Hybrid Precoding Based on Non-Uniform Quantization Codebook to Reduce Feedback Overhead in Millimeter Wave MIMO Systems
SSUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS , VOL. , NO. , 2018 1
Hybrid Precoding Based on Non-UniformQuantization Codebook to Reduce FeedbackOverhead in Millimeter Wave MIMO Systems
Yun Chen, Da Chen, and Tao Jiang
Abstract
In this paper, we focus on the design of the hybrid analog/digital precoding in millimeter wavemultiple-input multiple-output (MIMO) systems. To reduce the feedback overhead, we propose two non-uniform quantization (NUQ) codebook based hybrid precoding schemes for two main hybrid precodingimplementations, i.e., the full-connected structure and the sub-connected structure. Specifically, we firstlygroup the angles of the arrive/departure (AOAs/AODs) of the scattering paths into several spatial lobesby exploiting the sparseness property of the millimeter wave in the angular domain, which dividesthe total angular domain into effective spatial lobes’ coverage angles and ineffective coverage angles.Then, we map the quantization bits non-uniformly to different coverage angles and construct NUQcodebooks, where high numbers of quantization bits are employed for the effective coverage anglesto quantize AoAs/AoDs and zero quantization bit is employed for ineffective coverage angles. Finally,two low-complexity hybrid analog/digital precoding schemes are proposed based on NUQ codebooks.Simulation results demonstrate that, the proposed two NUQ codebook based hybrid precoding schemesachieve near-optimal spectral efficiencies and show the superiority in reducing the feedback overheadcompared with the uniform quantization (UQ) codebook based works, e.g., at least . feedbackoverhead could be reduced for a system with / transmitting/receiving antennas. Index Terms
Millimeter wave communication, hybrid precoding, feedback overhead, spatial lobe, non-uniformquantization.
Y. Chen, D. Chen, and T. Jiang are with School of Electronic Information and Communications, Huazhong University ofScience and Technology, Wuhan 430074, China (e-mail: chen [email protected]; [email protected]; [email protected]). a r X i v : . [ c s . I T ] M a r SUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS , VOL. , NO. , 2018
I. I
NTRODUCTION
The millimeter wave communication could achieve high data rates and has caused widespreadconcern owing to the large bandwidth [1]–[5]. However, millimeter wave signals suffer severepath-loss due to the use of very high carrier frequency on the order of 30-300 GHz. Fortunately,the small wavelength of millimeter wave signals enables the deployment of large antenna arraysin small physical dimensions and make massive multiple-input multiple-output (MIMO) practicalin wireless communications [6], [7]. The large antenna arrays could provide sufficient antennagains to compensate for the high path loss of millimeter wave signals. Therefore, millimeterwave MIMO has been a promising candidate for future cellular networks.The precoding is an important technology in MIMO systems since it could be utilized totransmit multiple data streams and cancel the interferences between different data streams. Inmillimeter wave MIMO systems, the precoding is essential and could further improve the spectralefficiency. However, the precoding in traditional MIMO systems is typically realized in thedigital domain and requires expensive radio frequency (RF) chains comparable in number tothe antennas, which will greatly increase the hardware cost of the millimeter wave MIMOsystem equipped with large antenna arrays [8], [9]. To address the above issue, the hybridanalog/digital precoding was proposed, where the number of the RF chains is much less thanthe number of antennas [10]. In the hybrid precoding, the signals are firstly precoded by a low-dimensional digital precoder to cancel the interference and allocate power, then, precoded by ahigh-dimensional analog precoder to produce high antenna gains.Hybrid precoding has two main implementation structures, i.e., full-connected structure andsub-connected structure [11]. The full-connected structure utilizes a large number of phaseshifters, where each RF chain is connected to all antennas to obtain full precoding gains [5]. Thesub-connected structure sacrifices some precoding gains, where each RF chain is only connectedto a subset of antennas or a subarray to reduce the required number of phase shifters [12].In the full-connected structure, the orthogonal matching pursuit (OMP) based hybrid precodingalgorithm was the first scheme proposed for millimeter wave MIMO systems to obtain thenear-optimal spectral efficiency [5]. Inspired by [5], there are many papers devoted to designinghybrid precoding algorithms mainly based on the alternative minimization, matrix decompositionand iterative searching [13]–[17]. In the sub-connected structure, the successive interferencecancelation (SIC) based hybrid precoding scheme was the first scheme proposed to obtain the
HEN et al. : HYBRID PRECODING BASED ON NUQ CODEBOOK TO REDUCE FEEDBACK OVERHEAD IN MILLIMETER WAVE MIMO SYSTEMS3 near-optimal spectral efficiency [12]. In [18], a near-optimal closed-form solution was proposedfor the sub-connected structure. Though the above methods proposed in [12]–[18] could achievegood spectral efficiency, the feedback overhead issue was not considered.For millimeter wave MIMO systems with large antenna arrays, the feedback overhead is verylarge and seriously affects the communication efficiency. Therefore, it is of great importance todesign hybrid precoding schemes and facilitate the limited feedback [5]. One of the effectivesolutions to the feedback problem is to quantize the analog precoding matrices. When the receiverobtains analog precoding matrices of which each column is selected from predefined quantizationcodebooks, it only needs to feed back the selected indexes rather than large dimensional precodingmatrices to the transmitter. Most prior hybrid precoding schemes designed for limited feedbackmillimeter wave MIMO systems were based on the uniform quantization (UQ) codebooks thatonly depend on the single parameter (the angle of arrive/departure (AOA/AOD)) quantizationand uniformly divide the total angular domain into b ( b is the number of quantization bits)quantized angle parts [19]–[22]. However, the sparseness property of the millimeter wave in theangular domain has not been fully utilized in existing works to reduce the feedback overhead.In this paper, we focus on the design of hybrid precoding for both full-connected and sub-connected structures to reduce the feedback overhead in millimeter wave MIMO systems. The keyidea is to construct novel non-uniform quantization (NUQ) codebooks that map the quantizationbits non-uniformly for different coverage angles. According to the measurement results ofNYU WIRELESS [23]–[25], the AOAs/AODs of the paths in the millimeter wave channelcould be grouped in several separated spatial lobes (SLs), which enables us to divide the totalangular domain into effective spatial lobes’ coverage angles and ineffective coverage angles.Therefore, we employ high numbers of quantization bits to quantize both AoAs and AoDs foreffective spatial lobes’ coverage angles to obtain the high spectral efficiency and employ zeroquantization bit for ineffective coverage angles to make the total feedback overhead lower. Themain contributions of this paper are summarized as follows. • By utilizing the sparseness property of the millimeter wave in the angular domain, weconstruct novel NUQ codebooks and propose a low-complexity NUQ-based hybrid precod-ing scheme for the full-connected structure, named NUQ-HYP-Full. This scheme requiressmaller feedback overhead than the UQ codebooks based schemes to maintain the near-optimal spectral efficiency. Moreover, we prove that narrowing the angle range that needsto be quantized is equivalent to increasing the quantization accuracy. When the quantization
SUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS , VOL. , NO. , 2018 accuracy is not high enough, increasing the quantization accuracy means improving thespectral efficiency. • We propose NUQ codebooks and a NUQ-based hybrid precodng scheme for the sub-connected structure, named NUQ-HYP-Sub. This scheme is the first one that utilizes beam-steering based quantization codebooks to design hybrid precoding matrices and greatlyreduces the feedback overhead compared with the existing hybrid precoding schemes in thesub-connected structure. • We observe that the required number of quantization bits in the sub-connected structureis smaller than that in the full-connected structure to obtain near-optimal spectral efficien-cies (more than of the spectral efficiencies achieved by the corresponding optimalunconstrained precoding schemes). Besides the spectral efficiency and power consumption,this observation provides a new insight when comparing full-connected and sub-connectedhybrid precoding implementations.Simulation results demonstrate that the proposed NUQ-HYP-Full scheme outperforms the UQbased OMP scheme and achieves similar spectral efficiency as the fully digital precoding scheme(whose spectral efficiency is the upper bound for the full-connected structure). Moreover, theproposed NUQ-HYP-Sub scheme achieves similar spectral efficiency as the SIC based hybridprecoding scheme and the optimal hybrid precoding scheme for the sub-connected structure.The rest of the paper is organized as follows. In Section II, the system model, channelmodel and the problem formulation are described. The non-quantization codebooks and thecorresponding NUQ codebook based hybrid precoding schemes for the full-connected and sub-connected structures are demonstrated in Section III and Section IV, respectively. Simulationresults are presented in Section V. Finally, we conclude this paper in Section VI.We use the following notations in this paper. a is a scalar, a is a vector, A is a matrix and A is a set. A ( i ) is the i th column of A and (cid:107) A (cid:107) F is the Frobenius norm of A . A T , A ∗ , A − denote the transpose, conjugate transpose and inverse of A respectively. diag( A ) is a vector thatconsists of diagonal elements of A and blkdiag( A , B ) is the block diagonal concatenation of A and B . [ A | B ] is the horizontal concatenation. | a | is the modulus of a . I N denotes a N × N identity matrix. L ( a ) denotes the length of a . CN ( a , A ) is a complex Gaussian vector withmean a and covariance matrix A . E [ A ] is the expectation of A . HEN et al. : HYBRID PRECODING BASED ON NUQ CODEBOOK TO REDUCE FEEDBACK OVERHEAD IN MILLIMETER WAVE MIMO SYSTEMS5
Digital
Baseband Precoder BB F RF ChainRF Chain RF F s N tRF N t N (a) Digital
Baseband Precoder BB F RF ChainRF Chain tRF N RF F s N tt RF / N N tt RF / N N (b)Fig. 1. Two block diagrams of the millimeter wave MIMO system models that use the hybrid precoding structure. (a) Full-connected structure, where each RF chain is connected to all antennas; (b) Sub-connected structure, where each RF chain isonly connected to a subset of antennas.
II. S
YSTEM M ODEL , C
HANNEL M ODEL AND P ROBLEM F ORMULATION
A. System Model
Consider two typical structures for the single user millimeter wave MIMO systems in Fig. 1,where Fig. 1(a) shows the transmitter of the full-connected structure and Fig. 1(b) shows thetransmitter of the sub-connected structure. In both structures, N t /N r antennas and N tRF /N rRF RFchains are equipped at the transmitter/receiver subject to the constrains N s ≤ N tRF ≤ N t and N s ≤ N rRF ≤ N r , where N s denotes the number of the data streams.At the transmitter, N s data streams are firstly transmitted to N tRF RF chains by an N tRF × N s baseband precoding matrix F BB . Then, an N t × N tRF analog precoding matrix F RF transformsthe data streams from RF chains to N t antennas. The discrete-time transmitted signal vector is X = F RF F BB s , (1)where s is the N s × signal vector with E [ ss ∗ ] = N s I N s . Since phase shifters are utilized toimplement F RF , each entry of F RF has constant amplitude constraint (cid:16) F ( i )RF F ( i ) ∗ RF (cid:17) l,l = 1 /N t ,where ( · ) l,l denotes the l th diagonal element of a matrix. In addition, the total power constrainis enforced by (cid:107) F RF F BB (cid:107) F = N s .The narrowband block-fading channel model is adopted as shown in [5], [21] and the signalvector observed by the receiver is r = √ ρ H F RF F BB s + n , (2) SUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS , VOL. , NO. , 2018 where H is the N r × N t millimeter wave channel matrix, ρ is the average received power, and n is the vector of independent and identically distributed (i.i.d.) CN (0 , σ ) noise.After the combining processing at the receiver, the received signal vector is given as y = √ ρ W ∗ BB W ∗ RF H F RF F BB s + W ∗ BB W ∗ RF n , (3)where W RF is the N r × N rRF RF combining matrix which should satisfy (cid:16) W ( i )RF W ( i ) ∗ RF (cid:17) l,l = 1 /N r and W BB is the N rRF × N s baseband digital combining matrix. When Gaussian symbols aretransmitted through the millimeter wave channel, the achievable spectral efficiency can be writtenas [26] R = log (cid:16)(cid:12)(cid:12)(cid:12) I N s + ρN s R − W ∗ BB W ∗ RF HF RF F BB × F ∗ BB F ∗ RF H ∗ W RF W BB (cid:12)(cid:12)(cid:12)(cid:17) , (4)where R n = σ W ∗ BB W ∗ RF W RF W BB is the N s × N s noise covariance matrix after combing. B. Channel Model
The high free-space path-loss of the millimeter wave signals leads to limited spatial scattering[5], [27]. Therefore, the geometric Saleh-Valenzuela model is usually used to represent themillimeter wave channel [28], which is given by H cl = (cid:115) N t N r N cl N ray N cl (cid:88) m =1 N ray (cid:88) n =1 α m,n a r ( θ r m,n ) a t ( θ t m,n ) ∗ , (5)where N cl is the number of scattering clusters and each cluster contributes N ray propagationpaths, α m,n denotes the complex gain of the n th path in the m th cluster, θ r m,n ∈ [0 , π ] and θ t m,n ∈ [0 , π ] are the AOA and AOD, respectively. We adopt uniform linear arrays (ULAs)at both the transmitter and the receiver. a r ( θ r m,n ) and a t ( θ t m,n ) are the antenna array responsevectors which can be written as a t ( θ t m,n ) = 1 √ N t (cid:104) , e j (2 π/λ ) dsin ( θ t m,n ) , ..., e j ( N t − π/λ ) dsin ( θ t m,n ) (cid:105) T , (6)and a r ( θ r m,n ) = 1 √ N r (cid:104) , e j (2 π/λ ) dsin ( θ r m,n ) , ..., e j ( N r − π/λ ) dsin ( θ r m,n ) (cid:105) T , (7)respectively, where λ is the wavelength of the signal and d = λ/ denotes the aperture domainsample spacing.According to the measurement results of NYU WIRELESS for the 28 GHz millimeter wavechannel in Manhattan that is a typical urban environment, the AOAs/AODs of the propagation HEN et al. : HYBRID PRECODING BASED ON NUQ CODEBOOK TO REDUCE FEEDBACK OVERHEAD IN MILLIMETER WAVE MIMO SYSTEMS7 paths could be grouped in several spatial lobes, which causes the sparseness property of themillimeter wave in the angular domain [23]–[25]. The polar plot of the millimeter wave channelmeasured in Manhattan at 28 GHz is shown in Fig. 2. It is observed that the angles of pathsin the five dominated spatial lobes are sufficiently separated. Moreover, the measurement resultsalso show that the paths in several clusters may arrive in the same spatial lobe, which meansthe AOAs/AODs of the paths in different clusters may not be sufficiently separated. Therefore,we reconstruct the channel from the spatial lobe’s perspective to fully utilize the sparsenessproperty of the millimeter wave in the angular domain. Since the millimeter wave channelconsists of several propagation paths, the equivalent spatial lobes millimeter wave channel couldbe expressed as H = (cid:114) N t N r P Q P (cid:80) p =1 Q (cid:80) q =1 α p,q a r ( θ r p,q ) a t ( θ t p,q ) ∗ , (8)where P is the number of spatial lobes, Q is the number of subpaths in one spatial lobe ( P Q = N cl N ray ), and α p,q is the path gains for the q th subpath in the p th spatial lobe, which obeys theRayleigh distribution.For convenience, the millimeter wave channel is rewritten in a more compact form as H = A r diag( α ) A t ∗ , (9)where α = (cid:113) N t N r P Q [ α , , α , , ..., α ,Q , ..., α P Q ] T contains the complex gains of all paths, and thematrices A r = (cid:2) A r1 , A r2 , ..., A r P (cid:3) = (cid:2) a r ( θ r1 , ) , a r ( θ r1 , ) , ..., a r ( θ r1 ,Q ) , ..., a r ( θ r P,Q ) (cid:3) (10)and A t = (cid:2) A t1 , A t2 , ..., A t P (cid:3) = (cid:2) a t ( θ t1 , ) , a t ( θ t1 , ) , ..., a t ( θ t1 ,Q ) , ..., a t ( θ t P,Q ) (cid:3) (11)contain the array response vectors. Inspired by the diagonal form of (9), we find that the totalnumber of paths is the upper bound of the rank of the millimeter wave channel matrix, whichmeans the number of data streams N s should satisfy N s ≤ P Q to maintain the good systemperformance.
SUBMITTED TO IEEE TRANSACTIONS ON COMMUNICATIONS , VOL. , NO. , 2018
Fig. 2. The polar plot of the millimeter wave channel measured in Manhattan at 28 GHz [23].
C. Problem Formulation
The target of designing the hybrid precoding matrices ( F RF , F BB , W RF , W BB ) is to maximizethe spectral efficiency. Therefore, the optimization problem could be expressed as ( F optRF , F optBB , W optRF , W optBB ) = arg max R s . t . F RF ∈ F RF , W RF ∈ W RF , (cid:107) F RF F BB (cid:107) F = N s , (12)where F RF and W RF are the sets of the feasible analog precoders and combiners induced bythe constant amplitude constraint, respectively. Directly optimizing the problem (12) is verynon-trivial due to the constant amplitude constraint on the analog precoding matrices. Basedon the mathematical derivations in [5], the design of the precoding matrices and combiningmatrices are firstly decoupled, which indicates that we could focus on the design of precodingmatrices ( F optRF , F optBB ) . The combining matrices ( W optRF , W optBB ) could be obtained in the similarway except that there is no extra power constraint [14]. Then, an equivalent sparse reconstructionproblem is formulated, which is aimed to minimize the Euclidean distance between the productof the analog and digital precoding matrices and the optimal unconstrained precoding matrix.The sparse reconstruction problem for the transmitter is given by ( F optRF , F optBB ) = arg min F BB , F RF (cid:107) F opt − F RF F BB (cid:107) F , s . t . F RF ∈ F RF , (cid:107) F RF F BB (cid:107) F = N s , (13) HEN et al. : HYBRID PRECODING BASED ON NUQ CODEBOOK TO REDUCE FEEDBACK OVERHEAD IN MILLIMETER WAVE MIMO SYSTEMS9 where F opt is the optimal unconstrained reference precoding matrix which could be obtainedfrom the singular value decomposition (SVD) of H .III. N ON -U NIFORM Q UANTIZATION C ODEBOOK B ASED H YBRID P RECODING FOR THE F ULL - CONNECTED S TRUCTURE
For the full-connected structure, each RF chain is connected to all antennas, as shown in Fig.1(a). In the design of the hybrid precoding scheme, similar to [14], we decouple the designof the analog precoding matrices and digital precoding matrices. Since there is no constantamplitude constraint on digital precoding mareices, the digital precoding matrices could beobtained by simply SVD for the effective channel (which is defined in Section III.C). Therefore,we mainly focus on the design of analog precoding matrices by using quantization codebooks,which means that the quantization codebooks are the key points to design the hybrid precodingmatrices. We will firstly introduce the classical uniform quantization beamsteering codebooks.Then, non-uniform quantization codebooks are proposed by exploiting the sparseness propertyof the millimeter wave in the angular domain. Finally, a low-complexity NUQ codebook basedhybrid precoding scheme for the full-connected structure is proposed.
A. Uniform Quantization Codebooks
Most of the prior works on the design of the hybrid precoding for limited feedback millimeterwave MIMO systems were based on the UQ beamsteering codebooks, since the UQ beamsteer-ing codebooks, which are of relatively small size, only depend on the single parameter (thebeamsteering angle) quantization and could provide good quantization performance [5], [19].The UQ beamsteering codebooks for the transmitter and receiver can be respectively written as A quantt = (cid:104) a quantt ( θ ) , a quantt ( θ ) , ..., a quantt ( θ b ) (cid:105) , (14)and A quantr = (cid:104) a quantr ( θ ) , a quantr ( θ ) , ..., a quantr ( θ b ) (cid:105) , (15)where the entries of A quantt and A quantr are a quantt ( θ i ) = 1 √ N t (cid:104) , e jπ sin( π ( i − b ) , ..., e j ( N t − π sin( π ( i − b ) (cid:105) T , (16) and a quantr ( θ i ) = 1 √ N r (cid:104) , e jπ sin( π ( i − b ) , ..., e j ( N r − π sin( π ( i − b ) (cid:105) T , (17) (cid:23) Uniform Quantization … b quantization bits (a) (cid:23) Non-Uniform Quantization0 High numbers of quantization bits for spatial lobes’ coverage angles
Zero quantization bit for ineffective coverage angles q Q q q Q q ,1 P q , P Q q … b quantization bits (b)Fig. 3. Two quantization schemes. (a) Uniform-quantization scheme, where b quantization bits are uniformly mapped to the totalangular domain; (b) Non-uniform quantization scheme, where b quantization bits are only mapped to spatial lobes’ coverageangles. respectively. Obviously, the larger the number of quantization bits b is, the higher the quantizationaccuracy of the UQ beamsteering codebooks is. B. Non-Uniform Quantization Codebooks for the Full-connected Structure
In this subsection, we present the non-uniform quantization codebooks for the full-connectedstructure. Since the angles of paths in different spatial lobes are sufficiently separated, we dividethe total angular domain into effective spatial lobes’ coverage angles and ineffective coverageangles. Therefore, we could employ high numbers of quantization bits to quantize effective spatiallobes’ coverage angles to obtain good performance and employ zero quantization bit to quantizeineffective coverage angles to maintain the average number of quantization bits unchanged. Thatis we only map the quantization bits to effective spatial lobes’ coverage angles. In the meantime,without loss of generality, we assume that the transmitter and receiver have the same spatial lobesdistribution, i.e., the mean angles of spatial lobes ( (cid:101) θ i , i = 1 , , ..., P ) are evenly distributed within [0 , π ] , and the angles of the subpaths in each spatial lobe are randomly distributed with a finiteangle spread ( ω i ) [23], [24]. Fig. 3 shows the details of the non-uniform quantization, where θ i,q HEN et al. : HYBRID PRECODING BASED ON NUQ CODEBOOK TO REDUCE FEEDBACK OVERHEAD IN MILLIMETER WAVE MIMO SYSTEMS11 is the angle of the q th subpaths in the i th spatial lobe and b quantization bits are non-uniformlymapped to different coverage angles.We define the beam coverage of the i th spatial lobe as CV ( SL i ) = (cid:91) j =1 , ,...,Q CV (cid:0) a ( θ i,j ) (cid:1) = [ θ i, , θ i,Q ]= (cid:104)(cid:101) θ i − ω i / , (cid:101) θ i + ω i / (cid:105) , i = 1 , , ..., P, (18)where CV (cid:0) a ( θ i,j ) (cid:1) is the beam coverage of the steering vector for the j th subpath in the i th spatial lobe.The quantized angles coverage for the i th spatial lobe is designed as CV quant i = [ (cid:101) θ i − θ range i / , (cid:101) θ i + θ range i / , (19) which should satisfy ω i ≤ θ range i , (20) (cid:91) i =1 , ,...,P CV quant i ⊆ [0 , π ] , (21)and (cid:92) i =1 , ,...,P CV quant i = ∅ , (22)where θ range i is the quantized angle range for the i th spatial lobe. (20) guarantees the codebooksare able to quantize the actual angles of all subpaths in each spatial lobe. (21) is an obviousconstraint. (22) comes from the sparseness property of the millimeter wave in the angular domainthat the angles of paths in different spatial lobes are sufficiently separated.Denoting by b the number of quantization bits, the quantized accuracy of the angle is definedas ∆ θ = 2 π/ (2 b ) , (23)and the vector of quantized angles for the i th spatial lobe could be obtained as Θ quant i = (cid:104)(cid:101) θ i − θ range i , (cid:101) θ i − θ range i θ , ..., (cid:101) θ i + θ range i (cid:105) . (24)For the full-connected structure employed with N t × N r antennas, the antenna indexes for thetransmitter and receiver have the form A tindex = [0 , , ..., ( N t − , (25) Algorithm 1
Non-Uniform Quantization (NUQ) Codebooks for the Full-connected Structure
Input: N t , N r , b Output: A quantt , A quantr A tindex = [0 , , ..., ( N t − ], A rindex = [0 , , ..., ( N r − for i ≤ P do CV quant i = [ (cid:101) θ i − θ range i / , (cid:101) θ i + θ range i / ∆ θ = 2 π/ (2 b ) Θ quant i = (cid:104)(cid:101) θ i − θ range i , (cid:101) θ i − θ range i + ∆ θ , ..., (cid:101) θ i + θ range i (cid:105) for m ≤ L (Θ quant i ) do A quantt i (: , m ) = (cid:112) /N t ∗ e ( jπ A tindex sin(Θ quant i ( m ))) A quantr i (: , m ) = (cid:112) /N r ∗ e ( jπ A rindex sin(Θ quant i ( m ))) end for end for A quantt = [ A quantt1 , A quantt2 , ..., A quantt P ] A quantr = [ A quantr1 , A quantr2 , ..., A quantr P ] and A rindex = [0 , , ..., ( N r − , (26)respectively. Therefore, the non-uniform quantization codebooks for the i th spatial lobe can bewritten as A quantt i (: , m ) = (cid:112) /N t ∗ e ( jπ A tindex sin(Θ quant i ( m ))) , (27)and A quantr i (: , m ) = (cid:112) /N r ∗ e ( jπ A rindex sin(Θ quant i ( m ))) , (28)where m = 1 , , ..., L (Θ quant i ) . The non-uniform quantization codebooks for the full-connectedstructure are summarized in Algorithm 1 . Remark 1:
In conventional MIMO systems where channels are relatively rich, many quan-tization beamforming codebooks are designed to satisfy some particular properties, e.g., theGrassmannian codebooks in which the property of maximizing the minimum distance betweenthe codebook vectors is adopted [29]. However, the codebooks designed for traditional MIMOsystems may not very suitable for millimeter wave MIMO systems where the channel has limited
HEN et al. : HYBRID PRECODING BASED ON NUQ CODEBOOK TO REDUCE FEEDBACK OVERHEAD IN MILLIMETER WAVE MIMO SYSTEMS13 spatial scattering and large antenna arrays are employed at both the BSs and the MSs, since theyare relatively complicated. Motivated by the good performance of the hybrid precoding schemeswhich are relied on RF beamsteering vectors, the proposed NUQ codebooks are still based onthe classic beamsteering codebooks. It is worthy to point out that the idea of NUQ is generalfor any codebooks. For future work, it is of interest to evaluate the performance of other NUQbased beamforming codebooks for millimeter wave MIMO systems.
Proposition 1.
Narrowing the angle range which needs to be quantized is equivalent to increas-ing the quantization accuracy when the number of quantization bits b is fixed. In particular,the quantization accuracy could be increased by 1 bit when the total quantized angle range isnarrowed by half.Proof. For the millimeter wave channel with P spatial lobes, the total effective beam coverageis (cid:102) CV = (cid:91) i =1 , ,...,P CV ( SL i ) . (29)According to (20)-(22), we have (cid:102) CV ⊆ (cid:91) i =1 , ,...,P CV quant i , (30)and P (cid:88) i =1 θ range i ≤ π. (31)Denoting by (cid:98) b the equivalent quantization bits, we have b = P (cid:80) i =1 θ range i ∆ θ = P (cid:80) i =1 θ range i π/ (cid:98) b , (32)therefore, we could obtain (cid:98) b = b − log P (cid:80) i =1 θ range i π . (33)According to (31), log P (cid:80) i =1 θ range i π ≤ , therefore, we have (cid:98) b ≥ b . Particularly, when P (cid:80) i =1 θ range i = π ,which means the total quantized angle range is narrowed by half, we could easily obtain that (cid:98) b = b − log π π = b + 1 , (34) Algorithm 2
NUQ Codebook Based Hybrid Precoding for the Full-connected Structure, i.e.NUQ-HYP-Full
Input: A t , A r , A quantt , A quantr Output: F RF , F BB , W RF , W BB First stage : Analog precoding matrices design for i ≤ P do F res = A t i Ψ = A quant ∗ t i F res for j ≤ Q do k = argmax l =1 ,..., Q diag( ΨΨ ∗ ) F RF i = (cid:104) F RF i (cid:12)(cid:12) A quantt i (: , k ) (cid:105) diag( ΨΨ ∗ )( k ) = 0 end for end for F RF = [ F RF , F RF , ..., F BB P ] We could obtain W RF in the similar way W RF = [ W RF , W RF , ..., W BB P ] Compute the effective channels H eq i = W ∗ RF i HF RF i , i = 1 , , ..., P Second stage : Digital precoding matrices design
Compute the SVD of each effective channel H eq i , W ∗ RF i HF RF i = U eq i Σ eq i V ∗ eq i F BB i = V eq i , W BB i = U eq i F BB = blkdiag( F BB , F BB , ..., F BB P ) , W BB = blkdiag( W BB , W BB , ..., W BB P ) Normalize the digital precoding matrix at the transmitter F BB = √ N s F BB (cid:107) F RF F BB (cid:107) F As we know, when b is not large enough, increasing the quantization accuracy means improvingthe spectral efficiency. Proposition 1 indicates that, when b is fixed, the spectral efficiency couldbe indirectly improved by narrowing the total angle range P (cid:80) i =1 θ range i that needs to be quantized. C. NUQ-Based Hybrid Precoding for the Full-connected Structure
We are committed to constructing a low-complexity hybrid precoding solution (i.e., NUQ-HYP-Full) which is based on the following two operations.
HEN et al. : HYBRID PRECODING BASED ON NUQ CODEBOOK TO REDUCE FEEDBACK OVERHEAD IN MILLIMETER WAVE MIMO SYSTEMS15 • The hybrid precoding matrix for each spatial lobe is designed one by one, since theAOAs/AODs of paths in different spatial lobes are sufficiently separated and thus the pathsin different spatial lobes could be considered approximately orthogonal. • Considering that the right and left singular matrices of H converge in chordal distance tothe antenna response matrices, when the number of paths is much smaller than number ofthe antennas [30]. We set the antenna response matrices A t i and A r i rather than the optimalunconstrained precoding matrices as the reference precoding matrices for the i th spatiallobe, since Q (cid:28) min ( N t , N r ) .Once we obtain the NUQ codebooks or the NUQ candidate matrices, the analog precodingmatrices could be obtained by searching all entries of the candidate matrices to find the vectorswhich are respectively closest to each entry of the reference precoding matrices in the l normsense.The analog precoding matrix design problem for the i th spatial lobe at the transmitter couldbe formulated as F optRF i = arg min (cid:107) F RF i − A t i (cid:107) F , s . t . F RF i ∈ A quantt i , (35)which could be equivalently solved by finding the Q vectors in A quantt i along which the referenceprecoding matrix has the largest Q projections. The combining matrix W RF i for the i th spatiallobe could be obtained similarly.After designing the analog precoding matrices, digital precoding matrices could be obtainedby simply SVD for the effective channel since there is no constant amplitude constraint on digitalprecoding mareices. The effective channel for each spatial lobe could be calculated as H eq i = W ∗ RF i HF RF i , = U eq i Σ eq i V ∗ eq i , i = 1 , , ..., P. (36)Therefore, the digital precoding matrices for the i th spatial lobe could be computed as F BB i = V eq i , (37)and W BB i = U eq i . (38)The total digital precoding matrices for the transmitter and receiver are the block diagonalconcatenation of F BB i and W BB i , respectively. Finally, we normalize the total digital precoding matrix to satisfy the power constraint at the transmitter. The proposed NUQ-HYP-Full schemefor the full-connected structure is summarized in Algorithm 2 .IV. N ON -U NIFORM Q UANTIZATION C ODEBOOK B ASED H YBRID P RECODING FOR THE S UB - CONNECTED S TRUCTURE
Different from the full-connected structure, each RF chain in the sub-connected structure isonly connected to a subset of antennas or a subarray, which dramatically reduces the numberof phase shifters, as shown in Fig.1(b) [12], [18], [22]. Therefore, the sub-connected structureis more energy efficient than the full-connected structure. For simplicity but without loss ofgenerality, we assume the transmitter and the receiver contain the same number of RF chains,i.e., N RF = N tRF = N rRF .Similar as the hybrid precoding design in the full-connected structure, we also decouple thedesign of the analog and digital precoding matrices. We focus on the design of the analogprecoding matrices with pre-defined quantization codebooks in the sub-connected structure, sincethe digital precoding matrices could be directly obtained by simply SVD for the effective channel.To the best of the authors’ knowledge, our work is the first attempt to utilize beamsteering basedquantization codebooks to design hybrid precoding matrices in the sub-connected structure. A. Non-Uniform Quantization Codebooks for the Sub-connected Structure
In the design of the NUQ codebooks for the sub-connected structure, the number of the totalRF chains equals the number of the subarray, i.e., N RF = N sub , which means the number of theantennas in each subarray N subt equals N t /N RF . At the transmitter, let the total antenna indexesbe [0 , , ..., N t − and Λ t k denote the partitioned subset of antenna indexes connected to the k th subarray, such as Λ t1 = [0 , ..., N subt − , Λ t2 = [ N subt , ..., N subt − , ... Λ t N RF = [( N RF − N subt , ..., N RF N subt − . (39)The total antenna index matrix at the transmitter for the sub-connected structure is (cid:98) A tindex = [Λ t1 ; Λ t2 ; ... ; Λ t N RF ] . (40) HEN et al. : HYBRID PRECODING BASED ON NUQ CODEBOOK TO REDUCE FEEDBACK OVERHEAD IN MILLIMETER WAVE MIMO SYSTEMS17
Algorithm 3
Non-Uniform Quantization (NUQ) Codebooks for the Sub-connected Structure
Input: N t , N r , N RF , b Output: (cid:98) A quantt , (cid:98) A quantr N sub t = N t /N RF , N sub r = N r /N RF for n ≤ N RF do (cid:98) A tindex ( n, :) = Λ t n = [( n − N sub t , ..., n ( N sub t ) − (cid:98) A rindex ( n, :) = Λ r n = [( n − N sub r , ..., n ( N sub r ) − end for for i ≤ P do CV quant i = [ (cid:101) θ i − θ range i / , (cid:101) θ i + θ range i / ∆ θ = 2 π/ (2 b ) Θ quant i = (cid:104)(cid:101) θ i − θ range i , (cid:101) θ i − θ range i + ∆ θ , ..., (cid:101) θ i + θ range i (cid:105) for q ≤ Q do for m ≤ L (Θ quant i ) do (cid:98) A quantt iq (: , m ) = (cid:113) N subt e ( jπ Λ t(( i − ∗ Q + q ) ∗ sin(Θ quant i ( m ))) (cid:98) A quantr iq (: , m ) = (cid:113) N subr e ( jπ Λ r(( i − ∗ Q + q ) ∗ sin(Θ quant i ( m ))) end for end for (cid:98) A quantt i = blkdiag( (cid:98) A quantt i , (cid:98) A quantt i , ..., (cid:98) A quantt iQ ) (cid:98) A quantr i = blkdiag( (cid:98) A quantr i , (cid:98) A quantr i , ..., (cid:98) A quantr iQ ) end for (cid:98) A quantt = blkdiag( (cid:98) A quantt1 , (cid:98) A quantt2 , ..., (cid:98) A quantt P ) (cid:98) A quantr = blkdiag( (cid:98) A quantr1 , (cid:98) A quantr2 , ..., (cid:98) A quantr P ) The total antenna index matrix at the receiver (cid:98) A rindex for the sub-connected structure could beobtained similarly.For the limited scattering millimeter wave channel, the number of total effective paths isusually very small [12], [21]. Thus, we make an assumption that the total number of effectivepaths is no more than the total number of RF chains, i.e., P Q ≤ N RF . In the meantime, sinceeach subarray is only connected to one RF chain, each subarray is arranged to precoding forone path separately. From (39) and (40), we could find the analog precoding matrices are in a block diagonal form, which could be depicted as (cid:98) F RF = (cid:98) F RF (cid:98) F RF . . . (cid:98) F RF P (cid:98) F RF ( N RF − PQ ) , (41)where (cid:98) F RF i = a i a i . . . a iQ , i = 1 , , ..., P, (42)is the analog precoding matrix for the i th spatial lobe and the elements of (cid:98) F RF ( N RF − PQ ) are 0.Similar to (19)-(24), the quantized angles coverage, quantized angle accuracy and quantizationangles for the i th spatial lobe are CV quant i , ∆ θ and Θ quant i , respectively. Therefore, the non-uniformquantization codebooks for the sub-connected structure can be designed as (cid:98) A quantt = (cid:98) A quantt1 (cid:98) A quantt2 . . . (cid:98) A quantt P , (43)where (cid:98) A quantt i = (cid:98) A quantt i (cid:98) A quantt i . . . (cid:98) A quantt iQ , i = 1 , , ..., P, (44)in which (cid:98) A quantt iq (: , m ) = (cid:115) N subt e ( jπ Λ t (( i − ∗ Q + q ) ∗ sin(Θ quant i ( m ))) , (45)where q = 1 , , ..., Q , m = 1 , , ..., L (Θ quant i ) . The non-uniform quantization codebooks forthe receiver (cid:98) A quantr could be designed similarly. In Algorithm 3 , we summarize the process ofconstructing the non-uniform quantization codebooks for the sub-connected structure.
HEN et al. : HYBRID PRECODING BASED ON NUQ CODEBOOK TO REDUCE FEEDBACK OVERHEAD IN MILLIMETER WAVE MIMO SYSTEMS19
Algorithm 4
NUQ Codebook Based Hybrid Precoding for the Sub-connected Structure, i.e.,NUQ-HYP-Sub
Input: A t , A r , (cid:98) A quantt , (cid:98) A quantr Output: (cid:98) F RF , (cid:98) F BB , (cid:99) W RF , (cid:99) W BB First stage : Analog precoding matrices design for both transmitter and receiver for i ≤ P do for q ≤ Q do F res = A t i (: , q ) Ψ = (cid:98) A quant ∗ t (: , L ( (cid:98) A quantt )( P Q ) (( i − ∗ Q + q −
1) + 1 : L ( (cid:98) A quantt )( P Q ) ∗ (( i − ∗ Q + q )) F res k = argmax l =1 ,..., Q diag( ΨΨ ∗ ) (cid:98) F RF iq = (cid:98) A quantt (: , ( L ( (cid:98) A quantt )( P Q ) ∗ (( i − ∗ Q + q −
1) + k ) end for (cid:98) F RF i = [ (cid:98) F RF i , (cid:98) F RF i , ..., (cid:98) F RF iQ ] end for (cid:98) F RF = [ (cid:98) F RF , (cid:98) F RF , ..., (cid:98) F RF P ] We could obtain (cid:99) W RF in the similar way (cid:99) W RF = [ (cid:99) W RF , (cid:99) W RF , ..., (cid:99) W RF P ] Compute the effective channels, (cid:98) H eq i = (cid:99) W ∗ RF i H (cid:98) F RF i , i = 1 , , ..., P Second stage : Digital precoding matrices design for both transmitter and receiver
Compute the SVD of each effective channel (cid:98) H eq i , W ∗ RF i HF RF i = U eq i Σ eq i V ∗ eq i F BB i = V eq i , W BB i = U eq i (cid:98) F BB = blkdiag( (cid:98) F BB , (cid:98) F BB , ..., (cid:98) F BB P ) , (cid:99) W BB = blkdiag( (cid:99) W BB , (cid:99) W BB , ..., (cid:99) W BB P ) Normalize the digital precoding matrix at the transmitter, (cid:98) F BB = √ N s (cid:98) F BB (cid:107) (cid:98) F RF (cid:98) F BB (cid:107) F B. NUQ-Based Hybrid Precoding for the Sub-connected Structure
The designing of hybrid precoding matrices for the energy-efficient sub-connected structureis an attractive topic for recent works in millimeter wave MIMO systems, e.g., the SIC-basedhybrid precoding scheme which decomposes the total optimization problem into several simplesub optimization problems and achieves the near-optimal performance. However, most of theprior works such as SIC-based hybrid precoding scheme did not consider the limited feedbackproblem which is also an important issue for the sub-connected structure. Moreover, to the best of our knowledge, there has been no beamsteering codebooks based hybrid precoding schemefor the sub-connected architecture in limited feedback millimeter wave MIMO systems. In thissubsection, based on the proposed non-quantization codebooks described in Section IV.A, wepropose a NUQ codebook based hybrid precoding scheme for the sub-connected structure, whichis summarized in
Algorithm 4 .In the analog precoding matrices designing stage, we quantize the array response vectorcorresponding to each path one by one to maintain the block diagonal form of the analogprecoding matrix. For the q th subpath in the i th spatial lobe, the reference precoding vector is A t i (: , q ) and the quantized candidate matrix is defined as (cid:98) A cant iq = (( i − Q + q − N subt ×L (Θ quant i ) (cid:98) A quantt iq ( N t − (( i − Q + q ) N subt ) ×L (Θ quant i ) , (46) which is actually the L ( (cid:98) A quantt )( P Q ) (( i − ∗ Q + q −
1) + 1 to L ( (cid:98) A quantt )( P Q ) ∗ (( i − ∗ Q + q ) columns of (cid:98) A quantt , where L ( (cid:98) A quantt )( P Q ) is equal to L (Θ quant i ) . Therefore, the analog precoding design problemfor the q th subpath in the i th spatial lobe can be formulated as (cid:98) F optRF iq = arg min (cid:13)(cid:13)(cid:13)(cid:98) F RF iq − A t i (: , q ) (cid:13)(cid:13)(cid:13) F , s . t . (cid:98) F RF iq ∈ (cid:98) A cant iq . (47)Similar to (35), (47) could be also directly solved by searching each entry of (cid:98) A cant iq to findthe column vector which has the maximum projection along A t i (: , q ) . Thus, the total analogprecoding matrices could be obtained by (cid:98) F RF = [ (cid:98) F RF , (cid:98) F RF , ..., (cid:98) F RF P ] , (48)where (cid:98) F RF i = [ (cid:98) F RF i , (cid:98) F RF i , ..., (cid:98) F RF iQ ] , i = 1 , , ..., P. (49)The analog combining matrix (cid:99) W RF for the receiver could be obtained similarly.Then, for the digital precoding designing stage, the steps are similar as has been describedfor the full-connected structure. Therefore, we omit the digital precoding designing steps for thesub-connected structure, which are presented in detail in steps 15-20 of Algorithm 4 . HEN et al. : HYBRID PRECODING BASED ON NUQ CODEBOOK TO REDUCE FEEDBACK OVERHEAD IN MILLIMETER WAVE MIMO SYSTEMS21
V. S
IMULATION R ESULTS
In this section, the performances of the proposed NUQ-HYP-Full and NUQ-HYO-Sub schemesare evaluated. The uniform quantization based OMP scheme (marked as UQ-OMP) and SIC-based hybrid precoding scheme are adopted as the benchmarks for the full-connected and sub-connected structures, respectively. We consider a single-user MIMO system, where ULAs with λ/ antenna spacing are equipped at both the transmitter and the receiver. According to themeasurements in downtown Manhattan environment which is a typical urban environment, thefrequency of the millimeter wave is set to be 28 GHz and the bandwidth is set to be 100MHz [23]–[25]. The spatial lobes millimeter wave channel (as shown in (8)) which has sparsityproperty in the angular domain is adopted. For P spatial lobes, the mean angles of spatial lobesare set to be (cid:101) θ i = θ CO + πP ( i − , i = 1 , , ..., P , where θ CO is a constant that is randomly selectedwithin [0 , π ] and the angle spreads are set as ω = ω = ... = ω P = πP . The angles of Q subpathsin one spatial lobe are assumed to be randomly distributed and the corresoponding power obeysthe Rayleigh distribution. In order to maximize the reduction in the feedback overhead, thequantized angle range of each spatial lobe is set as θ range i = ω, i = 1 , , ..., P , which means thetotal quantized angle range is narrowed by half. A. Full-connected hybrid precoding structure
Fig. 4 and Fig. 5 compare the spectral efficiencies of the proposed NUQ-HYP-Full scheme,OMP precoding scheme and fully digital precoding scheme with different numbers of spatiallobes and subpaths, respectively, where N t × N r = 144 × , N tRF = N rRF = 8 and b = 8 .Note that, the OMP precoding scheme we compare here is based on the uniform quantizationbeamsteering codebooks, which is slightly different from the Algorithm 1 in [5]. Since the totalangle range that needs to be quantized is narrowed by half, the quantization accuracy of theNUQ-HYP-Full scheme could be increased by 1 bit. We observe that, for different numbers ofspatial lobes and subpaths, the proposed NUQ-HYP-Full scheme always outperforms the OMPprecoding scheme and achieves more than of the spectral efficiency achieved by the fullydigital precoding scheme.In Fig. 6, the impact of the quantization bit on the spectral efficiency is presented, where N t × N r = 144 × , N tRF = N rRF = 4 , P = 2 and Q = 2 . We observe that the spectral efficiencies ofthe NUQ-HYP-Full scheme are always higher than the UQ-OMP scheme with different numbersof quantization bits. Moreover, the smaller the number of quantization bits is, the larger the -30 -25 -20 -15 -10 -5 0 5 10 SNR (dB) S pe c t r a l E ff i c i en cy ( bp s / H z ) NUQ-HYP-Full,P=2NUQ-HYP-Full,P=3NUQ-HYP-Full,P=4UQ-OMP,P=2UQ-OMP,P=3UQ-OMP,P=4Fully digital,P=2Fully digital,P=3Fully digital,P=4
Fig. 4. Spectral efficiencies of NUQ-HYP-Full, OMP and fully digital precoding schemes with different numbers of spatiallobes, where N s = P Q , N t = 144 , N r = 36 , N tRF = N rRF = 8 , Q = 2 , b = 8 . -30 -25 -20 -15 -10 -5 0 5 10 SNR (dB) S pe c t r a l E ff i c i en cy ( bp s / H z ) NUQ-HYP-Full,Q=1NUQ-HYP-Full,Q=2NUQ-HYP-Full,Q=3UQ-OMP,Q=1UQ-OMP,Q=2UQ-OMP,Q=3Fully digital,Q=1Fully digital,Q=2Fully digital,Q=3
Fig. 5. Spectral efficiencies of NUQ-HYP-Full, OMP and fully digital precoding schemes with different numbers of subpaths,where N s = P Q , N t = 144 , N r = 36 , N tRF = N rRF = 8 , P = 2 , b = 8 . performance gap becomes. Specially, we also observe that the NUQ-HYP-Full scheme achievessimilar spectral efficiencies as the UQ-OMP scheme, when the number of quantization bits isreduced by 1 (at least . feedback overhead reduction for the given number of antennas).This phenomenon is in consistent with Proposition 1 presented in Section III.B. Furthermore,we HEN et al. : HYBRID PRECODING BASED ON NUQ CODEBOOK TO REDUCE FEEDBACK OVERHEAD IN MILLIMETER WAVE MIMO SYSTEMS23 -30 -25 -20 -15 -10 -5 0 5 10
SNR (dB) S pe c t r a l E ff i c i en cy ( bp s / H z ) Fully DigitalNUQ-HYP-Full,b=6NUQ-HYP-Full,b=7NUQ-HYP-Full,b=8UQ-OMP,b=6UQ-OMP,b=7UQ-OMP,b=8
Fig. 6. Spectral efficiencies of NUQ-HYP-Fully, OMP and fully digital precoding schemes with different quantization bits,where N s = P Q , N t = 144 , N r = 36 , N tRF = N rRF = 4 , P = 2 , Q=2. Number of RF Chains S pe c t r a l E ff i c i en cy ( bp s / H z ) NUQ-HYP-FullUQ-OMPFully digital
Ns=6Ns=4Ns=2
Fig. 7. Spectral efficiencies of NUQ-HYP-Full, OMP and fully digital precoding schemes with different numbers of RF chains,where N tRF = N rRF , N t = 144 , N r = 36 , N s = P Q, Q = 2 , SNR=0 dB, b = 8 . observe that the number of quantization bits b should satisfy b ≥ max ( N t , N r ) to obtain goodspectral efficiency for the UQ-OMP scheme that is based on the vector-by-vector UQ-basedcodebooks.Fig. 7 shows the spectral efficiencies of different schemes with different numbers of RF chains,
20 40 60 80 100 120 140 160 180 200 N t S pe c t r a l E ff i c i en cy ( bp s / H z ) Fully DigitalNUQ-HYP-FullUQ-OMP
Fig. 8. Spectral efficiencies of NUQ-HYP-Full, OMP and fully digital precoding schemes with different numbers of transmitter’santennas N t , where N r = 36 , N tRF = N rRF = N s = P Q, P = Q = 2 , SNR=0 dB, b = 8 . where N t × N r = 144 × and the SNR is set to be 0 dB. We evaluate the cases when thenumber of subpaths Q = 2 and the number of spatial lobes varies from 1 to 3. Since we onlyutilize P Q
RF chains to transmit and receiver signals, we observe that the spectral efficiencies ofthe proposed NUQ-HYP-Full scheme remain unchanged but are always higher than the spectralefficiencies achieved by the uniform quantization based OMP scheme.Fig. 8 and Fig. 9 show the spectral efficiencies of different schemes with different numbersantennas at the transmitter and both the transmitter and receiver, respectively, where N tRF = N rRF = N s = P Q, P = Q = 2 , SNR=0 dB and the number of the quantization bit is setto be b = 8 . We observe that the proposed NUQ-HYP-Full scheme always outperforms theUQ-OMP scheme and achieves more than of the spectral efficiency achieved by the fullydigital precoding scheme at even very large number of antennas. Moreover, we also observe thatwhen the number of antennas becomes very large, the spectral efficiency of UQ-OMP schemedecreases, since the quantization bit b is relatively not large enough. B. Sub-connected hybrid precoding structure
For the sub-connected structure, the number of antennas are set as N t × N r = 144 × . Inaddition, the number of RF chains N RF is set equal to N s , which is actually the worst case HEN et al. : HYBRID PRECODING BASED ON NUQ CODEBOOK TO REDUCE FEEDBACK OVERHEAD IN MILLIMETER WAVE MIMO SYSTEMS25
20 40 60 80 100 120 140 160 180 200 N t =N r S pe c t r a l E ff i c i en cy ( bp s / H z ) Fully DigitalNUQ-HYP-FullUQ-OMP
Fig. 9. Spectral efficiencies of NUQ-HYP-Full, OMP and fully digital precoding schemes with different numbers of N t and N r where N t = N r , N tRF = N rRF = N s = P Q, P = Q = 2 , SNR=0 dB, b = 8 . -30 -25 -20 -15 -10 -5 0 5 10 SNR (dB) S pe c t r a l E ff i c i en cy ( bp s / H z ) Fully DigitalNUQ-HYP-FullUQ-OMPNUQ-HYP-SubSICOptimal-Sub
Ns=3Ns=2
Fig. 10. Spectral efficiencies comparison with different numbers of spatial lobes for the sub-connected structure, where N t =144 , N r = 36 , N RF = N s = P Q, Q = 1 , b = 6 . since N RF should satisfy N RF ≥ N s to ensure the system could simultaneously transmit N s datastreams.Fig. 10 and Fig. 11 compare the spectral efficiencies of NUQ-HYP-Sub scheme, SIC-based hy-brid precoding scheme, optimal unconstrained precoding scheme for the sub-connected structure -30 -25 -20 -15 -10 -5 0 5 10 SNR (dB) S pe c t r a l E ff i c i en cy ( bp s / H z ) Fully DigitalNUQ-HYP-FullUQ-OMPNUQ-HYP-SubSICOptimal-Sub
Ns=4Ns=2
Fig. 11. Spectral efficiencies comparison with different numbers of subpaths for the sub-connected structure, where N t =144 , N r = 36 , N RF = N s = P Q, P = 2 , b = 6 . (marked as Optimal-Sub), UQ-OMP scheme and fully digital precoding scheme with differ-ent numbers of spatial lobes and subpaths, respectively. The optimal unconstrained precodingscheme for the sub-connected structure was detailedly described in [12]. We observe that theproposed NUQ-HYP-Sub scheme achieves similar spectral efficiencies as the SIC and Optimal-Sub schemes, and the performance gaps are less than . In addition, Fig. 10 and Fig. 11also show that, the proposed NUQ-HYP-Sub scheme achieves more than of the spectralefficiencies achieved by the UQ-OMP scheme for all cases.In Fig. 12, the impact of the quantization bit on the spectral efficiency for the sub-connectedstructure is shown, where the number of the spatial lobes and subpaths are set as P = 3 and Q = 1 , respectively. We observe that as the number of quantization bits increases, thespectral efficiency gaps between the NUQ-HYP-Sub and Optimal-Sub schemes become smaller.Furthermore, we also observe that, to obtain the near-optimal spectral efficiency, the requirednumber of quantization bits for the proposed NUQ-HYP-Sub scheme is at least 6. However, theUQ-OMP scheme needs at least 8 quantization bits to maintain the near-optimal performanceat the full-connected structure. This is because only N subt = N t /N RF and N subr = N r /N RF antennas are utilized to transmit and receive the directional beams at the sub-connected structure,respectively. Therefore, we only need to make b approach max ( N subt , N subr ) to obtain goodperformance. This is a new advantage for the sub-connected structure in reducing the feedback HEN et al. : HYBRID PRECODING BASED ON NUQ CODEBOOK TO REDUCE FEEDBACK OVERHEAD IN MILLIMETER WAVE MIMO SYSTEMS27 -30 -25 -20 -15 -10 -5 0 5 10
SNR (dB) S pe c t r a l E ff i c i en cy ( bp s / H z ) Fully DigitalNUQ-HYP-FullUQ-OMPNUQ-HYP-Sub,b=6NUQ-HYP-Sub,b=5NUQ-HYP-Sub,b=4NUQ-HYP-Sub,b=3Optimal-SubSIC
Fig. 12. Spectral efficiencies comparison with different quantization bits for the sub-connected structure, where N t = 144 , N r =36 , N RF = N s = P Q, P = 3 , Q = 1 . overhead. VI. C ONCLUSIONS
In this paper, we proposed the NUQ-HYP-Full scheme and the NUQ-HYO-Sub schemefor the full-connected and the sub-connected structures in millimeter wave MIMO systems,respectively. The key idea of the proposed schemes is that the quantization bits are non-uniformlymapped to different coverage angles, according to the sparseness property of the millimeterwave in the angular domain. Both of the proposed schemes achieve at least . feedbackoverhead reduction for a system with 144/36 transmitting/receiving antennas. Simulation resultsdemonstrated that the proposed NUQ-HYP-Full scheme for the full-connected structure exhibitssimilar spectral efficiency as the fully digital precoding scheme and outperforms the UQ-OMPscheme. Simulation results also showed that the proposed NUQ-HYP-Sub scheme for the sub-connected structure achieves similar spectral efficiency as the optimal unconstrained precodingscheme. Furthermore, we also observed that, the required number of quantization bits in thesub-connected structure to obtain near-optimal spectral efficiency was smaller than that in thefull-connected structure, which provided a new insight to study low feedback overhead hybridprecoding schemes in millimeter wave MIMO systems. Our future work will focus on the wideband and time-varying millimeter wave channel scenarios, where the delay and dopplershift become key points to design hybrid precoding matrices.R EFERENCES [1] T. S. Rappaport, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G. N. Wong, J. K. Schulz, M. Samimi, and F. Gutierrez,“Millimeter wave mobile communications for 5G cellular: It will work!”
IEEE Access, vol. 1, pp. 335-349, May 2013.[2] Z. Pi and F. Khan, “An introduction to millimeter-wave mobile broadband systems,”
IEEE Commun. Mag., vol. 49, no. 6,pp. 101-107, Jun. 2011.[3] T.Bai, A.Alkhateeb, and R. Heath, “Coverage and capacity of millimeter-wave cellular networks,”
IEEE Commun. Mag .,vol. 52, no. 9, pp. 70-77, Sep. 2014.[4] Z. Gao, L. Dai, D. Mi, Z. Wang, M. A. Imran, and M. Z. Shakir, “MmWave massive MIMO based wireless backhaul for5G ultra-dense network,”
IEEE Wireless Commun ., vol. 22, no. 5, pp. 13-21, Oct. 2015.[5] O. E. Ayach, S. Rajagopal, S. Abu-Surra, Z. Pi, and R. W. Heath, Jr, “Spatially sparse precoding in millimeter wave MIMOsystems,”
IEEE Trans. Wireless Commun., vol. 13, no. 3, pp. 1499-1513, Mar. 2014.[6] Z. Gao, L. Dai, W. Dai, B. Shim and Z. Wang, “Structured Compressive Sensing-Based Spatio-Temporal Joint ChannelEstimation for FDD Massive MIMO,”
IEEE Trans. Commun., vol. 64, no. 2, pp. 601-617, Feb. 2016[7] A. Alkhateeb, J. Mo, N. Gonz´alez-Prelcic, and R. Heath, “MIMO precoding and combining solutions for millimeter-wavesystems,”
IEEE Commun. Mag ., vol. 52, no. 12, pp. 122-131, Dec. 2014.[8] X. Li, S. Jin, H. A. Suraweera, J. Hou and X. Gao, “Statistical 3-D Beamforming for Large-Scale MIMO DownlinkSystems Over Rician Fading Channels,”
IEEE Trans. Commun., , vol. 64, no. 4, pp. 1529-1543, April 2016.[9] Y. Wu, C. Xiao, Z. Ding, X. Gao and S. Jin, “Linear precoding for finite-alphabet signaling over MIMOME wiretapchannels,”
IEEE Trans. Veh. Technol., vol. 61, no. 6, pp. 2599-2612, Jul. 2012.[10] W. Xu, J. Liu, S. Jin and X. Dong, “Spectral and Energy Efficiency of Multi-Pair Massive MIMO Relay Network WithHybrid Processing,”
IEEE Trans. Commun., vol. 65, no. 9, pp. 3794-3809, Sept. 2017.[11] S. Han, C.-L. I, Z. Xu, and C. Rowell, “Large-scale antenna systems with hybrid precoding analog and digital beamformingfor millimeter wave 5G,”
IEEE Commun. Mag ., vol. 53, no. 1, pp. 186-194, Jan. 2015.[12] X. Gao, L. Dai, S. Han, I. Chih-Lin, and R. W. Heath, “Energy-efficient hybrid analog and digital precoding for mmWaveMIMO systems with large antenna arrays,”
IEEE J. Sel. Areas Commun ., vol. 34, no. 4, pp. 998-1009, Apr. 2016.[13] W. Ni and X. Dong, “Hybrid block diagonalization for massive multiuser MIMO systems,”
IEEE Trans. Commun ., vol.64, no. 1, pp. 201-211, Jan. 2016.[14] X. Yu, J.-C. Shen, J. Zhang, and K. B. Letaief, “Alternating minimization algorithms for hybrid precoding in millimeterwave MIMO systems,”
IEEE J. Sel. Topics Signal Process ., vol. 10, no. 3, pp. 485-500, Apr. 2016[15] C. E. Chen, “An iterative hybrid transceiver design algorithm for millimeter wave MIMO systems,”
IEEE Wireless Commun.Lett .,vol.4, no. 3, pp. 285-288, Jun. 2015.[16] C. Rusu, R. Mndez-Rial, N. Gonzlez-Prelcic and R. W. Heath, “Low complexity hybrid precoding strategies for millimeterwave communication systems,”
IEEE Trans. Wireless Commun., vol. 15, no. 12, pp. 8380-8393, Dec. 2016.[17] X. Gao, L. Dai, C. Yuen, and Z. Wang, “Turbo-like beamforming based on tabu search algorithm for millimeter-wavemassive MIMO systems,”
IEEE Trans. Veh. Technol ., vol. 65, no. 7, pp. 5731-5737, Jul. 2016.[18] S. Park, A. Alkhateeb, and R. W. Heath, “Dynamic Subarrays for Hybrid Precoding in Wideband mmWave MIMO Systems,”
IEEE Trans. Wireless Commun ., vol. 16, no. 5, pp. 2907-2920, May 2017.
HEN et al. : HYBRID PRECODING BASED ON NUQ CODEBOOK TO REDUCE FEEDBACK OVERHEAD IN MILLIMETER WAVE MIMO SYSTEMS29 [19] A. Alkhateeb, G. Leus, and Heath Jr. Robert W, “Limited feedback hybrid precoding for multi-user millimeterwavesystems,”
IEEE Trans. Wireless Commun ., vol. 14, no. 11, pp. 6481-6494, Nov. 2015.[20] Y. P. Lin, “On the quantization of phase shifters for hybrid precoding systems,”
IEEE Trans. Signal Process., vol. 65, no.9, pp. 2237-2246, May 2017.[21] A. Alkhateeb, O. El Ayach, G. Leus, and R. W. Heath, “Channel estimation and hybrid precoding for millimeter wavecellular systems,”
IEEE J. Sel. Topics Signal Process., vol. 8, no. 5, pp. 831-846, Oct. 2014[22] J. Singh and S. Ramakrishna, “On the feasibility of codebook-based beamforming in millimeter wave systems with multipleantenna arrays,”
IEEE Trans. Wireless Commun ., vol. 14, no. 5, pp. 2670-2683, May 2015[23] M. K. Samimi and T. S. Rappaport, “Ultra-wideband statistical channel model for non line of sight millimeter-wave urbanchannels,” in
Proc. IEEE Global Commun. Conf. (GLOBECOM),
Austin, TX, Dec. 2014, pp. 3483-3489.[24] M. K. Samimi and T. S. Rappaport, “3-D millimeter-wave statistical channel model for 5G wireless system design,”
IEEETrans. Microw. Theory Techn., vol. 64, no. 7, pp. 2207-2225, Jul. 2016.[25] T. S. Rappaport, G. R. MacCartney, M. K. Samimi, and S. Sun, “Wideband millimeter-wave propagation measurements andchannel models for future wireless communication system design,”
IEEE Trans. Commun., vol. 63, no. 9, pp. 3029-3056,Sept. 2015.[26] A. Goldsmith, S. Jafar, N. Jindal, and S. Vishwanath, “Capacity limits of MIMO channels,”
IEEE J. Sel. Areas Commun., vol. 21, no. 5, pp. 684-702, Jun. 2003.[27] T. Rappaport, F. Gutierrez, E. Ben-Dor, J. Murdock, Y. Qiao, and J. Tamir, “Broadband millimeter-wave propagationmeasurements and models using adaptive-beam antennas for outdoor urban cellular communications,”
IEEE Trans. AntennasPropag., vol. 61, no. 4, pp. 1850-1859, Apr. 2013.[28] T. S. Rappaport, R. W. Heath, R. C. Daniels, and J. N. Murdock,
Millimeter Wave Wireless Communications . New York,NY, USA: Pearson Education, 2014.[29] D. Love, R.W. Heath, Jr., and T. Strohmer, “Grassmannian beamforming for multiple-input multiple-output wirelesssystems,”
IEEE Trans. Inf. Theory , vol. 49, no. 10, pp. 2735-2747, Oct. 2003.[30] O. El Ayach, R. Heath, S. Abu-surra, S. Rajagopal, and Z. Pi, “The capacity optimality of beam steering in large millimeterwave MIMO systems ,” in