## Linear precoding via conic optimization for fixed MIMO receivers (2004)

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Venue: | IEEE Trans. Signal Processing |

Citations: | 52 - 3 self |

### BibTeX

@ARTICLE{Wiesel04linearprecoding,

author = {Ami Wiesel and Yonina C. Eldar and Shlomo Shamai (Shitz)},

title = {Linear precoding via conic optimization for fixed MIMO receivers},

journal = {IEEE Trans. Signal Processing},

year = {2004},

volume = {54},

pages = {161--176}

}

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### Abstract

We consider the problem of designing linear precoders for fixed multiple input multiple output (MIMO) receivers. Two di#erent design criteria are considered. In the first, we minimize the transmitted power subject to signal to interference plus noise ratio (SINR) constraints. In the second, we maximize the worst case SINR subject to a power constraint. We show that both problems can be solved using standard conic optimization packages. In addition, we develop conditions for the optimal precoder for both of these problems, and propose two simple fixed point iterations to find the solutions which satisfy these conditions. The relation to the well known downlink uplink duality in the context of joint downlink beamforming and power control is also explored. Our precoder design is general, and as a special case it solves the beamforming problem. In contrast to most of the existing precoders, it is not limited to full rank systems. Simulation results in a multiuser system show that the resulting precoders can significantly outperform existing linear precoders. 1

### Citations

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Citation Context ...s can significantly outperform existing linear precoders. 1 INTRODUCTION Multiple input multiple output (MIMO) systems arise in many modern communication channels, such as multiple user communication =-=[3]-=-, and/or multiple antennas channels [4]. It is well known that the use of multiple antennas promises substantial capacity gains when compared to traditional single antenna systems. In order to exploit... |

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Citation Context ... casting of the precoder design problems as standard conic optimization packages. The power optimization can be formulated as a Second Order Cone Program (SOCP) [52], or a semi definite program (SDP) =-=[53]-=- (otherwise known as a linear matrix inequalities (LMI) program). This procedure has already been addressed in [42] in the context of beamforming. We generalize it for the case of precoding. Moreover,... |

734 | Using SeDuMi 1.02, A Matlab ∗ toolbox for optimization over symmetric cones (updated for Version
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Citation Context ...among all the sub channels. Our precoders design is based on the powerful framework of convex optimization theory [18], which allows efficient numerical solutions using standard optimization packages =-=[51]-=-. A brief review of such programs and their standard forms is provided in Section 3. The first contribution of this paper is the casting of the precoder design problems as standard conic optimization ... |

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Citation Context ... if P(γo) is feasible, then the above iteration will converge from any λ (0) i to a set λ (n) i > 0 that satisfies (40). The convergence proof is based on the standard function approach introduced =-=in [56], which can -=-be summarized as follows. Consider the fixed point iteration λ (n+1) i = fi(Λ (n) ), i = 1, · · · K, (63) where Λ (n) = diag � λ (n)� . If the functions fi(Λ) obey the following properties... |

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Citation Context ...on to (40), and therefore the inequality in (49) must be strict. We now show that the arguments of the squared roots in (42) are non negative. Using a series expansion for the matrix inversion yields =-=[55] ⎡ ⎢ ⎣ The elements of δ 2 1 . δ 2 K ⎤ ⎥ ⎦ = � γo I -=-− F 1 + γo � −1 ⎡ ⎢ ⎣ λ1σ 2 1 . λKσ 2 K ⎤ ⎥ ⎦ = 1 + γo γo ∞� j=1 � γo F 1 + γo � j ⎡ ⎢ ⎣ λ1σ 2 1 . λKσ 2 K ⎤ ⎥ . (52) ⎦ γo F are nonnegative. Theref... |

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Citation Context ...k noise vector. 4.4 INTERPRETATION USING DOWNLINK UPLINK DUALITY In this section, we provide an alternative solution for the power optimization problem based on the well known downlink uplink duality =-=[43, 44]-=-. As explained in the previous sections, the power optimization can be solved efficiently without the use of duality. However, previous attempts for solving the downlink beamforming problem, which is ... |

133 | A vector-pertubation technique for near-capacity multi-antenna multiuser communications,” submitted to
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Citation Context ... linear precoders have also been derived. Among them are the non linear Tomlinson Harashima MIMO 3sprecoder [6–10,31], the well known “Dirty Paper” precoder [32], and the vector perturbation pre=-=coder [33]. -=-Another non linear precoder which optimizes the transmitted symbol’s vector itself was derived in [34]. The problem of precoder design is highly related to other problems in the literature. As state... |

128 | Joint tx-rx beamforming design for multicarrier mimo channels: A unified framework for convex optimization
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Citation Context ...], whereas signature design for matched decision feedback receivers was explored in [39]. One of the interesting properties of these joint designs is that maximizing SINR is related to minimizing MSE =-=[15]-=-. Thus, although different criteria have been explored, most of the research was dedicated to variants of the MMSE criterion. Another related problem is the rank one transmit beamforming design and op... |

99 | Generalized linear precoder and decoder design for MIMO channels using the weighted MMSE criterion
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Citation Context ...ptimization. Recently, the quest for better performance with lower complexity led researchers to also optimize the transmitter [3]–[11], and even to jointly optimize the transmitter and receiver [12]–=-=[20]-=-. This, as well as new results and algorithms in convex optimization theory [21], have significantly improved state of the art communication systems. Manuscript received July 4, 2004; revised February... |

98 |
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Citation Context ...resting duality between downlink and uplink beamforming [43–45]. The uplink beamforming problem has been solved before in [46, 47]. Using the duality, the downlink SINR problem can be handled as well =-=[41,48,49]-=-. Recently, a non linear Tomlinson Harashima version of these papers was presented in [50]. In this paper, we integrate the ideas above in the context of MIMO precoding for fixed receivers. As explain... |

88 | Joint optimal power control and beamforming in wireless networks using antenna arrays
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Citation Context ...thematically more difficult than MSE optimization. It was solved using an interesting duality between downlink and uplink beamforming [43–45]. The uplink beamforming problem has been solved before i=-=n [46, 47]-=-. Using the duality, the downlink SINR problem can be handled as well [41,48,49]. Recently, a non linear Tomlinson Harashima version of these papers was presented in [50]. In this paper, we integrate ... |

83 | Robust optimization – methodology and applications
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Citation Context ...ems, the transmitter does not have access to perfect CSI, and needs to resort to noisy channel estimates, and/or delayed feedback. In this case, robust optimization algorithms should be applied as in =-=[59]-=-. Another possible direction is to consider fixed non linear receivers, such as the successive cancelling receiver. It is well known that such receivers outperform the linear receivers explored in our... |

79 |
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Citation Context ... i i = 1, · · · , K; σ2 i λi ≤ Po. ⎪⎬ ⎪⎭ 5 until convergence The solution for the optimization in line 3 remains as in (67). The optimization in line 4 is more complicated. It is based =-=on the work of [13], which showe-=-d that this optimization can be carried out using eigenvalue optimization. As already discussed, at the optimal solution, all the constraints are active: or � 1 + 1 � λi � �wH i HHei �2 �... |

79 | Optimal sequences, power control, and user capacity of synchronous CDMA systems with lin169 MMSE multiuser receivers
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Citation Context ...the receiver, which has been treated, e.g., in [9,11–17,35,36]. The design of the optimal signatures (which can be considered as a linear precoding scheme) for matched MMSE receivers was discussed i=-=n [37, 38]-=-, whereas signature design for matched decision feedback receivers was explored in [39]. One of the interesting properties of these joint designs is that maximizing SINR is related to minimizing MSE [... |

70 | Precoding and Signal Shaping for Digital Transmission - Fischer - 2002 |

69 |
Transmitter precoding in synchronous multiuser communications
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Citation Context ... fedback from the receiver. Therefore, following prior work, we assume that the CSI is perfect and errorless. One of the first results on optimizing a precoder for a fixed MIMO linear model is due to =-=[5]-=- in the context of multiuser transmission. In that work, two precoders were derived. The precoders applied a linear transformation on the transmitted symbols prior to the spreading. The first precoder... |

69 | Iterative construction of optimum signature sequence sets in synchronous cdma systems
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Citation Context ...the receiver, which has been treated, e.g., in [9,11–17,35,36]. The design of the optimal signatures (which can be considered as a linear precoding scheme) for matched MMSE receivers was discussed i=-=n [37, 38]-=-, whereas signature design for matched decision feedback receivers was explored in [39]. One of the interesting properties of these joint designs is that maximizing SINR is related to minimizing MSE [... |

60 |
A transmit preprocessing technique for multiuser MIMO systems using a decomposition approach
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Citation Context ...m likelihood approach and maximum asymptotic multiuser efficiency with different power constraints were derived in [28]. A linear precoding technique based on a decomposition approach was proposed in =-=[29]-=-. A linear precoder design for non linear maximum likelihood (ML) receivers was discussed in [30]. Moreover, non linear precoders have also been derived. Among them are the non linear Tomlinson Harash... |

60 | E1 Ghaoui. Method of centers for minimizing generalized eigenvalues. Linear Algebra and its Applications
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Citation Context ...forming. We generalize it for the case of precoding. Moreover, we show that the SINR optimization can also by formulated as a standard conic program known as the generalized eigenvalue problem (GEVP) =-=[54]-=-. Another contribution of this paper is the derivation of the optimality conditions for both of the design problems. By analyzing the Karush-Kuhn-Tucker (KKT) conditions for conic programs we present ... |

59 |
Introduction to convex optimization with engineering applications. Course Notes
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Citation Context ...]. The work was supported in part by the NewCom project. 1sand even to jointly optimize the transmitter and receiver [11–17]. This, as well as new results and algorithms in convex optimization theor=-=y [18], have significantly improved stat-=-e of the art communication systems. b ✲ T ✲ HTx Fixed! ✲ HCh,1 ✲ ✲ HCh,2 ✲ w1 w2 ✲ HCh,M ✲ ✲ HRx,M ✻ ✲ . yM wM ✲ ✻ HRx,1 ✲ y1 ✲ ✻ HRx,2 ✲ y2 Figure 1: Block diagram o... |

51 | Optimal beamforming using transmit antenna arrays
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(Show Context)
Citation Context ...resting duality between downlink and uplink beamforming [43–45]. The uplink beamforming problem has been solved before in [46, 47]. Using the duality, the downlink SINR problem can be handled as wel=-=l [41,48,49]-=-. Recently, a non linear Tomlinson Harashima version of these papers was presented in [50]. In this paper, we integrate the ideas above in the context of MIMO precoding for fixed receivers. As explain... |

42 | Transceiver optimization for multiuser MIMO systems - Serbetli, Yener - 2004 |

37 |
Minimum BER block precoder for zero-forcing equalization
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- 2003
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Citation Context ...ver optimization. Recently, the quest for better performance with lower complexity led researchers to also optimize the transmitter [3]–[11], and even to jointly optimize the transmitter and receiver =-=[12]-=-–[20]. This, as well as new results and algorithms in convex optimization theory [21], have significantly improved state of the art communication systems. Manuscript received July 4, 2004; revised Feb... |

31 | Joint transmitter receiver diversity for efficient space division multiple access
- Chang, Tassiulas, et al.
- 2002
(Show Context)
Citation Context ...have been explored, most of the research was dedicated to variants of the MMSE criterion. Another related problem is the joint design of rank-one transmit beamforming design and optimal power control =-=[37]-=-–[39]. This problem is equivalent to precoding when there is no transmitter, i.e., the precoder itself is the transmitter. At first glance, it seems that the precoding problem can be solved by address... |

26 |
Pre-rake diversity combining in time division duplex CDMA mobile communications
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Citation Context ... rakes, which perform better in low SNR. In addition, transmit minimum mean-squared error (MMSE) 1 precoders that tried to compensate for the performance in the different SNR regions were derived [5]–=-=[10]-=-. Other precoders using different kinds of optimization criteria were also derived. Variants of the previous precoders were discussed in [22]–[27]. Linear precoders based on an approximate maximum-lik... |

26 | Optimal downlink beamforming using semidefinite optimization,” presented at the 37th Annu
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Citation Context ...mulated as a Second Order Cone Program (SOCP) [52], or a semi definite program (SDP) [53] (otherwise known as a linear matrix inequalities (LMI) program). This procedure has already been addressed in =-=[42]-=- in the context of beamforming. We generalize it for the case of precoding. Moreover, we show that the SINR optimization can also by formulated as a standard conic program known as the generalized eig... |

24 | Optimum linear joint transmit-receive processing for MIMO channels with QoS constraint
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Citation Context ...rage transmitted power subject to QoS constraints. This criterion is interesting from a system level perspective. Given the required QoS, the system tries to satisfy it with minimum transmitted power =-=[16], [41]: ⎧ ⎪⎨ minT -=-Tr P(γo) = ⎪⎩ � THHH � TxHTxT |[HT] s.t. i,i| mini 2 where γo > 0 is the given worst SINR constraint. � i�=j|[HT] i,j| 2 +σ 2 i ≥ γo, The second strategy is maximizing the minimal SI... |

19 | Transmit wiener filter for the downlink of TDD DS-CDMA systems
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Citation Context ... [3]. Inverting the channel decreases the output SNR. Therefore, in low SNR, the preferable receiver is the conventional MF which maximizes the SNR. This motivated the design of transmit MF precoders =-=[7] a-=-nd transmit rakes [19–21]. Clearly, just as in the receiver case, in high SNR, the interference dominates the noise, and the performance of these precoders degrades. Recently, transmit scaled MMSE p... |

16 |
Optimal downlink power assignment for smart antenna systems
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Citation Context ...gence where is a matrix with elements is a length vector with elements , and is a length vector with elements . Line 2 optimizes the matrix for fixed . Line 3 optimizes the weights for fixed based on =-=[53]-=-. Clearly, this solution is much less appealing than the fixed point iteration in (72) and (73). VI. SPECIAL CASES In this section, we examine a few interesting cases in which the problems and have si... |

15 |
A general duality theory for uplink and downlink beamforming
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(Show Context)
Citation Context ...k noise vector. 4.4 INTERPRETATION USING DOWNLINK UPLINK DUALITY In this section, we provide an alternative solution for the power optimization problem based on the well known downlink uplink duality =-=[43, 44]-=-. As explained in the previous sections, the power optimization can be solved efficiently without the use of duality. However, previous attempts for solving the downlink beamforming problem, which is ... |

14 | Joint transmission: Advantageous downlink concept for CDMA mobile radio systems using time division duplexing,” Electron - Meurer, Baier, et al. - 2000 |

13 | Transceiver optimization for block-based multiple access through ISI channels - Luo, Davidson, et al. |

13 | Joint signal precoding in the downlink of spread spectrum systems - Barreto, Fettweis |

11 | On transmitter-based interference mitigation in TDD-downlink with frequency selective fading environment,” presented at the Asia–Pacific
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Citation Context ... � T = c H H H + Kσ2 Po 1 H H TxHTx � † i,i ; (93) − 1 H H , (94) where c is a constant that scales the matrix to satisfy the power constraint. This particular precoder has been previously de=-=rived in [7, 8]-=- and recently in [9, 10] through scaled MMSE considerations. However, it is easy to verify that in general, i.e., in nonsymmetric channels, it does not necessarily satisfy the conditions in Theorem 1,... |

11 | transmitter precoding, and joint transmission techniques for time division duplex CDMA - Georgoulis, Cruickshank, et al. - 2001 |

10 | Downlink-uplink duality and effective bandwidths - Tse, Viswanath |

10 |
Applications of second order cone programming. Linear Algebra and its Applications 284
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Citation Context ...first contribution of this paper is the casting of the precoder design problems as standard conic optimization packages. The power optimization can be formulated as a Second Order Cone Program (SOCP) =-=[52]-=-, or a semi definite program (SDP) [53] (otherwise known as a linear matrix inequalities (LMI) program). This procedure has already been addressed in [42] in the context of beamforming. We generalize ... |

8 |
Joint optimal power control and beamforming in wireless networks using antenna arrays
- Farrokhi, Tassiulas, et al.
- 1998
(Show Context)
Citation Context ...thematically more difficult than MSE optimization. It was solved using an interesting duality between downlink and uplink beamforming [43–45]. The uplink beamforming problem has been solved before in =-=[46, 47]-=-. Using the duality, the downlink SINR problem can be handled as well [41,48,49]. Recently, a non linear Tomlinson Harashima version of these papers was presented in [50]. In this paper, we integrate ... |

8 |
Applications of second order cone programming
- Lobo, Vandenberghe, et al.
- 1998
(Show Context)
Citation Context ...I. We then cast the precoder design problems as standard conic optimization packages. Specifically, we show that the power optimization problem can be formulated as a second-order cone program (SOCP) =-=[47]-=- or a semidefinite program (SDP) [48] [otherwise known as a linear matrix inequalities (LMI) program]. The SINR optimization can also be formulated as a standard conic program known as the generalized... |

7 | Optimal minimum distance based precoder for MIMO spatial multiplexing systems
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- 2004
(Show Context)
Citation Context ...s were derived in [28]. A linear precoding technique based on a decomposition approach was proposed in [29]. A linear precoder design for non linear maximum likelihood (ML) receivers was discussed in =-=[30]. More-=-over, non linear precoders have also been derived. Among them are the non linear Tomlinson Harashima MIMO 3sprecoder [6–10,31], the well known “Dirty Paper” precoder [32], and the vector perturb... |

6 | Transmit MMSE pre-RAKE preprocessing with simplified receivers for the downlink of MISO TDDCDMA systems - Choi, Murch |

6 | An optimized transmitter precoding scheme for synchronous DS-CDMA
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- 2002
(Show Context)
Citation Context ...der [6–10,31], the well known “Dirty Paper” precoder [32], and the vector perturbation precoder [33]. Another non linear precoder which optimizes the transmitted symbol’s vector itself was der=-=ived in [34]-=-. The problem of precoder design is highly related to other problems in the literature. As stated, we investigate the design of linear precoders for fixed linear receivers. A related problem is the pr... |

5 |
Precoding and power loading for multi-antenna broadcast channels
- Shi, Schubert
- 2004
(Show Context)
Citation Context ...has been solved before in [46, 47]. Using the duality, the downlink SINR problem can be handled as well [41,48,49]. Recently, a non linear Tomlinson Harashima version of these papers was presented in =-=[50]-=-. In this paper, we integrate the ideas above in the context of MIMO precoding for fixed receivers. As explained before, the design of most of the previous precoders is based on minimizing variants of... |

4 |
Transmit-preprocessing techniques with simplified receivers for the downlink of MISO TDD-CDMA systems
- Choi, Murch
- 2004
(Show Context)
Citation Context ... 1 H H TxHTx � † i,i ; (93) − 1 H H , (94) where c is a constant that scales the matrix to satisfy the power constraint. This particular precoder has been previously derived in [7, 8] and recent=-=ly in [9, 10]-=- through scaled MMSE considerations. However, it is easy to verify that in general, i.e., in nonsymmetric channels, it does not necessarily satisfy the conditions in Theorem 1, and is therefore subopt... |

4 | Optimal designs for space time linear precoders and decoders - Scaglione, Stoica, et al. - 2002 |

4 | Transmitter based multiuser interference rejection for the down-link of a wireless CDMA system in a multipath environment - Brandt-Pearce, Dharap - 2000 |

4 | Transmitter-based inverse filters for reducing MAI and ISI in CDMA-TDD downlink - Georgoulis, Cruickshank - 2004 |

3 | Shitz), “Multiuser precoders for fixed receivers
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- 2004
(Show Context)
Citation Context ...formance with lower complexity led researchers to also optimize the transmitter [5–10], ∗ e-mails: amiw@tx.technion.ac.il, yonina@ee.technion.ac.il and sshlomo@ee.technion.ac.il. Presented in part=-= in [1, 2]. -=-The work was supported in part by the NewCom project. 1sand even to jointly optimize the transmitter and receiver [11–17]. This, as well as new results and algorithms in convex optimization theory [... |

3 |
New transmit schemes and simplified receivers for MIMO wireless communications systems
- Choi, Murch
- 2003
(Show Context)
Citation Context ... 1 H H TxHTx � † i,i ; (93) − 1 H H , (94) where c is a constant that scales the matrix to satisfy the power constraint. This particular precoder has been previously derived in [7, 8] and recent=-=ly in [9, 10]-=- through scaled MMSE considerations. However, it is easy to verify that in general, i.e., in nonsymmetric channels, it does not necessarily satisfy the conditions in Theorem 1, and is therefore subopt... |