A limited feedback scheme based on spatially correlated channels for coordinated multipoint systems
 Fernando Domene^{1}Email author,
 Gema Piñero^{1},
 Carmen Botella^{2} and
 Alberto Gonzalez^{1}
https://doi.org/10.1186/168714992012176
© Domene et al; licensee Springer. 2012
Received: 7 December 2011
Accepted: 18 May 2012
Published: 18 May 2012
Abstract
High spectral efficiency can be achieved in the downlink of multiantenna coordinated multipoint systems provided that the multiuser interference is appropriately managed at the transmitter side. For this sake, downlink channel information needs to be sent back by the users, thus reducing the rate available at the uplink channel. The amount and type of feedback information required has been extensively studied and many limited feedback schemes have been proposed lately. A common pattern to all of them is that achieving low rates of feedback information is possible at the cost of increasing complexity at the user side and, sometimes, assuming that some statistics of the channel are known. In this article, we propose a simple and versatile limited feedback scheme that exploits the spatial correlation at each multiantenna base station (BS) without requiring any previous statistical information of the channel and without adding significant computational complexity. It is based on the separate quantization of the channel impulse response modulus and phase and it shows better mean square error performance than the standard scheme based on quantization of real and imaginary parts. In order to evaluate the performance of the downlink regarding multiuser interference management, different precoding techniques at the BSs, such as zeroforcing (ZF), TomlinsonHarashima precoding (THP) and lattice reduction Tomlinson Harashima precoding (LRTHP), have been evaluated. Simulations results show that LRTHP and THP present a higher robustness than ZF precoding against channel quantization errors but at the cost of a higher complexity at the BS. Regarding sumcapacity and bit error rate performances, our versatile scheme achieves better results than the standard one in the medium and high SNR regime, that is, in the region where quantization errors are dominant against noise, for the same feedback cost measured in bits per user.
1. Introduction
In the last decades, precoding techniques allowing spatial multiplexing of several users have been proposed to improve the spectral efficiency of multiuser multipleinput multipleoutput (MUMIMO) communication systems. Dirty paper coding (DPC) [1] is a theoretical scheme which allows to precancel the noncasually known interference at the transmitter without entailing a power penalty. For a given user ordering, DPC is serially applied over the users allowing to presubstract the interference caused by users with lower indices [2]. Although it has been proved that DPC achieves the whole capacity region of the MIMO broadcast channel [2, 3], it suffers from a high level of complexity when implemented in practical systems. Due to this, precoding schemes requiring lower complexity are usually employed. Linear schemes, such as zeroforcing (ZF) [4], and nonlinear schemes, such as TomlinsonHarashima precoding (THP) [5] or latticereduction TomlinsonHarashima precoding (LRTHP) [6], are mostly used for singleantenna receivers. In interferencepredominant scenarios, nonlinear techniques achieve better performance at the cost of higher complexity [7]. For multipleantenna receivers, however, a linear technique called block diagonalization has been proposed showing good performance when such optimizations as in [8] are considered.
Multiuser MIMO precoding and scheduling techniques require an accurate knowledge of the CSI at the transmitter to achieve full multiuser multiplexing gain [3, 9, 10]. In frequency division duplex (FDD) systems, CSI at the receivers is obtained through an estimation of the channel using reference signals (RS) and it is subsequently sent back to the transmitters via a lowrate feedback channel. Thus, designing limited feedback schemes to reduce the amount of necessary feedback information plays an important role to achieve efficient communication systems. MIMO techniques can also enhance the performance of orthogonal frequency division multiplexing (OFDM) by exploiting the spatial domain. These systems are known as MIMOOFDM systems. OFDM is a technique used to mitigate the effects of intersymbol interference in frequency selective channels, turning a broadband frequency selective channel into a set of parallel narrowband frequency flat subchannels [11]. For these systems, multiuser precoding techniques can be carried out independently in each one of the subchannels.
In MIMOOFDM systems, the amount of CSI that the user equipments (UEs) need to feed back to the transmitter is related to the number of subcarriers or the length of the channel impulse response (CIR). For instance, long term evolution (LTE) Rel. 8 supports a scalable bandwidth up to 20 MHz [12], but it does not satisfy the International Mobile TelecommunicationsAdvanced (IMTAdvanced) requirements defined by the International Telecommunication Union. Due to this, LTEAdvanced presents some new radio features [13], such as carrier aggregation (CA), in order to improve the peak data rate. CA allows a contiguous or noncontiguous aggregation of bandwidth up to 100 MHz [14], which stands for 6,000 modulated LTE subcarriers. Not only an increase in user data rates is provided but also a more flexible and optimal utilization of frequency resources. However, since the UE is using a higher number of subcarriers, the amount of information that needs to be fed back is larger too.
In terms of feedback, the simplest generalization of MIMO systems to MIMOOFDM systems would require feeding back independent CSI information per subcarrier. However, this solution is inefficient, since it neglects the frequency correlation between subcarriers. In systems allowing CA, it would mean a large amount of feedback overhead. In order to reduce the feedback information, some frequencydomain techniques take advantage of the channel frequency correlation, grouping adjacent subcarriers. This approach has been adopted for LTE and LTEAdvanced, where groups of 12 adjacent subcarriers are known as resource blocks (RBs) [13]. A common approach in limited feedback schemes consists on assuming that the channel is constant for the subcarriers within a RB. This assumption holds under some conditions based on the channel coherence bandwidth and the feedback rate [15].
Limited feedback schemes for MIMOOFDM systems have been widely proposed in the literature, and we will comment the most representative as in [16–19]. A frequencydomain limited feedback scheme is presented in [16]. The beamforming matrix for the pilot subcarrier within each RB is calculated, quantized through random vector quantization (RVQ) and fed back by the receiver. The beamforming matrices for nonpilot subcarriers are obtained through a spherical interpolation at the transmitter. In [17], the frequency correlation is exploited by dividing the channel frequency response (CFR) into smaller vectors and performing a RVQ over them. The length of these vectors is related to the frequency correlation properties of the channel (i.e., the channel coherence bandwidth). However, correlation between subcarriers can be difficult to exploit and computationally expensive. In order to avoid complex frequency interpolation operations, a timedomain channel quantized feedback scheme is presented in [18], comparing it with two different frequencydomain channel quantization schemes: an analog feedback scheme and a direction quantized feedback scheme. It is shown that the scheme based on timedomain channel quantization outperforms frequencydomain schemes in terms of system sumrate, requiring lower complexity. In [19], the amount of information to feed back is reduced by exploiting temporal and spatial correlation through rank reduction. However, statistical channel information, such as the channel covariance matrix, has to be estimated and also fed back, but it allows for a robust precoder design at the transmitter as an advantage.
Spectral efficiency is one of the targets of IMTAdvanced. High spectral efficiency can be achieved by means of high or full frequency reuse. However, intercell interference (ICI) increases, limiting the system throughput especially at the cell edge. In LTEAdvanced, coordinated multipoint (CoMP) transmission/reception has been considered as a key technique to mitigate ICI and, thus, to improve the spectral efficiency [20–22]. Joint processing (JP), also known as network MIMO, is one of the techniques falling under the umbrella of CoMP. This technique consists of several coordinated cells acting as a single and distributed antenna array, simultaneously transmitting to the different UEs. With JP, ICI can be reduced applying MUMIMO techniques in the distributed antenna array. However, one of its drawbacks is the large amount of required feedback information, since users need to send back CSI of every coordinated cell. In addition, a large signaling overhead is required for the intercell information exchange [23]. In order to alleviate these requirements, the system is usually divided into clusters of cells (coordinated clusters) and JP is performed by the cells within each cluster [21]. In this framework, limited feedback schemes could contribute to further reduce the feedback overhead, bringing CoMP techniques close to practical systems.
In this article, we propose a lowcomplexity limited feedback scheme based on timedomain channel quantization for a cluster allowing JP. The limited feedback scheme exploits the spatial correlation between the different antennas of each base station (BS) without requiring any previous statistical knowledge of the channel. In our system, UEs are assumed to perfectly estimate their channels. The reduction of feedback information is achieved by means of a differential quantization (DQ) of the CIR coefficients. The contributions of this article can be summarized as follows:

A proper pilot symbol allocation grid based on LTEAdvanced allowing the pilot channel estimation in the cluster under consideration has been proposed.

Different strategies regarding feedback bit allocation for the proposed feedback scheme have been analyzed. A practical expression of the error introduced by this scheme has been obtained and compared to the error of the standard quantization scheme.

The effect of imperfect CSI on some multiuser precoding techniques at the downlink, such as ZF, THP and LRTHP, has been investigated. An expression that relates the achieved sumrate and the amount of feedback information needed has also been obtained for a general case.
The article is organized as follows. In Section 2, the system model and the pilot symbol allocation scheme for the cluster layout under consideration is presented. The main contribution of this article is presented in Section 3, where the limited feedback scheme is described. The evaluation of the impact of the limited feedback scheme on the downlink using different precoding techniques is carried out in Section 4. The simulation environment and numerical results are presented in Section 5. Finally, conclusions are stated in Section 6.
The following notation is used throughout the article: boldface uppercase letters denote matrices, A, boldface lowercase letters denote vectors, a, and italics denote scalars, a. Superscripts (·)^{ T }, (·)^{ H }, (·)^{1}, (·)^{†} stand for matrix transpose, Hermitian transpose, inversion and pseudoinverse operations, respectively. The Frobenius norm of a matrix is denoted by ∥ · ∥_{ F }, and ∥ · ∥ stands for the Euclidean norm of a vector. We use $\left\cdot \right,\angle \left(\cdot \right),\mathcal{R}\left\{\cdot \right\}$ and $\mathcal{I}\left\{\cdot \right\}$ to refer to absolute value, phase, real part and imaginary part of a complex value, respectively. We use ℂ^{m×n}to denote the set of m × n complex matrices. Regarding quantization, ${\mathcal{Q}}_{B}^{X}\left(\cdot \right)$ with X being G, U or L denote a scalar quantization using B bits and an optimal nonuniform codebook for an input signal with a Gaussian, Uniform or Laplacian probability density function (PDF), respectively. For simplicity, we will refer to them as Gaussian, Uniform or Laplacian quantization, respectively. The rest of calligraphic letters denote sets and denotes the cardinality of the set . Finally, $\mathcal{E}\left[\cdot \right]$ denotes the expectation operator.
2. System model
where $\text{H}\left[k\right]=\left[{\text{H}}_{1}\left[k\right],\dots ,{\text{H}}_{\left\mathcal{B}\right}\left[k\right]\right]\in {\u2102}^{J\times \left(\left\mathcal{B}\right\cdot {N}_{t}\right)}$ is the aggregated channel matrix and ${\text{H}}_{b}\left[k\right]={\left[{\text{h}}_{1,b}\left[k\right]\dots {\text{h}}_{J,b}\left[k\right]\right]}^{T}\in {\u2102}^{J\times {N}_{t}}$. Vector y[k] ∈ ℂ^{ J× }^{1} and n[k] ∈ ℂ^{ J× }^{1} collect the received symbols and the noise components, respectively, for the J UEs in the system. Vector $\text{x}\left[k\right]={\left[{\text{x}}_{1}^{T}\left[k\right],\dots ,{\text{x}}_{\left\mathcal{B}\right}^{T}\left[k\right]\right]}^{T}\in {\u2102}^{\left(\left\mathcal{B}\right\cdot {N}_{t}\right)\times 1}$ collects the precoded signal of the different sectors, which is obtained from the precoding techniques analyzed in Section 4. Vector n is the received circular complex additive white Gaussian noise with zero mean and variance ${\sigma}_{n}^{2}$.
2.1. Pilot symbol allocation
The LTE slot, also used in LTEAdvanced, is composed by seven OFDM symbols with a duration of 0.5 ms [12], whereas the LTE subframe consists of two LTE slots. In each one of the OFDM symbols, there are N_{IFFT} subcarriers. The subcarrier spacing is Δf = 15 kHz and it remains constant for the different bandwidth configurations. The sampling frequency f_{ s }is proportional to N_{IFFT}. However, not all the subcarriers are modulated. Only K over N_{IFFT} subcarriers are used, that are placed around the zero frequency in the baseband spectrum. Unmodulated subcarriers are placed at the edges as a guard band. In LTEAdvanced, channel state information reference signals (CSIRSs) have been introduced for the use of up to eight transmit antennas [25]. However, for backward compatibility, the CSIRSs must be placed in resource elements (REs) which do not contain cellspecific reference signals (CRSs) or user equipment specific reference signals (UERSs) [25].
More advanced pilot allocation schemes using combinations of FDM, TDM and CDM are presented in [26–28]. However, the evaluation of the different pilot allocation schemes is out of the scope of the article, since the main objective is the design of a limited feedback scheme for the coordinated cluster presented in Section 2. In the remaining of the article, we assume that the UE obtains an errorfree channel estimation through a simple least square (LS) estimation [29]. It should be noticed that the presence of a guard band with unmodulated subcarriers causes an ill conditioning problem in the LS estimation. Thus, different solutions, such as the ones presented in [30, 31], need to be applied in order to achieve an accurate estimation.
3. Limited feedback scheme
As stated in Section 1, a reliable CSI plays an important role in wireless communication systems. Limited feedback schemes for MIMO and MUMIMO systems have been extensively studied in the literature [32]. However, despite the fact that MUMIMO and CoMP MUMIMO channel representations are quite similar, some important differences between them should be pointed out [21]. In coordinated clusters, users can experiment different path loss coefficients in channels from the different BSs. Due to this, in [33] different percell codebooks are used. Another important difference is that channel information in CoMP systems is usually larger, since there can be up to $\left\mathcal{B}\right\cdot {N}_{t}$ transmit antennas instead of N_{ t }. In [24, 34], cluster techniques are proposed to reduce the overhead requirements. The slow variances of the channel within the coherence time can also be exploited in limited feedback schemes. In [35], a hierarchical codebook design method which makes use of the temporal correlation is proposed to reduce the feedback overhead in coordinated clusters.
One of the characteristics of the SCM channel is that the channel is generated without explicitly setting any spatial correlation parameter. A more detailed analysis of the spatial correlation in the SCM channel can be found in [38]. This study shows that the spatial crosscorrelation function of the SCM is related to the joint distribution of the angle of arrival (AoA) and the AoD through the different paths and subpaths.
DQ feedback scheme
A. UE: channel quantization and feedback  

1. ${\stackrel{\u0303}{h}}_{{s}^{\prime}n}={\mathcal{Q}}_{{B}_{R}}^{\text{G}}\left(\mathcal{R}\left\{{h}_{{s}^{\prime}n}\right\}\right)+j{\mathcal{Q}}_{{B}_{R}}^{\text{G}}\left(\mathcal{I}\left\{{h}_{{s}^{\prime}n}\right\}\right)$  (1.A) 
2. ${\stackrel{\u0303}{h}}_{sn}^{\text{dif}}={\mathcal{Q}}_{{B}_{M}}^{\text{L}}\left(\left{h}_{sn}/{\stackrel{\u0303}{h}}_{{s}^{\prime}n}\right\right)$  (1.B) 
$\stackrel{\u0303}{\gamma}={\mathcal{Q}}_{{B}_{P}}^{\text{U}}\left(\angle \left({h}_{sn}\right)\right)$  (1.C) 
B. BS: channel reconstruction  
1. ${\stackrel{\u0303}{h}}_{sn}={\stackrel{\u0303}{h}}_{sn}^{\text{dif}}\left{\stackrel{\u0303}{h}}_{{s}^{\prime}n}\right\text{exp}\left(j\stackrel{\u0303}{\gamma}\right)$  (1.D) 
One important point to note here is that, in order to reduce the quantization error, ${\stackrel{\u0303}{h}}_{sn}^{\text{dif}}$ is obtained from the quantized version of h_{ s'n }, that is, ${\stackrel{\u0303}{h}}_{{s}^{\prime}n}$. Thus, the BS can reconstruct the parameter with lower quantization error. On the other hand, the parameter ∠(h_{ sn }) presents a uniform distribution in [π, π), hence it is quantized through uniform quantization. The fact of quantizing ∠(h_{ sn }/h_{ s'n }) instead of ∠(h_{ sn }) does not have any benefit since both variances are similar. Therefore, additional mathematical operations can be avoided by quantizing ∠(h_{ sn }) directly. The reconstructed coefficients at the BS once ${\stackrel{\u0303}{h}}_{sn}^{\text{dif}}$ and $\stackrel{\u0303}{\gamma}$ have been received are expressed in Equation (1.D) in Table 1.
Finally, in order to improve the stability of $\left{h}_{sn}/{\stackrel{\u0303}{h}}_{{s}^{\prime}n}\right$ quantization, the UE can choose the reference antenna as the central one (s' = 2, 3) showing a greater $\left{\stackrel{\u0303}{h}}_{{s}^{\prime}n}\right$, using only one additional bit for sending back this information.
4. Impact on precoding techniques
where $\stackrel{\u0303}{\text{H}}\left[k\right]$ and H[k] are the estimated aggregated channel matrix at the BS and the true aggregated channel, respectively, for the k th subcarrier. Matrix E[k], whose entries are i.i.d. and follow a $\text{CN}\left(o,{\sigma}_{e}^{2}\right)$ distribution, represents the additive error in the channel matrix due to the channel quantization. Thus, the precoding design at the BS is obtained from $\stackrel{\u0303}{\text{H}}\left[k\right]$ instead of H[k].
In the following subsections, different precoding techniques, such as ZF, THP and LRTHP, are described and the effect of quantization error on them is analyzed. For the sake of simplicity, the frequency indexes k are omitted since the precoding process is performed over each subcarrier separately.
4.1. Zeroforcing
4.2. TomlinsonHarashima precoding
4.3. Latticereduction TomlinsonHarashima precoding
where matrix T is a J × J unimodular matrix with integer elements and W is a matrix with the same dimensions but better orthogonality properties than the original channel matrix $\stackrel{\u0303}{\text{H}}$. Since the traditional LLL algorithm originally worked with a real lattice basis, most authors use the realvalued equivalent matrix of the complexvalued channel matrix. However, this approach doubles the channel matrix dimension and can be avoided by using a complex version of the LLL algorithm [46].
These expressions are the same that in Equations (24) and (25), except from the fact that l_{ jj }comes from matrix L which has been obtained through an LQ decomposition of the reduced channel W instead of the original one $\stackrel{\u0303}{\text{H}}$. Since W shows better orthogonality properties than $\stackrel{\u0303}{\text{H}}$, a better performance is obtained with this scheme [6]. It should be noticed that the overall computational cost of this scheme increases considerably due to the lattice reduction process. Some efficient computational algorithms to reduce the overall cost of this scheme can be found in the literature, for example, in [47].
5. Numerical results
Channel and system parameters
Parameter  Value 

Intersite distance  500 m 
Channel model  3GPP SCME 
Channel scenario  Suburban macro 
Number of paths (N)  6 
Carrier frequency  2 GHz 
Sampling frequency  30.72 MHz 
Bandwidth  20 MHz 
CP length (μ s/samples)  4.69/144 
Shadowing standard deviation  8 dB 
Number of subcarriers (N_{IFFT})  2,048 
Number of used subcarriers (K)  1,200 
Number of used RB  100 
Number of coordinated sectors in the cluster $\left(\left\mathcal{B}\right\right)$  3 
BS antennas per sector (N_{ t })  4 
BS antenna spacing  λ/2 
UE number (J)  8 
Signal constellation  64QAM 
5.1. Performance of limited feedback schemes
In this subsection, we discuss the relation between the number of bits employed to quantize the different parameters in our DQ limited feedback scheme and its performance. Therefore, Gaussian quantization is used for the real and imaginary parts of h_{ s'n }(Equation (1.A)) and separate modulo and phase quantization is performed for h_{ sn }with s ≠ s' (Equations (1.B) and (1.C)).
Configurations of the differential quantizer
♯  B _{ R }  B _{ M }  B _{ P }  Cost  MSE 

1  7  3  5  643  0.451 
2  7  3  6  687  0.279 
3  7  3  7  732  0.233 
4  7  3  8  776  0.222 
5  7  4  5  687  0.316 
6  7  4  6  732  0.144 
7  7  4  7  777  0.099 
8  7  4  8  821  0.089 
9  7  5  5  732  0.277 
10  7  5  6  776  0.101 
11  7  5  7  821  0.057 
12  7  5  8  866  0.049 
13  7  6  5  776  0.263 
14  7  6  6  821  0.089 
15  7  6  7  865  0.046 
16  7  6  8  910  0.037 
17  8  4  5  717  0.308 
18  8  4  6  763  0.132 
19  8  4  7  806  0.089 
20  8  4  8  850  0.077 
21  8  5  5  761  0.265 
22  8  5  6  806  0.090 
23  8  5  7  850  0.048 
24  8  5  8  894  0.037 
25  8  6  5  806  0.249 
26  8  6  6  850  0.078 
27  8  6  7  895  0.034 
28  8  6  8  939  0.025 
29  8  7  5  849  0.248 
30  8  7  6  895  0.075 
31  8  7  7  939  0.029 
32  8  7  8  985  0.018 
where ${\sigma}_{e}^{2}$ was introduced in Equation (11).
From a practical point of view, precoding techniques are much more sensitive to phase errors than magnitude errors. Therefore, for a given whole number of bits, for instance B_{ T }= 12, GQ must use B_{ G }= 6 bits in quantizing real and imaginary parts of h_{sn}, obtaining an MSD of 10^{}^{3} for the phase, whereas DQ can allocate B_{ P }= 7 bits in the phase and B_{ M }= 5 bits in the modulus ratio, achieving a reduced MSD in the reconstructed phase of 2·10^{}^{4}. It should be noted that the feedback bit allocation strategy has been evaluated through the simulation of different configurations of the DQ scheme. However, ratedistortion theory could provide a framework for deriving the optimal feedback bit allocation.
Summarizing, DQ outperforms GQ in terms of MSE with respect to the same number of bits (Figure 9) and shows a higher flexibility regarding feedback bit allocation.
5.2. Performance of precoding techniques
In the previous subsection we have analyzed different feedback bit allocations for the DQ scheme, carrying out a performance comparison between this scheme and GQ feedback schemes in terms of MSE. However, from a practical point of view, it is more interesting to evaluate the cluster performance in terms of bit error rate (BER) and sumrate. In this subsection, we compare how the different precoding techniques can deal with imperfect channel information due to quantization when using different limited feedback schemes. Note that either BER or sumrate could be further improved by means of power allocation like loading strategies or waterfilling. Nevertheless, these techniques may result in some users being dropped due to their channel condition. In this article, we are interested in comparing both feedback schemes and the performance of precoding techniques under quantized channels. Thus, we do not use any particular power allocation technique.
where C represents the cost expressed as the number of bits per UE and k and p are fitting parameters. For the GQ scheme, k ≈ 1.860 and p ≈ 3.712 · 10^{3}, whereas for the DQ scheme, k ≈ 1.465 and p ≈ 4.013 · 10^{3}. From this result, we can state that both feedback schemes have a quite similar slope and, therefore, the difference between them remains almost constant.
Figure 12 also shows an interesting tradeoff between processing complexity and the amount of feedback information. Given a certain DQ configuration with a certain precoding technique, there are two choices to reduce the BER. The first one would be to increase the amount of feedback information per user, that is, to increase the cost. This choice would involve an increase of the signaling overhead and would reduce the system efficiency. The second choice would be to increase the complexity of the precoding technique. Substituting ZF precoding by THP or LRTHP, the BER would decrease at the cost of increasing the computational cost of the precoding stage.
Regarding the different precoding schemes, Figure 14 shows that LRTHP can provide a gain around 4 dB over THP, whereas THP outperforms ZF with a gain around 2 dB. It is also interesting to point out that the different precoding techniques also achieve different levels of error floor for SNRs higher than 20 dB. In Figure 15, we can see that LRTHP is the best precoding technique to deal with the noise and interference due to the quantized channel information. However, it is important to realize that THP performs closer to LRTHP than to ZF. In the band of SNRs that is not completely limited by the interference (from 15 to 25 dB), LRTHP provides a gain around 3 dB over THP, whereas the gain of THP over ZF increases up to more than 5 dB. In this figure, it also can be observed the tradeoff stated before. A system using a GQ scheme and THP precoding with a system SNR of 30 dB provides a system sumrate of 6.3 bps/Hz approximately. If we want to increase the sumrate, we could use DQ instead of GQ or we could use LRTHP precoding instead of THP. Both choices offer the same sumrate (see Figure 15). The first option will increase slightly the complexity of quantization at the UE whereas the second option will increase considerably the computational complexity at the BS [7].
6. Conclusion
In this article, a lowcomplexity limited feedback scheme based on timedomain channel quantization for a coordinated cluster allowing JP has been presented. The channel estimation is performed using the proposed pilot symbol allocation grid for a coordinated cluster and the CSI is fed back through the proposed scheme. This scheme takes advantage of the spatial correlation between antennas without requiring a statistical knowledge of the channel or a higher computational complexity, carrying out a DQ over the CIR. Its performance has been compared with the standard quantization of the CIR in a CoMP scenario. The simulation results show that the proposed scheme outperforms the scheme based on standard quantization in terms of MSE, offering a higher flexibility regarding feedback bit allocation.
The effect of imperfect CSI due to the limited feedback scheme has been evaluated on different precoding schemes: ZF, THP and LRTHP. An expression that relates the sumrate with the number of feedback bits for a general precoding case has been obtained. The proposed scheme achieves a higher sumrate than the scheme based on standard quantization for the same number of feedback bits. Simulation results also show that the proposed scheme achieves a better performance in terms of sumrate and BER when ZF, THP or LRTHP techniques are used.
Among the evaluated precoding techniques, numerical results show that the highest robustness against imperfect CSI is achieved with LRTHP at the cost of a higher complexity. An interesting tradeoff between the precoding technique complexity and the amount of feedback information has been stated. Given a performance requirement, the amount of feedback information can be reduced by means of using a higher complexity precoding technique and vice versa.
Declarations
Acknowledgement
The authors would like to thank the anonymous reviewers for their valuable comments and suggestions, which have greatly helped to improve the quality of this work. This work has been supported by the Spanish Ministry of Science and Innovation through CICYT Grant TEC200913741, Regional Government Generalitat Valenciana through Grant PROMETEO/2009/013, and by Universitat Politècnica de València through its PAIDFPI program. C. Botella's work is supported by the Spanish MEC Grants CONSOLIDERINGENIO 2010 CSD200800010 "COMONSENS" and COSIMA TEC201019545C0401. The authors are within VLC/campus "Sustainable Communications and Computing (COCOS)" cluster.
Authors’ Affiliations
References
 Costa M: Writing on dirty paper (Corresp.). IEEE Trans Inf Theory 1983, 29(3):439441. 10.1109/TIT.1983.1056659View ArticleGoogle Scholar
 Goldsmith A, Jafar S, Jindal N, Vishwanath S: Capacity limits of MIMO channels. IEEE J Sel Areas Commun 2003, 21(5):684702. 10.1109/JSAC.2003.810294View ArticleGoogle Scholar
 Caire G, Shamai S: On the achievable throughput of a multiantenna gaussian broadcast channel. IEEE Trans Inf Theory 2003, 49(7):16911706. 10.1109/TIT.2003.813523MathSciNetView ArticleGoogle Scholar
 Peel C, Hochwald B, Swindlehurst A: A vectorperturbation technique for nearcapacity multiantenna multiuser communicationpart I: channel inversion and regularization. IEEE Trans Commun 2005, 53: 195202. 10.1109/TCOMM.2004.840638View ArticleGoogle Scholar
 Windpassinger C, Fischer R, Vencel T, Huber J: Precoding in multiantenna and multiuser communications. IEEE Trans Wirel Commun 2004, 3(4):13051316. 10.1109/TWC.2004.830852View ArticleGoogle Scholar
 Xu D, Huang Y, Yang L: Improved nonlinear multiuser precoding using lattice reduction. Signal Image Video Process 2009, 3: 4752. 10.1007/s1176000800583View ArticleGoogle Scholar
 Roger S, Domene F, Gonzalez A, Almenar V, Piñero G: An evaluation of precoding techniques for multiuser communication systems. In Proceedings of the 7th IEEE International Symposium on Wireless Communication Systems. York; 2010:295299.Google Scholar
 Shu F, Gang W, ShaoQian L: Optimal multiuser MIMO linear precoding with LMMSE receiver. EURASIP J Wirel Commun Netw 2009, 2009: 110.View ArticleGoogle Scholar
 Gesbert D, Kountouris M, Heath R, Chae CB, Salzer T: Shifting the MIMO paradigm. IEEE Signal Process Mag 2007, 24(5):3646.View ArticleGoogle Scholar
 Conte E, Tomasin S, Benvenuto N: A comparison of scheduling strategies for MIMO broadcast channel with limited feedback on OFDM systems. EURASIP J Wirel Commun Netw 2010, 2010: 112.View ArticleGoogle Scholar
 Stuber G, Barry J, McLaughlin S, Li Y, Ingram M, Pratt T: Broadband MIMOOFDM wireless communications. Proc IEEE 2004, 92(2):271294. 10.1109/JPROC.2003.821912View ArticleGoogle Scholar
 3GPP TS 36.211 V8.9.0: Evolved universal terrestrial radio access (EUTRA); Physical channels and modulation. 2009.Google Scholar
 3GPP TS 36.201, V10.0.0: Evolved universal terrestrial radio access (EUTRA); Physical layer  general description. 2010.Google Scholar
 Pedersen K, Frederiksen F, Rosa C, Nguyen H, Garcia L, Wang Y: Carrier aggregation for LTEadvanced: functionality and performance aspects. IEEE Commun Mag 2011, 49(6):8995.View ArticleGoogle Scholar
 Trivellato M, Tomasin S, Benvenuto N: On channel quantization and feedback strategies for multiuser MIMOOFDM downlink systems. IEEE Trans Commun 2009, 57(9):26452654.View ArticleGoogle Scholar
 Choi J, Heath R: Interpolation based transmit beamforming for MIMOOFDM with limited feedback. IEEE Trans Signal Process 2005, 53(11):41254135.MathSciNetView ArticleGoogle Scholar
 Domene F, Piñero G, de Diego M, Gonzalez A: Performance of multiuser MIMOOFDM precoding techniques with quantized channel information. In Proceedings of the 7th IEEE International Symposium on Wireless Communication Systems. York; 2010:5660.Google Scholar
 ShiraniMehr H, Caire G: Channel state feedback schemes for multiuser MIMOOFDM downlink. IEEE Trans Commun 2009, 57(9):27132723.View ArticleGoogle Scholar
 Castro P, Joham M, Castedo L, Utschick W: Robust MMSE linear precoding for multiuser MISO systems with limited feedback and channel prediction. In Proceedings of the 19th IEEE International Symposium on Personal, Indoor and Mobile Radio Communications. Cannes; 2008:15.Google Scholar
 3GPP TR 36.814, V9.0.0: Evolved universal terrestrial radio access (EUTRA); Further advancements for EUTRA physical layer aspects. 2010.Google Scholar
 Gesbert D, Hanly S, Huang H, Shamai Shitz S, Simeone O, Yu W: Multicell MIMO cooperative networks: a new look at interference. IEEE J Sel Areas Commun 2010, 28(9):13801408.View ArticleGoogle Scholar
 Irmer R, Droste H, Marsch P, Grieger M, Fettweis G, Brueck S, Mayer HP, Thiele L, Jungnickel V: Coordinated multipoint: concepts, performance, and field trial results. IEEE Commun Mag 2011, 49(2):102111.View ArticleGoogle Scholar
 Björnson E, Zakhour R, Gesbert D, Ottersten B: Cooperative multicell precoding: rate region characterization and distributed strategies with instantaneous and statistical CSI. IEEE Trans Signal Process 2010, 58(8):42984310.MathSciNetView ArticleGoogle Scholar
 Papadogiannis A, Gesbert D, Hardouin E: A dynamic clustering approach in wireless networks with multicell cooperative processing. In Proceedigns of the 2008 IEEE International Conference on Communications. Beijing; 2008:40334037.View ArticleGoogle Scholar
 3GPP TS 36.211, V10.0.0: Evolved universal terrestrial radio access (EUTRA); Physical channels and modulation. 2010.Google Scholar
 Minn H, AlDhahir N: Optimal training signals for MIMOOFDM channel estimation. IEEE Trans Wirel Commun 2006, 5(5):11581168.View ArticleGoogle Scholar
 Thomas T, Mondal B, Ghosh A: CSI reference signal designs for enabling closedloop MIMO feedback. In Proceedings of the 72nd IEEE Vehicular Technology Conference Fall. Ottawa; 2010:15.Google Scholar
 Koivisto T, Schober K, Kuosmanen T, Roman T, Enescu M: Reference signal design for flexible MIMO operation in LTEadvanced downlink. In Proceedings of the 71st IEEE Vehicular Technology Conference Spring. Taipei; 2010:15.Google Scholar
 Sesia S, Toufik I, Baker M: LTE: the UMTS Long Term Evolution: From Theory to Practice. John Wiley & Sons, Chichester; 2009.View ArticleGoogle Scholar
 Doukopoulos X, Legouable R: Robust channel estimation via FFT interpolation for multicarrier systems. In Proceedings of the 65th IEEE Vehicular Technology Conference Spring. Dublin; 2007:18611865.Google Scholar
 Ancora A, Bona C, Slock D: Downsampled impulse response leastsquares channel estimation for LTE OFDMA. In Proceedings of the 32nd IEEE International Conference on Acoustics, Speech and Signal Processing. Volume 3. Honolulu; 2007:293296.Google Scholar
 Love D, Heath R, Lau V, Gesbert D, Rao B, Andrews M: An overview of limited feedback in wireless communication systems. IEEE J Sel Areas Commun 2008, 26(8):13411365.View ArticleGoogle Scholar
 Cheng Y, Lau V, Long Y: A scalable limited feedback design for network MIMO using percell product codebook. IEEE Trans Wirel Commun 2010, 9(10):30933099.View ArticleGoogle Scholar
 Piñero G, Botella C, Gonzalez A, de Diego M, Cardona N: Downlink power control and beamforming for a cooperative wireless system. In Proceedings of the 15th IEEE International Symposium on Personal, Indoor and Mobile Radio Communications. Volume 2. Barcelona; 2004:974978.Google Scholar
 Kim JH, Zirwas W, Haardt M: Efficient feedback via subspacebased channel quantization for distributed cooperative antenna systems with temporally correlated channels. EURASIP J Adv Signal Process 2008, 2008: 113.Google Scholar
 Baum D, Hansen J, Salo J: An interim channel model for beyond3G systems: extending the 3GPP spatial channel model (SCM). In Proceedings of the 61st IEEE Vehicular Technology Conference Spring. Volume 5. Stockholm; 2005:31323136.Google Scholar
 3GPP Technical Report 25.996 V10.0.0: Spatial channel model for multiple input multiple output (MIMO) simulations. 2011.Google Scholar
 Wang CX, Hong X, Wu H, Xu W: Spatialtemporal correlation properties of the 3GPP spatial channel model and the kronecker MIMO channel model. EURASIP J Wirel Commun Netw 2007, 2007: 19.View ArticleGoogle Scholar
 3GPP Technical Report 25.996 V10.0.0: MATLAB implementation of the 3GPP spatial channel model. 2011.Google Scholar
 Max J: Quantizing for minimum distortion. IRE Trans Inf Theory 1960, 6: 712. 10.1109/TIT.1960.1057548MathSciNetView ArticleGoogle Scholar
 Sullivan G: Efficient scalar quantization of exponential and laplacian random variables. IEEE Trans Inf Theory 1996, 42(5):13651374. 10.1109/18.532878View ArticleGoogle Scholar
 Zhang R: Cooperative multicell block diagonalization with perbasestation power constraints. IEEE J Sel Areas Commun 2010, 28(9):14351445.View ArticleGoogle Scholar
 Kaviani S, Krzymień W: Optimal multiuser zero forcing with perantenna power constraints for network MIMO coordination. EURASIP J Wirel Commun Netw 2011, 2011: 112.View ArticleGoogle Scholar
 Windpassinger C: Detection and precoding for multiple input multiple output channels. PhD thesis. University ErlangenNurnberg, Germany; 2004.Google Scholar
 Lenstra A, Lenstra H, Lovász L: Factoring polynomials with rational coefficients. Math Ann 1982, 261: 515534. 10.1007/BF01457454MathSciNetView ArticleGoogle Scholar
 Gan YH, Ling C, Mow WH: Complex lattice reduction algorithm for lowcomplexity fulldiversity MIMO detection. IEEE Trans Signal Process 2009, 57(7):27012710.MathSciNetView ArticleGoogle Scholar
 Roger S, Gonzalez A, Almenar V, Vidal AM: Extended LLL algorithm for efficient signal precoding in multiuser communication systems. IEEE Commun Lett 2010, 14(3):220222.View ArticleGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.