increase linearly with the numbers of antennas for both transmission and reception. Thirdly, the robust approaches are insensitive to the estimation errors of the channel statistics (e.g., to channel correlations and path loss). This important feature makes robust approaches practical. Fourthly, the system performances derived under the more practical per-antenna power constraint are very similar to those with the per-cell power constraint. Thus, the practical per-antenna power constraint inflicts little performance losses compared to the optimum per-cell power constraint. Fifthly, the performance gain of the robust approaches over the non-robust approaches is more profound in larger MIMO systems. Sixthly, the performance gain of the robust approaches over the non-robust approaches is reduced if the channel correlations increase.

In short, we have herein proposed, for joint MMSE transceiver designs, three novel robust approaches: the

The authors would like to thank InterDigital Communications Corporation for its financial support.