I Introduction
Wireless communications systems exploiting millimeter wave (mmWave) frequencies are thought to be a core technology that will enable the deployment of the fifth generation (5G) cellular system. Broadband mmWave systems promise significant increase in the data rates due to the extremely wide bandwidths available in the mmWave spectrum. The adverse channel conditions at mmWave frequencies make the communication a hard challenge, this can however be compensate by means of using a large number of antennas that results in large array gain [4, 17]. Nevertheless, the conventional fully digital precoding techniques require a dedicated radio frequency (RF) chain for each antenna element, which is impractical at present due to the high cost and power consumption. Hence, it is desirable to design economical hardware that will utilize the potential gain from a large number of cheap antenna elements using a small number of expensive RF chains. Thus, processing schemes with less RF chains than antenna elements a.k.a. hybrid processing have been proposed [5].
Several approaches have been considered for hybrid processing. The work in [4] is one of the most popular in the literature, the authors exploit the spatial structure of mmWave channels to formulate the precoding/combining problem as a sparse reconstruction problem and they proposed the principle of basis pursuit as tool for its solution. This idea motivated other authors to continue developing hybrid precoders based on sparse reconstruction, e.g., [1, 10, 13]. Similar approaches were taken in [2, 12, 8], where a digital precoder/combiner is defined and then a complex algorithm is proposed to find a hybrid approximation, e.g., gradient descent, considering a weighted sum mean square error (WSMSE) minimization problem or using the orthogonal matching pursuit algorithm. In [18], a algorithm based on manifold optimization is proposed. In each iteration of the algorithm, it assumes a given digital precoder and develops a conjugate gradient method to find an analog precoder that is a local minimizer of the approximation gap from the fullydigital one. Next, the digital precoder is computed using a least squares solution. This method achieves good performance but suffers from high complexity and run time. More recently works as [11, 5, 14] lead to more successful methodologies, which are described briefly in this paper.
We propose four hybrid combiner/precoder for downlink mmWave massive MUMIMO systems. The designing of a hybrid combiner/precoder is divided in two parts, analog and digital. The system model in baseband shows that the signal processed by the mobile station can be interpreted as a received signal in the presence of colored Gaussian noise, therefore, since the digital part of the combiner in the receiver and in the precoder do not have constraints for their generation, they can be designed using any traditional signal processing approaches that take into account this kind of noise. However, to the best of our knowledge, previous works do not consider colored noise, e.g., [9, 11, 5]. A more realistic and appropriate design is described in this paper. Two of our proposals consist in the improvement of the digital part of the works in [11, 14] using that stated before.
On the other hand, typical approaches for the designing of the analog parts of both the combiner and the precoder are focused on increasing the detection signaltonoise ratio (SNR) without trying to reduce the inter user/symbol interference. In this paper we propose two simple solutions that are able to decrease the inter symbol interference while keeping SNR large. The first proposal consists in the improvement of the iterative algorithm proposed in [5] through a recursive algorithm. The second is based on single value decomposition (SVD), and reaches excellent performance with much less complexity. In addition one of the the proposed hybrid combiners reaches the maximum value of our objective function according with the Hadamard’s inequality. Numerical results in different environments show the improvement obtained through our proposals in relation to the considered hybrid combiner/precoder [11, 5, 14].
The remaining of this paper is organized as follows: Section II and Section III describe the system model and channel model, respectively; Section IV resumes the hybrid design approaches described in [11, 5, 14]; Section V, VI and VII are dedicated to describe our proposals. Section VIII presents four suboptimal data detection approaches. In Section IX simulations results are shown; and finally, in Section X some conclusions wrap up this paper.
The following notation is used throughout the paper: denotes the field of complex numbers; is a set; is a matrix;
is a vector;
is a scalar; , , , denote the th entry, th row, and th column of the matrix , respectively; is the x all ones matrix; is the xidentity matrix; returns the trace of matrix ; is the norm, for the euclidean norm case, , the underindex is avoided; represents the determinant function;returns the product of the nonzero eigenvalues of the square matrix
; is the Kronecker product; and denote the transpose and conjugate transpose, respectively; is the expectation operator;denotes a complex Gaussian random variable with mean
and variance
; and the function returns the entries of the matrix normalized to magnitude 1, i.e.,Ii System model
We consider downlink mmWave MUMIMO systems using HB in the base station (BS) and in each mobile station (MS). The HB in the BS can be represented by the product between the RF beamformer, , and the baseband beamformer, . There are users equipped with antennas and RF chains to process streams. The BS has antennas and sends streams simultaneously using RF chains, where satisfies . If is equal to , the BS performs digital beamformer [9].
Power normalization is satisfied such that . Then the received signal by the user , , is expressed as
(1) 
where denotes the channel matrix from the BS to the user satisfying ; is a complex Gaussian noise vector with zeromean and covariance matrix , i.e., ; is the data stream vector expressed as the concatenation of the user’s stream vectors such that with and whose entries belong to a constellation . The analog part of the precoder, , is implemented by phase shifters, satisfying .
The receiver uses its RF chains and analog phase shifters to obtain the processed received signal
(2) 
where is the RF combining matrix and denotes the baseband combining matrix of the user . Similarly to the RF precoder, is implemented using phase shifters and therefore [4].
Equation (2) can be rewritten in baseband terms as follows
(3) 
where represents the equivalent baseband channel of the user and in a similar way is the equivalent baseband noise vector, with covariance matrix . Since the combiner and precoder matrix, and , do not have constraints for its generation, they can be designed from signal processing approaches that take into account the colored Gaussian noise. Therefore, the problem is in the selection of the analog matrix and , such that an equivalent baseband channel that facilitates the digital processing is obtained.
The signaltonoise ratio (SNR) is defined as
(4)  
where represents the total energy available at the BS for transmission.
Iii Channel model
The mmWave channel can be described as follows [9]
(5)  
where is the number of multipath components in the channel; is the complex gain of the th multipath component in the channel for the th user, whereas () and () are its azimuth (elevation) angles of arrival and departure, respectively [4]. We consider the use of an uniform planar array (UPA) formed by () antennas, () antennas in the horizontal side and () antennas in the vertical side, with the antenna spacing of half wave length at the transmitter (receiver)[17], whose response is given by:
(6) 
with ; ; and
(7) 
Iv Hybrid designing approaches
This section presents three different approaches for designing the HB for both the transmitter and the receivers. The common factor in theses designs is that they use a HB generation divided in two stages, where the first (second) stage obtains the analog (digital) part of the precoder and of the combiner. In order to stress the main characteristic of each stage in a HB design, we considered the following notation to refer to them hereafter: []SS, where the first term indicates the reference number and the description of S (S) is related to the first (second) stagecharacteristic. An asterisk appearing as an upper index, , in a given characteristic means that it has been modified. In addition, the replacement of the reference number by the letter P is used to refer to our proposals.
Iva [11]SVDMMSE algorithm
The design of the HB in [11]
is based on channel knowledge by each user, the analog combiner of each user is independently designed based on the singular value decomposition (SVD), while the analog precoder is obtained by conjugate transposition to maximize the effective channel gain. Then, with the resulting equivalent baseband channel, low dimensional baseband precoders can be efficiently applied, e.g., MMSE or ZF filter. The Algorithm
1 resumes the steps for the HB designing in [11]. Note that the considered MMSE filter in the step 4 is a pseudo MMSE linear precoder. A more appropriate expression for the MMSE filter is demonstrated in the Appendix and used in the proposals described in sections V, VI and VII.IvB [5]CIABD algorithm
In [5], the authors focused on the design of the equivalent baseband channel, i.e., the analog part of the combiner and precoder, and eliminated the interference through baseband block diagonalization (BD) precoding.
The analog part of the combiner is obtained through the optimization problem:
(8)  
where . The solution of (8) is reached using an column iterative algorithm (CIA) defined in [15] and presented in Algorithm 2 [5].
The analog part of the precoder is obtained using the Algorithm 2 over the objective function in (9).
(9)  
IvC [14]CIAMMSE algorithm
Our previous work in [14], [14]CIAMMSE, consists in changing the digital beamforming part of [5]CIABD for the pseudo MMSE filter precoder defined in step 4 of the Algorithm 1 instead of a BD filter. In this approach the hybrid combiner complexity decreases because , which means that just analog beamforming is used in the receivers.
V Hybrid precoder/combiner proposal I
Our first proposal, PCIAMMSE, is described as follows. According to [5] the analog part of the combiner can be designed from the maximization of the determinant of , where , which means consider the following problem
(10)  
Considering an ideal case with no multiuser interference, (10) can be simplified to
(11)  
where is the submatrix of corresponding to the analog precoder of the user . To solve this nonconvex optimization problem, we can use the column iterative algorithm used in [5] (see Algorithm 2) and to alleviate the dependence between and , a recursive algorithm is considered as illustrated in Algorithm 5.
The procedure described in Algorithm 5 is performed by the BS. For the generation of the analog part of the combiner by the MS side, it is necessary that the receivers obtain an estimate of the product which can be simpler than obtaining an estimate of just . An option to obtain this estimate is by sending pilot symbols to a single user per time without the digital precoder part, such that the signal vector received by the user be
(12) 
Then the MS can compute and finally . Note that considering an errorfree estimation, the analog part of the precoder computed in the BS and in the MB are the same. In order not to increase the complexity of the hybrid combiner generation, the digital part of the hybrid combiner is considered as an identity matrix, i.e., .
For the digital part of the hybrid precoder the received signal vector processed by the user (see equation (2)) can be rewritten as
(13) 
where and with . For this model we propose the MMSE linear filter as derived in Appendix A.
(14) 
where with , and with [7].
Vi Hybrid precoder/combiner proposal II and III
Our second and third proposal consist in the improvement of the digital part of the HB [11]SVDMMSE and [14]CIAMMSE, which are referred hereafter as PSVDMMSE and PCIAMMSE, respectively. This improvement is obtained through the replacement of the pseudo MMSE defined in the step 4 of Algorithm 1 by expression (14).
Vii Hybrid precoder/combiner proposal IV
As previously mentioned the problem in the hybrid processing is the selection of the analog matrix and . Since the goal is to reduce the inter user/symbol interference and also keeping SNR large, the ideal equivalent baseband channel of each user is a diagonal matrix with large entries. Consequently, obtaining an approximation for this diagonal matrix is a good approach for the design of the analog parts. The following subsections describe our fourth proposal for the hybrid combiner/precoder construction, which is referred hereafter as PSVDMMSE.
Viia Hybrid combiner proposal
Based on (2) the mutual information between the information signal sent by the BS and the user can be written as follows
(15) 
where and . One approach for the designing of the hybrid combiner is to find a couple of matrix, and , that maximize (15) under the hardware constraints considered in mmWave scenarios. We then consider the optimization problem expressed by
(16)  
Many methods have been proposed in the literature to solve (16), e.g., [4, 5, 6], however, these methods involve complicated mathematical developments leading to complex solutions. In our approach we consider the approximation of the mutual information for large SNR.
(17) 
Then, using the properties of the determinant and taking (4) into account, (17) can be rewritten as:
(18) 
where and . Looking for a suboptimum manageable solution for large SNR we concentrate in the channel dependent first term of (18) and disregard the second term. Furthermore, for practical issues the combiner design should not depend on the precoder knowledge, because the users do not have access to the whole precoder matrix, even with estimation procedures they can obtain only an estimate of their precoder part, . Therefore, we consider that . This same assumption has been considered by others authors, e.g., [5]. With these simplifying assumptions the optimization problem is formulated as
(19)  
Note that a similar expression is considered in [5] (see equation (8)). The authors in [5] proposed a solution based on an iterative algorithm, whose complexity is considerable due to the matrix inversion required in each iteration (see Algorithm 2). From the Hadamard’s inequality which states that if an arbitrary square matrix a positive definite then
(20) 
with equality iff is a diagonal matrix [3], it is desirable that the product in (19) be a diagonal matrix with large entries. From the literature on traditional MIMO systems the above can easily be reached through single value decomposition (SVD) [16]. However, in mmWave scenarios this can not be used directly due to the constraints on the number of RF chains. To proceed with our proposed design, let us introduce the following SVD:
(21) 
(22) 
Note that is an approximation of when only phase shifters are used. Then, rewriting the problem in (19) in terms of the product between the user channel and analog combiner part, , we have
(23) 
which represents a baseband problem similar to (19) but without constraints on the matrix construction, therefore, it can be solved using
(25) 
where . Algorithm 6 summarizes the steps for the realization of the proposed hybrid combiner.
ViiB Hybrid precoder proposal
Our hybrid precoder design approach is divided in two parts, analog and digital. As stated in the previous subsection, the construction of the combiner’s (precoder’s) analog part based on the analog approximation to the eigenvectors of the channel user (entire channel) can benefit to diagonalization of the equivalent baseband user channel (entire equivalent baseband channel), which reduces the intersymbol (user) interference when is applied in the MS (BS). Thus, for the analog part of the precoder we consider the mutual information maximization problem of the entire channel, , when only analog processing is available in the transmitter, and we solve it through the same methodology used for the hybrid combiner realization.
Considering only analog processing in the transmitter with large SNR, the mutual information maximization problem of the entire channel can be reduced to the following problem
(26)  
or equivalently as^{1}^{1}1Consider an arbitrary complex matrix with . If is full ranking, then , where is the th eigenvalue of ..
(27)  
where . To construct we use the same methodology of the hybrid combiner described in Subsection VIIA. Therefore, the suboptimum analog part of the precoder is obtained as
(28) 
where .
Viii Data detection approaches
This section presents four suboptimal approaches to obtain by each user requiring different levels of parameter knowledge (or estimation) as follows:

Minimum distance detection (MDD)
(29) where , and is the, unknown to the receiver, submatrix of corresponding to the hybrid precoder of user .

Approximate MDD (AMDD) (assumes that )
(30) 
Noise whitening operation followed by MDD (NWMDD)
(31) where .

Noise whitening operation followed by approximate MDD (NWAMDD)
(32) 
Noise and interference whitening operation followed by MDD (NWIMDD)
(33) where .
Ix Numerical results
In the simulations, the users’ channels are generated with
multipaths components, the azimuth and elevation departure angles values are given by a random variable with uniform distribution in the interval of (0;
) and (0;), respectively. The UPAs have square formats for both transmitter and receivers, i.e., and . The maximum allowed setting for RF chains number is used for both the BS and for each MS, so that and . The results are averaged over channels generations for each user.Reference [5] proposed an optimization involving the effective baseband user channel, , to find the analog part of the combiner/precoder. Their authors used the value of , where is the th single value of the entire effective baseband channel, , as the metric to ilustrate the advantages of their proposal. Figure 1 presents a comparison using this metric corresponding to the obtained by [11, 5] and our proposals. The simulation settings are , , and ().
From Figure 1 it can be observed that [11]SVDMMSE and the hybrid designs proposed here yield the largest eigenvalues of the entire effective baseband channel. Despite the high complexity of PCIAMMSE there is not a relevant gain if this metric is considered as a measure of performance.
Figure 2 shows a comparison in terms of sum rate computed by the expression
(34) 
where . In this figure the BS has antennas and sends streams to and users equipped with .
Comments
There are no comments yet.