Hybrid precoding codebook design in millimetre-wave massive MIMO systems with low-resolution phase shifters

Millimetre-wave (mmWave) massive multiple-input multiple-output (MIMO) is one of the promising techniques for 5G wireless communications and beyond. Low-resolution hybrid precoding using low-resolution phase shifters (PSs) is considered to be promising for mmWave massive MIMO, since it can realize an acceptable performance with significantly reduced energy consumption. However, to realize accurate channel state information acquisition, the traditional channel feedback codebooks that quantize the channel with high resolution are not suitable for low-resolution hybrid precoding. To solve this problem, angle-based codebook is proposed here. In the proposed codebook, the analog codebook is designed based on the channel angle-of-departures (AoDs) and the digital codebook is generated by the random vector quantization. Speciﬁcally, the analog codewords are optimized by a neighbour search algorithm under the constraint of low-resolution PSs. These analog codewords are designed to be aligned with channel AoDs. In this


INTRODUCTION
Millimetre-wave (mmWave) massive multiple-input multipleoutput (MIMO) has been widely considered as one of the promising techniques to provide a very high data rate for the 5G communications and beyond [1][2][3][4][5][6]. To realize the multiplexing gain of mmWave massive MIMO systems, channel-adaptive techniques such as precoding should be used. However, the traditional fully digital precoding requires an equal number of radio-frequency (RF) chains as the large antenna number at the base station (BS). This will cause unacceptable power consumption and hardware complexity in mmWave massive MIMO systems. To solve this challenging problem, hybrid precoding has been proposed to significantly reduce the number of RF chains [7][8][9][10][11][12][13][14][15]. The main idea of hybrid precoding is to decompose the large-size fully digital precoder into a large-size analog beamformer realized by phase shifters (PSs) and a small-size digital precoder realized by a small number of RF chains. Due to the sparsity of mmWave channels, it has been proved that hybrid precoding is able to achieve the near-optimal achievable rate performance [14].

Prior works
Accurate channel state information at the transmitter (CSIT) is essential to realize the near-optimal achievable rate performance for massive MIMO systems [16][17][18][19]. To this end, various channel feedback schemes have been proposed to realize accurate CSIT for massive MIMO. Specifically, the authors in [20] proposed a distributed compressive sensing-based channel feedback scheme to exploit the common support of the channel to achieve better channel quantization accuracy. In [21], a codebook that linearly combines traditional discrete Fourier transform (DFT) codewords was designed to quantize the channel more accurately. In addition, an adaptive codebook was proposed by quantizing the low-dimensional channel subspace in [22]. These schemes work well for massive MIMO using the fully digital precoding. However, the inevitable channel quantization error in these schemes will cause an obvious performance loss for mmWave massive MIMO using hybrid precoding, where the number of RF chains is much smaller. One possible solution to this problem is to directly quantize the optimal precoder to avoid the error propagation caused by the channel quantization error. Specifically, a two-stage hybrid precoding scheme with limited feedback was proposed in [23]. The beamsteering codebook and random vector quantization (RVQ) codebook were used for analog beamformer and digital precoder, respectively [23]. In [24,25], a hierarchical codebook was designed to form the analog beamformer for hybrid precoding via the deactivation antenna processing and the multi-RF-chain sub-array technique. However, these existing codebooks only consider high-resolution PSs (e.g. 8 bits or more [25]). When large numbers of high-resolution PSs are used for hybrid precoding, the system may suffer from high energy consumption (e.g. 30 − 60 mW for 8 bits PSs in mmWave frequencies [26][27][28][29], and 8 × 1024 = 8192 PSs will be used at the BS with 1024 antennas for the fully connected hybrid precoding to serve 8 users) [30]. Compared with the high-resolution hybrid precoding utilizing high-resolution PSs (e.g. 8 bits or more), the recently proposed low-resolution hybrid precoding using low-resolution PSs (e.g. 1-2 bits) is attractive due to its acceptable achievable rate performance and higher energy efficiency [31][32][33]. In [31], the near-optimal low-resolution hybrid precoder was designed by projecting the optimal precoder to the subspace determined by the constraint of low-resolution PSs. In [32], two types of lowresolution hybrid precoding methods which respectively quantize real parts and imaginary parts of the optimal precoder were proposed and analysed. A machine learning-based lowresolution hybrid precoding method was proposed for 1-bit PSs realized by switches and inverters in [33]. Moreover, an iterative phase matching algorithm was introduced to design optimal low-resolution precoder and combiner in [34]. However, all of these low-resolution hybrid precoding schemes [31][32][33][34] were designed under the ideal assumption of perfect CSIT at the base station. Unfortunately, the traditional channel feedback schemes [16,35] which quantize the channel cannot realize the perfect CSIT, especially with the large antenna number in massive MIMO systems. Therefore, the channel quantization error will propagate to the design of low-resolution hybrid precoding, and result in an obvious achievable rate performance loss. In addition, the existing channel feedback schemes in [23,25] are designed to quantize the optimal precoder with high resolution. They are not suitable for low-resolution hybrid precoding. Therefore, it is of great importance to design a codebook suitable for low-resolution hybrid precoding. This has not been well addressed in the literature to the best of our knowledge.

Contributions
To fill in this gap, here, we propose an angle-based codebook to realize low-overhead channel feedback for low-resolution hybrid precoding. The main contributions are summarized as follows: • We propose an angle-based codebook for low-resolution hybrid precoding. Unlike existing codebooks which quantize the channel [16,21], the proposed codebook directly quantizes the analog beamformer and digital precoder to avoid error propagation. Its basic idea is to design the highdimensional analog codebook based on channel angle-ofdepartures (AoDs), while the traditional RVQ codebook is utilized for the low-dimensional digital precoder. To be suitable for low-resolution hybrid precoding, each element of the analog codeword in the proposed codebook satisfies the PSs resolution constraint. • We formulate an optimization problem to design the highdimensional analog codebook under the PSs resolution constraint. The target of the optimization problem is to make analog codewords have directional array gain aligned with channel AoDs. A neighbour search algorithm (NSA) is utilized to solve the optimization problem. Under the framework of angle-coherence time [36], the analog codewords designed by solving the optimization problem can remain unchanged for a much longer time than the channelcoherence time. Therefore, the feedback frequency of the analog codewords can be significantly reduced, which leads to a low channel feedback overhead. • We analyse the required channel feedback overhead and the achievable rate of the proposed angle-based codebook. These analyses show that the proposed angle-based codebook can achieve an acceptable achievable rate performance with a much lower channel feedback overhead. Numerical simulation results illustrate that the proposed angle-based codebook can form narrow and directional beams. Also, it can achieve a better tradeoff between the achievable rate and power consumption with a low channel feedback overhead for lowresolution hybrid precoding in mmWave massive MIMO systems.

Organization and notations
The remainder of this paper is organized as follows. Section 2 introduces the channel model of mmWave massive MIMO and low-resolution hybrid precoding. The proposed angle-based codebook and the corresponding analysis are provided in Section 3. The numerical simulation results are shown in Section 4. Finally, in Section 5, the conclusions are drawn. Notation: Lower-case and upper-case boldface letters represent vectors and matrices, respectively; (⋅) T , (⋅) H , ‖ ⋅ ‖ F , and ‖ ⋅ ‖ k denote the transpose, conjugate transpose, Frobenius norm, and k-norm of a vector or matrix, respectively; (⋅) denotes the expectation; | ⋅ | denotes the absolute operator; I N represents the identity matrix of size N × N ; ( n k ) denotes the number of k-combinations of a set with n elements;  ( , ) and  (a, b) denote the Gaussian distribution with mean and covariance , and the uniform distribution between a and b, respectively. Finally, A[i, j ] denotes the element of matrix A at the ith row and the j th column, respectively.

SYSTEM MODEL
We consider the typical mmWave massive MIMO systems utilizing hybrid precoding in this paper. The BS and the user deploy the widely used uniform linear array (ULA). The antenna number is M for the BS and N for the user. Considering hybrid precoding, M RF RF chains are applied at the BS, where M RF ≪ M [23]. Therefore, the received signal vector y of size N × 1 at the user can be represented as The  Specifically, we exploit the classical ray-based channel model for the channel matrix H to describe the mmWave massive MIMO channel characteristics as [1,2] where L represents the number of the resolvable paths of the channel, g denotes the complex gain of the th path following the distribution  (0, 1), and represent the th path's spatial AoD and angle-of-arrival (AoA), respectively, with , ∈ (−1, 1). For instance, a( , M ) is the steering vector of size M × 1 for the spatial AoD with the following form where d is antenna spacing, denotes the signal wavelength, and = sin̄wherēis the physical AoD. Usually, at mmWave frequencies, the distance d is set as ∕2 [1].
In the existing hybrid precoding structure for mmWave massive MIMO shown in Figure 1a, every element of the analog beamformer F RF is realized by a PS. As shown in Figure 1a, high-resolution PSs (8 bits or more) are utilized to guarantee good achievable rate performance [14]. Nevertheless, the power consumption of high-resolution PSs is high (e.g. 30 − 60 mW for 8 bits PSs in mmWave frequencies [26][27][28][29]). This will result in the high power consumption of hybrid precoding, since a large number of PSs will be utilized. To alleviate the high power consumption problem, substituting lowresolution PSs (1-2 bits, e.g. P = 1, 2) for high-resolution PSs (8 bits or more, e.g. P = 8) is considered to be a feasible solution without an obvious loss in the achievable rate [31][32][33][34]. The lowresolution PSs can be realized by power-efficient components, for example, one possible implementation is to use switches and inverters as shown in Figure 1b. The power consumption of one inverter or one switch is merely 5 mW at mmWave frequencies [37,38]. Therefore, the low-resolution hybrid precoding can realize a better tradeoff between the power consumption and the system achievable rate performance. However, the existing low-resolution hybrid precoding methods such as [33] assume perfect CSIT at the BS. In practical scenarios, due to the large antenna number in massive MIMO systems, the quantization error for channel feedback will cause a serious performance loss for these methods. On the other hand, in existing codebooks like the beamsteering codebook [23] and the hierarchical codebook [24], which directly quantize and feedback the optimal precoder, the constraint of low-resolution PSs has not been considered. Therefore, it is necessary to design a codebook to quantize the precoder under the constraint of low-resolution PSs for low-resolution hybrid precoding. To fill in this gap, we will propose an angle-based codebook that directly quantizes the optimal precoder for low-resolution hybrid precoding. The proposed angle-based codebook will be discussed in detail in Section 3.

PROPOSED ANGLE-BASED CODEBOOK
Here, the codebook-based hybrid precoding is first introduced. Then, we propose an angle-based codebook for low-resolution hybrid precoding including the analog codebook and the digital codebook. In our proposed codebook, the analog codebook is composed of codewords that have array gains aligned with different AoDs. These codewords are designed by an NSA. Due to the channel characteristic that the path angles vary much slower than the path gains, the channel feedback overhead of the proposed angle-based codebook can be reduced significantly. This will be verified through the theoretical analyses of channel feedback overhead and achievable rate.

Codebook-based hybrid precoding
In codebook-based hybrid precoding, the BS usually acquires the analog beamformer F RF and the digital precoder F BB through limited feedback. Specifically, the user selects F RF and F BB in an analog codebook  and a digital codebook , respectively, and then feeds the indexes of the chosen codewords back to the BS. Generally, the optimal pair of analog beamformer and digital precoder can be selected by maximizing the achievable rate R as [1] {F RF , F BB } = arg max where ) .
(6) Due to the mmWave channel characteristics, the beamsteering codebook has been widely used as the analog codebook [23,39]. The codewords in the beamsteering codebook are steering vectors with quantized AoDs. Specifically, the beamsteering codebook  b can be denoted as In (7), the quantized AoD i for i = 1, 2, … , 2 N A satisfies i = −1 + (2i − 1)∕2 N A and N A is the quantization bits of AoDs. The analog beamformer F RF is composed of the codewords in  b [23,39], which can be represented as For the digital precoder, several codebooks have been proposed. Since the digital precoders are realized by baseband signal processing for both low-and high-resolution PSs, the existing digital codebooks can be also used for low-resolution hybrid precoding. We utilize the classical RVQ codebook [16] to quantize the digital precoder here, which will be discussed in detail later in Section 3.2. However, the existing analog codebooks such as beamsteering codebook do not consider the low-resolution of PSs, and cannot be utilized in low-resolution hybrid precoding due to the quantization error. In Section 3.2, we will solve this problem by the proposed angle-based codebook.

Angle-based codebook
Here, we will propose an angle-based codebook to directly quantize and feed back the low-resolution hybrid precoder. At first, the overall design of the angle-based codebook is introduced. Then, we formulate an optimization problem to design analog codewords. An NSA is utilized to solve the optimization problem. Finally, the design of the digital codebook is illustrated.

The overall design of the angle-based codebook
The basic idea of the proposed angle-based codebook is to directly quantize the hybrid precoder. Considering that the hybrid precoder is composed of two parts, that is, analog beamformer and digital precoder, the proposed codebook also has two parts as analog codebook  and digital codebook .
Inspired by the beam steering codebook [23], the analog codebook  is designed to contains analog codewords that have array gains aligned with different AoDs under PSs resolution constraint. Each codeword quantizes one column of the analog beamformer. In this way, the array gain of each channel path can be sufficiently achieved. Specifically, the analog codebook  can be designed as N A is the feedback bits of each analog codeword, and f i of size M × 1 is the codeword designed to have the directional array gain aligned with angle i = −1 + (2i − 1) Considering these angles i distribute in the angle domain uniformly, the analog codebook can cover the whole angle domain (−1, 1) uniformly with near-optimal array gains. On the other hand, the digital codebook is designed to contain codewords that directly quantize the digital precoder. In specific, the digital codebook  can be denoted as N D denotes the feedback bits of each digital codeword, and D i of size M RF × M s is the ith digital codebook. Next, we will explain how to design analog codebook  and digital codebook  in detail.

Analog codebook design
The major problem of analog codebook design is how to design codewords f i and make f i have directional array gain. Such analog codewords are hard to design due to the PSs resolution constraint in low-resolution hybrid precoding. To solve this problem, we formulate an optimization problem under the PSs resolution constraint to design analog codewords. The key idea is to make analog codewords have directional array gains similar to steering vectors. To simplify the expression, we denote f of size M × 1 as the analog codeword that has the directional array gain on AoD under PSs resolution constraint.
First, we give a definition of the array gain. Specifically, denote (a( , M ), ) as the array gain of the steering vector a( , M ) on the AoD ∈ (−1, 1). It can be denoted as is the Dirichlet function [40]. It should be noticed that due to the definition of steering vectors in (4), the array gain has a value range as (a( , M ), ) ∈ [0, 1] 1 . From (11), the following two observations on the array gain of a( , M ) can be obtained [40] to provide instructions for analog codewords design: • Due to the power-focusing property of Dirichlet function, the power of (a( , M ), ) concentrates on the mainlobe of , the power leakage of a( , M ) will induce interference between different analog codewords when they form the analog beamformer F RF , which leads to achievable rate loss. Therefore, for a specific angle , the analog codeword which has similar directional array gain as the steering vector should have high array gain on ( − ). Following the above idea, we propose to formulate the design of analog codeword f as an optimization problem In (12), the numerator ‖ [ ]‖ 2 2 represents the array gain on the angle range ( − ). Specifically,  denotes the set including all possible analog codewords that satisfies the PSs resolution constraint (2). represents the array gain vector of f over the whole angle domain as It should be pointed out that when P = 1, the elements of array gain vector is symmetric as (f, i ) = (f, − i ). This is because where (a) comes from the PSs resolution constraint with P = 1 as f[i] ∈ 1 , and (b) comes from the form of steering vectors as shown in (4). Therefore, for the case P = 1, we slightly modify the definition of to take this symmetry property of the codeword f into consideration as By maximizing the array gain ratio, (12) can guarantee a nearoptimal directional array gain of f , which is aligned with the AoD . The next question is how to resolve the optimization problem (12). We can find that (12) is a non-convex problem Output: analog codeword f with directional array gain aligned with ; 2: Randomly generate f ,(0) with PSs constraint that each element belongs to P ; 3: because of the PSs resolution constraint in (2). Note that for the codebook-based hybrid precoding, the codebook is designed in advance so that both the BS and the user know the codebook. Therefore, the exhausted search (ES) algorithm can be adopted to solve (12). However, although the PSs resolution P is small, the number of candidates in  is still quite large (e.g. 2 64 when P = 1, M = 64). Such a large number of candidates will cause unacceptable search complexity. To avoid the high search complexity in the ES algorithm, we utilize a NSA to solve (12) in a near-optimal manner. The main idea of the NSA algorithm is to search the neighbour set iteratively to find out the codeword with the maximum array gain ratio. In the NSA, to make the neighbour set suitable for the low-resolution hybrid precoding, we design the neighbour set elaborately to meet the constraint of low-resolution PSs in (2). Specifically, the neighbour set D of the codeword f ∈  is defined as The pseudo-code of the NSA is shown in Algorithm 1. Specifically, G is first calculated in step 1, and in step 2, the ini-tial codeword f ,(0) of size M × 1 whose elements belong to P is randomly generated. Then, in step 4, a neighbour set D,(i ) is constructed by selecting all the codewords which have D elements different from f ,(i ) for the ith iteration. After that, in step 5, the new near-optimal codeword f opt,(i ) is searched out from the neighbour set D,(i ) by finding the best candidate that maximizes the array gain ratio as shown in (12). Finally, we can obtain the near-optimal codeword f with directional array gain on when f ,(i ) equals f opt,(i ) in step 6, or the iteration number I is approached in step 9. Due to low-resolution PSs, the size of the neighbour set D,(i ) is small. Under these circumstances, the search range in each iteration in step 5 is limited. Hence, the NSA possesses favourable convergence. The convergence of the NSA will be verified later by simulation results in Section 4.
By utilizing the NSA, the analog codebook  in (9) can be easily obtained. For example, by setting the target angle = i , we can obtain the ith codeword f i in the analog codebook .

Digital codebook design
After the analog beamformer F RF has been determined, the optimal digital precoder F BB can be computed by the singular value decomposition (SVD) principle as discussed in [16]. Specifically, we define H eq = HF RF as the equivalent channel and the ordered SVD of H eq can be defined as H eq = U eq eq V H eq . Then, we have the optimal digital precoder F BB as V eq,1 is of size M RF × N , and = √ M s ∕‖F RF B‖ F is the power normalization factor. Therefore, the digital codebook  should quantize the digital precoder F BB . Regarding that in hybrid precoding the number of RF chains M RF is usually much smaller than the number of antennas M (e.g. M = 64, M RF = 4), the digital precoder F BB is low dimensional. Therefore, to quantize the low-dimensional randomly distributed digital precoder, we utilize the traditional RVQ codebook which can guarantee an acceptable quantization accuracy [16]. Specifically, the digital codebook  can be generated as  = {D 1 , D 2 , … , D 2 N D }. D i is an M RF × M s matrix with all randomly distributed elements, and N D is the feedback bits of a digital codeword.

Complexity and performance analysis
Here, we analyse the computational complexity of the NSA algorithm and the ES. From Algorithm 1, we can notice that steps 4 and 5 occupies the major part of the complexity of the NSA. Specifically, we construct the neighbour set D,(i ) in step 4. Since we select f ∈  that has D elements different from f ,(i ) , the size of the neighbour set D,(i ) can be easily obtained  . }. In summary, after I iterations of searching, the total complexity of the NSA C NSA is By contrast, the complexity C ES of the ES scheme can be denoted as Comparing (20) and (21), we can notice that the computational complexity of the NSA increases with the square of the antenna number M , while the complexity of the ES is an exponential function of M . This indicates that the NSA enjoys much lower complexity when M is large in mmWave massive MIMO systems. Specifically, Table 1 shows the complexity comparison between the NSA and the ES, where we set T = 200, I = 50, D = 2 in the NSA. From Table 1, we can observe that the NSA has much lower complexity than the ES especially for a large antenna number. For example, when M = 64, P = 1, the searching complexity of the NSA is only 2.58 × 10 9 . However, the complexity of the ES is 1.18 × 10 21 , which is unacceptable in practice.
It should be pointed out that the selection of the parameter D may simultaneously influence the complexity and performance of the NSA algorithm. Specifically, on the one hand, as illustrated in (20), when D increases, ( M D ) will increase and in contrast the iteration number I will decrease because of the enlarged search range. Hence, the overall impact on the complexity of an increased D may be a larger complexity or a smaller complexity. On the other hand, since the search range becomes larger when D grows up, the NSA may fall into local optimum with a larger probability. This means a large D may cause performance decline to some degree. Therefore, we can conclude that the selection of the parameter D may depend on the system requirements and the system parameters such as antenna number M . Specifically, if the system requires a lower complexity, a large D may be promising due to the fast convergence speed. While, if the performance, that is, array gain ratio, is more important to the system, a small D such as D = 1 may be better.
An example on how to select the parameter D will be shown in Figure 5 in Section 4.

Channel feedback overhead analysis
Here, we will analyse the channel feedback overhead of the proposed angle-based codebook. Under the assumption that efficient channel estimation is carried out, the channel matrix H and the AoDs are known at the user [19,20,41]. Without loss of generality, we assume that the AoDs are sorted in a descending order according to the power of the path gain g . Then, the analog beamformer ] and the digital precoder F BB can be selected out from analog codebook  with size of 2 N A and digital codebook  with size of 2 N D , respectively. Specifically, the analog beamformer F RF is composed of M RF codewords which are selected out as In (23), we define N A ∕{̂1,̂2, … ,̂i −1 } = { | ∈ N A , ∉ {̂1,̂2, … ,̂i −1 }}. In (22) and (23), the analog codewords aligned with the nearest angle to the AoDs are selected out one by one, to guarantee the near-optimal directional array gain. After the analog beamformer has been determined, the digital precoder can be chosen in digital codebook  to maximize the achievable rate R as After the analog codewords and digital codeword are selected out at the user, their indexes are fed back to the BS through the uplink channel. Then, the hybrid precoding is carried out based on these indexes. Since M RF analog codewords and one digital codeword are selected, the channel feedback overhead of the proposed angle-based codebook is M RF N A bits for the analog beamformer, and N D bits for the digital precoder. Therefore, the total channel feedback overhead is M RF N A + N D bits. We can observe that the channel feedback overhead consists of two parts. One part is M RF N A which is for the selected analog codewords to indicate analog beamformer, and the other part is N D which is for the selected digital codeword to indicate digital precoder.
We first analyse the channel feedback overhead for analog beamformer M RF N A . Because the scatters surrounding the BS do not change their positions frequently, the path angles which are mainly determined by these scatters, vary much slower than the path gains [36]. Therefore, the angle-coherence time, during which the AoDs can be regarded as unchanged, is much longer than the channel-coherence time [42,43]. For example, [36] shows that the angle-coherence time is over 20 times longer than the channel-coherence time. Recalling that in angle-based codebook the analog codewords are designed to have directional array gains, which can be seen determined by the channel AoDs, the analog codewords selected in (22) and (23) will remain unchanged for a much longer time than the digital codeword. Thus, we can conclude that the average channel feedback overhead can be considered mainly decided by the part of the digital precoder. That is, the average channel feedback overhead of the proposed codebook is not M RF N A + N D bits but close to N D bits.
Fortunately, due to the low-dimensional property of the digital precoder, the channel feedback overhead N D could be very small. Specifically, to achieve common quantization accuracy, the required channel feedback overhead should linearly increase with the dimension of the target matrix of the feedback [17]. For instance, when M = 64, M RF = 2, M s = 2, the dimension of the digital precoder, for example, M RF M s = 2 × 2, is much smaller than the dimension of massive MIMO channel, for example, MM s = 64 × 2. Consequently, the channel feedback overhead N D can be significantly reduced by utilizing the proposed anglebased codebook. This is because a low-dimensional digital precoder instead of a large-dimensional massive MIMO channel is fed back.

Achievable rate performance analysis
Under the codewords selection principle in (22), (23), and (24), the achievable rate of our proposed angle-based codebook can be analysed. Specifically, the achievable rate R can be rewritten by the SVD of H as d denotes the rank of H. i , i = 1, 2, … , d denotes the ith largest singular value of H. Without losing generality, we assume that the system parameters for hybrid precoding (M , M RF , M s ) are dedicatedly designed so that the multiplexing gain from the multi-path channel H can be fully utilized [7]. This assumption, specifically, can be easily satisfied in practical massive MIMO systems as: Under this assumption, according to [7], (24) can be rewritten as where Notice that in (3), the steering vectors of AoDs a( , M ) form an orthogonal basis of H H H due to the approximate orthogonality of steering vectors with large number of antennas [22]. Con-sidering that the columns of V f are also an orthogonal basis of H H H, we can approximate the columns of the matrix V by the linear combination of a( l , M ) as In (27) ) .
(28) According to (22) and (23), the analog precoder F RF is composed of the codewords selected from the proposed angle-based codebook. Because the codewords in the analog codebook  are designed to have directional array gains aligned with channel AoDs in (12), we can expect that f in  is near-optimally directional and approximately orthogonal to each other, just as the steering vectors. This property of the analog codewords in the proposed angle-based codebook will be verified in Section 4. Hence, we can obtain that the matrix A H t F RF is approximately diagonal and satisfies where (f̂i , i ), i = 1, 2, … , M RF denotes the array gain of f̂i on the angle i . For the analog codebook , the size N A should be sufficiently large such that all the codewords can provide near-optimal array gain over the whole angle domain. This can be easily realized since the feedback of the analog codewords is infrequent based on the discussion in Section 3.4. Thus, we have (f̂1 , 1 ) 2 ≈ (f̂2 , 2 ) 2 ≈ ⋯ ≈ (f̂M RF , M RF ) 2 ≈ ( (f , ) 2 ). Then, from substituting (29) into (28), we can obtain Notice that the high effective signal-to-noise rate (SNR) assumption can be easily realized in mmWave system due to the high array gain. (30) can be reformulated according to [7] under high SNR situation as In (31), we can observe that the achievable rate performance based on our proposed angle-based codebook is decided by two terms. The first one log 2 (|I M s + N s 2 ( (f , ) 2 ) 2 |) reveals the array gain loss caused by the low-resolution analog codewords, and the second term (M s − ‖D H opt F BB ‖) reflects the quantization error of the digital precoder.
We first analyse the performance of the analog codebook, which is mainly determined by the expectation of the array gain achieved by the codewords in the analog codebook ( (f , ) 2 ). However, due to the low resolution of the PSs, this expectation cannot be easily computed. To this end, we utilize the array gain of the random generated codeword f when the antenna number tends to infinity as M → ∞ to replace ( (f , ) 2 ). We denote this array gain as lim M →∞ (f, ) 2 . As shown in Appendix, Lemma 1 proves the property of lim M →∞ (f, ) 2 satisfies (33) reveals that the achievable rate is influenced by the resolution of PSs P, where PSs with lower resolution induces achievable rate loss. However, we can notice from (33) that although the application of low-resolution PSs induces achievable rate loss, this loss is acceptable and maintains asymptotically constant as the number of BS antennas M increases to infinity.
For the digital precoder, we utilize a RVQ codebook to quantize and feedback. In (24), the digital precoder is chosen to maximize the achievable rate. From (31), we can notice that to maximize the achievable rate is to maximize ‖D H opt F BB ‖ 2 F . This means the digital precoder F BB is the codeword closest to D opt in . Therefore, in the RVQ framework, the expectation of the maximum of ‖D H opt F BB ‖ 2 F can be bounded based on the dimension of F BB and the channel feedback overhead N D . Specifically, according to [16], we can obtain .
(35) reflects that the achievable rate achieved by the analog beamformer and digital precoder selected from our proposed angle-based codebook is mainly determined by the resolution of PSs P and the feedback overhead N D . On the one hand, the achievable rate loss caused by the low-resolution PSs is limited and acceptable. On the other hand, since the dimension of digital precoder M s M RF is much smaller than the antenna number M , a small N D is able to guarantee a sufficiently small In conclusion, our proposed angle-based codebook can achieve acceptable achievable rate performance with a low channel feedback overhead. Recalling the lower power consumption of low-resolution PSs, our proposed angle-based codebook for low-resolution hybrid precoding can achieve a better tradeoff between the achievable rate performance and the power consumption. This conclusion will be confirmed by simulation results in Section 4.

Extension to multi-user scenario
Here, we will introduce how to apply the proposed angle-based codebook into multi-user scenarios. In this case, we consider K single-antenna users are served by the BS. Without loss of generality, we assume the number of users are equal to the number of RF chains as K = M RF so that each user is served by one RF chain. Under these circumstances, the received signals of K users y = [ȳ 1 ,ȳ 2 , … ,ȳ K ] T can be denoted as y =HF RF F BBs +n.
F RF of size M RF × M is the analog beamformer which is realized by PSs and follows the constraint in (2). F BB of size M RF × K is the digital precoder.H of size K × M represents the multiuser channel matrix whereH = [h T 1 ,h T 2 , … ,h T K ] withh T k representing the channel between the BS and the kth user.s = [s 1 , s 2 , … , s K ] T is the transmitted signals for K users.n of size K × 1 represents the Gaussian noise vector. Besides, the channel of the kth userh k can be denoted by the mmWave massive MIMO channel model [1] as where L is the number of channel paths, g l ,k denotes the path gain of the l -path for the kth user, and l ,k denotes the AoD of the l th path for the kth user.
In multi-user cases, the digital precoder cannot be decided only by the channel of one user, but should be designed based on channels of all users to mitigate the inter-user interference.
To this end, we follow the idea of the two-stage multi-user precoder [23] and propose a multi-user hybrid precoding scheme by utilizing the proposed angle-based codebook. In the proposed scheme, the analog beamformer is composed of codewords which are aligned with different users AoDs to realize sufficient array gain. Then, the digital precoder is designed based on the equivalent channel. Specifically, the proposed multi-user hybrid precoding scheme has two stages as follows.
1. Each user select a codeword from the analog codebook  based on the channel h k . The label i k of the selected codeword for the kth user can be denoted as Then, each user feed the label i k , k = 1, 2, … , K back to the BS. After that, the BS can determine the analog beamformer by The BS acquires the equivalent channel H eq =HF RF through traditional channel estimation methods [15] 2 . Then, the digital precoder F BB can be designed based on the equivalent channel H eq by using existing method, for example, zero forcing [33].
By utilizing the above two-stage hybrid precoding scheme, the array gain can be sufficiently obtained by stage 1 and the inter-user interference can be mitigated by stage 2. In this way, an acceptable achievable rate performance can be expected, which will be verified by simulation results in Section 4.

SIMULATION RESULTS
Numerical  Figure 2 shows the array gain of the codewords designed by the NSA where both 1-bit and 2-bit PSs are considered. The array gain is calculated according to (6). The array gain of the ideal steering vector with infinite-resolution PSs is also shown as the benchmark. For the NSA, the parameters are set as: target angle = , 64), which can be seen as the benchmark. The array gain of the proposed anglebased codeword with 1-bit and 2-bit PSs are shown in Figure 2b and 2c, respectively. We can observe from Figure 2b,c that the codewords designed by the NSA with 1-bit and 2-bit PSs can achieve equal beam width together with about 70% and 90% array gain compared to the ideal steering vector with infinite resolution. This reveals that the proposed angle-based codebook can generate narrow enough beams to distinguish different AoDs, and supports the analysis in Sections 3.4 and 3.5. On the other hand, we can notice that when P = 1, the codeword may induce interference in direction = − . This interference may cause achievable rate performance loss when there are two users locate in a symmetric location. Fortunately, the probability of this case is quite small due to the narrow beam width, and performance loss caused by this interference can be avoided by efficient user scheduling scheme [44].
In Figure 3, we simulate the convergence of the NSA with different search interval D = 1, D = 2, and D = 3. Specifically, the array gain ratio in (12) against the number of iterations I is illustrated in Figure 3, where M = 16, P = 1. The NSA parameters are the same as those in Figure 2. The array gain ratio of the codeword searched by ES is also shown for comparison. From Figure 3, we can observe that the NSA can converge with a small number of iterations, for example, I = 10. The NSA can achieve close array gain ratio performance to ES with D = 1, D = 2, and D = 3. On the other hand, we can see that with the search interval D growing up, the array gain ratio performance decreases, while the required iteration number I becomes smaller. Hence, to determine how to select D, the specific complexity and performance of the NSA with parameters M = 16, P = 1 should be considered. Specifically, considering The convergence of the NSA with different D.

FIGURE 4
Achievable rate performance comparison for the low-resolution hybrid precoding with 1-bit PSs and the iteration number shown in Figure 3, we can know that the case D = 1 has smaller complexity than the cases D = 2 and D = 3 according to (20). Moreover, since Figure 3 shows the case D = 1 can obtain better array gain ratio performance than the cases D = 2 and D = 3. We can conclude that when M = 16, P = 1, D is preferred to be D = 1 to obtain a better performance with a smaller complexity. Figures 4 and 5 present the comparison of the achievable rate performance for the low-resolution hybrid precoding with 1bit and 2-bit PSs, respectively. The SNR is defined as 2 . For different codebooks, we set the channel feedback overheads of them are the same to make a fair comparison. Specifically, for the proposed angle-based codebook, we set N A = 6, N D = 7 to directly quantize hybrid precoder. This leads to a 7-bits channel feedback overhead as described in Section 3.4; for the FIGURE 5 Achievable rate performance comparison for the low-resolution hybrid precoding with 2-bit PSs DFT-combined codebook [21] and the RVQ codebook [16], the channel feedback overheads to quantize the channel are 7 bits, and low-resolution hybrid precoding method in [33] is carried out. In addition, the high-resolution hybrid precoding with 8-bit PSs [7] and fully digital hybrid precoding with perfect CSIT [1] are also considered as benchmarks for comparison. From Figures 4 and 5, We can observe that the proposed angle-based codebook outperforms both of the DFT-combined codebook and the RVQ codebook with low-resolution hybrid precoding [33], under equal channel feedback overhead. Moreover, the performance gap between the proposed angle-based codebook and the traditional high-resolution hybrid precoding is small, for example, the 1-bit and 2-bit angle-based codebook can approach 85% and 90% rate performance of high-resolution hybrid precoding when SNR is 10 dB. On the other hand, the performance gap between fully digital hybrid precoding and the angle-based codebook keeps constant when the SNR increases, which is consistent with Lemma 1.
We provide the achievable rate performance of the proposed angle-based codebook versus different antenna number M in Figure 6. The parameters are set the same as those in Figure 4. The performance of fully digital precoding and high-resolution hybrid precoding with perfect CSIT are also illustrated. We can observe from Figure 6 that the achievable rate performance achieved by the proposed angle-based codebook increases as the antenna number grows. Moreover, the performance ratio between the proposed angle-based codebook and fully digital precoding with perfect CSIT maintains constant when the antenna number M increases. This reflects that the proposed angle-based codebook can realize acceptable achievable rate performance with different antenna number M .
To further compare the effect of ES and NSA on the codebook design, we provide an achievable rate comparison between the angle-based codebook designed by the ES and the NSA. The parameters are set as M = 16 and P = 1. The other parameters Achievable rate performance comparison between the angle-based codebooks designed by the ES and the NSA are the same as these in Figure 4. We generate two angle-based codebooks by solving the optimization problem in (12) with ES and NSA. From Figure 7, we can see that the ES can obtain a very small performance gain compared with the NSA. Considering the huge complexity of the ES, this small performance gain can be ignored. Therefore, we can conclude that the NSA can achieve nearly the same performance as the ES with a much smaller complexity. The NSA is effective to be utilized to design the proposed angle-based codebook. Figure 8 illustrates the achievable sum-rate performance of the proposed angle-based codebook in multi-user scenarios. We set M = 64, P = 1, and K = 4. The size of the analog codebook is N A = 7. The overhead for estimating the equivalent channel is set as 12 bits. For the RVQ codebook [16], the channel feedback overhead is also set as 12 bits to quantize the chan-FIGURE 8 Achievable sum-rate performance for multi-user scenario nel, and the low-resolution hybrid precoding method in [33] is carried out at the BS. Just as described in Section 3.4, the channel feedback overhead of the analog codewords can be ignored due to the long angle-coherence time. Hence, the total channel feedback overheads of the proposed angle-based codebook and the RVQ codebook are the same. In addition, fully digital precoding with perfect CSIT is also provided for comparison. We can observe from Figure 8 that the proposed anglebased codebook can achieve much better performance than the RVQ codebook. This is because the RVQ codebook suffers from the propagation error between channel estimation and precoder design. In contrast, the proposed angle-based codebook avoids this performance decline since it directly quantizes and feeds back the precoder. Besides, we also find that the proposed angle-based codebook can approach 70% sum-rate performance of the fully digital precoding with perfect CSIT. The performance gap keeps constant as the SNR increases. This indicates that the proposed angle-based codebook can also realize acceptable achievable sum-rate performance in multi-user scenarios. Figure 9 illustrates the energy efficiency comparison of lowresolution hybrid precoding with 1-bit PSs using the proposed angle-based codebook, the existing high-resolution hybrid precoding with 8-bit PSs [7] and perfect CSIT, and the fully digital precoding [1] with perfect CSIT. We set M = 64, SNR = 10 dB, and M RF = N varies from 1 to 16. The parameters of the angle-based codebook and high-resolution hybrid precoding are set the same as those in Figure 4. According to [30], the energy efficiency can be defined as the ratio between the achievable rate and the energy consumption. Specifically, the energy efficiency of fully digital precoding P FD , high-resolution hybrid precoding P HH , and low-resolution hybrid precoding P LH , can be respectively represented as P LH = P t + M RF P RF + M RF P IV + MM RF P SW + P BB , (41) where P t , P RF , P PS , and P BB denote the transmitted power, the energy consumption of RF chain, the energy consumption of 4-bit PS, and the energy consumption of baseband signal processing. P IV and P SW are the energy consumption of inverter and switch, which can realize 1-bit PS [33]. For simulation, we adopt the practical values [31,33]: P t = 30 mW, P RF = 300 mW, P BB = 200 mW, P PS = 40 mW (from [26][27][28][29][30], we consider a reference power consumption for 8-bit PSs) and P IV = P SW = 5 mW. As shown in Figure 6, we can observe that the energy efficiency of 1-bit hybrid precoding with the proposed anglebased codebook is much higher than both of the fully digital precoding [1] and high-resolution hybrid precoding [7]. Particularly, when M RF is smaller than 8 which is the common case for mmWave massive MIMO systems with hybrid precoding structure [33], the advantage of energy efficiency is more significant. On the other hand, we can find that the energy efficiency of hybrid precoding decreases as the number of M RF grows. This is owing to the fact that when M RF increases, the number of PSs utilized for hybrid precoding increases rapidly. Therefore, we can conclude that Figure 9 supports the motivation to reduce the huge energy consumption of the PSs by using low-resolution hybrid precoding in mmWave massive MIMO systems.

CONCLUSIONS
Here, the codebook design for low-resolution hybrid precoding in mmWave massive MIMO systems was investigated. We proposed an angle-based codebook, where the analog codebook was designed based on slow-varying channel path AoDs. An NSA was utilized to optimize the analog codewords for having directional array gain. Based on the observation that the channel vectors are composed of slow-varying path AoDs and fast-varying path gains, the analog beamformer selected from the proposed angle-based codebook can maintain unchanged for a long time. Thus, the channel feedback overhead can be reduced significantly. Theoretical analysis and numerical simulation results verified that our proposed angle-based codebook can achieve the acceptable achievable rate performance and higher energy efficiency for low-resolution hybrid precoding in mmWave massive MIMO systems, with low channel feedback overhead. The proposed codebook also has the potential to be applied in THz massive MIMO systems [46] and reflectable intelligent surface-assisted systems [47,48], which may be investigated in the future work.