Hybrid Precoder and Combiner Design with Low Resolution Phase Shifters in mmWave MIMO Systems
Millimeter wave (mmWave) communications have been considered as a key technology for next generation cellular systems and Wi-Fi networks because of its advances in providing orders-of-magnitude wider bandwidth than current wireless networks. Economical and energy-efficient analog/digial hybrid precoding and combining transceivers have been often proposed for mmWave massive multiple-input multiple-output (MIMO) systems to overcome the severe propagation loss of mmWave channels. One major shortcoming of existing solutions lies in the assumption of infinite or high-resolution phase shifters (PSs) to realize the analog beamformers. However, low-resolution PSs are typically adopted in practice to reduce the hardware cost and power consumption. Motivated by this fact, in this paper, we investigate the practical design of hybrid precoders and combiners with low-resolution PSs in mmWave MIMO systems. In particular, we propose an iterative algorithm which successively designs the low-resolution analog precoder and combiner pair for each data stream, aiming at conditionally maximizing the spectral efficiency. Then, the digital precoder and combiner are computed based on the obtained effective baseband channel to further enhance the spectral efficiency. In an effort to achieve an even more hardware-efficient large antenna array, we also investigate the design of hybrid beamformers with one-bit resolution (binary) PSs, and present a novel binary analog precoder and combiner optimization algorithm with quadratic complexity in the number of antennas. The proposed low-resolution hybrid beamforming design is further extended to multiuser MIMO communication systems. Simulation results demonstrate the performance advantages of the proposed algorithms compared to existing low-resolution hybrid beamforming designs, particularly for the one-bit resolution PS scenario.
Millimeter wave (mmWave) communications, hybrid precoder, multiple-input multiple-output (MIMO), phase shifters, one-bit quantization.
The past decade has witnessed the exponential growth of data traffic along with the rapid proliferation of wireless devices. This flood of mobile traffic has significantly exacerbated spectrum congestion in current frequency bands, and therefore stimulated intensive interest in exploiting new spectrum bands for wireless communications. Millimeter wave (mmWave) wireless communications, operating in the frequency bands from 30-300 GHz, have been demonstrated as a promising candidate to fundamentally solve the spectrum congestion problem -.
However, challenges always come along with opportunities. MmWave communications still need to overcome several technical difficulties before real-world deployment. As a negative result of the ten-fold increase of the carrier frequency, the propagation loss in mmWave bands is much higher than that of conventional frequency bands (e.g. 2.4 GHz) due to atmospheric absorption, rain attenuation, and low penetration . From a positive perspective, the smaller wavelength of mmWave signals allows a large antenna array to be packed in a small physical dimension . With the aid of pre/post-coding techniques in massive multiple-input multiple-output (MIMO) systems, the large antenna array can provide sufficient beamforming gain to overcome the severe propagation loss of mmWave channels. It also enables simultaneous transmission of multiple data streams resulting in significant improvements to spectral efficiency.
For MIMO systems operating in conventional cellular frequency bands, the full-digital precoder and combiner are completely realized in the digital domain by adjusting both the magnitude and phase of the baseband signals. However, these conventional full-digital schemes require a large number of expensive and energy-intensive radio frequency (RF) chains, analog-to-digital converters (ADCs), and digital-to-analog converters (DACs). Since mmWave communication systems operate at much higher carrier frequencies and wider bandwidths, the enormous cost and power consumption of the required RF chains and ADCs/DACs make the adoption of full-digital precoding and combining schemes impractical for mmWave systems. Recently, economical and energy-efficient analog/digital hybrid precoders and combiners have been advocated as a promising approach to tackle this issue. The hybrid precoding approaches adopt a large number of phase shifters (PSs) to implement high-dimensional analog precoders to compensate for the severe path-loss at mmWave bands, and a small number of RF chains and DACs to realize low-dimensional digital precoders to provide the necessary flexibility to perform advanced multiplexing/multiuser techniques.
The investigation of hybrid precoder and combiner design has attracted extensive attention in recent years because of its potential energy efficiency for mmWave MIMO communications. The major challenges in designing hybrid precoders are the practical constraints associated with the analog components, such as the requirement that the analog precoding be implemented with constant modulus PSs. Thus, hybrid precoder design typically requires the solution of various matrix factorization problems with constant modulus constraints. In particular, a popular solution to maximize the spectral efficiency of point-to-point transmission is to minimize the Euclidean distance between the hybrid precoder and the full-digital precoder -. Hybrid precoder design for partially-connected architectures are also studied in -. Due to the special characteristics of mmWave channels, codebook-based hybrid precoder designs are commonly proposed -, in which the columns of the analog precoder are selected from certain candidate vectors, such as array response vectors of the channel and discrete Fourier transform (DFT) beamformers. Extensions of the hybrid beamformer design to multiuser mmWave MIMO systems have also been investigated in -.
The aforementioned existing hybrid precoder and combiner designs generally assume that infinite or high-resolution PSs are used for implementing the analog beamformers in order to achieve satisfactory performance close to the full-digital scheme. However, implementing infinite/high-resolution PSs at mmWave frequencies would significantly increase the energy consumption and complexity of the required hardware circuits , . Obviously, it is impractical to employ infinite/high-resolution PSs for mmWave systems and real-world analog beamformers will be implemented with low-resolution PSs. Consequently, an important research direction is the exploration of signal processing techniques for hybrid analog/digital architectures that can mitigate the loss of beamforming accuracy due to the low-resolution PSs.
A straightforward approach to obtain the finite-resolution beamformer is to design the infinite-resolution analog beamformer first, and then directly quantize each phase term to a finite set . However, this solution becomes inefficient when the PSs have very low resolution. An alternative solution for hybrid beamforming with finite-resolution PSs is codebook-based design -. However, for low-resolution PSs, the size of the codebook is very small and the resulting performance is not satisfactory. In , , Sohrabi and Yu proposed to iteratively design the low-resolution hybrid precoder to maximize the spectral efficiency. However, the performance of this algorithm often suffers when one-bit quantized PSs are applied.
In this paper, we first consider the problem of designing hybrid precoders and combiners with low-resolution PSs for a point-to-point mmWave MIMO system. The objective of the proposed algorithm is to minimize the performance loss caused by the low-resolution PSs while maintaining a low computational complexity. To achieve this goal, we propose to successively design the low-resolution analog precoder and combiner pair for each data stream, aiming at conditionally maximizing the spectral efficiency. An iterative phase matching algorithm is introduced to implement the low-resolution analog precoder and combiner pair. Then, the digital precoder and combiner are computed based on the obtained effective baseband channel to further enhance the spectral efficiency.
Note that the power consumption and cost of the PS are proportional to its resolution. For example, a 4-bit (i.e. ) resolution PS at mmWave frequencies requires 45-106 mW, while a 3-bit (i.e. ) resolution PS needs only 15 mW . In an effort to achieve maximum hardware efficiency, we also investigate the design of hybrid beamformers with one-bit resolution (binary) PSs. Inspired by the findings in , we present a binary analog precoder and combiner optimization algorithm under a rank-1 approximation of the interference-included equivalent channel. This algorithm has quadratic complexity in the number of antennas and can achieve almost the same performance as the optimal exhaustive search method. Finally, our investigation of low-resolution hybrid precoders and combiners is extended to multiuser mmWave MIMO systems. Numerical results in the simulation section demonstrate that the proposed algorithms can offer a performance improvement compared with existing low-resolution hybrid beamforming schemes, especially for the one-bit resolution PS scenario.
Notation: The following notation is used throughout this paper. Boldface lower-case and upper-case letters indicate column vectors and matrices, respectively. and denote the transpose and transpose-conjugate operations, respectively. represents statistical expectation. extracts the real part of a complex number; denotes the sign operator; represents the phase of a complex number. indicates an identity matrix. denotes the set of complex numbers. denotes the determinant of matrix . denotes the cardinality of set . and are the magnitude and norm of a scalar and vector , respectively. denotes the Frobenius norm of matrix . Finally, we adopt a Matlab-like matrix indexing notation: denotes the -th column of matrix ; denotes the element of the -th row and the -th column of matrix ; denotes the -th element of vector .
Ii System Model and Problem Formulation
Ii-a Point-to-Point mmWave MIMO System Model
We first consider a point-to-point mmWave MIMO system using a hybrid precoder and combiner with low-resolution PSs, as illustrated in Fig. 1. The transmitter employs antennas and RF chains to simultaneously transmit data streams to the receiver which is equipped with antennas and RF chains. To ensure the efficiency of the mmWave communication with a limited number of RF chains, the number of data streams and the number of RF chains are constrained as .
The transmitted symbols are first processed by a baseband digital precoder , then up-converted to the RF domain via RF chains before being precoded with an analog precoder of dimension . While the baseband digital precoder enables both amplitude and phase modifications, the elements of the analog precoder , which are implemented by the PSs, have a constant amplitude and quantized phases: , in which the phase is quantized as , and is the number of bits to control the phase. We denote the constraint set of the analog precoder as follows: . Obviously, a larger number of bits leads to finer resolution for the PSs and potentially better performance, but also results in higher hardware complexity and power consumption.
The discrete-time transmitted signal can be written in the following form
where is the symbol vector, , represents transmit power and this power constraint is enforced by normalizing such that .
We consider a narrow-band slow-fading propagation channel, which yields the following received signal
where is the received signal vector, is the channel matrix, and is the complex Gaussian noise vector corrupting the received signal.
The receiver employs an analog combiner implemented by the PSs and a digital combiner using RF chains to process the received signal. The signal after the spatial processing has the form
where is the analog combiner whose elements have the same constraint as , i.e. , and thus , is the digital baseband combiner and the combiner matrices are normalized such that .
Ii-B Problem Formulation
We consider the practical and hardware-efficient scenario in which the PSs have very low-resolution (e.g. ) to reduce the power consumption and complexity. Under this hardware constraint, we aim to jointly design the hybrid precoder and combiner for a mmWave MIMO system. When Gaussian symbols are transmitted over the mmWave MIMO channel, the achievable spectral efficiency is given by
where is the noise covariance matrix after combining. We aim to jointly design the digital beamformers , as well as the low-resolution analog beamformers , to maximize the spectral efficiency:
Obviously, the optimization problem (5) is a non-convex NP-hard problem. In the next section, we attempt to decompose the original problem into a series of sub-problems and seek a sub-optimal solution with low-complexity and satisfactory performance.
Iii Low-Resolution Hybrid Precoder and Combiner Design
To simplify the joint hybrid precoder and combiner design, the objective problem is decomposed into two separate optimizations. We first focus on the joint design of the analog precoder and combiner . Then, having the effective baseband channel associated with the obtained optimal analog precoder and combiner, the digital precoder and combiner are computed to further maximize the spectral efficiency.
Iii-a Low-Resolution Analog Precoder and Combiner Design
We observe that under the assumption of high signal-to-noise-ratio (SNR), the achievable spectral efficiency in (4) can be approximated as
While the per-antenna SNR in mmWave systems is typically low, the post-combining SNR should be high enough to justify this approximation. In addition, it has been verified in  that for large-scale MIMO systems, the optimal analog beamformers are approximately orthogonal, i.e. . This enables us to assume when , where is a normalization factor. Similarly, we have and . Let , then (6) can be further simplified as
where follows since when and are both square matrices. Therefore, the analog precoder and combiner design with low-resolution PSs can be approximately reformulated as:
Unfortunately, the optimization problem (9) is still NP-hard and has exponential complexity . Therefore, we propose to further decompose this difficult optimization problem into a series of sub-problems, in which each transmit/receive RF chain pair is considered one by one and the analog precoder and combiner for each pair are successively designed.
In particular, we define the singular value decomposition (SVD) of as
where is an unitary matrix, is an unitary matrix, and is a rectangular diagonal matrix of singular values. Due to the sparse nature of the mmWave channel, the matrix is typically low rank. In particular, the effective rank of the channel serves as an upper bound for the number of data streams that the channel can support. Thus, we assume that the channel can be well approximated by retaining only the strongest components , where , , and . Then, the objective in (9) can be converted to
Next, we write the analog precoding and combining matrices as and , respectively, where and , , are the analog precoder and combiner pair for the -th data stream. Furthermore, we denote as the precoding matrix excluding the -th precoder vector and as the combining matrix excluding the -th combiner vector . Then, the formulation (11) can be further transformed to (12)-(16), which are presented at the top of following page,
where is a very small scalar to assure invertibility. Thus, the objective in (9) can be reformulated as:
where we define the interference-included channel matrix as
This motivates us to propose an iterative algorithm, which starts with appropriate initial RF precoding and combining matrices then successively designs and according to (19) with an updated as in (18) until the algorithm converges.
The complexity of obtaining an optimal solution to (19) for each iteration is now reduced to , which is still too high. To practically solve the problem (19), in what follows we present an iterative phase matching algorithm, which searches the conditionally optimal phase of each element of the analog precoder and combiner . Specifically, we first design the analog precoder assuming the analog combiner is fixed. Let be the phase of the -th element of the analog precoder and let be the phase of the -th element of the analog combiner . If we temporarily remove the discrete phase constraint, the optimal continuous phase of the -th element of the analog precoder is given by the following proposition, whose proof is provided in Appendix A.
Proposition 1: Given the phases of the analog combiner and the phases , , of the analog precoder , the optimal continuous phase of the -th element of analog precoder is
Then, after finding the optimal continuous phase by (20), we reconsider the discrete phase constraint and find the optimal low-resolution phase by quantization:
Similarly, if the analog precoder is determined, the optimal continuous phase of the -th element of is
and the optimal low-resolution phase is obtained by
Motivated by (20)-(23), the iterative procedure to design the precoder and combiner as in (19) is straightforward. With appropriate initial , , we design the precoder by finding the conditionally optimal phases as in (20) and (21). Then, with the obtained , , we design the combiner by finding the conditionally optimal phases as in (22) and (23). We alternate the designs of and iteratively until the obtained phase of each element of and does not change and the convergence is achieved. Note that since in each precoder and combiner design step, the objective function of (19) is monotonically non-decreasing, and thus our proposed algorithm is guaranteed to converge to at least a locally optimal solution.
We summarize the proposed joint low-resolution analog precoder and combiner design in Algorithm 1.
Iii-B Digital Precoder and Combiner Design
After all analog precoder-combiner pairs have been determined, we can obtain the effective baseband channel as
where and . For the baseband precoder and combiner design, we define the SVD of the effective baseband channel as
where and are unitary matrices, is an diagonal matrix of singular values. Then, to further enhance the spectral efficiency, an SVD-based baseband digital precoder and combiner are employed:
Finally, the baseband precoder and combiner are normalized
Iv One-Bit Resolution Analog Precoder and Combiner Design
In the previous section, we proposed a novel hybrid beamformer design for maximizing the spectral efficiency of a mmWave MIMO system, in which the analog precoder and combiner are implemented with low-resolution PSs. In order to achieve maximum hardware efficiency, in this section we focus on the design of analog precoders and combiners using “one-bit” resolution (binary) PSs, which can maximally reduce the power consumption and simplify the hardware complexity. Although the iterative phase matching algorithm proposed in the previous section can also be applied, a simpler approach is possible in the one-bit case. Therefore, in this section, we present an efficient one-bit resolution analog beamformer design, which can achieve good performance with much lower complexity.
We follow the procedure of the hybrid beamforming design proposed in the previous section, but only modify the optimization problem (19), which attempts to determine the -th analog precoder and combiner pair. Particularly, we reformulate this analog beamformer design problem (19) with the constraint of one-bit resolution PSs as
The optimization problem (30) can be solved through exhaustive search with exponential complexity , which would not be possible with large antenna arrays. Therefore, in the following we attempt to develop an efficient one-bit resolution beamformer design with polynomial complexity in the number of antennas.
We first define the SVD of as
where and are the -th left and right singular vectors of , respectively, and is the -th largest singular value, . Then, the objective in (30) can be rewritten as
If we utilize a rank-1 approximation by keeping only the strongest term, i.e. , the optimization function in (30) can be approximated by
Now, the joint optimization problem (33) can be decoupled into individually designing the analog precoder and combiner :
These two optimization problems (34) and (35) require only the singular vectors and associated with the largest singular value, which can be quickly obtained by the power iteration algorithm  instead of the complete SVD calculation. However, solving (34) and (35) by exhaustive search still has exponential complexity in the number of antennas. In order to further reduce the complexity without a significant loss of performance, we propose to construct a smaller dimension candidate beamformer set, from which the optimal beamformer can be found with linear complexity. In the following, we present this algorithm for the precoder design (34) as an example, while the combiner design (35) follows the same procedure.
We introduce an auxiliary variable and we reformulate the optimization problem (34) as:
where denotes the phase of . Obviously, given any , the corresponding binary precoder that maximizes (37) is
With the conditionally optimal for any given shown in (38), we will now show that we can always construct a set of candidate binary precoders and guarantee . Then, the maximization in (34) can be carried out over a set of only candidates without loss of performance.
We first define the angles , , as
so that . Then, we map the angles to , , which are rearranged in ascending order, i.e. . Because of the periodicity of the cosine function, the maximization problem (37) with respect to can be carried out over any interval of length . If we construct non-overlapping sub-intervals , then the optimal must be located in one of sub-intervals since the full interval is of length . Therefore, the optimization problem (37) can be solved by examining each sub-interval separately.
Assuming the optimal is in the -th sub-interval, the corresponding optimal binary precoder can be obtained by (38) as , , and has the form
After that, given the inverse sorting that maps to , we rearrange the corresponding elements of and obtain . Then, based on the relationship between and defined in (39), we can achieve the conditionally optimal precoder by
for the case that is in the -th sub-interval.
Since the optimal must be located in one of sub-intervals, we can obtain conditionally optimal precoders by examining all sub-intervals and construct a candidate precoder set as
which must contain the optimal precoder . Therefore, without loss of performance, the problem in (34) can be transformed to an equivalent maximization task over only the set
which has linear complexity . Similarly, we can also construct a candidate analog combiner set and obtain by the same procedure.
The rank-1 solution returned by (43) is based on the rank-1 approximation of the interference-included equivalent channel . The approximation of may cause a performance degradation when we revisit the original problem (30). Therefore, in order to enhance the performance, we propose to jointly select the precoder and combiner over candidate sets and as
which may return the rank-1 or a better solution with quadratic complexity . This low-complexity analog beamformer design with one-bit resolution PSs is summarized in Algorithm 2.
V Hybrid Precoder and Combiner Design for Multiuser mmWave MIMO Systems
In this section, we consider a mmWave multiuser MIMO uplink system and extend the low-resolution hybrid precoder and combiner designs proposed in the previous sections to the multiuser system.
V-a System Model and Problem Formulation
We consider a multiuser mmWave MIMO uplink system as presented in Fig. 2, where a base-station (BS) is equipped with antennas and RF chains and simultaneously serves mobile users. Due to power consumption and hardware limitations, each mobile user has antennas and a single RF chain to transmit only one data stream to the BS. We further assume the number of RF chains at the BS is equal to the number of users, i.e. .
Let be the analog precoder of the -th user, where each element of has a constant magnitude and low-resolution discrete phases, i.e. , . The transmitted signal of the -th user after precoding can be formulated as
where is the symbol of the -th user, , and is the -th user’s transmit power.
Let , denote the uplink channel from the -th user to the BS. The received signal at the BS can be written as
where is complex Gaussian noise. The BS first applies an analog combining matrix to process the received signal, in which the analog combiner corresponding to the -th user is also implemented by low-resolution PSs, i.e. , . Then, a baseband digital combiner is employed to retrieve the information of the -th user. Let denote the hybrid combiner corresponding to the -th user. After the combining process at the BS, the estimated symbol of the -th user can be expressed as
Given the received signal at the BS in (47), the signal-to-interference-plus-noise ratio (SINR) of the -th user can be expressed as
and the achievable sum-rate of the multiuser uplink system is
We aim to jointly design the analog precoders and combiners implemented by low-resolution PSs as well as the digital combiners to maximize the sum-rate of the uplink multiuser system:
V-B Low-Resolution Hybrid Precoder and Combiner Design
Obviously, the optimization problem (50) cannot be directly solved. Thus, we adopt an approach similar to  and propose to successively design the low-resolution analog beamformer pair for each user, aiming at enhancing the channel gain as well as suppressing the inter-user interference. Then, the baseband combiner at the BS is calculated to further mitigate the interference and maximize the sum-rate.
In particular, for the first user, the analog precoder and combiner pair is designed to maximize the corresponding channel gain, which can be formulated as follows:
This analog precoder and combiner design problem can be efficiently solved by the algorithm presented in Sec. III-A when low-resolution PSs are utilized, or the algorithm proposed in Sec. IV if only one-bit resolution PSs are available. Then, the analog precoders and combiners , , for the remaining users are successively designed by an iterative procedure. In each iteration, we attempt to find the analog beamformer pair that suppresses the interference from the users whose analog beamformers have already been determined. To achieve this goal, the channel of the user whose combiner is to be calculated is projected onto the space orthogonal to the collection of previously designed analog combiners. This approach leads to orthogonal analog combiners that suppress the inter-user interference.
Specifically, to design the -th user’s analog beamformer pair, we first extract the orthonormal components of the previously determined analog combiners , by the Gram-Schmidt procedure: