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## Lowcomplexity polynomial channel estimation in large-scale MIMO with arbitrary statistics (2014)

Venue: | IEEE J. Select. Topics in Signal Process |

Citations: | 4 - 2 self |

### Citations

1573 |
Fundamentals of Statistical Signal Processing: Estimation Theory
- Kay
- 1993
(Show Context)
Citation Context .... The instantaneous channel matrix is acquired from the received pilot signal by applying an appropriate estimation scheme. The Bayesian MMSE estimator is optimal if the 2channel statistics are known =-=[12]-=-–[16], while the minimumvariance unbiased (MVU) estimator is applied otherwise [12]. These channel estimators basically solve a linear system of equations, or equivalently multiply the received pilot ... |

785 |
Matrix Analysis
- Bhatia
- 1997
(Show Context)
Citation Context ...attenuation to the receiver. In general, a low-complexity method to approximate the extreme eigenvalues of any arbitrary covariance matrix was proposed in [21], based on the Gershgorin circle theorem =-=[34]-=-. This approach exploits the structure of the matrix imposed by the system setup to improve the convergence speed. For more details on how to choose α with low-complexity and compute the extreme eigen... |

550 | Fading correlation and its effect on the capacity of multielement antenna systems
- Shiu, Foschini, et al.
- 2000
(Show Context)
Citation Context ...loss of generality, we assume zero-mean channel and disturbance, since the non-zero mean assumption has no impact on the MSE performance as shown earlier in Section III. We follow the Kronecker model =-=[41]-=- to describe correlation among antennas 12 of the desired and disturbance MIMO channels. In the simulation, the covariance matrix of a MIMO channel is modeled as R = Rt ⊗Rr, where Rt ∈ CNt×Nt and Rr ∈... |

518 |
Gaussian elimination is not optimal
- Strassen
- 1969
(Show Context)
Citation Context ...lexity scaling of the classical inversion algorithms, such as Gaussian elimination and inversion based on Cholesky decomposition [31]. The exponent is reduced to O(M2.8074) by Strassen’s algorithm in =-=[32]-=-, which is a divide-an-conquer algorithm that exploits that 2×2 matrices can be multiplied efficiently. Using the complexity expressions in [32], it is easy to show that the algorithm is only computat... |

290 |
A well-conditioned estimator for large-dimensional covariance matrices
- Ledoit, Wolf
- 2004
(Show Context)
Citation Context ...variance matrix is not even invertible if the number of samples is smaller than the matrix dimension. Instead of using the pure sample covariance matrix, we suggest to follow a similar approach as in =-=[36]-=- and use a new estimator Ĉ which is an affine function of the sample covariance matrix Csample. In [36], the authors have shown that this estimator is a better fit for large-dimensional covariance ma... |

276 |
Noncooperative cellular wireless with unlimited numbers of base station antennas,”
- Marzetta
- 2010
(Show Context)
Citation Context ...unication standards, such as LTE-Advanced [3], there is an increasing interest from academy and industry to equip base stations (BSs) with much larger arrays with several hundreds of antenna elements =-=[4]-=-– [9]. Such large-scale MIMO, or “massive MIMO”, techniques can give unprecedented spatial resolution and array gain, thus enabling a very dense spatial reuse that potentially can keep up with the rap... |

219 | Scaling up MIMO: Opportunities and challenges with very large arrays
- Rusek, Persson, et al.
- 2013
(Show Context)
Citation Context ...l covariance matrices are diagonal, but large-scale MIMO channels typically have a distinct spatial channel correlation due to insufficient antenna spacing and richness of the propagation environment =-=[7]-=-. The spatial correlation decreases the estimation errors [15], but only if an appropriate estimator is applied. Moreover, the necessary pilot reuse in cellular networks creates spatially correlated i... |

150 | Multiplying matrices faster than Coppersmith–Winograd
- Williams
- 2012
(Show Context)
Citation Context ...head computations. It also has other drawbacks, such as lower computational accuracy and that the matrix dimensions must be M = 2k for some integer k. The exponent can be further reduced to O(M2.373) =-=[33]-=-, but at the cost of more overhead that pushes the breaking point to even higher values of M . In this paper, we propose new estimators with the complexity scaling O(M2), which both is a asymptoticall... |

116 | Channel capacity of MIMO architecture using the exponential correlation matrix,”
- Loyka
- 2001
(Show Context)
Citation Context ...mited case, while βi = 1 represents the case when the useful channel and the ith interfering channel are equally strong. To generate covariance matrices, we use the exponential correlation model from =-=[42]-=-. All the covariance matrices have diagonal elements equal to one which results in tr(R) = NtNr and tr(Σi) = βiNtNr. We assume that there are two dominating interfering cells, i = 1, 2. The correlatio... |

109 | Massive MIMO in the UL/DL of cellular networks: How many antennas do we need?”
- Hoydis, Brink, et al.
- 2013
(Show Context)
Citation Context ...if R and Σi have very different span, or if tr(Σi) is weak altogether—this is easily observed if P̃HP̃ is a scaled identity matrix. Similar observations were recently made in the capacity analysis of =-=[6]-=- and when developing a pilot allocation algorithm in [10]. Under certain conditions, the subspaces of the useful channel and pilot contamination can be made orthogonal by coordinated allocation of pil... |

88 |
Massive MIMO for next generation wireless systems
- Larsson, Edfors, et al.
- 2013
(Show Context)
Citation Context ...y efficiency. The antenna elements in large-scale MIMO can be either collocated in one- or multi-dimensional arrays or distributed over a larger area (e.g., on the facade or the windows of buildings) =-=[8]-=-. Apart from increasing the spectral efficiency of conventional wireless systems, which operate at carrier frequencies of one or a few GHz, the use of massive antenna configurations is also a key enab... |

76 | Pilot contamination and precoding in multi-cell TDD systems - Jose, Ashikhmin, et al. - 2011 |

67 |
Transmit signal design of optimal estimation of correlated MIMO channels
- Kotecha, Sayeed
- 2004
(Show Context)
Citation Context ...finite. Observe that R is generally not a scaled identity matrix, but describes the spatial propagation environment. In order to estimate the channel coefficients, we exploit pilot signals similar to =-=[13]-=-–[15]. This means that the transmitter sends the columns of a fixed predefined pilot matrix P ∈ CNt×B over B channel uses; see Fig. 1. The integer B is the length of the pilot sequence and usually sat... |

57 |
D.L.: Multistage linear receivers for DS-CDMA systems
- Moshavi, Kanterakis, et al.
- 1996
(Show Context)
Citation Context ...ich reduces the estimation performance and spectral efficiency [5]–[7], [10], [11]. Polynomial expansion (PE) is a well-known technique to reduce the complexity of large-dimensional matrix inversions =-=[17]-=-. Similar to classic Taylor series expansions for scalar functions, PE approximates a matrix function by an L-degree matrix polynomial. PE has a long history in the field of signal processing for mult... |

55 | Performance of reduced-rank linear interference suppression,”
- Xiao
- 2001
(Show Context)
Citation Context ...The degree L basically describes the accuracy to which the inversion of each eigenvalue is approximated, thus the degree needs not scale with the system dimensions to achieve near optimal performance =-=[20]-=-. Instead, L is simply selected to balance between computational complexity and detection performance. A main problem is to select the coefficients of the polynomial to achieve high performance at sma... |

47 | Argos: Practical many-antenna base stations.
- Shepard, Yu, et al.
- 2012
(Show Context)
Citation Context ... can easily be parallelized and implemented using efficient integrated circuits. On the contrary, the matrix inversions in the MMSE/MVU estimators are known to be complicated to implement in hardware =-=[39]-=-. Consequently, whenever the PEACH estimators and MMSE/MVU estimators are similar in terms of FLOPs, the computational delays and energy consumption are probably lower when implementing the proposed P... |

46 | A coordinated approach to channel estimation in large-scale multiple-antenna systems,”
- Yin, Gesbert, et al.
- 2013
(Show Context)
Citation Context ...ly be exploited if the propagation environment is precisely known. CSI is typically acquired by transmitting predefined pilot signals and estimating the channel coefficients from the received signals =-=[10]-=-–[15]. The pilot overhead is proportional to the number of transmit antennas, thus it is commonly assumed that the pilots are sent from the array with the smallest number of antennas and used for tran... |

44 | A competitive minimax approach to robust estimation of random parameters
- Eldar, Merhav
- 2004
(Show Context)
Citation Context ...ved for ĥMVUPEACH and ĥMVUW-PEACH in a similar way. Remark 4 (Other PEACH estimators). The PE technique can be applied to any type of channel estimators that involve matrix inversions. For example, =-=[35]-=- derives a robust estimator, the minimax regret estimator, under certain uncertainty and statistical assumptions. This estimator has a similar expression as the MMSE estimator, but involves other matr... |

41 | Design and analysis of low-complexity interference mitigation on vector channels
- Mueller, Verdú
- 2001
(Show Context)
Citation Context ... the polynomial to achieve high performance at small L; the optimal coefficients are expensive to compute [17], but alternatives based on appropriate scalings [18], [21], [23] and asymptotic analysis =-=[19]-=-, [22] exist. Recently, PE has also been used to reduce the precoding complexity in large-scale MIMO systems [24]–[26], and high performance was achieved by optimizing the matrix polynomials using asy... |

39 |
A framework for training-based estimation in arbitrarily correlated Rician MIMO channels with Rician disturbance,”
- Bjornson, Ottersten
- 2010
(Show Context)
Citation Context ... exploited if the propagation environment is precisely known. CSI is typically acquired by transmitting predefined pilot signals and estimating the channel coefficients from the received signals [10]–=-=[15]-=-. The pilot overhead is proportional to the number of transmit antennas, thus it is commonly assumed that the pilots are sent from the array with the smallest number of antennas and used for transmiss... |

24 | Transmit diversity vs. spatial multiplexing in modern MIMO systems,”
- Lozano, Jindal
- 2010
(Show Context)
Citation Context ...8 from The Swedish Research Council. This research has been supported by the ERC Starting Grant 305123 MORE (Advanced Mathematical Tools for Complex Network Engineering). through spatial multiplexing =-=[2]-=-. While 8×8 MIMO transmissions have found its way into recent communication standards, such as LTE-Advanced [3], there is an increasing interest from academy and industry to equip base stations (BSs) ... |

19 |
Blind pilot decontamination,”
- Muller, Cottatellucci, et al.
- 2014
(Show Context)
Citation Context ...ry pilot reuse in cellular networks creates spatially correlated inter-cell interference, known as pilot contamination, which reduces the estimation performance and spectral efficiency [5]–[7], [10], =-=[11]-=-. Polynomial expansion (PE) is a well-known technique to reduce the complexity of large-dimensional matrix inversions [17]. Similar to classic Taylor series expansions for scalar functions, PE approxi... |

17 | Training signal design for estimation of correlated MIMO channels with colored interference. - Liu, Wong, et al. - 2007 |

16 |
Inter-cell interference in noncooperative TDD large scale antenna systems,”
- Fernandes, Ashikhmin, et al.
- 2013
(Show Context)
Citation Context ...ronized with the desired pilot and that the delays between cells are negligible. These are, essentially, worst-case assumptions and alternative unsynchronized scenarios have recently been analyzed in =-=[40]-=-. 11 When (40) is substituted into the PEACH and W-PEACH estimator expressions in (10) and (14) we get contaminated disturbance terms of the form RP̃HP̃ΣiP̃H . These terms are small if R and Σi have v... |

13 |
Downlink beamforming avoiding DOA estimation for cellular mobile communications
- Asté, Forster, et al.
- 1998
(Show Context)
Citation Context ...ing can, for example, be achieved by exploiting the pilot signals on multiple flat-fading subcarriers since the large-scale fading properties can be transformed between different adjacent subcarriers =-=[28]-=-, [29]. Interestingly, the coherence time of the long-term statistics is relatively short; the measurements in [30] observe coherence times of 5–23 seconds, depending on the propagation environment. H... |

12 | Asymptotic Moments for Interference Mitigation in Correlated Fading Channels
- Hoydis, Debbah, et al.
(Show Context)
Citation Context ...e low-complexity approach to compute the weights, which is done in the next subsection. Note that the weights cannot be optimized by random matrix theory (as was done for multiuser detection in [19], =-=[22]-=- and precoding in [24]–[26]) due to lack of randomness in the MMSE estimation expression in (2). Remark 3 (Low-Complexity Classical PEACH Estimators). Following the same approach as used to derive low... |

10 |
Evolving wireless communications: Addressing the challenges and expectations of the future
- Baldemair, Dahlman, et al.
- 2013
(Show Context)
Citation Context ...tion standards, such as LTE-Advanced [3], there is an increasing interest from academy and industry to equip base stations (BSs) with much larger arrays with several hundreds of antenna elements [4]– =-=[9]-=-. Such large-scale MIMO, or “massive MIMO”, techniques can give unprecedented spatial resolution and array gain, thus enabling a very dense spatial reuse that potentially can keep up with the rapidly ... |

9 |
Low complexity polynomial expansion multiuser detector for CDMA systems,”
- Sessler, Jondral
- 2005
(Show Context)
Citation Context ...em is to select the coefficients of the polynomial to achieve high performance at small L; the optimal coefficients are expensive to compute [17], but alternatives based on appropriate scalings [18], =-=[21]-=-, [23] and asymptotic analysis [19], [22] exist. Recently, PE has also been used to reduce the precoding complexity in large-scale MIMO systems [24]–[26], and high performance was achieved by optimizi... |

7 |
Robust uplink to downlink spatial covariance matrix transformation for downlink beamforming
- Chalise, Haering, et al.
- 2004
(Show Context)
Citation Context ...n, for example, be achieved by exploiting the pilot signals on multiple flat-fading subcarriers since the large-scale fading properties can be transformed between different adjacent subcarriers [28], =-=[29]-=-. Interestingly, the coherence time of the long-term statistics is relatively short; the measurements in [30] observe coherence times of 5–23 seconds, depending on the propagation environment. High us... |

6 |
Lowcomplexity channel estimation in large-scale mimo using polynomial expansion,”
- Shariati, Bjornsson, et al.
- 2013
(Show Context)
Citation Context ..., France (e-mail: {emil.bjornson, merouane.debbah}@supelec.fr). This work was presented in part at IEEE Symposium on Personal, Indoor, Mobile and Radio Communications (PIMRC), London, UK, Sept. 2013. =-=[1]-=- E. Björnson is funded by the International Postdoc Grant 2012-228 from The Swedish Research Council. This research has been supported by the ERC Starting Grant 305123 MORE (Advanced Mathematical Too... |

5 | Simplified polynomial-expansion linear detectors for DS-CDMA systems
- Lei, Lim
- 1998
(Show Context)
Citation Context ... problem is to select the coefficients of the polynomial to achieve high performance at small L; the optimal coefficients are expensive to compute [17], but alternatives based on appropriate scalings =-=[18]-=-, [21], [23] and asymptotic analysis [19], [22] exist. Recently, PE has also been used to reduce the precoding complexity in large-scale MIMO systems [24]–[26], and high performance was achieved by op... |

5 |
Lowcomplexity Linear Precoding for Downlink Large-Scale
- Zarei, Gerstacker, et al.
- 2013
(Show Context)
Citation Context ...ut alternatives based on appropriate scalings [18], [21], [23] and asymptotic analysis [19], [22] exist. Recently, PE has also been used to reduce the precoding complexity in large-scale MIMO systems =-=[24]-=-–[26], and high performance was achieved by optimizing the matrix polynomials using asymptotic analysis. The optimization of the polynomial coefficients is the key to high performance when using PE. S... |

3 |
Linear precoding based on truncated polynomial expansion: Reducing complexity in massive MIMO,”
- Muller, Kammoun, et al.
- 2014
(Show Context)
Citation Context ...ternatives based on appropriate scalings [18], [21], [23] and asymptotic analysis [19], [22] exist. Recently, PE has also been used to reduce the precoding complexity in large-scale MIMO systems [24]–=-=[26]-=-, and high performance was achieved by optimizing the matrix polynomials using asymptotic analysis. The optimization of the polynomial coefficients is the key to high performance when using PE. Since ... |

3 | Spatial long-term variations in urban, rural and indoor environments,”
- Viering, Hofstetter, et al.
- 2002
(Show Context)
Citation Context ...ge-scale fading properties can be transformed between different adjacent subcarriers [28], [29]. Interestingly, the coherence time of the long-term statistics is relatively short; the measurements in =-=[30]-=- observe coherence times of 5–23 seconds, depending on the propagation environment. High user velocity or 4rapid scheduling decisions in neighboring systems can further reduce the coherence time. More... |

3 |
Recursive MMSE estimation of wireless channels based on training data and structured correlation learning
- Matz
- 2005
(Show Context)
Citation Context ...red for the matrix-matrix product AB, matrix-vector product Ax, and vector-vector product xHy, respectively. In the special case of M = P and C = AB 8The MMSE estimator can be implemented recursively =-=[37]-=-, which is suitable for tracking variations in the covariance matrices. The complexity of each recursion is O(M2), but we need more than M recursions (per longterm statistics coherence time) to obtain... |

2 | Efficient series expansion for matrix inversion with application to MMSE equalization
- Josse, Laot, et al.
- 2008
(Show Context)
Citation Context ...to select the coefficients of the polynomial to achieve high performance at small L; the optimal coefficients are expensive to compute [17], but alternatives based on appropriate scalings [18], [21], =-=[23]-=- and asymptotic analysis [19], [22] exist. Recently, PE has also been used to reduce the precoding complexity in large-scale MIMO systems [24]–[26], and high performance was achieved by optimizing the... |

1 |
Robust training sequence design for correlated MIMO channel estimation
- Shariati, Wang, et al.
- 2014
(Show Context)
Citation Context ...ned as MSEtr(R) , as the performance measure. In all the figures, we compare the performance of the proposed estimators with the conventional MMSE and MVU estimators. The pilot matrix is P = √PtI. In =-=[16]-=-, it has been shown that this choice of pilot matrix, i.e., the scaled identity, performs (in the MSE sense) almost identical to the optimally robust designed pilot when the channel covariance matrix ... |

1 |
Low complexity channel estimation in TDD coordinated multi-point transmission systems
- Chen, Hou, et al.
(Show Context)
Citation Context ...LATION We consider a MIMO channel where the receiver and the transmitter are equipped with Nr and Nt number of antennas, 1After the submission of this paper, we became aware of the concurrent work of =-=[27]-=- which also applies PE to reduce the complexity of MMSE estimation. However, orthogonal frequency division multiplexing (OFDM) systems with a large number of subcarriers are considered in [27], while ... |

1 |
Receive beamforming design based on a multiple-state interference model
- Yang, Björnson, et al.
- 2011
(Show Context)
Citation Context ...nsmitter/receiver travel with high velocity. Similarly, the disturbance statistics can change rapidly if it contains interference from other systems (particularly if adaptive scheduling is performed) =-=[38]-=-. For a given total time Ttot, the computational complexity for each of the estimators consists of two parts: one part which can be precomputed once per coherence time of the statistics (i.e., ks = Tt... |