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IEEE COMMUNICATIONS LETTERS, VOL. 21, NO. 3, MARCH 2017
CSI Partitioning Method with PCA-Based Compression for Low-Complexity Feedback of Large-Dimensional Channels Jingon Joung, Senior Member, IEEE, and Ernest Kurniawan, Senior Member, IEEE Abstract— This letter considers a codebook-based channel state information (CSI) feedback technique for large-dimensional channels, e.g., massive multiple-input multiple-output (MIMO) channels. We propose a CSI partitioning method, focus on the search complexity required in order to obtain the best beamforming vector, and investigate the tradeoff between the complexity and the performance. Then, we present an analytical framework that provides design guidelines for balancing the complexity, performance, and feedback overhead of practical largescale MIMO systems. Index Terms— Codebook, massive MIMO.
compression,
PCA,
feedback,
I. I NTRODUCTION
M
ASSIVE multiple-input multiple-output (MIMO) promises a significant increase in spectral efficiency (SE) and has been considered as a candidate for next-generation wireless systems. As with other multiantenna systems, the benefit of massive MIMO can only be realized when the channel state information (CSI) is available, particularly at a transmitter. Because massive MIMO transmitters can have orders of magnitude more antennas, e.g., 100 or more, the vast CSI feedback overhead might diminish the benefit of the massive antennas. Various approaches have been investigated in order to resolve this issue, such as exploiting the CSI sparseness that results from the spatial-and-frequency correlation of the channels [1], [2], grouping highly correlated antennas [3], using a codebook [4], and adopting a two-stage beamforming strategy [5]. Combining the CSI dimensionality reduction with channel tracking has also proven to be beneficial in multiuser scenarios [6]. However, most of these studies have only focused on reducing the feedback overhead without considering consideration to the complexity incurred at a receiver. In this work, we propose a joint feedback method in which the CSI is reduced to a sparse vector by using principal component analysis (PCA); the sparse CSI is further reduced through selecting a codeword from a codebook and sending the index as feedback to the transmitter. Here, we focus on the complexity required to search for the best codeword. Assuming a randomly generated codebook, we investigate Manuscript received September 27, 2016; accepted November 8, 2016. Date of publication November 22, 2016; date of current version March 8, 2017. This research was supported by Basic Science Research Program through the National Research Foundation (NRF) funded by the Korean government (2016R1D1A1B03930250). The associate editor coordinating the review of this letter and approving it for publication was F. Gao. J. Joung is with the School of Electrical and Electronics Engineering, Chung-Ang University, Seoul 06974, South Korea (e-mail:
[email protected]). E. Kurniawan is with the Institute for Infocomm Research (I2 R), A STAR, Singapore 138632 (e-mail:
[email protected]). Digital Object Identifier 10.1109/LCOMM.2016.2631472
the tradeoff among complexity, performance (i.e. SE), and feedback overhead, which is a good reference for designing massive MIMO systems. II. S YSTEM M ODEL We consider a transmitter (Tx) and receiver (Rx) with Nt and Nr antennas, respectively, communicating over N f -frequency subbands (or subcarriers). Let h ∈ C N×1 be a channel vector between the Tx and Rx that consists of N (N = Nt Nr N f ) spatial-and-frequency domain channel samples as follows: h = vec h1 · · · h N f , (1) where vec( A) denotes the vectorization of an m-by-n matrix A to form an mn-by-1 column vector obtained via stacking the columns of A on top of one another; hn is the spatial-domain channel vector at subcarrier n that is modeled as hn = vec (H(n)) ∈ C Nr Nt ×1 , n ∈ {1, . . . , N f }; H(n) ∈ C Nr ×Nt is the spatially correlated MIMO channel at subcarrier n. In order to feed back the CSI h with Q-bit quantization for a real value, 2Q N bits are required. In order to reduce the feedback amount, PCA-based compression is used [1], [2]. From N correlated components, PCA extracts M uncorrelated principal components (M N) using a signal transformation, i.e. the dimensionality reduction of a set with correlated components [7]. This holds even in the asymptotic regime of a large number of antennas as described in [5]. For the transformation, PCA uses a Karhunen-Loève transform (KLT) matrix that consists of the eigenvectors of a CSI covariance matrix C h = E hh H , such that the original CSI can be efficiently compressed to the uncorrelated CSI. That is, the KLT matrix, i.e. a sparse representation matrix, is obtained as follows [7]: = eig (C h ) ∈ C N×N , where eig( A) takes U H from an eigenvalue decomposition such that A = U DU H . Here, D is a diagonal matrix whose diagonal elements are the eigenvalues of A, and U’s column vectors are the eigenvectors of A. Using , an Rx can generate a sparse vector s representing h through extracting the principal components from h as s = h ∈ C N×1 .
(2)
The sparse structure of s is a consequence of the low rank covariance matrix C h , which occurs in practice particularly when the antenna elements are placed close to one another [8]. Then, using a binary selection matrix S ∈ R M×N , the Rx selects M most significant elements and sends the vector g to a Tx, where g = Ss ∈ C M×1 .
1558-2558 © 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
(3)
JOUNG AND KURNIAWAN: CSI PARTITIONING METHOD WITH PCA-BASED COMPRESSION
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If the nth element of s is selected to be fed back, it will be the mth element of g through designing S such that the nth element is “1” and other elements are “0”s in the mth row of S. Because each element of s can be selected at most once, each column and row of S includes at most one “1”. Once a Tx has the compressed CSI feedback information g, it can recover the original CSI through rearranging and inversely transforming g with S and , respectively, as follows [1], [2]: h = −1 ST g = −1 ST Ss = −1 ST S h.
(4)
Providing that the parameter M (selected principal components) is not smaller than the rank of C h , a perfect reconstruction at the Tx is possible, i.e. h = h, following a robust uncertainty principle [9]. Otherwise, M must be appropriately selected in order to retain all significantly large components in order to ensure that h is sufficiently close to h [2]. The PCA-based method feeds back g, which is part of the sparse CSI s; henceforth, this is referred to as the sparse-domain CSI feedback (SCF) method [1], [2]. III. C ODEBOOK D IMENSION R EDUCTION M ETHOD Using the feedback of g, i.e. SCF, the original 2Q N bits for h feedback can be reduced to 2Q M bits. Instead of quantizing each real element with Q bits, a codebook is considered for g in order to further reduce the feedback amount. The codebook cardinality, i.e. the number of codewords, is denoted by C. The codeword length is the same as the dimension of g, i.e. M. In order to simplify the codebook design, a random codebook is considered that is an M-by-C complex matrix W, in which each column vector represents an M-by-1 unit-norm codeword vector. An Rx selects the best codeword based on the following rule: k ∗ = arg max{|w[k] H g|},
(5)
k∈[1,C]
where w[k] ∈ C M×1 is the kth codeword, and the Rx feeds back the codeword index k ∗ , which requires log2 C-bit feedback. The codebook cardinality C must satisfy C ≤ 22Q M ; otherwise, there is no additional feedback amount reduction. For a given g, the cumulative distribution function (CDF) of the effective channel magnitude |w[k ∗ ] H g| using the selected beamforming vector can be calculated as follows [10]: C Pr |w[k ∗ ] H g| ≤ τ = 1 − I(1−(τ/||g||)2 ) (M − 1, 1) , where τ ∈ 0, |g| and Ia (b, c) is a regularized incomplete beta function. The above probability calculation is related to the area ratio of a hyper-spherical cap and an N-sphere. A comparison of the above formula with the empirical distribution for ||g|| = 1 is given in Fig. 1. For a fixed M, increasing the codebook size C is favorable because the probability of finding a codeword that is close to g is higher. However, the increased C will, in turn, increase the total feedback amount as well as the search complexity in (5), which scales according to O(C M). Similarly, for a fixed C, selecting a smaller M will increase the likelihood of finding a codeword that is close to g, because the space covered by the codebook has a lower dimension.
Fig. 1.
CDF of effective channel magnitude.
Fig. 2. Illustration of M-by-C codebook matrix W and its reduced M -by-C codebook matrix W , when α = C −0.5 and β = 0.5 + logC 2.
The foundation of our proposed scheme is to simultaneously reduce the codebook cardinality (size) from C to C = αC where 0 ≤ α ≤ 1 and to reduce the codeword length, i.e. dimension, from M to M = β M (where 0 < β ≤ 1) for searching each of the β1 sub-vectors of g. The motivation for combining these two approaches is to balance the degradation and improvement of the effective channel gain distribution, which are respectively caused by the reduction of the codebook cardinality and the reduction in the codeword length (as seen in the trend of the channel magnitude CDF in Fig. 1). In terms of complexity, both approaches will significantly reduce the computational load. In order to address the performance loss caused by the non-represented principal components, the total number of components is maintained equal to the number of elements in g. Assuming that β1 is an integer, g is partitioned into L = β1 sub-vectors g for ∈ {1, . . . , L}, and the reduced dimension codebook W ∈ C M ×C is used to select the best codeword index according to k∗ = arg max{|w [k] H g |}, ∀ ∈ {1, . . . , L}, k∈[1,C ]
(6)
as illustrated in Fig. 2. Appropriate normalization is applied to ensure w [k], ∀k ∈ 1, C has a unit norm. Then, the Rx will feed back all the indexes K = {k∗ } to the Tx for reconstruction. In this manner, the total feedback amount formally becomes L log2 αC and the complexity of selecting the optimal codewords becomes O(LC M ) = O( β1 αCβ M) = O(αC M).
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After the Rx feeds back K , the Tx will reconstruct the compressed CSI g through stacking g into g = √1 [ gT · · · g TL ]T , L 1 where g = [W ]k∗ for ∈ {1, . . . , L}. Here, [ A]n denotes the nth column vector of a matrix A. Once the Tx has g, it can reconstruct the estimated original CSI from (4). An important consideration when reconstructing the compressed CSI g is the polarity of both the real and imaginary parts of the individual −0.5 vector g . To illustrate this, consider the case Twhere T α = C T and β = 0.5 (L = 2), and denote g = g1 g2 . From (6), it is apparent that the best index k∗ (and correspondingly the sub-vector g = [W ]k∗ ) is obtained based on the absolute inner product |w [k] H g |. Meanwhile, the effective channel magnitude is given by √ √ H H g1 g1 / 2+ g g = g2H g2 / 2
√ H = | g1H g1 |2 +| g2H g2 |2 +2Re g1 g1 g2H g2 2. (7) Therefore, the selection criteria for g is only favorable for the first two terms inside the square root in (7). In order to accommodate the last term, it is necessary to consider the polarity of both the real and imaginary part of g , which requires two extra feedback bits for each sub-vector. In general, the best sub-vector is chosen as g∗ ∈ g {±Re ( g ) ± j Im ( g )}. The choices of g∗ , ∀, are determined according to ∗ L (8) g1 · · · g L ]} , g =1 = arg max g H vec {[ g ∈ g ,∀
which requires O(4 L M) = O(41/β M) complexity. In total, this requires an additional β2 -bits, and the required feedback amount with this proposed CSI dimension reduction is 1 2 β log2 αC + β . In order to restrain the feedback amount to be no greater than the original scheme without dimension reduction, i.e. β1 log2 αC + β2 ≤ log2 C, β ≥ 1 + logC (4α) was set, or equivalently α ≤ C β−1 /4. That is, as the codebook cardinality was reduced through decreasing α to reduce computational complexity, the codeword length (dimension) was also reduced up to a certain point. For example, as illustrated in Fig. 2, when √ the codebook cardinality was reduced from C to C = C with α = C −0.5 , the codeword length was reduced up to approximately half the original codeword length by setting β = 0.5 + logC 4 ≈ 0.5. The proposed CSI partitioning method is summarized in Algorithms 1 and 2. IV. N UMERICAL R ESULTS In order to perform the numerical analysis, the following equivalent received signal model is considered as y = h H wx + n, where h is the stacked channel model as given in (1) and w is the beamforming vector based on the reconstructed sparse channel h according to (4). The transmitted symbol x has unit energy E |x|2 = 1, and n is the additive white Gaussian noise with a zero mean and variance N0 . The normalized signal-to-noise ratio (SNR) is defined as 10 log10 (1/N0 ), which is set to 15dB throughout the analyses. The space-frequency correlation of channel h
Algorithm 1 Parameter Design for CSI Partitioning 1. Set up M and C: Choose a compression ratio N/M based on the eigenvalues of a channel covariance matrix and construct the original codebook with a cardinality and codeword length of C and M, respectively. 2. Design α, β, and L: Determine the reduced codebook cardinality C = αC, where 0 < α ≤ 1; set the codeword length reduction factor β = 1 + logC (4α), where 0 < β ≤ 1, and set the number of sub-dimensions L = β −1 . The choice of the design parameters is based on the acceptable complexity O(αC M + 41/β M).
Algorithm 2 Proposed Feedback Method 1. /*PCA-based dimensionality reduction*/ Rx estimates the original CSI h and computes the compressed CSI g from (2) and (3). 2. /*codebook based reduction*/ for = 1 : L do Rx selects the best codebook index k∗ for the th subdimension of g from (6). end for 3. Rx designs the best polarity of each sub-dimension of g based on (8). 4. /*feedback and reconstruction of CSI*/ Rx feeds back K = {k∗ } together with the β2 bits of polarity information. 5. Tx reconstructs compressed CSI g = [( g1∗ )T· · · ( g ∗L )T ]T . 6. Tx estimates the original CSI using (4) and uses it to perform either beamforming or MU-MIMO. 7. The above steps can be iterated until the desired performance and complexity is achieved. Because the trend is monotonic (decreasing α and β results in complexity reduction yet performance degradation), the search for a desired value can be performed using a bisection method.
is designed such that more than 99% of the channel power is contained within M = 24 of its principal components. Correspondingly, the effective channel magnitude is given by g H g|2 /N0 . | g H g|, and SE can be calculated as log2 1 + | Figure 3 presents the average SE with different codebook sizes. The full-dimensional codebook is chosen to have a cardinality (codebook size) of {C 1/5 , C 1/4 , C 1/3 , C 1/2 , C}, which corresponds to different values of α, namely C −4/5 , C −3/4 , C −2/3 , C −1/2 , and 1, respectively, where C = 220 . In the reduced dimension case, due to the two extra feedback bits for each sub-vector, the average SE is evaluated for different codebook sizes (e.g. quarter-full codebook size), which shifts the x-axis of logC (1/4) = −0.1 to between the full dimensional case and the reduced dimension case. From the results in Fig. 3, it can be seen that for a fixed codebook size, the average SE increased as the channel vector was partitioned into additional parts. This is a consequence of being able to select a good beamforming vector, which achieves a normalized effective channel gain equal to one, with a higher probability for smaller dimension sub-vectors.
JOUNG AND KURNIAWAN: CSI PARTITIONING METHOD WITH PCA-BASED COMPRESSION
Fig. 5.
Fig. 3.
Average SE evaluation for different sub-dimensional splits.
Fig. 4. Search complexity (in logarithmic scale) when M = 24 and C = 220 .
This is true despite two bits being used to convey the sign information (the codebook size is a factor of the four smaller than the full-dimensional one). However, it should be noted that the overall number of required feedback bits increases with the number of channel vector partitions. As well as the better average SE, the overall complexity of searching for the best beamforming vector is also lower with more partitions (direction A). Note that the conventional system’s tradeoff is achieved in directions B and C. In order to compare the different performances for the same amount of feedback bits, the different points are highlighted using a dashed line in Fig. 3. As highlighted by the dotted lines in Fig. 3, it is apparent that for the same overall feedback bits, the search complexity was reduced to O(C β M) from O(C M) through increasing the number of partitions, i.e. decreasing β < 1, with the sacrifice of the SE. Figure 4 clearly demonstrates the significant reduction in the overall complexity as the number of partitions is increased. For a more complete description on how the SE is affected by this dimension partitioning, the probability density function (PDF) of the SE is plotted under the same total feedback setup (20 bits) in Fig. 5. The results demonstrate that, in addition to the reduction in the average SE, the channel dimension partition also caused the variation in the SE. More specifically, the partitioning caused an approximately twofold increase in the variance. That is, the proposed method to break the channel vector into multiple partitions resulted in an SE that was more spread further compared with the
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PDF of spectral efficiency (SE) for different sub-dimensional splits.
full-dimensional case. This is attributed to the effective channel gain increasing the amounts of several random quantities, which generally expand the support of the overall distribution, with channel partitioning. As such, when implementing the proposed method in practice, it is important to consider this extra variation in SE. V. C ONCLUSION In this letter, we have proposed a low-complexity codebook-based feedback scheme that is suitable for massive MIMO systems. In addition to the dimensional reduction achieved using the PCA-based compression method, the proposed scheme considers a smaller codebook and divides the channel vector into several smaller sub-vectors. This dimensional splitting significantly reduces the overall complexity, at the expense of a slight degradation in beamforming gain. In practice, the parameters to determine the number of sub-dimensions must be tuned according to the desired performance and the acceptable level of complexity. R EFERENCES [1] J. Joung and S. Sun, “SCF: Sparse channel-state-information feedback using Karhunen–Loève transform,” in Proc. IEEE Global Commun. Conf. (GLOBECOM), Austin, TX, USA, Dec. 2014, pp. 399–404. [2] J. Joung, E. Kurniawan, and S. Sun, “Channel correlation modeling and its application to massive MIMO channel feedback reduction,” IEEE Trans. Veh. Technol., to be published. [3] B. Lee, J. Choi, J.-Y. Seol, D. J. Love, and B. Shim, “Antenna grouping based feedback compression for FDD-based massive MIMO systems,” IEEE Trans. Commun., vol. 63, no. 9, pp. 3261–3274, Sep. 2015. [4] J. Nam. (2014). “A codebook-based limited feedback system for largescale MIMO.” [Online]. Available: http://arxiv.org/abs/1411.1531 [5] A. Adhikary, J. Nam, J. Y. Ahn, and G. Caire, “Joint spatial division and multiplexing—The large-scale array regime,” IEEE Trans. Inf. Theory, vol. 59, no. 10, pp. 6441–6463, Oct. 2013. [6] E. Kurniawan, J. Joung, and S. Sun, “Limited feedback scheme for massive MIMO in mobile multiuser FDD systems,” in Proc. IEEE Int. Conf. Commun. (ICC), London, U.K., Jun. 2015, pp. 1710–1715. [7] I. T. Jolliffe, Principal Component Analysis, 2nd ed. New York, NY, USA: Springer-Verlag, 1986. [8] H. Xie, F. Gao, S. Zhang, and S. Jin, “A unified transmission strategy for TDD/FDD massive MIMO systems with spatial basis expansion model,” IEEE Trans. Veh. Technol., to be published. [9] E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 489–509, Feb. 2006. [10] C. K. Au-yeung and D. J. Love, “On the performance of random vector quantization limited feedback beamforming in a MISO system,” IEEE Trans. Wireless Commun., vol. 6, no. 2, pp. 458–462, Feb. 2007.
