Channel Alignment for Hybrid Beamforming in Millimeter ... - IEEE Xplore

Channel Alignment for Hybrid Beamforming in Millimeter Wave Multi-User Massive MIMO Nuan Song, Huan Sun, Qingchuan Zhang, and Tao Yang Nokia Bell Labs China Nokia Shanghai Bell, Shanghai, China Email: {Nuan.Song, Huan.a.Sun, Tao.a.Yang}@nokia-sbell.com, [email protected] Abstract—To assist hybrid beamforming in multi-user massive Multiple Input Multiple Output (MIMO) systems at millimeter wave frequencies, we propose Auxiliary Processing Network (APN) based multi-phase channel alignment techniques. The APN, which is realized in parallel to the main beamforming network, have two implementation architectures, namely the hybrid analog-digital APN based on antenna switching as well as the fully digital APN with low-resolution Analog-to-Digital Converters (ADCs). For different APN architectures, we design various channel estimation algorithms for the channel alignment procedure and evaluate the corresponding performance.

I. I NTRODUCTION Millimeter Wave (mmWave) communications play a significant role in the fifth generation mobile networks (5G), as more spectrum and accordingly a larger channel bandwidth are available at frequencies above 28 GHz [1]. Massive Multiple Input Multiple Output (MIMO) systems at mmWave bands utilize hybrid array architectures to combine a large number of antennas, which reduces the implementation complexity, cost, as well as the power consumption [2], [3], [4]. Downlink hybrid beamforming techniques have been widely studied for single-user MIMO [5], [6], [7] and Multi-User MIMO (MU-MIMO) systems [8], [9], [10], [11], [12], [13]. Most MU scenarios assume that single-antenna users are deployed [8], [9], [13] or analog-only combining is applied using multiple antennas at users [10]. For a much more general case, we consider the MU-MIMO system, where both the Base Station (BS) and users utilize hybrid array architectures, and propose a coordinated hybrid beamforming technique to support the transmission of multiple data streams for each user [14], [15]. The proposed technique takes advantage of both analog beamforming by means of the coordination between the BS and users as well as digital precoding, to achieve a large array gain and multiplexing gain at the same time. Compared to the scheme in [11] which solves a similar problem, our technique has no dimensionality constraint on the number of transceiver antennas and only requires the information of the composite channel, instead of the complete channel matrix that is assumed to be known in most existing solutions. The hybrid beamforming design can be split into analog and digital domains for the ease of solving the related optimization problem. Analog beamforming mainly aims at aligning the transmit and receive beams so that the effective array gain can be maximized. Two approaches are identified, i.e., beam

training without the explicit channel knowledge and the beam determination based on the estimation of the explicit channel knowledge. For the former, beam training can be carried out by means of beam sweeping over a codebook consisting of beam patterns with various resolutions [16], [17], [18], [19]. It usually applies an iterative procedure to exchange the information between the BS and users, using an exhaustive search over narrow beam patterns or a hierarchical search over progressively narrower patterns, and as a result the best beam combination can be obtained. All of these discussions are restricted to the purely analog beamforming case, which are not trivial to be applied to hybrid precoding especially in the aforementioned general MU-MIMO scenarios. Reference [6] proposes a beam training scheme for the hybrid beamforming design based on the full channel estimation but only assumes the single-user case. Hybrid receive beamforming with the online channel covariance estimation is studied in [20] for traditional uplink MIMO. In this paper, we propose Auxiliary Processing Network (APN) based channel alignment techniques according to the estimated composite channels for our proposed coordinated hybrid beamforming scheme in MU massive MIMO systems. The APN, which is separated from the main hybrid beamforming network, provides a high flexibility in the system design with respect to the number of transceiver units and bandwidth, as well as assists MU uplink MIMO communications in terms of channel acquisition and scheduling. Two architectures are considered, namely a hybrid APN using antenna switching and a fully digital APN with low-resolution Analog-to-Digital Converters (ADCs). To support the channel alignment in hybrid beamforming, we develop both the direct and quasitransparent channel estimation algorithms for corresponding APN architectures. Simulations are carried out to evaluate and compare the performance of proposed techniques. Notations: The superscripts ∗, 𝑇, 𝐻, + represent complex conjugate, transpose, Hermitian transpose, and pseudo inverse, respectively. The Kronecker product is ⊗. The vec{⋅} operator stacks columns of a matrix into a vector. The operations trace{⋅}, Re{⋅}, Im{⋅}, and ∥⋅∥𝐹 denote the trace of a matrix, taking the real/imaginary parts, and Frobenius norm.

978-1-5090-5019-2/17/$31.00 ©2017 IEEE

We consider hybrid array architectures for both the BS and users in a MU-MIMO system. The BS is mounted with 𝑀𝑇 transmit antennas that fully connect to 𝑁𝑇 RF chains, where 𝑁𝑇 < 𝑀𝑇 holds for the hybrid design. Similarly, for the 𝑘-th user (𝑘 = 1, ⋅ ⋅ ⋅ , 𝐾), the number of receive antennas is denoted by 𝑀𝑅𝑘 and the corresponding number of RF chains satisfies 𝑁𝑅𝑘 < 𝑀𝑅𝑘 . Accordingly, we∑ denote the total 𝐾 number of antennas for all users by 𝑀𝑅 = 𝑘=1 𝑀𝑅𝑘 and ∑𝐾 that of the RF chains by 𝑁𝑅 = 𝑘=1 𝑁𝑅𝑘 . For an efficient design, we split hybrid beamforming into analog and digital parts, where the array gain and the multiplexing gain can be exploited respectively. For analog beamforming, it can be considered that the analog precoder 𝑭RF ∈ (𝑘) ℂ𝑀𝑇 ×𝑁𝑇 and the combiner 𝑾RF ∈ ℂ𝑀𝑅𝑘 ×𝑁𝑅𝑘 perform a low-rank transformation on the 𝑘-th user’s full channel 𝑯𝑘 ∈ ℂ𝑀𝑅𝑘 ×𝑀𝑇 , 𝑘 = 1, ⋅ ⋅ ⋅ , 𝐾. The mmWave channel is generally sparse and the effective (or transformed) channel ˇ 𝑘 = 𝑾 (𝑘)𝐻 𝑯𝑘 𝑭RF should keep the rank characteristic of 𝑯 RF 𝑯𝑘 as much as possible. Therefore, the analog beamformer design by solving the maximization of the effective array gain in (1) can be equivalent to the channel alignment. 𝐾  2 ∑  (𝑘)𝐻  𝑾RF 𝑯𝑘 𝑭RF 

max

(𝑘)

𝐹

(1)

= 𝑰𝑁𝑅 , ∀𝑘. 𝑘

Main Beamforming Network 1 2

MT

Channel Alignment & Analog Beamforming

𝑭RF

RF Chain 1

Digital Beamforming

...

⎟ ⎜ 𝐾 ⎟ ⎜ 𝐻 ∑ (𝑘) (𝑘)𝐻 𝐻 ⎟ max trace ⎜ 𝑭 𝑯 𝑾 𝑾 𝑯 𝑭 RF 𝑘 𝑘 RF RF ⎟ ⎜ RF 𝑭RF ⎠ ⎝ 𝑘=1  

(2)

RF Chain NT

⎜ trace ⎜ ⎝

𝐾 ∑ 𝑘=1

(𝑘)𝐻

𝑾RF

(𝑘)𝐻

s. t. 𝑾RF

⎞

(𝑘) ⎟ 𝐻 𝑯𝑘 𝑭RF 𝑭RF 𝑯𝑘𝐻 𝑾RF ⎟ ⎠,  

(𝑘)

Auxiliary Processing Network

Fig. 1. Block diagram of the APN in parallel to the main network for hybrid beamforming.

𝑷

𝐻 s. t. 𝑭RF 𝑭RF = 𝑰𝑁𝑇 ⎛

max

To fully recover the composite channel knowledge, e.g., (𝑘)𝐻 𝑾RF 𝑯𝑘 , based on the access to the beamformed data at the port level, we apply the APN in parallel to the main processing network. Figure 1 shows the block diagram of the APN based hybrid beamforming architecture, where the main network performs hybrid beamforming and the channel alignment or analog beamforming part is adjusted by the APN.

⎞

⎛

(𝑘) 𝑾RF ,∀𝑘

A. Auxiliary Processing Network

...

In [14], we propose an iterative Generalized Low Rank Approximation of Matrices (GLRAM) algorithm to solve (1) and calculate analog beamformers. The GLRAM scheme can be implemented by an initial channel alignment procedure and a (𝑘) multi-phase refinement. It iteratively calculates 𝑭RF and 𝑾RF by solving two reformulated optimization problems in (2) and (3) respectively with their fixed counterparts until convergence is achieved. Such an iterative analog beamforming procedure, which requires the coordination between the BS and users, can be considered as the multi-phase transmit-receive coordinated channel alignment.

III. S YSTEM D ESCRIPTIONS

...

s. t.

𝑭RF , 𝑾RF ,∀𝑘 𝑘=1 (𝑘)𝐻 (𝑘) 𝐻 𝑭RF 𝑭RF = 𝑰𝑁𝑇 , 𝑾RF 𝑾RF

ii) The BS applies the determined 𝑭RF (𝑖) and sends downlink pilots. Each user measures its composite channel (𝑘) 𝑯𝑘 𝑭RF (𝑖) and updates its receive beamformer 𝑾RF (𝑖) by solving (3). (𝑘) Until 𝑭RF (𝑖), 𝑾RF (𝑖) formed transceiver beams are aligned. Otherwise, continue Step 2) and set 𝑖 = 𝑖 + 1. We can see from Step (b) that the estimation of the com(𝑘)𝐻 posite channel is required, i.e., 𝑾RF 𝑯𝑘 to compute 𝑭RF (𝑘) and 𝑯𝑘 𝑭RF to obtain 𝑾RF . However, as the number of RF chains is smaller than the number of antennas in the hybrid architecture (i.e., 𝑁𝑇 < 𝑀𝑇 for the BS), the full channel information on the antenna element level cannot be directly obtained. Thus, the crucial part in the channel alignment lies in the recovery of the composite channel.

...

II. C HANNEL A LIGNMENT

(3)

𝑸𝑘

𝑾RF = 𝑰𝑁𝑅 , ∀𝑘. 𝑘

The multi-phase channel alignment procedure is described as follows for the Time Division Duplex (TDD) case. (a) Coarse channel alignment: The BS sends downlink broadcast pilots and each user initializes its receive beamformer (𝑘) 𝑾RF (0). Set 𝑖 = 1. (b) Refined channel alignment: Repeat (𝑘)∗ i) Each user applies 𝑾RF (𝑖 − 1) as the transmit beamformer and sends uplink pilots. The BS estimates the com(𝑘)𝐻 posite channel 𝑾RF (𝑖)𝑯𝑘 and determines the current downlink beamformer 𝑭RF (𝑖) by solving (2).

We identify two possible architectures for the APN implementation, i.e., hybrid and fully digital, shown in Figure 2 (a) and (b) respectively. For the fully digital case, each antenna is connected to an ADC and the channel can be directly estimated. Due to a high power consumption and hardware complexity at mmWave frequencies, the hybrid architecture is preferred. It has been widely discussed on hybrid array architectures for mmWave massive MIMO systems, where both analog phase shifters and switches can be considered for analog beamforming [21]. In the light of reducing the complexity, cost, as well as power consumption, we apply a hybrid architecture based on a network of switches to the APN. The switching network performs antenna selection in the analog processing stage, with 𝑁𝐷 antennas chosen out of 𝑀𝑇 (𝑁𝐷 < 𝑀𝑇 ). Thereby, only 𝑁𝐷 ADCs are connected to the selected antennas as well

...

2

MT

LNA LNA

LNA

ADC ADC

...

1

ADC

Channel Estimation & Analog Beam Design

MT

LNA

ND

MS

MT-MS+1

MT

full-array based (a)

...

LNA

1

LNA

LNA ND

output to ADCs

Digital

2

1

LNA

...

(a)

LNA

output to ADCs

ADC

LNA

1

...

MT

ND

1

...

LNA

Channel Estimation & Analog Beam Design

Beam Weights

2

ADC

...

LNA

...

1

1

Beam Weights

Digital

Analog

LNA

sub-array based (b)

Fig. 3. The (a) full-array and (b) sub-array based analog antenna selection/switching architectures.

(b)

Fig. 2. Two APN architectures: (a) hybrid and (b) fully digital.

as to the subsequent channel estimation part. Since we have to obtain information of a dimension 𝑀𝑇 according to the data of a smaller dimension 𝑁𝐷 , a range of beamformers 𝑭𝑖 ∈ 𝒞 𝑀𝑇 ×𝑁𝐷 , 𝑖 = 1, . . . , 𝐿 carried out by antenna switching should be applied. From the implementation perspective, the 𝐿 beamformers or antenna switching sub-networks can be carried out in a sequential fashion using 𝐿 time slots or in a partially parallel way for a reduced latency. Choices of implementations depend on the required latency and hardware complexity of the system. Figure 3 shows two possibilities of the antenna switching architectures, i.e., (a) full-array based and (b) sub-array based. The full-array based antenna switching network makes sure that only 𝑁𝐷 antennas are active at a time and the switches can connect to any element. In the sub-array based antenna switching network only one antenna per sub-array is active and each switch is assigned to its corresponding sub-array. Therefore, the full-array one has more degrees of freedom to design the beamforming matrices 𝑭𝑖 and the implementation complexity of the sub-array case is lower. Alternatively, high-rate but low-resolution ADCs are suggested in the APN. Since mmWave often utilizes a wide bandwidth (e.g. ≥ 500 MHz), an ADC for wideband processing has to operate at a sampling rate on the order of GHz, which is power demanding and very expensive. The power dissipation (in Watt) of an ideal ADC can be estimated by 𝑃𝐴𝐷𝐶 = 𝛾⋅𝑓𝑠2𝑏 , where 𝑓𝑠 is the sampling rate, 𝑏 is the ADC resolution, and 𝛾 is a constant [22]. To maintain a low power consumption, one idea is to apply the Nyquist sampling rate ADC with a very limited resolution such as one-bit. As the power dissipation increases linearly with 𝑓𝑠 while exponentially with 𝑏, oversampling seems more promising than increasing the the resolution. Thus, the sigma-delta ADC is also an option. The rapid advances in semiconductor technology have shown the availability and potential of implementing high-rate and lowresolution ADCs [23]. In this paper, for simplicity we only consider one-bit ADCs operating at a Nyquist rate used in wideband processing.

Remarks: Most existing hybrid beamforming schemes rely on the main processing network used for both beamforming and channel estimation. In such a system, a sequence of time-varying wideband beamformers 𝑭𝑖 should be applied to recover the full channel information. With a separate APN, the number of RF chains or ports 𝑁𝐷 can be designed differently from that of using the main network with a fixed 𝑁𝑇 , and combing with low-resolution ADCs is also possible. Additionally, a different and narrower transceiver bandwidth can be applied so that a fully digital APN using regular ADCs is feasible. Therefore, compared with the scheme using the main beamforming network, the APN based channel alignment has a potential to provide a higher flexibility as well as achieve a reduced latency and a smaller system overhead. B. Data Model (𝑘)𝐻 We take the composite channel estimation 𝑾RF 𝑯𝑘 as an example and the downlink counterpart 𝑯𝑘 𝑭RF can be carried out in a similar fashion. Based on the TDD reciprocity, the es(𝑘) (𝑘)∗ timation of the uplink composite channel 𝑯𝑢 = 𝑯𝑘𝑇 𝑾RF is performed at the BS. In the case of hybrid APN, if we define the uplink pilots to be sent as 𝑿𝑘 ∈ ℛ𝑁𝑅𝑘 ×𝑁𝑅𝑆 , the received signal at the BS at the 𝑖-th time slot can be written as 𝒀𝑖 = 𝑭𝑖𝐻 𝑯𝑢 𝑿 + 𝑭𝑖𝐻 𝑵 ∈ 𝒞 𝑁𝐷 ×𝑁𝑅𝑆 ,

(4)

where 𝑁𝑅𝑆 is the pilot length, 𝑭𝑖 is the receive analog beamforming matrix or the antenna switching matrix at the 𝑖-th time slot, and 𝑵 ∈ 𝒞 𝑀𝑇 ×𝑁𝑅𝑆 is the Additive White Gaussian Noise (AWGN) with a zero mean and a power spectral density 𝑁0 . The total composite channel matrix and the total pilot matrix are defined as 𝑯𝑢 = [ 𝑯𝑢(1) , 𝑯𝑢(2) , . . . , 𝑯𝑢(𝐾) ] ∈ 𝒞 𝑀𝑇 ×𝑁𝑅 and 𝑿 = 𝑇 𝑇 ] ∈ 𝒞 𝑁𝑅 ×𝑁𝑅𝑆 , respectively. A [ 𝑿1𝑇 , 𝑿2𝑇 , . . . , 𝑿𝐾 range of 𝐿 receive analog beamformers 𝑭𝑖 are applied at the BS to acquire enough measurements and to reconstruct the composite channel state information. For the fully digital APN, there are no analog beamformers and by setting 𝑁𝐷 = 𝑀𝑇 , 𝑭𝑖 = 𝑰𝑀𝑇 , we rewrite the received measurements as 𝒀 = 𝑯𝑢 𝑿 + 𝑵 ∈ 𝒞 𝑀𝑇 ×𝑁𝑅𝑆 .

(5)

By stacking all the vectors vec (𝑹𝑖 ), we obtain

IV. C HANNEL E STIMATION A LGORITHMS As discussed earlier, the key part of the channel alignment lies in the acquisition of the channel state informa(𝑘)𝐻 tion 𝑾RF 𝑯𝑘 or 𝑷 in (2) to obtain the downlink analog beamforming 𝑭RF at the BS. This section proposes several composite channel estimation algorithms for both the hybrid APN and fully digital APN architectures. A. Hybrid APN based Channel Estimation Techniques 1) Direct Estimation: In order to estimate 𝑯𝑢 , we reformulate the uplink data model (4) into a vector form as ( ( ) ) vec (𝒀𝑖 ) = 𝑿 𝑇 ⊗ 𝑭𝑖𝐻 vec (𝑯𝑢 ) + vec 𝑭𝑖𝐻 𝑵 ,

⎢ ⎣

⎤ ⎡ 𝑿𝑇 vec (𝒀1 ) ⎥ ⎢ . .. ⎦=⎣ vec (𝒀𝐿 ) 𝑿𝑇

( ) ⎤ ⎡ ⎤ vec 𝑭1𝐻 𝑵 ⊗ 𝑭1𝐻 ⎢ ⎥ ⎥ . . .. .. ⎦ vec (𝑯𝑢 ) + ⎣ ⎦, ( ) ⊗ 𝑭𝐿𝐻 vec 𝑭𝐿𝐻 𝑵 (7)

which can be rewritten as 𝒚 = 𝑺𝒉𝑢 + 𝒏,

(8)

where 𝒚, 𝒏 ∈ 𝒞 𝐿𝑁𝐷 𝑁𝑅𝑆 , 𝒉𝑢 ∈ 𝒞 𝑀𝑇 𝑁𝑅 , and 𝑺 ∈ 𝒞 𝐿𝑁𝐷 𝑁𝑅𝑆 ×𝑀𝑇 𝑁𝑅 . To fully recover 𝒉𝑢 , the necessary condition is that 𝑺 should be full column rank and 𝐿 ≥ 𝑀𝑇 𝑁𝑅 /𝑁𝐷 𝑁𝑅𝑆 . One simple solution is 1) to design 𝑿 that contain orthogonal columns satisfying 𝑿𝑿 𝐻 = 𝛼𝑰𝑁𝑅 , 𝛼 > 0, meaning that orthogonal pilots should be used for different ports; 2) to construct antenna switching matrices 𝑭𝑖 whose columns can be chosen from those of the identity matrix 𝑰𝑀𝑇 and set 𝐿 = 𝑀𝑇 /𝑁𝐷 . Thus, the vectorized composite channel can be estimated by Least Squares (LS), i.e., ˆ 𝑢 = 𝑺 + 𝒚 = 𝑺 𝐻 𝒚. 𝒉

(9)

It can be easily shown that 𝑺 also has an orthogonal design. As a result, the computational complexity of the solution in (9) is dominated by the matrix multiplication 𝒪(𝑀𝑇2 𝑁𝑅2 ), which is much lower than that of the pseudo inverse. The sub-array based antenna switching architecture can be applied. Please note that this LS algorithm can be applied to the main RF analog beamforming network in the system to recover the channel, where 𝑁𝐷 = 𝑁𝑇 and the beamformer vectors in 𝑭𝑖 can be chosen from the Discrete Fourier Transform (DFT) based codebook. 2) Quasi-Transparent Estimation: As the determination of 𝑭RF by solving (2) for the channel alignment in Step 2) requires the knowledge of either the composite channel (𝑘)𝐻 𝑾RF 𝑯𝑘 or its covariance form 𝑷 , we could also estimate the matrix 𝑷 instead of the composite channel. From (6), we calculate 𝑹𝑖 = 𝒀𝑖 𝒀𝑖𝐻 = 𝑭𝑖𝐻 𝑯𝑢 𝑿𝑿 𝐻 𝑯𝑢𝐻 𝑭𝑖 + 𝑭𝑖𝐻 𝑵 𝑵 𝐻 𝑭𝑖 .

(10)

To ensure a good estimate, uplink signals are better to satisfy 𝑿𝑿 𝐻 = 𝛼𝑰𝑁𝑅 . Similarly as in the previous case, pilots should be repeated within 𝐿 time slots. Then we can rewrite (10) by the matrix vectorization vec (𝑹𝑖 )

=

( ) ( ) vec 𝑭𝑖𝐻 𝑷 ∗ 𝑭𝑖 + vec 𝑭𝑖𝐻 𝑵 𝑵 𝐻 𝑭𝑖

=

(𝑭𝑖∗

⊗ 𝑭𝑖 )

𝐻

∗

vec (𝑷 ) + 𝒏𝑖 .

(11) (12)

⎤

⎤ ⎡ 𝒏1 ⎥ ⎢ . ∗ ⎥ vec (𝑷 ) + ⎣ . ⎥ , ⎦ . ⎦ 𝒏𝐿

(13)

which can also be rewritten as 𝒓 = 𝑺𝑐 𝒑 + 𝒏𝑐 , 2 𝐿𝑁𝐷

𝑀𝑇2

(14) 2 𝐿𝑁𝐷 ×𝑀𝑇2

where 𝒓, 𝒏𝑐 ∈ 𝒞 ,𝒑 ∈ 𝒞 and 𝑺𝑐 ∈ 𝒞 . To make sure that 𝒑 can be fully recovered, 𝑺𝑐 should has a full 2 . As a result, the LS algorithm column rank and 𝐿 ≥ 𝑀𝑇2 /𝑁𝐷 can be applied to obtain 𝒑 directly as

(6)

according to the property of the Kronecker product. By stacking all 𝒀𝑖 , 𝑖 = 1 . . . , 𝐿,we can obtain the vectorized data model as ⎡

)𝐻 ⎤ ⎡ ( ∗ 𝑭1 ⊗ 𝑭1 vec (𝑹1 ) ⎢ ⎥ ⎢ ⎢ . . . . ⎦=⎣ ⎣ . . ( ∗ )𝐻 vec (𝑹𝐿 ) 𝑭𝐿 ⊗ 𝑭𝐿 ⎡

𝒑ˆ = 𝑺𝑐+ 𝒓.

(15)

It can be seen that this channel covariance estimation scheme is receiver transparent, since no knowledge on pilots 𝑿 is required at the BS. We can thus consider it as a quasitransparent estimation method. A key point is the design of 𝑭𝑖 so that the matrix 𝑺𝑐 has a full column rank. It is actually a Compressive Covariance Sensing (CCS) problem [24]. The dynamic sampling scheme can be applied to the 𝑀𝑇 antennas but only a subset of antennas is activated at each time slot. The full rank condition is satisfied if every possible antenna pair is active within at least one time slot. We apply the greedy algorithm for the dynamic array design [25] to achieve the full rank requirement. As any combination of an antenna pair is possible, we resort to the full-array based antenna switching architecture. This antenna selection algorithm can be carried out offline once the number of antennas as well as that of the RF chains is fixed. The computational complexity of the algorithm )is mainly determined by the LS operation, i.e., ( 𝒪 𝑀𝑇6 /6 . B. Fully Digital APN based Channel Estimation Technique In the fully digital APN case, we reformulate the received signal in (5) into a vector form as vec (𝒀 )

=

𝒚¯

=

⇓

(

𝑿 𝑇 ⊗ 𝑰𝑀𝑇

)

vec (𝑯𝑢 ) + vec (𝑵 ) ,

(16)

¯ 𝑢 + 𝒏, ¯𝒉 ¯ 𝑺

¯ ∈ ℂ𝑀𝑇 𝑁𝑅 and 𝑺¯ ∈ ¯ 𝒏 ¯ ∈ ℂ𝑀𝑇 𝑁𝑅𝑆 , 𝒉 where 𝒚, ℂ𝑀𝑇 𝑁𝑅𝑆 ×𝑀𝑇 𝑁𝑅 . We consider low-resolution ADCs (one-bit for the simplest case) are applied in the fully digital APN and obtain the quantized complex-valued observations as ) ( ¯𝑢 + 𝒏 ¯ , ¯ = sgn 𝑺¯𝒉 (17) 𝒛¯ = sgn (𝒚) which can be reformulated in the real-valued expression as ( ) ˜𝑢 + 𝒏 ˜ = sgn 𝑺˜𝒉 ˜ , 𝒛 = sgn (𝒚) (18) with

[

] ¯ Re (𝒛) ¯ [ ( )Im (𝒛) ( ) ] ¯ ¯ Re (𝑺 −Im( 𝑺 ˜ ) ) 𝑺= ¯ ¯ Im 𝑺 Re 𝑺 𝒛=

[ ] ¯ Re (𝒏) ˜ = 𝒏 ¯ ) ] [ Im((𝒏) ¯𝑢 Re (𝒉 ˜ ) . 𝒉𝑢 = ¯𝑢 Im 𝒉

(19)

˜ 𝑢 should be estimated based on the quantized The channel 𝒉 observation 𝒛, which is similar to the one-bit compressive

sensing problem and can be solved by the Expectation Maximization (EM) algorithm [26], [27] as follows. ˆ (0) (a) Set the estimated channel 𝒉 𝑢 to a certain initial vector and 𝑞 = 1. (b) Repeat Expectation step: to estimate the 𝑖-th component of 𝒚˜ (20)

5 0 −5 NMSE

} { ˆ (𝑞−1) 𝑦ˆ𝑖 = 𝔼 𝑦˜𝑖 ∣𝒛, 𝒉

10

Maximization step: to apply the maximum likelihood criterion )−1 ( ˆ (𝑞) = 𝑺 ˜ ˜𝐻 𝑺 𝒉

˜𝒚ˆ 𝑺

−10 −15 −20 −25

(21)

−30

ˆ 𝑢 converges, otherwise continue Step (b) and set (c) Until 𝒉 𝑞 = 𝑞 + 1.

V. S IMULATION R ESULTS In this section we evaluate the performance of the proposed APN based channel estimation and alignment algorithms for MU-MIMO systems using Monte-Carlo simulations. The mmWave channel is generated by the widely used clustered model [5] with 8 clusters and 10 rays per cluster, where the channel gain follows the complex normal distribution and the angles of arrival/departure are assumed to follow uniform distribution within [−𝜋, 𝜋] and [−𝜋/6, 𝜋/6], respectively. Uniform linear arrays with a half wavelength for the inter-element distance are considered at both the BS and users. It is assumed that the BS has 𝑀𝑇 = 64 antennas and 𝑁𝑇 = 8 RF chains. There are in total 𝐾 = 4 users, where each user is mounted with 𝑀𝑅𝑘 antennas connecting to 𝑁𝑅𝑘 = 2 RF chains. We denote the training based LS algorithm using hybrid APN as “APN-Hybrid-LS”, the CCS based quasi-transparent estimation scheme using hybrid APN as “APN-Hybrid-CCS”, and the EM algorithm using fully digital APN with one-bit ADCs as “APN-Digital-EM-1b”. For simplicity, it is assumed that the number of RF chains in the hybrid APN is the same as that in the main beamforming network, which has no impact on the performance. A. NMSE performance The Normalized is defined } { Mean Square Error (NMSE) ˆ 𝑢 ∥2 /∥𝑯𝑢 ∥2 . We plot NMSE as NMSE = 𝔼 ∥𝑯𝑢 − 𝑯 𝐹 𝐹 performance of various algorithms in Figure 4 as a function of the pilot Signal-to-Noise Ratio (SNR) 10 log10 𝛼/𝑁0 , where each user has 𝑀𝑅𝑘 = 2 antennas. It can be observed that the “APN-Hybrid-LS” and “APN-Hybrid-CCS” schemes have the same performance, once the constructed matrices 𝑺 and 𝑺𝑐 have full column rank. Even though the “APN-Digital-EM1b” scheme is based on the fully digital APN, a performance loss due to quantization is still obvious, especially when the pilot SNR is large.

−15

−10

−5 0 Pilot SNR (dB)

5

10

15

Fig. 4. NMSE performance versus the pilot SNR.

B. Throughput performance We apply the proposed channel estimation algorithms in the multi-phase channel alignment procedure (c.f. SectionII) and evaluate the corresponding throughput performance. For the baseband precoding in the hybrid beamforming, block diagonalization is considered. The throughput performance for different pilot SNRs is depicted in Figure 5, where each user has a hybrid array with 𝑀𝑅𝑘 = 8 antennas and the channel estimation is carried out at both the BS and users. We can see that the throughput performance of channel alignment techniques with estimated channels varies with the pilot SNR. Both “APN-Hybrid-LS” as well as “APN-Hybrid-CCS” exhibit only a small performance loss compared to the case with the perfect channel knowledge and “APN-Digital-EM-1b” is slightly worse. Pilot SNR

120

100

Throughput (bps/Hz)

The computational complexity of the EM algorithm in the fully digital one-bit APN is 𝒪(𝑞 ⋅ 4𝑀𝑇2 𝑁𝑅2 ), where 𝑞 is the number of iterations used in the EM algorithm.

−35 −20

APN−Hybrid−LS APN−Hybrid−CCS APN−Digital−EM−1b

80

60

40

Perfect Channel APN−Hybrid−LS APN−Hybrid−CCS APN−Digital−EM−1b APN−Hybrid−LS APN−Hybrid−CCS APN−Digital−EM−1b Perfect Channel APN−Hybrid−LS APN−Hybrid−CCS APN−Digital−EM−1b APN−Hybrid−LS APN−Hybrid−CCS APN−Digital−EM−1b

0dB 𝑟𝑘 = 2

-5dB

0dB 𝑟𝑘 = 1

-5dB

20

0 −20

−10

0

10

20

30

SNR (dB)

Fig. 5. Throughput performance versus the signal SNR with 𝑟𝑘 = 1, 2.

VI. D ISCUSSIONS AND C ONCLUSIONS In this paper, we propose APN based channel estimation and alignment techniques for hybrid beamforming in MU massive

TABLE I S UMMARY AND C OMPARISON Schemes “Main-Hybrid-LS” “APN-Hybrid-LS” “APN-Hybrid-CCS” “APN-Digital-EM-1b”

Time Slots 𝑀𝑇 /𝑁𝑇 𝑀𝑇 /𝑁𝐷 2 > 𝑀𝑇2 /𝑁𝐷 1

Hardware reuse main RF network 𝑁𝐷 switches & RF chains 𝑁𝐷 switches & RF chains 𝑀𝑇 RF chains with 1-b ADCs

Computational Complexity 2) 𝒪(𝑀𝑇2 𝑁𝑅 2) 𝒪(𝑀𝑇2 𝑁𝑅 6 𝒪(𝑀𝑇 /6) 2 2) 𝒪(𝑞 ⋅ 4𝑀𝑇 𝑁𝑅

MIMO system at mmWave bands. The APN is connected to existing antennas, where two architectures are considered, i.e., hybrid APN using low-cost RF switches and fully digital APN with low-resolution ADCs. For the hybrid architecture, we develop the LS channel estimation method to directly recover the channel matrix as well as the CCS algorithm using dynamic sampling to reconstruct the channel covariance matrix in a quasi-transparent manner. For the fully digital APN, the EM algorithm is applied to solve the one-bit compressive sensing problem. We compare and summarize those techniques in Table I, where “Main-Hybrid-LS” corresponds to the scheme when we apply the LS algorithm in Section IV-A1 using the main analog beamforming network. We can observe that even though “APN-Hybrid-CCS” has a good performance but is disadvantageous in terms of the implementation complexity and required time slots for estimation. The “APN-DigitalEM-1b” significantly reduces the physical layer latency and system overhead but has a certain performance degradation and a potentially higher hardware complexity. The “APNHybrid-LS” is a good trade-off between the performance and complexity. It provides a higher flexibility than the “MainHybrid-LS”, which has a similar performance but requires complicated adjustments of all phase shifters. ACKNOWLEDGMENT This work was supported by the National Science and Technology Major Projects under Grants 2017ZX03001011. R EFERENCES [1] W. Roh, J-Y. Seol, J. Park, B. Lee, J. Lee, Y. Kim, J. Cho, K. Cheun, and F. Aryanfar, “Millimeter-wave beamforming as an enabling technology for 5G cellular communications: theoretical feasibility and prototype results,” IEEE Communications Magazine, vol. 52, no. 2, pp. 106–113, 2014. [2] S. Sun, T. S. Rappaport, R. W. Heath, A. Nix, and S. Rangan, “MIMO for millimeter-wave wireless communications: beamforming, spatial multiplexing, or both?,” IEEE Communications Magazine, vol. 52, no. 12, pp. 110–121, 2014. [3] R. W. Heath, N. Gonzalez-Prelcic, S. Rangan, W. Roh, and A. Sayeed, “An overview of signal processing techniques for millimeter wave MIMO systems,” IEEE Journal of Selected Topics in Signal Processing, vol. 99, Feb. 2016. [4] S. Han, I. Chih-Lin, Z. Xu, and C. Rowell, “Large-scale antenna systems with hybrid analog and digital beamforming for millimeter wave 5G,” IEEE Communications Magazine, vol. 53, no. 1, pp. 186–194, 2015. [5] O. El Ayach, S. Rajagopal, S. Abu-Surra, Z. Pi, and R. W. Heath, “Spatially sparse precoding in millimeter wave MIMO systems,” IEEE Transactions on Wireless Communications, vol. 13, no. 3, pp. 1499– 1513, 2014. [6] A. Alkhateeb, O. El Ayach, G. Leus, and R. W. Heath, “Channel estimation and hybrid precoding for millimeter wave cellular systems,” IEEE Journal of Selected Topics in Signal Processing, vol. 8, no. 5, pp. 831–846, 2014. [7] A. Alkhateeb, O. El Ayach, G. Leus, and R. W. Heath, “Hybrid precoding for millimeter wave cellular systems with partial channel knowledge,” in Information Theory and Applications Workshop (ITA), 2013, 2013, pp. 1–5.

[8] F. Sohrabi and W. Yu, “Hybrid digital and analog beamforming design for large-scale MIMO systems,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2015, pp. 2929– 2933. [9] A. Sayeed and J. Brady, “Beamspace MIMO for high-dimensional multiuser communication at millimeter-wave frequencies,” in 2013 IEEE Global Communications Conference (GLOBECOM), 2013, pp. 3679– 3684. [10] A. Alkhateeb, G. Leus, and R. W. Heath, “Limited feedback hybrid precoding for multi-user millimeter wave systems,” IEEE Transactions on Wireless Communications, vol. 14, no. 11, pp. 6481–6494, 2015. [11] W. Ni and X. Dong, “Hybrid Block Diagonalization for Massive Multiuser MIMO Systems,” IEEE Transactions on Communications, vol. 64, no. 1, pp. 201–211, 2016. [12] R. A. Stirling-Gallacher and Md. S. Rahman, “Linear MU-MIMO pre-coding algorithms for a millimeter wave communication system using hybrid beam-forming,” in IEEE International Conference on Communications (ICC), 2014, pp. 5449–5454. [13] A. Adhikary, J. Nam, J.-Y. Ahn, and G. Caire, “Joint spatial division and multiplexing − the large-scale array regime,” IEEE Transactions on Information Theory, vol. 59, no. 10, pp. 6441–6463, 2013. [14] N. Song, H. Sun, and T. Yang, “Coordinated Hybrid Beamforming for Millimeter Wave Multi-User Massive MIMO Systems,” IEEE Global Communications Conference (GLOBECOM), Dec. 2016. [15] N. Song, T. Yang, and H. Sun, “Overlapped Sub-Array based Hybrid Beamforming for Millimeter Wave Multi-User Massive MIMO,” IEEE Signal Processing Letters, vol. 24, no. 5, pp. 550–554, 2017. [16] J. Wang, “Beam codebook based beamforming protocol for multi-Gbps millimeter-wave WPAN systems,” IEEE Journal on Selected Areas in Communications, vol. 27, no. 8, pp. 1390–1399, 2009. [17] “ISO/IEC/IEEE International Standard for Information technology Telecommunications and information exchange between systems Local and metropolitan area networksSpecific requirements-part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications Amendment 3: Enhancements for Very High Throughput in the 60 GHz Band (adoption of IEEE Std 802.11ad-2012),” ISO/IEC/IEEE 8802-11:2012/Amd.3:2014(E), march 2014. [18] Y. M. Tsang, A. SY Poon, and S. Addepalli, “Coding the beams: Improving beamforming training in mmwave communication system,” in IEEE Global Telecommunications Conference (GLOBECOM 2011), 2011, pp. 1–6. [19] K. Hosoya, N. Prasad, K. Ramachandran, N. Orihashi, S. Kishimoto, S. Rangarajan, and K. Maruhashi, “Multiple sector ID capture (MIDC): A novel beamforming technique for 60-GHz band multiGbps WLAN/PAN systems,” IEEE Transactions on Antennas and Propagation, vol. 63, no. 1, pp. 81–96, 2015. [20] V. Venkateswaran and A.-J. Van der Veen, “Analog beamforming in MIMO communications with phase shift networks and online channel estimation,” IEEE Transactions on Signal Processing, vol. 58, no. 8, pp. 4131–4143, 2010. [21] R. M´endez-Rial, C. Rusu, A. Alkhateeb, N. Gonz´alez-Prelcic, and R. W. Heath, “Channel estimation and hybrid combining for mmwave: Phase shifters or switches?,” in Information Theory and Applications Workshop (ITA), 2015, 2015, pp. 90–97. [22] B. Le, T. W. Rondeau, J. H. Reed, and C. W. Bostian, “Analog-todigital converters,” IEEE Signal Processing Magazine, vol. 22, no. 6, pp. 69–77, 2005. [23] I. D. O’Donnell and R. W. Brodersen, “An ultra-wideband transceiver architecture for low power, low rate, wireless systems,” IEEE Transactions on vehicular technology, vol. 54, no. 5, pp. 1623–1631, 2005. [24] D. Romero, D. D. Ariananda, Z. Tian, and G. Leus, “Compressive covariance sensing: Structure-based compressive sensing beyond sparsity,” IEEE Signal Processing Magazine, vol. 33, no. 1, pp. 78–93, 2016. [25] D. D. Ariananda and G. Leus, “Direction of arrival estimation for more correlated sources than active sensors,” Signal Processing, vol. 93, no. 12, pp. 3435–3448, 2013. [26] M. T Ivrlac and J. A Nossek, “On MIMO channel estimation with single-bit signal-quantization,” in ITG Smart Antenna Workshop, 2007. [27] J. Mo, P. Schniter, N. G. Prelcic, and R. W. Heath, “Channel estimation in millimeter wave MIMO systems with one-bit quantization,” in 48th Asilomar Conference on Signals, Systems and Computers, 2014, pp. 957–961.