Design of Millimeter Wave Hybrid Beamforming Systems Girim Kwon, Yeonggyu Shim, and Hyuncheol Park
Hyuck M. Kwon
Department of Electrical Engineering Korea Advanced Institute of Science and Technology Daejeon 305-701, KOREA E-mail: {girim88, ygshim, and parkhc}@kaist.ac.kr
Department of Electrical Engineering and Computer Science Wichita State University Wichita, Kansas 67260 E-mail:
[email protected]
Abstract—Millimeter wave (mmWave) signals experience a significant path-loss in free space. To overcome this weakness, a large number of antennas are needed to obtain a high beamforming gain. Although a large number of antennas can be implemented in small area due to the short wavelength, the digital beamforming techniques cannot be implemented easily due to the high complexity of hardwares. To solve this problem, the hybrid beamforming systems which have smaller number of radio frequency (RF) chains are proposed in the literature. Although the hybrid beamforming systems may achieve the spectral efficiencies of the digital beamforming systems closely, the spectral efficiency cannot be monotonically increase along with the number of data streams due to the limited scattering in mmWave channel. In this paper, we provide a guide for the design of mmWave hybrid beamforming systems. We find the optimal number of streams, and present the spectral efficiency achievable region in which the system guarantees the reliable communications with the lowest cost.
I.
I NTRODUCTION
There is a significantly increasing data traffic due to the popular use of wireless devices. To solve the problem, many communications techniques are being developed by researchers [1], [2]. Millimeter wave (mmWave) technology is one of the leading candidates for the future wireless communications system technology thanks to the following two merits [2]. First, the large unlicensed bandwidth (30 to 300GHz) allows us to communicate with multi-Gbps of data rate. Second, the short wavelength of mmWave enables the devices to have a compact size with a lot of antennas. Although mmWave signals have high path-loss in free space, a number of antennas can provide the beamforming gain which comes from the spatial directivity [3]. For a reliable communications with high data rate in mmWave system, there are many mmWaverelated standardization works such as ECMA TC48, WiGig, WirelessHD, IEEE 802.15.3c, and IEEE 802.11ad. In multiple-input multiple-output (MIMO) systems, the beamforming techniques can be implemented by using many precoding methods [4], [5]. The digital signal processing (DSP) is required to implement the theoretical precoding methods. In this digital beamforming system, a lot of radiofrequency (RF) chains including digital to analog converters (DACs) / analog to digital converters (ADCs) are needed as many as antennas. Therefore, there exists the difficulty of
implementation due to the high complexity of hardware. To reduce the complexity of hardware, the hybrid beamforming techniques are recently proposed by combining the analog beamforming system with the digital beamforming system [6]–[8]. The work in [6] develops a joint transmit-receive spatial processing algorithm based on mixed analog/digital beamforming. In [7], the hybrid beamforming system is treated as the digital beamforming system with hardware constraints, and the results show that the spatially sparse precoding via orthogonal matching pursuit allows the hybrid beamforming system to approach their unconstrained theoretical limits on spectral efficiency. In this paper, we provide a guide for the design of the hybrid beamforming systems based on the hybrid beamforming system in [7]. Due to the limited scattering of mmWave signals, the channel cannot provide sufficient spatial degrees of freedom (d.o.f) even if the number of antennas are large enough. Therefore, to send many streams with a lot of antennas may not be the solution to achieve the large spectral efficiency. Based on this fact, we find the optimal number of streams and the minimum number of RF chains to guarantee the reliable communications. This guide allows us to reduce the cost of implementation in mmWave hybrid beamforming systems. Our system model is motivated by the work in [7], and we present the more detailed system block diagram. The rest of this paper is organized as follows. System description is described in Section II. Precoding and combining method for the hybrid beamforming system is described in Section III. A guide for the design of hybrid beamforming system and the simulation results are presented in Section IV and V. The conclusion follows in the Section VI. II.
S YSTEM D ESCRIPTION
A. System Model We consider the single user MIMO mmWave system shown in Fig. 1. In this hybrid beamforming system [7], the hardware constraint is considered. To reduce the complexity of hardware, optimal precoder is separated into baseband precoder and RF precoder. There are Nt antennas in transmitter and Nr antennas in receiver. The transmitter can send Ns streams to the receiver with NtRF RF chains and DACs such that
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Fig. 1. Hardware block diagram of mmWave single user hybrid beamforming system [7]. In the transmitter, baseband precoder using digital signal processing and constrained RF precoder using RF phase shifters are implemented. Similarly, RF combiner and baseband combiner are implemented in the receiver.
Ns ≤ NtRF ≤ Nt . If NtRF is equal to Nt , the transmitter becomes a digital beamformer. Also, if Ns is equal to NtRF , the transmitter becomes an analog beamformer. The same is true of the receiver. In this paper, we only consider the hybrid beamformer with following conditions. Ns < NtRF < Nt , Ns < NrRF < Nr .
(1)
The transmitter has an NtRF × Ns baseband precoder, FBB , followed by an Nt × NtRF RF precoder, FRF . The transmit signal is presented as follows [7], x = FRF FBB s,
(2)
where x is Nt ×1 vector and s is the Ns ×1 stream vector such that E[ss∗ ] = N1s INs . Since the RF precoder is equiped with analog phase shifters, its elements are constrained to satisfy (i) (i)∗ FRF FRF = Nt−1 , where (.)l,l denotes the lth diagonal l,l
element, i.e. all elements of FRF have equal norm. The total power constraint is given by FRF FBB 2F = Ns , and there is no constraint on the baseband precoder. We assume the H as a narrowband block-fading propagation channel, and both transmitter and receiver have perfect channel state information (CSI). Therefore, we can represent the received signal as follows [7], √ y = ρHx + n √ (3) = ρHFRF FBB s + n, where y is the Nr×1 vector, H is the Nr ×Nt channel matrix which satisfies E H2F = Nt Nr , ρ is the average received power, and n is the Nr × 1 vector of i.i.d CN (0, σn2 ) noise. At the receiver, the RF combiner WRF and the baseband combiner WBB are used to recover the transmitted data stream s from the received signal y. After combining, the received stream ˆs can be written as [7] √ ∗ ∗ ∗ ∗ ˆs = ρWBB WRF HFRF FBB s + WBB WBB n, (4) where WRF is the Nr × NrRF matrix with unit norm entries, and WBB is the NrRF × Ns matrix.
B. Channel Model The mmWave signal experiences the high path loss in free space which leads to limited spatial scattering. Therefore, the condition number (CN) of channel matrix is larger than the spatially uncorrelated Rayleigh fading channel. Due to large CN, spatial multiplexing is limited in mmWave systems. In this paper, we consider a narrowband clustered channel model based on the extended Saleh-Valenzuela model that can describe the mmWave channel [9], [10], [11]. The channel matrix H is composed of Ncl clusters which are distributed uniformly in space, and Nray rays have a truncated Laplacian distribution in each cluster [12]. Using this model, the channel matrix H can be written as [7] H=
Ncl N ray Nt Nr r t )Λt (φtil , θil ) αil Λr (φril , θil Ncl Nray i=1 l=1 r t ∗ ar (φril , θil )at (φtil , θil ) ,
(5)
where αil is the complex gain of the lth ray in the ith r t scattering cluster, whereas φril (θil ) and φtil (θil ) are its azimuth (elevation) angles of arrival and departure, respectively. The r t Λr (φril , θil ) and Λt (φtil , θil ) represent the sector gain of the transmit and receive antenna element at the corresponding angles of departure and arrival [7]. In addition, we consider the uniform planar array (UPA) to enable beamforming in both elevation and azimuth. Therefore, we use the following array steering and response vector of UPA [7] in yz-plane with Wt (Wr ) and Ht (Hr ) elements at transmitter (receiver) on the y and z axes, 2πd 1 at (φ, θ) = √ 1, · · · , , ej λ (msin(φ)sin(θ)+ncos(θ)) , · · · , Nt T 2πd · · · , ej λ ((Wt −1)sin(φ)sin(θ)+(Ht −1)cos(θ) , (6)
2πd 1 ar (φ, θ) = √ 1, · · · , , ej λ (lsin(φ)sin(θ)+kcos(θ)) , · · · , Nr T 2πd · · · , ej λ ((Wr −1)sin(φ)sin(θ)+(Hr −1)cos(θ) , (7) where 0 < m < Wt and 0 < n < Ht are the y and z indices of transmit antennas, 0 < l < Wr and 0 < k < Hr are the y and z indices of receive antennas. III.
H YBRID P RECODER /C OMBINER D ESIGN
In this section, the spatially sparse precoding/combining algorithm for the hybrid beamformer is decribed which was proposed in [7]. A. Precoder Design In digital beamforming systems, the singular value decomposition (SVD) of H can be used to design the precoder and combiner, and we consider the right singular vectors as the optimal precoder Fopt . In hybrid beamforming systems, Fopt cannot be implemented due to the hardware constraints. In [7], the precoding algorithm is proposed by solving the optimization problem which is to find the best low dimensional presentation of Fopt using the basis vectors at (φtk , θkt ), 1 ≤ k ≤ Ncl Nray . This algorithm is based on the well known concept of basis pursuit, and the pseudo-code is given in [7]. B. Combiner Design The combining algorithm is similar to the precoding algorithm. The pseudo-code for the combining algorithm is given in [7]. Note that when we design both precoder and combiner, it is better to start at the side with less RF chains, i.e. the more constrained side. For example, consider the case where a receiver only has a single RF chain and thus is restricted to applying a single response vector ar (φrl , θlr ). In such a situation, separately designing the precoder to approximate Fopt and radiate power in several directions leads to a loss in actual received power (since the receiver can only form a beam in one direction). As a result, the transmitter must always consider the constraints at the receiver, and vice versa. IV.
L OW COST HYBRID BEAMFORMER DESIGN
We introduce the method and parameters which are used for the simulations in Section V. This guide can be used to decide the optimal number of RF chains when the number of data streams and the number of antennas for the transmitter and receiver are given. In hybrid beamforming systems, the achieved spectral efficiency assuming Gaussian signaling is given by [7],
ρ −1 ∗ ∗ R = log2
INs + R WBB WRF HFRF FBB Ns n
(8)
F∗BB F∗RF H∗ WRF WBB
, ∗ ∗ where Rn = σn2 WBB WRF WRF WBB is the noise covariance matrix after combining. Note that in digital beamforming
system, we can replace WRF WBB with WM M SE in (8) [13], [14]. In uncorrelated Rayleigh fading channel, we can expect the spectral efficiency will be increased by increasing Ns . However, for mmWave systems, we cannot expect the same effect in the clustered channel model because the limited scattering cannot provide sufficient d.o.f. Therefore, there is optimal number of streams which can maximize the spectral efficiency in the clustered channel. Motivated by this fact, we present the guide for design of the low cost hybrid beamformer. First, we find the optimal Ns by spectral efficiency simulation using (8) when Nt , Nr , NtRF , NrRF , Ncl , and Nray are given. From the simulation data, we can calculate the ratio of the spectral efficiency for the hybrid beamformer to the spectral efficiency for the digital beamformer when the optimal Ns is chosen. The spectral efficiency ratio can be used to select the number of RF chains to achieve the target spectral efficiency. Second, we define the metrics βt and βr which are the ratio of the number of the RF chains to the number of antennas at the transmitter and receiver, respectively; NtRF , Nt N RF βr r . Nr βt
(9)
Along the values of βt and βr , we present the region in which the target spectral efficiency ratio can be achieved for the different Ns . Using this region, the optimal βt and βr to reduce the cost of hardware implemetation can be found easily. The simulation results of the above procedures are shown in the next section. V.
S IMULATION R ESULTS
We present the simulation results for the single user hybrid beamforming system. We assume the channel model as the clustered channel with Ncl = 8, Nray = 10, and angle spread = 7.5o [7]. All clusters have equal power, i.e., 2 σα,i = σα2 ∀i. Also, the angle spread at both the transmitter and receiver are equal in the azimuth and the elevation domains, i.e., σφt = σφr = σθt = σθr . The transmitter is likely to be sectorized to reduce interference and increase beamforming gain [7]. Therefore, we consider that the antenna arrays in transmitter and receiver are UPAs in (6) and (7), and the transmitter’s sector angle is assumed to be 60o -wide in the azimuth domain and 20o -wide in elevation [7]. In contrast, we assume that the receivers have omni-directional antennas [7] because receivers must be able to steer beams in any direction. The antenna spacing is assumed to be half-wavelength. In the simulation of spectral efficiency, we assume that the signal-tonoise ratio (SNR) is equal to 12dB where SNR σρ2 . n
A. The Optimal Number of Streams Theoretically, the maximum number of spatial d.o.f is the rank of channel matrix which is equal to min(Nt , Nr ) for the rich scattering channel. Therefore, we only consider the case that Ns does not exceed the min(Nt , Nr ). Due to the limited
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Fig. 2. Spectral efficiency of the digital beamformer with Nt = 64 at SNR = 12dB. Due to the limited scattering, the spectral efficiency does not monotonically increase along with Ns . Instead, there is optimal Ns which maximizes the spectral efficiency when Nt and Nr are given.
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Fig. 3. Spectral efficiency of the hybrid beamformer with (Nt , Nr ) = (64, 16) at SNR = 12dB. The trend of the curves is similar to that of the digital beamformer when NtRF and NrRF are large enough. Therefore, there is optimal Ns which maximizes the spectral efficiency.
scattering of mmWave, however, the effective spatial d.o.f of the channel is much less than the rank of the channel matrix. Therefore, the optimal Ns of the digital beamformer, i.e., Ns,d has following condition. Ns,d = min(Nt , Nr ) for rich scattering channel Ns,d < min(Nt , Nr ) for sparse scattering channel
(10)
In sparse scattering channel, we need to find the optimal number of streams which maximizes the spectral efficiency of the beamforming system. The optimal Ns can be found by simulation, and the results are shown in Fig. 2. Also, we can see that the spectral efficiencies are strictly concave functions. When Nt and Nr are given, the optimal Ns can be selected by using this result. For example, if (Nt , Nr ) is given as (64, 16), the optimal Ns of the digital beamformer becomes 8. This trend can be also observed in the hybrid beamformer; the spectral efficiencies of the hybrid beamformer are shown in Fig. 3 when (Nt , Nr ) is given as (64, 16). In the hybrid beamformer, we consider the restrictions (1). For example, if NtRF (NrRF ) is equal to Nt (Nr ), then the hybrid beamformer performs as the digital beamformer with an unnecessarily large number of phase shifters. Also, if Ns is equal to min(NtRF , NrRF ), the hybrid beamformer performs as the analog beamformer. From Fig. 2 and 3, the optimal Ns of the hybrid beamformer, i.e., Ns,h is same with Ns,d when both NtRF and NrRF are larger than Ns,d . However, if NtRF or NrRF is smaller than Ns,d , Ns,h becomes min(NtRF , NtRF ) − 1. In summary, we can select the optimal Ns of the hybrid beamformer by using the expression (11). Ns,d , if Ns,d < min(NtRF , NrRF ) Ns,h = RF RF min(Nt , Nr ) − 1, if Ns,d > min(NtRF , NrRF ) (11) B. Spectral Efficiency Achievable Region Fig. 4 shows the regions in which the hybrid beamformer can achieve the 99% spectral efficiency of the digital beam-
Fig. 4. 99% spectral efficiency achievable region of the hybrid beamformer. we assume that Nt = 64, Nr = 16, and SNR = 12dB. This region guides to design of the low cost hybrid beamformer when Ns is given. The markers present the minimum βt and βr when the target Ns is 8.
former. Note that to achieve the same spectral efficiency with the digital beamformer, both βt and βr are required to be 100%. However, the hybrid beamformers can be designed with smaller number of RF chains. In the case of (Nt , Nr ) = (64, 16), the minimum βt and βr can be found according to the target Ns . Since the optimal Ns which maximizes the spectral efficiency is equal to 8, we select the target Ns as 8. Then, we can achieve the 99% spectral efficiency of the digital beamformer with the lowest cost when (βt , βr ) is equal to point A(26.56%, 62.50%) or point B(23.43%, 68.75%). Therefore, the minimum (NtRF , NrRF ) is equal to (17, 10) or (15, 11). Fig. 5 presents the spectral efficiency achievable regions of the hybrid beamformer for different target spectral efficiency ratio, i.e, 95.0%, 98.0%, 99.0%, and 99.5%. If we select the target spectral efficiency ratio as 95%, the required (βt , βr ) is
VI.
C ONCLUSION
We present a guide for the design of the hybrid beamformer. Because of the limited scattering in mmWave channel, the spectral efficiency cannot monotonically increase along with the number of streams. Therefore, there is optimal number of streams which maximizes the spectral efficiency. When the target number of streams is given, we can obtatin the minimum ratio of the number of RF chains to the number of antennas to achieve the target spectral efficiency of the hybrid beamformer. Therefore, we can design the low cost hybrid beamforming systems with the minimum number of RF chains. ACKNOWLEDGMENT
Fig. 5. Spectral efficiency achievable regions of the hybrid beamformer for four different target spectral efficiency ratios. We assume that Nt = 64, Nr = 16, Ns = 8, and SNR=12dB.
This research was supported by ‘The Cross-Ministry Giga KOREA Project’ of The Ministry of Science, ICT and Future Planning, Korea. (GK13N0100, 5G mobile communication system development based on mmWave). R EFERENCES
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Fig. 6. BER curves of different beamformers. We assume that (Nt , Nr ) = (64, 16) and Ns = 8. The hybrid beamformer is designed to achieve 99% spectral efficiency of the digital beamformer with the lowest cost, i.e., (NtRF , NrRF ) = (17, 10)
reduced to (14.06%, 56.25%). In addition, the BER curves of different beamformers are presented in Fig. 6. The modulation and channel coding schemes are assumed to be a binary phase shift keying and a 1/2-rate convolutional code with generator polynomial of (133, 171)(8) , respectively. For the analog beamformer without channel coding, there exists an error floor. In this case, the reliable communications are impossible. On the other hand, compared to the digital beamformer, the hybrid beamformer has a SNR loss of only 1dB at the target BER of 10−5 . In coded schemes, the hybrid beamformer achieves almost same BER performance with the digital beamformer, but the analog beamformer has a SNR loss of 3dB for target BER = 10−5 . Note that SNR = 12dB is assumed for the 99% spectral efficiency achievable region. Therefore, we can expect the reliable communications with the lowest cost hybrid beamformer.
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