A limited feedback scheme for massive MIMO systems based on principal component analysis
 Tiankui Zhang^{1}Email author,
 Anmeng Ge^{1},
 Norman C. Beaulieu^{1},
 Zhirui Hu^{2} and
 Jonathan Loo^{3}
DOI: 10.1186/s1363401603649
© Zhang et al. 2016
Received: 28 May 2015
Accepted: 12 May 2016
Published: 26 May 2016
Abstract
Massive multipleinput multipleoutput (MIMO) is becoming a key technology for future 5G cellular networks. Channel feedback for massive MIMO is challenging due to the substantially increased dimension of the channel matrix. This motivates us to explore a novel feedback reduction scheme based on the theory of principal component analysis (PCA). The proposed PCAbased feedback scheme exploits the spatial correlation characteristics of the massive MIMO channel models, since the transmit antennas are deployed compactly at the base station (BS). In the proposed scheme, the mobile station (MS) generates a compression matrix by operating PCA on the channel state information (CSI) over a longterm period, and utilizes the compression matrix to compress the spatially correlated highdimensional CSI into a lowdimensional representation. Then, the compressed lowdimensional CSI is fed back to the BS in a shortterm period. In order to recover the highdimensional CSI at the BS, the compression matrix is refreshed and fed back from MS to BS at every longterm period. The information distortion of the proposed scheme is also investigated and a closedform expression for an upper bound to the normalized information distortion is derived. The overhead analysis and numerical results show that the proposed scheme can offer a worthwhile tradeoff between the system capacity performance and implementation complexity including the feedback overhead and codebook search complexity.
Keywords
Massive MIMO Limited feedback Principal component analysis Information distortion analysis1 Introduction
The massive multipleinput multipleoutput (MIMO) system which deploys large numbers of transmit antennas at the base station (BS) has been listed as one of the key techniques for fifth generation (5G) cellular networks [1]. The deployment of numerous antennas enables massive MIMO systems to achieve not only higher system capacity, but also higher spectrum and energy efficiency than conventional MIMO systems [2, 3].
The superior performance of the massive MIMO systems relies on the spatial multiplexing and the minor multiuser interference. As is the case for conventional MIMO systems, this in turn requires the BS to have perfect knowledge of the downlink channel state information (CSI) [4]. In a time division duplexing (TDD) system, the channel reciprocity can be exploited to acquire the downlink CSI at the BS [5]. However, things become more challenging when the system operates in a frequency division duplexing (FDD) mode, where the channel reciprocity no longer holds. Therefore, a mobile station (MS) needs to feedback the downlink CSI through a ratelimited uplink channel. The authors in [6] drew the conclusion that the required feedback rate per user should be increased in proportion to the number of the transmit antennas for the sake of obtaining the full multiplexing gain. Therefore, feedback overhead turns into a key challenge in the massive MIMO systems.
The foundation of the works on feedback overhead reduction for MIMO systems is the correlation feature of MIMO channels. Limited feedback techniques for correlated MIMO channels were designed in [7–9]. A modified Grassmannian line packing codebook was proposed in [7], and the authors in [8, 9] rotated the codebook for i.i.d. channels with a unitary matrix to obtain the codebook for correlated MIMO channels. A systematic codebook was designed for quantized beamforming in [10], which was implemented by maps that can rotate and scale spherical caps on the Grassmannian manifold.
Furthermore, a codebook for uniform rectangular arrays (URA) for massive MIMO antennas was designed in [11, 12]. It was derived by the Kronecker product of two ULA codebooks. The authors in [13] proposed a feedback framework for FDD massive MIMO systems that divides the coverage area into subsectors, where each subsector is formed by a set of narrow beams that covers a preassigned area in azimuth and elevation. Noncoherent trelliscoded quantization and trellisextended codebooks for massive MIMO systems were proposed in [14, 15], which exploited a Viterbi decoder for CSI quantization and a convolutional encoder for CSI reconstruction. A projection based feedback compression was utilized to project the highdimensional channel space into a lower dimensional subspace [16]. However, [16] did not explain how to feedback the projection matrix.
The compressive sensing (CS)based limited feedback schemes for massive MIMO were proposed to reduce the feedback overhead by exploiting the spatial correlation of CSI [17–20]. The authors in [17] introduced CS to massive MIMO for limited feedback. A unique insight was provided that strong spatial correlations are exhibited in massive closelypacked antenna arrays, so channel vectors can be represented in sparse form in the spatialfrequency domain. Subsequently, a compressed analog feedback strategy for spatially correlated massive MIMO channels was proposed in [18]. In contrast to the strategy in [18], the lowdimensional CSI was quantized with a codebook and the preferred index was fed back in [19] and [20].
The choice of orthogonal basis, which is intended for the sparse representation of the original signal, plays an important role in the recovery of the original highdimensional signal at the BS. Two such kinds of orthogonal basis construction, the discrete cosine transform (DCT) and the KarhunenLoeve transform (KLT), are usually employed [21]. If the channel correlation matrix is neither known at the MS nor the BS, the signalindependent DCT basis is a better option. On the one hand, because of its signalindependent nature, the utilization of the DCT basis does not require the MS to inform the BS of the channel correlation matrix. On the other hand, this makes the DCT basis incapable of tracking the realtime change of channel state, which has a negative effect on system capacity. In contrast to the DCT basis, the KLT basis can excellently adapt to CSI change. Therefore, when MS and BS both know the instantaneous channel correlation matrix, the KLT basis can provide the optimal sparse representation, which promises accurate recovery even if only a small number of measurements are available. Unfortunately, the signaldependent nature of the KLT basis requests the MS to feedback channel correlation matrix instantaneously [22]. This can hardly be implemented in practical systems because of the heavy feedback overhead.
In this case, principal component analysis (PCA) can offer a tradeoff between system capacity and practical implementation [23, 24]. Compared with a DCT basis, PCA can be more adaptive to the change of the channel state, since PCA is signaldependent [23]. This guarantees PCA better system capacity than a DCT basis. Compared with a KLT basis, PCA only needs that the MS and the BS have knowledge of the channel correlation matrix in a longterm period. This makes PCA achieve feedback overhead reduction much better than a KLT basis. What is more, the most attractive characteristic of PCA is that it is effective for dimensionality reduction of highdimensional data [24], whose elements are correlated. Inspired by this, PCA has great potential to be applied to the compression of highdimensional CSI with strong spatial correlation to reduce feedback overhead in massive MIMO systems. To the best of our knowledge, there have not existed any works addressing a practical feedback scheme based on PCA.

A PCAbased feedback scheme for FDD massive MIMO systems is proposed. The operation procedures at the BS and the MS are divided into two types, which are longterm period operations and shortterm period operations. In more detail, the exact operation procedures both at the BS and the MS, as well as the derivation of the compression matrix at the MS, are presented. The distortion of the proposed scheme is analyzed. An upper bound to the normalized distortion is derived.

System performance comparisons of the PCAbased feedback scheme, the DCTbased CS scheme and the KLTbased CS scheme are presented. The feedback overhead and the codebook search complexity are analyzed and the system capacity performance is simulated. Looking at the simulation results and the feedback overhead analysis comprehensively, we draw the conclusion that our proposed scheme can achieve a compromise between system capacity and implementation complexity (feedback overhead and codebook search complexity).
The remainder of this paper is organized as follows. In Section 2, the massive MIMO system model is described. Section 2 first reviews the PCA method itself in Subsection and then provides the details of the proposed scheme in Subsection. Moreover, distortion of the proposed scheme is analyzed in Subsection. The feedback overhead as well as codebook search complexity comparison and numerical results follow in Section 4 and Section 5, respectively. Finally, the conclusion of this paper is presented in Secton 6.
Notation: Throughout this paper, upper and lower case boldfaces are used to describe matrix A and vector a, respectively. We denote the transpose and the conjugate transpose of matrix A or vector a by A ^{ T }(a ^{ T }) and A ^{ H }(a ^{ H }). In addiction, A ^{−1} denotes the inverse of a square matrix.
2 System model
We consider a downlink massive MIMO system, where there is a single cell, in which the BS equipped with N _{t} antennas serves K singleantenna MSs.
2.1 Spatially correlated massive MIMO channel
where \(\textbf {R}_{\text {Tx}}^{\frac {1}{2}}\) is the square root of the correlation matrix at the transmitter depicting the impact of insufficient interelement spacing and h _{one−ring} is derived from the onering model describing the spatial correlation caused by a scattering environment. Note that the correlation of the channel is time varying, due to the change of both the relative positions of scatterers and the correlation matrix at the transmitter.
where d _{ uv } is the distance between the two antenna elements and λ denotes the carrier wavelength.
where α is a constant and γ is the path loss exponent.
2.2 Downlink signal model
where P _{t} is the total transmit power of the BS. Equal power allocation is assumed with \({\frac {{{P_{\mathrm {t}}}}}{K}}\) being the power distributed to each MS.
3 Feedback scheme for massive MIMO
3.1 Review of principal component analysis
in which, \({\bar {\boldsymbol {\Psi }}}\) is composed of ldominating eigenvectors, the socalled principal components, which are selected from all b eigenvectors of X.
For the sake of determining which components are to be selected, the concept of contribution rate is introduced. Consider a descending ordering of the b eigenvalues λ _{1},λ _{2}…,λ _{ b }. Then, the contribution rate of the g ^{ t h } eigenvalue λ _{ g } is defined as \({\frac {{{\lambda _{g}}}}{{{\sum \nolimits }_{g = 1}^{b} {{\lambda _{g}}} }}}\), while the cumulative contribution rate of the top l eigenvalues can be expressed by \({\frac {{{\sum \nolimits }_{g = 1}^{l} {{\lambda _{g}}} }}{{{\sum \nolimits }_{g = 1}^{b} {{\lambda _{g}}} }}}\). Generally, when the cumulative contribution rate of the chosen l principal components exceeds a certain level, the information loss is acceptable.
3.2 Proposed PCAbased feedback scheme
A PCAbased feedback scheme for massive MIMO is proposed in this subsection. In the proposed scheme, different operations at the MS and the BS have different time periods, longterm period T _{l} and shortterm period T _{s}. Every longterm period T _{l} contains several shortterm periods T _{s}. In every T _{s}, the MS utilizes the compression matrix to compress highdimensional CSI into lowdimensional representation. Then, the compressed lowdimensional CSI is quantized by the RVQ codebook and the index of the preferred codeword is fed back to the BS. Because of the signaldependent nature of PCA, the compression matrix is derived by executing PCA on the CSI which is obtained through continuous channel estimation during a whole longterm period.
3.2.1 Compression matrix derivation
Procedure of deriving the compression matrix in each longterm period with the PCA method at the MS
Initialization  
S:  Number of the shortterm period T _{s} in every 
longterm period T _{l}.  
M:  Number of the dominating eigenvectors chosen 
to form the compression matrix.  
Procedures  
1)  Channel estimation: perform channel 
estimation continuously in T _{l} ^{(n)} to obtain  
S highdimensional channel vectors  
\({\textbf {h}_{k}^{\left ({n,1} \right)},\textbf {h}_{k}^{\left ({n,2} \right)} \ldots, \textbf {h}_{k}^{\left ({n,S} \right)}}\).  
2)  Computation of covariance matrix: \({\tilde {\mathbf {H}}^{\left (n \right)} =}\) 
\({{\left [ {{{\left ({\textbf {h}_{k}^{\left ({n,1} \right)}} \right)}^{H}},{{\left ({\textbf {h}_{k}^{\left ({n,2} \right)}} \right)}^{H}}\ldots,{{\left ({\textbf {h}_{k}^{\left ({n,S} \right)}} \right)}^{H}}} \right ]^{H}}}\)  
\({\in {\mathbb {C}^{S \times {N_{\mathrm {t}}}}}}\) and compute its covariance matrix.  
3)  Eigendecomposition: perform eigen 
decomposition on the above covariance matrix  
\({\text {Cov}\left ({{{\tilde {\mathbf {H}}}^{\left (n \right)}},{{\tilde {\mathbf {H}}}^{\left (n \right)}}} \right) = {\textbf {U}^{\left (n \right)}}{\textbf {D}^{\left (n \right)}}{\left ({{\textbf {U}^{\left (n \right)}}} \right)^{H}}}\).  
4)  Formation of compression matrix: choose M 
dominating eigenvectors to form compression  
matrix \({{\bar {\mathbf {U}}^{\left (n \right)}} \in {\mathbb {C}^{{N_{\mathrm {t}}} \times M}}}\). 
The compression matrix obtained in the longterm period T _{l} ^{(n)} will be used by the MS to compress 1×N _{t} channel vectors into 1×M vectors, as well as by the BS to perform recovery in the period T _{l} ^{(n+1)}.
3.2.2 MS operation
The main operations at the MS can be classified into two types: longterm period operations and shortterm period operations.
In the longterm period \({T_{\mathrm {l}}^{\left (n \right)}}\), the MS performs the continuous channel estimation to obtain S highdimensional channel vectors, and derives the compression matrix \({{\bar {\mathbf {U}}^{\left (n \right)}}}\). Then, each column of the compression matrix is quantized by another RVQ codebook. After quantization, the compression matrix \({{\bar {\mathbf {U}}^{\left (n \right)}}}\) is fed back to the BS at the end of \({T_{\mathrm {l}}^{\left (n \right)}}\).
Operation in the s ^{th} (s=1,2…,S) shortterm period of the n ^{th} longterm period, \({T_{\mathrm {s}}^{\left ({n,s} \right)}}\), is described as follows:
Step 1. Channel estimation is performed to obtain a 1×N _{t} channel vector \({\textbf {h}_{k}^{\left ({n,s} \right)}}\).
By this step, the original highdimensional CSI (1×N _{t}) is compressed into a lowdimensional representation (1×M). The compression ratio is \({\frac {M}{{{N_{\mathrm {t}}}}}}\).
where c _{ j } is the j ^{th} codeword of the codebook. Compared with the quantizing highdimensional CSI directly, the RVQ codebook used above can be designed to be much smaller. This not only reduces the feedback overhead, but also decreases the codebook search complexity.
Step 4. The index of the preferred codeword is fed back to the BS.
3.2.3 BS operation
Similarly, the main operation at the BS can also be classified into longterm period operation and shortterm period operation. At the end of \({T_{\mathrm {l}}^{\left (n \right)}}\), the BS receives the compression matrix \({{\bar {\mathbf {U}}^{\left (n \right)}}}\) to perform highdimensional CSI recovery in the next longterm period \({T_{\mathrm {l}}^{\left ({n + 1} \right)}}\). Meanwhile, the shortterm period operation follows the steps below:
Step 1. The codeword index j ^{(n,s)} is received in each shortterm period;
Step 2. As the BS and the MS share the same codebook, it is easy for the BS to find the quantized lowdimensional CSI \({\hat {\mathbf {h}}_{k}^{\left ({n,s} \right)}}\) by letting \({\hat {\mathbf {h}}_{k}^{\left ({n,s} \right)} = {\textbf {c}_{{j^{\left ({n,s} \right)}}}}}\);
where \({{\bar {\mathbf {U}}^{\left ({n  1} \right)}}}\) is derived from the period \({T_{\mathrm {l}}^{\left ({n  1} \right)}}\).
3.3 Distortion analysis of proposed scheme
Different representations of CSI in different stages
Representations  Implications 

h  Original highdimensional CSI estimated by MS 
Lowdimensional CSI after normalization  
\({\bar {\mathbf {h}}}\)  and compression, that is \({\bar {\mathbf {h}} = \frac {{{\mathbf {h}\bar {\mathbf {U}}}}}{{\left \ {{\mathbf {h}\bar {\mathbf {U}}}} \right \}}}\) 
\({\hat {\mathbf {h}}}\)  Quantized lowdimensional CSI 
h⌢  Highdimensional CSI recovered from \({\hat {\mathbf {h}}}\) 
Highdimensional CSI recovered from \({\bar {\mathbf {h}}}\)  
\({\tilde {\mathbf {h}}}\)  (with no quantization error of lowdimensional CSI) 
3.3.1 Distortion analysis of PCA
where u _{ i } denotes the i ^{th} basis vector.
where S denotes the number of shortterm periods in a longterm period.
Proposition.
Proof.
See Appendix A.
3.3.2 Distortion analysis of quantization
where B _{2} denotes the number of feedback bits of the compression matrix.
Before analyzing the distortion between the original highdimensional CSI h and the reconstructed h⌢, we first focus on how to express h⌢ in terms of h.
Proposition.
where \({{\textbf {I}_{M}} = \left [ {\begin {array}{*{20}{c}} {{\textbf {I}_{M \times M}}}&{{{\textbf {0}}_{M \times \left ({{N_{\mathrm {t}}}  M} \right)}}}\\ {{{\textbf {0}}_{\left ({{N_{\mathrm {t}}}  M} \right) \times M}}}&{{{\textbf {0}}_{\left ({{N_{\mathrm {t}}}  M} \right) \times \left ({{N_{\mathrm {t}}}  M} \right)}}} \end {array}} \right ]}\).
Proof.
See Appendix B.
Having derived an expression for h⌢ in terms of h, our purpose is to analyze the distortion of the proposed scheme. We derive an upper bound to the normalized distortion (denoted by δ) between h⌢ and h. Instead of calculating δ directly, we first calculate the normalized similarity (denoted by ρ) between h⌢ and h. Then, δ can be conveniently obtained by δ=1−ρ.
Before calculating the normalized similarity ρ between h⌢ and h, it is insightful to look at the nonnormalized similarity \({\mathbb {E}\left [ {\left  {\textbf {h}{{\stackrel {\frown }{\mathbf {h}}}^{H}}} \right } \right ]}\).
Proposition.
where \(A={\left ({{N_{\mathrm {t}}}  \sum \limits _{i = M + 1}^{{N_{\mathrm {t}}}} {{\lambda _{i}}}} \right)}\).
Proof.
which are proved in Appendix C. Moreover, the upper boundary of \({\mathbb {E}\left [ {{D^{2}}} \right ]}\) and \({\mathbb {E}\left [ {{D^{2}}} \right ]}\) are \({{2^{ \frac {{{B_{2}}/M}}{{{N_{\mathrm {t}}}  1}}}}}\) and \({{2^{ \frac {{{B_{1}}}}{{M  1}}}}}\), respectively. Consequently, we obtain (20).
4 Implementation complexity analysis
This section analyzes the feedback overhead and the codebook search complexity of the proposed scheme. For comparison, the existing CSbased schemes utilizing the KLT basis and the DCT basis are also taken into account.
where ρ denotes the received signaltonoise ratio (SNR) in decibels at the MS.
where M is the number of the principal components, the first term denotes the number of feedback bits for quantizing lowdimensional CSI, and the second term is caused by the quantization of the compression matrix.
For the DCTbased CS scheme, there is no need for the MS to inform the BS of the channel correlation matrix, due to the signalindependent nature of the DCT basis. So the number of feedback bits for DCTbased CS is given by \({{B_{\text {DCT}}} = S \cdot \frac {{\left ({M  1} \right)\rho }}{3}}\).
As for the codebook search complexity, it is proportional to the number of conjugate multiplications when searching for the best codeword. So the search complexity of the proposed scheme, DCTbased CS and KLTbased scheme in a longterm period can be expressed as \({S \cdot M \cdot {2^{\frac {{\left ({M  1} \right)\rho }}{3}}} + M \cdot {N_{\mathrm {t}}} \cdot {2^{\frac {{\left ({{N_{\mathrm {t}}}  1} \right)\rho }}{3}}}}\), \({S \cdot M \cdot {2^{\frac {{\left ({M  1} \right)\rho }}{3}}}}\) and \({S \cdot M \cdot {2^{\frac {{\left ({M  1} \right)\rho }}{3}}} + S \cdot {N_{\mathrm {t}}}^{2} \cdot {2^{\frac {{\left ({{N_{\mathrm {t}}}  1} \right)\rho }}{3}}}}\), respectively.
5 Simulation results
Simulation parameters
Parameters  Assumption 

Antenna configuration of BS  ULA 0.5λ spaced 
Feedback channel  Lossless & without delay 
Carrier frequency  2.6 GHz 
Bandwidth  10 MHz 
Cell radius  200 m 
Shortterm period  1 ms 
Longterm period  10 ms 
Radius of scatterer ring r  10 m 
Number of scatterers Q  10 
Path loss exponent γ  2.5 
Constant α in channel model  10^{7} 
5.1 Feasibility validation
5.2 Evaluations of the proposed scheme
Figure 4 shows the effect of the compression ratio \(\left ({\eta = \frac {M}{{{N_{\mathrm {t}}}}}}\right)\) on the system capacity. The comparison is among the proposed scheme, DCTbased CS and KLTbased CS. We can observe that whether in low or high compression ratio regimes, the KLTbased CS has the best performance, while the DCTbased CS performs the worst. To be emphasized, the best performance of the KLTbased CS is at a sacrifice of increased feedback overhead, as shown in Table 3. In this sense, the proposed scheme can offer a useful tradeoff. Additionally, as Fig. 4 shows, the proposed scheme performs much better than DCTbased CS in low compression ratio regimes.
Figure 5 illustrates the recovery performance of the highdimensional CSI at the BS under the circumstances that the BS has perfect knowledge of the lowdimensional CSI without quantization. We take the proposed scheme and DCTbased CS for comparison. The 1×N _{t} original CSI is compressed into 1×M _{CS} lowdimensional information, where M _{CS}=20 and the compression ratio is \({{\eta _{\text {CS}}} = \frac {{{M_{\text {CS}}}}}{{{N_{\mathrm {t}}}}} \approx 0.16}\), while in the proposed scheme, the number of principal components is M _{PCA}=4 with the compression ratio being \({{\eta _{{\text {PCA}}}} = \frac {{{M_{\text {PCA}}}}}{{{N_{\mathrm {t}}}}} \approx 0.03}\).
As can be seen, the reconstructed highdimensional CSI is considerably close to the original data when the PCA is utilized. But there still exists distortion because the PCA itself inevitably introduces information loss. However, the recovery performance gets poorer in the case of the DCTbased CS. The reason is that the proposed scheme takes advantage of the signaldependent nature of PCA, which makes it possible for the compression matrix to change adaptively in every longterm period according to the variation of the original data.
When we utilize the RVQ codebook to quantize lowdimensional CSI, the system capacity decreases in both cases because the quantization error must be taken into account. Based on the results in Fig. 6 and the feedback overhead analysis in Subsection 3.3, we can draw the conclusion that our proposed scheme can offer a worthwhile design tradeoff between system capacity and feedback overhead.
6 Conclusions
In this paper, a PCAbased feedback scheme for massive MIMO was proposed. In the proposed scheme, two kinds of feedback information, the quantized lowdimensional CSI index and the compression matrix utilized to perform both compression and recovery, are fed back hierarchically. Moreover, we obtained a closedform expression for an upper bound to the normalized information distortion. We analyzed the feedback overhead and codebook search complexity of the proposed scheme. Simulation results showed that without considering the lowdimensional CSI quantization, the proposed scheme outperforms the existing DCTbased CS scheme in terms of compression ratio. When a RVQ codebook is adopted to quantize the lowdimensional CSI, worthwhile system capacity and recovery performance can be achieved. Finally, we draw the conclusion that the proposed scheme can achieve a useful performance tradeoff between system capacity and feedback overhead, which gives it high potential to be implemented in practical massive MIMO systems.
7 Appendix
7.1 A Proof of Proposition 1
7.2 B Proof of Proposition 2
7.3 C Proof of Eqs. (23) and (24)
Since each element of channel vector h obeys the Gaussian distribution with unit variance, then \({\mathbb {E}\left [ {{{\left \ \textbf {h} \right \}^{2}}} \right ] = {N_{t}}}\). So we can easily obtain that \({\mathbb {E}\left [ {{{\left \ {{\mathbf {h}\bar {\mathbf {U}}}} \right \}^{2}}} \right ] = {N_{t}}  J}\).
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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