© Springer Nature Switzerland AG 2019
Tho Le-Ngoc and Ruikai MaiHybrid Massive MIMO Precoding in Cloud-RANWireless Networkshttps://doi.org/10.1007/978-3-030-02158-0_4

4. Hybrid Precoding/Combining for Massive MIMO with Hybrid ARQ

Tho Le-Ngoc1  and Ruikai Mai1
(1)
Department of Electrical and Computer Engineering, McGill University, Montréal, QC, Canada
 

4.1 Introduction

Because of poor channel conditions, correct reception at a certain target data rate can sometimes become impossible. Therefore, packet retransmission protocols are used in modern wireless communications systems to improve the reliability of data transmission. Specifically, when a one-bit feedback link is available from the receiver to the transmitter, the simple scheme of hybrid automatic repeat request (ARQ) with Chase combining (HARQ-CC) can be employed, where the same packet is retransmitted in the event of decoding failure.

In this chapter, we would like to address the design of hybrid RF-baseband precoding and combining suitable for massive multiple-input multiple-output (MIMO) with hybrid ARQ. In particular, based on the perfect channel state information (CSI), we consider a progressive approach to the hybrid precoder and combiner design in an attempt to exploit the temporal diversity inherent in packet retransmissions. Conditioned on the knowledge of previous retransmissions, the hybrid precoder and combiner are sequentially optimized for the current ARQ round without considering potential future retransmissions. To this end, we propose a two-step strategy to optimize the hybrid RF-baseband precoders/combiners with the objective of maximizing the spectral efficiency. On the heuristic assumption that the linear minimum mean square error (MMSE) filter is perfectly realizable by the two-stage receive combiner, we separate the derivation of the hybrid precoding from the hybrid combining. At the transmitter, during each ARQ round, we choose the RF precoder either from the set of transmit array response vectors or from a discrete Fourier transform (DFT)-based codebook. Built upon the selected RF precoder, the optimal beamformer at baseband is analytically shown to be a function of the generalized eigenvectors of the effective channel and RF precoder for the current ARQ retransmission, while the transmit power is allocated based on the precoding solutions from the previous and current packet retransmissions. At the receiver, a novel hybrid combining structure is proposed to address the issue of increased computational and storage complexity caused by repeated packet retransmissions. To minimize the performance loss of the decoupled precoding-combining optimization, the two-step strategy is further applied to derive the hybrid combining solution as an approximation of the optimal linear digital counterpart in terms of error performance. Through numerical simulations, we validate the efficacy of the proposed progressive approach to hybrid precoding and combining from (1) its comparable performance with the fully digital optimal progressive scheme, and (2) its performance advantage over other hybrid baseline that is oblivious to the presence of time diversity.

Although previous works, e.g., [13], have proposed various solution techniques for hybrid precoding and combining in the context of massive MIMO, they do not directly carry over when hybrid ARQ is incorporated. The study in this chapter is motivated by the observation that sequential precoding optimization is unique to ARQ systems as data retransmission occurs only if the previous transmissions fail. Since it is not possible to alter the previous transmission attempts of a data packet and another retransmission attempt may not be required after the current transmission, the transmitter can only optimize the current transmission for each ARQ round with respect to the desired performance metric. In theory, the formulation of matrix reconstruction as in Chap. 3 can be exploited to solve for the hybrid design as an approximation of the optimal solution. This nonetheless requires the knowledge of the fully digital solutions, which is generally nontrivial to obtain in the first place. Furthermore, such an approach tends to lead to suboptimal power loading schemes at baseband. In contrast, the proposed progressive precoding solution takes into account power allocation for each round of packet retransmission, and also gives insights into the relation between the precoder for the current ARQ transmission and the previous ones. Mathematically, the presence of the RF stage in the power constraint also necessitates a different solution approach.

The rest of this chapter is organized as follows. In Sect. 4.2, we describe the system model of massive MIMO with hybrid ARQ where hybrid precoding and combining are employed. We develop the proposed methods of progressive optimization of hybrid precoding and combining in Sects. 4.3 and 4.4, respectively. Illustrative results are provided in Sect. 4.5, where the proposed progressive hybrid solution is numerically compared with various baselines. Concluding remarks are made in Sect. 4.6.

4.2 System Model and Problem Statement

We consider a point-to-point flat-fading massive MIMO system with N t antennas at the transmitter and N r antennas at the receiver, where the large-scale transmit and receive antenna arrays are driven by n t ≪ N t and n r ≪ N r RF chains, respectively. Let 
$$N_{\mathrm {s}}\leq \min (n_{\mathrm {t}},n_{\mathrm {r}})$$
denote the number of data streams to be transmitted. As a result of a limited number of RF chains, a hybrid RF-baseband precoder/combiner in place of a single-stage, fully digital precoder/combiner is employed. As illustrated in Fig. 4.1, the data streams are first linearly transformed using a low-dimensional baseband precoder 
$${\mathbf {F}}_{\mathrm {BB}}\in \mathbb {C}^{n_{\mathrm {t}}\times N_{\mathrm {s}}}$$
, the output of which is then precoded by a high-dimensional RF precoder 
$${\mathbf {F}}_{\mathrm {RF}}\in \mathbb {C}^{N_{\mathrm {t}}\times n_{\mathrm {t}}}$$
. The RF precoder is assumed to be implemented using analog phase-shifters, i.e., the elements are constrained to satisfy 
$$\vert [{\mathbf {F}}_{\mathrm {RF}}]_{i,j}\vert =\frac {1}{\sqrt {N_{\mathrm {t}}}},1\leq i\leq N_{\mathrm {t}},1\leq j\leq n_{\mathrm {t}}$$
.
../images/470489_1_En_4_Chapter/470489_1_En_4_Fig1_HTML.png
Fig. 4.1

Hybrid RF-baseband precoding and combining for M rounds of packet retransmission in a point-to-point massive MIMO hybrid ARQ system

At the receiver, a single-bit feedback is leveraged to inform the transmitter of the reception status. In particular, the receiver sends an acknowledgment (ACK) for successful decoding and a negative ACK (NACK) signal otherwise. For the case of HARQ-CC which is of interest to this work, the transmitter simply resends the same signal upon receiving a NACK feedback. This makes it possible for the receiver to combine the same transmitted signal across all transmission attempts, as shown in Fig. 4.1. Denoting the RF and baseband precoders during the mth ARQ round of transmission by F RF,m and F BB,m, respectively, the received signal during the mth ARQ transmission is given by

$$\displaystyle \begin{aligned} {\mathbf{y}}_{m}=\sqrt{\rho}{\mathbf{H}}_{m}{\mathbf{F}}_{\mathrm{RF},m}{\mathbf{F}}_{\mathrm{BB},m}\mathbf{s}+{\mathbf{n}}_{m}, \end{aligned}$$
where ρ represents the average received power, the data vector s is assumed to be encoded by an independently and identically distributed (i.i.d.) Gaussian codebook, i.e., 
$$\mathbf {s}\sim \mathcal {CN}(\mathbf {0},\frac {1}{N_{\mathrm {s}}}{\mathbf {I}}_{N_{\mathrm {s}}})$$
, 
$${\mathbf {H}}_{m}\in \mathbb {C}^{N_{\mathrm {r}}\times N_{\mathrm {t}}}$$
denotes the massive MIMO channel between the transmitter and receiver during the mth retransmission of the data vector s, and is assumed to change independently from transmission to transmission, and 
$${\mathbf {n}}_{m}\sim \mathcal {CN}\left (\mathbf {0},\sigma _{n}^{2}{\mathbf {I}}_{N_{\mathrm {r}}}\right )$$
is the spatially white Gaussian noise with variance 
$$\sigma _{n}^{2}$$
. In particular, we characterize H m, ∀m by a parametric clustered channel model [1, 4], in which the channel matrix is a sum of contributions of N cl scattering clusters, each with N ray propagation paths. Assuming uniform linear arrays (ULA) at both link ends, the MIMO channel is generically expressed as1

$$\displaystyle \begin{aligned} \mathbf{H}=\gamma\sum_{i=1}^{N_{\mathrm{cl}}}\sum_{l=1}^{N_{\mathrm{ray}}}\alpha_{il}{\mathbf{a}}_{\mathrm{r}}\left(\phi_{il}^{\mathrm{r}}\right){\mathbf{a}}_{\mathrm{t}}\left(\phi_{il}^{\mathrm{t}}\right)^{\mathrm{H}}, \end{aligned}$$
where 
$$\gamma =\sqrt {N_{\mathrm {t}}N_{\mathrm {r}}/N_{\mathrm {cl}}N_{\mathrm {ray}}}$$
is a normalization factor such that 
$$\mathbb {E}[\left \Vert \mathbf {H}\right \Vert _{\mathrm {F}}^{2}]=N_{\mathrm {r}}N_{\mathrm {t}}$$
, and 
$$\alpha _{il}\sim \mathcal {CN}\left (0,1\right )$$
is the i.i.d. complex path gain of the lth ray in the ith scattering cluster. The vector 
$${\mathbf {a}}_{\mathrm {r}}\left (\phi _{il}^{\mathrm {r}}\right )\in \mathbb {C}^{N_{\mathrm {r}}}$$
(
$${\mathbf {a}}_{\mathrm {t}}\left (\phi _{il}^{\mathrm {t}}\right )\in \mathbb {C}^{N_{\mathrm {t}}}$$
) represents the normalized receive (transmit) array response vector at an azimuth angle of arrival (AoA) 
$$\phi _{il}^{\mathrm {r}}$$
(angle of departure (AoD) 
$$\phi _{il}^{\mathrm {t}}$$
). For a generic N-element ULA on the y-axis, the array response is expressed as [1]

$$\displaystyle \begin{aligned} {\mathbf{a}}_{\mathrm{ULA}}\left(\phi\right)=\frac{1}{\sqrt{N}}\left[1,\,e^{\jmath2\pi d_{\lambda}\sin\left(\phi\right)},\cdots,\,e^{\jmath\left(N-1\right)2\pi d_{\lambda}\sin\left(\phi\right)}\right]^{\mathrm{T}},{} \end{aligned} $$
(4.1)
where d λ is the inter-antenna spacing normalized by the wavelength. The perfect knowledge of H m, ∀m is assumed to be available at the transmitter and receiver. While it is challenging to acquire the high-dimensional CSI in closed-loop frequency-division duplexing (FDD) systems which depend on the mechanism of channel training and feedback, such an issue can be simplified by exploiting channel reciprocity in time-division duplexing (TDD) systems.2
After M retransmissions of the data vector s, the received signals can be aggregately written as
../images/470489_1_En_4_Chapter/470489_1_En_4_Equc_HTML.png
As the noise vectors during different retransmissions are independent, it holds that 
$${\mathbf {R}}_{\tilde {n}}\triangleq \mathbb {E}[\tilde {\mathbf {n}}_{M}\tilde {\mathbf {n}}_{M}^{\mathrm {H}}]=\sigma _{n}^{2}{\mathbf {I}}_{MN_{\mathrm {r}}}$$
. The fact that the same signal is transmitted in the event of decoding failure enables the coherent combining of all the received signals across different ARQ rounds. Ideally, the combining should take place in both the RF and baseband domains for the optimal performance, which would require storage of the high-dimensional received signals 
$$\{{\mathbf {y}}_{m}\}_{m=1}^{M}$$
from all ARQ rounds and applying to the aggregate received signal 
$$\widetilde {\mathbf {y}}_{M}$$
an RF combiner of size MN r × Mn r followed by a baseband combiner of size Mn r × N s. Unfortunately, these high-dimensional combiners pose nontrivial computational and storage complexity, especially when N r is large. Furthermore, additional RF phase-shifters are needed to implement the joint combining in the RF domain.3 Out of these concerns, in this work, we propose to first perform independent RF combining of the received signals from each ARQ round, and then perform joint baseband combining of the RF-processed received signals across all ARQ rounds, as illustrated in Fig. 4.1. Accordingly, after the Mth ARQ round, the signal at the output of the hybrid combiner can be written as

$$\displaystyle \begin{aligned} \hat{\mathbf{y}}_{M}={\mathbf{W}}_{\mathrm{BB},M}^{\mathrm{H}}\mathbb{W}_{\mathrm{RF},M}^{\mathrm{H}}\tilde{\mathbf{y}}_{M}=\sqrt{\rho}{\mathbf{W}}_{\mathrm{BB},M}^{\mathrm{H}}\mathbb{W}_{\mathrm{RF},M}^{\mathrm{H}}\mathbb{H}\mathbf{s}+{\mathbf{W}}_{\mathrm{BB},M}^{\mathrm{H}}\mathbb{W}_{\mathrm{RF},M}^{\mathrm{H}}\tilde{\mathbf{n}}_{M}, \end{aligned}$$
where the block diagonal matrix 
$$\mathbb {W}_{\mathrm {RF},M}=\mathrm {blkdiag}\{{\mathbf {W}}_{\mathrm {RF},1},\ldots ,{\mathbf {W}}_{\mathrm {RF},M}\}$$
is the aggregate RF combiner with the mth component 
$${\mathbf {W}}_{\mathrm {RF},m}\in \mathbb {C}^{N_{\mathrm {r}}\times n_{\mathrm {r}}},$$
1 ≤ m ≤ M, as the RF combiner with respect to the received signal y m, and 
$${\mathbf {W}}_{\mathrm {BB},M}\in \mathbb {C}^{Mn_{\mathrm {r}}\times N_{\mathrm {s}}}$$
is a baseband combiner used during the Mth ARQ round. The achievable rate per channel use, i.e., spectral efficiency, with the proposed precoding and combining strategy after M ARQ rounds is therefore expressed as

$$\displaystyle \begin{aligned} R_{ARQ}^{M}=\frac{1}{M}\log_{2}\left|{\mathbf{I}}_{N_{\mathrm{s}}}+\beta{\mathbf{R}}_{{\mathbf{W}}_{n}}^{-1}{\mathbf{W}}_{\mathrm{BB},M}^{\mathrm{H}}\mathbb{W}_{\mathrm{RF},M}^{\mathrm{H}}\mathbb{H}\mathbb{H}^{\mathrm{H}}\mathbb{W}_{\mathrm{RF},M}{\mathbf{W}}_{\mathrm{BB},M}\right|,{} \end{aligned} $$
(4.2)
where we define 
$$\beta \triangleq \frac {\rho }{N_{\mathrm {s}}\sigma _{n}^{2}}$$
and 
$${\mathbf {R}}_{{\mathbf {W}}_{n}}\triangleq {\mathbf {W}}_{\mathrm {BB},M}^{\mathrm {H}}\mathbb {W}_{\mathrm {RF},M}^{\mathrm {H}}\mathbb {W}_{\mathrm {RF},M}{\mathbf {W}}_{\mathrm {BB},M}$$
for notational simplicity.

4.3 Progressive Hybrid Precoding Design

In this section, we formulate the progressive hybrid precoding problem during the Mth ARQ round of data retransmission. Built upon the hybrid precoders 
$$\{{\mathbf {F}}_{\mathrm {RF},m}^{\star },{\mathbf {F}}_{\mathrm {BB},m}^{\star }\}_{m=1}^{M-1}$$
and hybrid combiners 
$$\{{\mathbf {W}}_{\mathrm {RF},m}^{\star },{\mathbf {W}}_{\mathrm {BB},m}^{\star }\}_{m=1}^{M-1}$$
used for the previous retransmission attempts, the objective is to find the hybrid precoder {F RF,M, F BB,M} and hybrid combiner {W RF,M, W BB,M} during the Mth ARQ round such that 
$$R_{\mathrm {ARQ}}^{M}$$
in (4.2) is maximized.

Finding the global optimum generally requires joint hybrid precoding and combining optimization, which is unfortunately mathematically intractable especially in the presence of the nonconvex modulus constraints on F RF,M and W RF,M. Instead, we seek an approach where the hybrid precoding can be decoupled from the hybrid combining optimization. In particular, we assume that the fully digital linear MMSE combiner is perfectly realizable by the hybrid RF-baseband counterpart. Following a similar line of reasoning as in [6], it can be shown that in this case, the achievable rate 
$$R_{\mathrm {ARQ}}^{M}$$
in (4.2) is reduced to the mutual information between s and 
$$\widetilde {\mathbf {y}}_{M}$$
, i.e., 
$$\mathcal {I}(\mathbf {s};\widetilde {\mathbf {y}}_{M})$$
, as given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \mathcal{I}\left(\mathbf{s};\tilde{\mathbf{y}}_{M}\right)=\log_{2}\bigg|{\mathbf{I}}_{N_{\mathrm{s}}}+\beta\bigg(\sum_{m=1}^{M-1}{\mathbf{F}}_{\mathrm{BB},m}^{\star\mathrm{H}}{\mathbf{F}}_{\mathrm{RF},m}^{\star\mathrm{H}}{\mathbf{H}}_{m}^{\mathrm{H}}{\mathbf{H}}_{m}{\mathbf{F}}_{\mathrm{RF},m}^{\star}{\mathbf{F}}_{\mathrm{BB},m}^{\star}\\ &\displaystyle &\displaystyle \qquad  \qquad  \qquad  \qquad  +{\mathbf{F}}_{\mathrm{BB},M}^{\mathrm{H}}{\mathbf{F}}_{\mathrm{RF},M}^{\mathrm{H}}{\mathbf{H}}_{M}^{\mathrm{H}}{\mathbf{H}}_{M}{\mathbf{F}}_{\mathrm{RF},M}{\mathbf{F}}_{\mathrm{BB},M}\bigg)\bigg|, \end{array} \end{aligned} $$
where we note that optimization of the precoders F BB,M and F RF,M for the Mth ARQ round depends on the previous precoders 
$$\{{\mathbf {F}}_{\mathrm {RF},m}^{\star },{\mathbf {F}}_{\mathrm {BB},m}^{\star }\}_{m=1}^{M-1}$$
. The problem of interest is accordingly formulated as

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle \underset{{\scriptstyle {\mathbf{F}}_{\mathrm{RF},M},{\mathbf{F}}_{\mathrm{BB},M}}}{\max} &\displaystyle \mathcal{I}\left(\mathbf{s};\tilde{\mathbf{y}}_{M}\right) \\ &\displaystyle \mathrm{s.t.} &\displaystyle {\mathbf{F}}_{\mathrm{RF},M}\in\mathcal{F}_{\mathrm{RF}},\,\left\Vert {\mathbf{F}}_{\mathrm{RF},M}{\mathbf{F}}_{\mathrm{BB},M}\right\Vert _{\mathrm{F}}^{2}=N_{\mathrm{s}}.{} \end{array} \end{aligned} $$
(4.3)
In view of the difficulty with jointly solving for {F RF,M, F BB,M} subject to the constraint 
$${\mathbf {F}}_{\mathrm {RF},M}\in \mathcal {F}_{\mathrm {RF}}$$
, we propose a two-step solution technique. In the first step, we choose the RF precoder F RF,M to be one of the feasible solutions from the set 
$$\mathcal {F}_{\mathrm {RF}}$$
, say G RF,M, and conditioned on this choice, we derive the closed-form solution for the optimal baseband precoder 
$${\mathbf {F}}_{\mathrm {BB},M}^{\star }\left ({\mathbf {G}}_{\mathrm {RF},M}\right )$$
. This procedure is repeated for each feasible G RF,M to generate a set of 
$$\{{\mathbf {G}}_{\mathrm {RF},M},{\mathbf {F}}_{\mathrm {BB},M}^{\star }({\mathbf {G}}_{\mathrm {RF},M})\}$$
, from which the pair that yields the maximum mutual information in (4.3) is declared as the solution to the original problem. We remark that unlike the one-shot approach based on matrix reconstruction in [1], the proposed two-step technique obviates the need for knowing the fully digital solution, and enjoys the flexibility of performing waterfilling-based power loading at baseband.

Clearly, the set of constant-modulus RF precoders 
$$\mathcal {F}_{\mathrm {RF}}$$
contains an unlimited number of elements. To facilitate the RF precoding optimization, we consider two suboptimal but effective alternatives, where the columns of the RF precoder F RF,M are assumed to be chosen either from the set of N cl N ray transmit array response vectors 
$$\mathcal {A}_{\mathrm {t}}\triangleq \{{\mathbf {a}}_{\mathrm {t}}(\phi _{il}^{\mathrm {t}})\}_{i,l}$$
[1] or from the N t columns of the N t-dimensional DFT matrix 
$$\mathcal {D}_{\mathrm {t}}\triangleq \left [\frac {1}{\sqrt {N_{\mathrm {t}}}}e^{-\frac {\jmath 2\pi mn}{N_{\mathrm {t}}}}\right ],0\leq m,n\leq N_{\mathrm {t}}-1$$
[2]. The rationale for constraining the columns of the RF precoder to the finite sets 
$$\mathcal {A}_{\mathrm {t}}$$
and 
$$\mathcal {D}_{\mathrm {t}}$$
for the ARQ retransmission is threefold: (1) on one hand, the optimal progressive digital precoder has been known to be a function of a unitary matrix whose columns span the row space of the channel H M [7]. On the other hand, it has been observed in [1] that under certain conditions, the set of transmit array response vectors 
$$\mathcal {A}_{\mathrm {t}}$$
serves as another basis for the row space of the channel H M; (2) as indicated by the definition (4.1), the transmit array response vector 
$${\mathbf {a}}_{\mathrm {t}}(\phi _{il}^{\mathrm {t}}),\forall i,l$$
, consists of constant-modulus entries; (3) when angle-domain quantization is considered as an option to reduce the overhead of estimating the complete AoD information, the array response vector-based RF precoding design can still be directly applied. One extreme case is the DFT-based codebook, which represents blind uniform quantization of the azimuth AoDs. Interestingly, in the large-scale array regime, the DFT matrix becomes an asymptotically good approximation for the channel eigen-space [8]. Our proposed approach to generate the hybrid RF-baseband precoder for the Mth ARQ retransmission is summarized in Algorithm 1.

Algorithm 1 Two-step progressive hybrid RF-baseband precoding design

../images/470489_1_En_4_Chapter/470489_1_En_4_Figa_HTML.png
In order to derive the optimal baseband solution 
$${\mathbf {F}}_{\mathrm {BB},M}^{\star }\left ({\mathbf {G}}_{\mathrm {RF},M}\right )$$
to (4.4), we observe that given the eigenvalue decomposition

$$\displaystyle \begin{aligned} \sum_{m=1}^{M-1}{\mathbf{F}}_{\mathrm{BB},m}^{\star\mathrm{H}}{\mathbf{F}}_{\mathrm{RF},m}^{\star\mathrm{H}}{\mathbf{H}}_{m}^{\mathrm{H}}{\mathbf{H}}_{m}{\mathbf{F}}_{\mathrm{RF},m}^{\star}{\mathbf{F}}_{\mathrm{BB},m}^{\star}=\mathbf{U}\boldsymbol{\Lambda}_{M-1}{\mathbf{U}}^{\mathrm{H}}, \end{aligned}$$
the objective function in (4.4) can be reduced to

$$\displaystyle \begin{aligned} & \log_{2}\left|{\mathbf{I}}_{N_{\mathrm{s}}}+\beta\mathbf{U}\boldsymbol{\varLambda}_{M-1}{\mathbf{U}}^{\mathrm{H}}+\beta{\mathbf{F}}_{\mathrm{BB},M}^{\mathrm{H}}{\mathbf{G}}_{\mathrm{RF},M}^{\mathrm{H}}{\mathbf{H}}_{M}^{\mathrm{H}}{\mathbf{H}}_{M}{\mathbf{G}}_{\mathrm{RF},M}{\mathbf{F}}_{\mathrm{BB},M}\right|\\ =\, & \log_{2}\left|\mathbf{\widetilde{\varLambda}}_{M-1}+\beta\widetilde{\mathbf{F}}_{\mathrm{BB},M}^{\mathrm{H}}{\mathbf{G}}_{\mathrm{RF},M}^{\mathrm{H}}{\mathbf{H}}_{M}^{\mathrm{H}}{\mathbf{H}}_{M}{\mathbf{G}}_{\mathrm{RF},M}\widetilde{\mathbf{F}}_{\mathrm{BB},M}\right|, \end{aligned} $$
where 
$$\boldsymbol {\widetilde {\varLambda }}_{M-1}\triangleq {\mathbf {I}}_{N_{\mathrm {s}}}+\beta \boldsymbol {\varLambda }_{M-1}=\mathrm {diag}\{\widetilde {\lambda }_{1,M-1},\ldots ,\widetilde {\lambda }_{N_{\mathrm {s}},M-1}\}$$
, and 
$$\widetilde {\mathbf {F}}_{\mathrm {BB},M}\triangleq {\mathbf {F}}_{\mathrm {BB},M}\mathbf {U}$$
. On the other hand, following the orthonormality of U, the power constraint in (4.4) is equivalent to ../images/470489_1_En_4_Chapter/470489_1_En_4_IEq48_HTML.gif. In other words, we can reformulate the problem (4.4) as

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle \underset{{\scriptstyle {\scriptstyle \widetilde{\mathbf{F}}_{\mathrm{BB},M}}}}{\max} &\displaystyle \log_{2}\left|\widetilde{\boldsymbol{\varLambda}}_{M-1}+\beta\widetilde{\mathbf{F}}_{\mathrm{BB},M}^{\mathrm{H}}{\mathbf{G}}_{\mathrm{RF},M}^{\mathrm{H}}{\mathbf{H}}_{M}^{\mathrm{H}}{\mathbf{H}}_{M}{\mathbf{G}}_{\mathrm{RF},M}\widetilde{\mathbf{F}}_{\mathrm{BB},M}\right| \\ &\displaystyle \mathrm{s.t.} &\displaystyle \big\Vert{\mathbf{G}}_{\mathrm{RF},M}\widetilde{\mathbf{F}}_{\mathrm{BB},M}\big\Vert_{\mathrm{F}}^{2}=N_{\mathrm{s}},{} \end{array} \end{aligned} $$
(4.5)
to which the optimal solution can be analytically established, as shown in the following theorem.

Theorem 4.1

Let A M jointly diagonalize the pair of positive semi-definite matrices 
$${\mathbf {G}}_{\mathrm {RF},M}^{\mathrm {H}}{\mathbf {H}}_{M}^{\mathrm {H}}{\mathbf {H}}_{M}{\mathbf {G}}_{\mathrm {RF},M}$$
and 
$${\mathbf {G}}_{\mathrm {RF},M}^{\mathrm {H}}{\mathbf {G}}_{\mathrm {RF},M}$$
as

$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathbf{A}}_{M}^{\mathrm{H}}{\mathbf{G}}_{\mathrm{RF},M}^{\mathrm{H}}{\mathbf{H}}_{M}^{\mathrm{H}}{\mathbf{H}}_{M}{\mathbf{G}}_{\mathrm{RF},M}{\mathbf{A}}_{M}=\boldsymbol{\varSigma}_{1M},\\ {\mathbf{A}}_{M}^{\mathrm{H}}{\mathbf{G}}_{\mathrm{RF},M}^{\mathrm{H}}{\mathbf{G}}_{\mathrm{RF},M}{\mathbf{A}}_{M}=\boldsymbol{\varSigma}_{2M}, \end{array} \end{aligned} $$
where 
$$\boldsymbol {\varSigma }_{iM}=\mathrm {diag}\{\sigma _{1,iM},\ldots ,\sigma _{n_{\mathrm {t}},iM}\},\,i=1,2.$$
Let 
$$k_{1},\ldots ,k_{N_{\mathrm {s}}}$$
denote the indices of the largest N s elements of the set 
$$\{\psi _{i,M}\triangleq \frac {\sigma _{i,1M}}{\sigma _{i,2M}}\}_{i=1}^{n_{\mathrm {t}}}$$
such that 
$$\psi _{k_{1},M}\geq \cdots \geq \psi _{k_{N_{\mathrm {s}}},M}$$
. Denote 
$$\mathbf {\boldsymbol {\varPsi }}_{M}\triangleq \mathrm {diag}\{\psi _{k_{1},M},\ldots ,\psi _{k_{N_{\mathrm {s}}},M}\}.$$
Let P 1,M and P 2,M be the permutation matrices such that the diagonal elements of 
$${\mathbf {P}}_{1,M}\boldsymbol {\varPsi }_{M}{\mathbf {P}}_{1,M}^{\mathrm {H}}$$
are in non-decreasing order while those of 
$${\mathbf {P}}_{2,M}\widetilde {\boldsymbol {\varLambda }}_{M-1}{\mathbf {P}}_{2,M}^{\mathrm {H}}$$
are in non-increasing order. Then the optimal solution to (4.5) is given by

$$\displaystyle \begin{aligned} \widetilde{\mathbf{F}}_{\mathrm{BB},M}={\mathbf{A}}_{M}{\mathbf{S}}_{M}{\mathbf{P}}_{1,M}^{\mathrm{H}}\widehat{\mathbf{F}}_{\mathrm{BB},M}{\mathbf{P}}_{2,M} \end{aligned}$$
where 
$$\widehat {\mathbf {F}}_{\mathrm {BB},M}=\mathrm {diag}\{f_{M}^{1},\cdots ,f_{M}^{N_{\mathrm {s}}}\}$$
with

$$\displaystyle \begin{aligned} f_{M}^{j}=\sqrt{\frac{1}{\sigma_{k_{j},2M}}\left(c-\frac{\widehat{\lambda}_{1,M-1}}{\beta\psi_{k_{j},M}}\right)^{+}},1\leq j\leq N_{\mathrm{s}}, \end{aligned}$$

and c chosen such that 
$$\mathrm {tr}(\widehat {\mathbf {F}}_{\mathrm {BB},M}\widehat {\mathbf {F}}_{\mathrm {BB},M}^{\mathrm {H}})=N_{\mathrm {s}}$$
, and S M selects the N s columns of A M indexed by 
$$k_{1},\ldots ,k_{N_{\mathrm {s}}}$$
.

Proof

See Appendix 1.

Remark 4.1

The optimal baseband precoder, conditioned on the selected RF precoder G RF,M, consists of the following features:
  1. 1.

    The optimal beamforming subspace is spanned by the generalized eigenvectors A M of the Gram matrices of the effective channel H M G RF,M and the RF precoder G RF,M.

     
  2. 2.

    The optimal beamforming directions are determined by the selection matrix S M, which picks the N s largest indices of the set 
$$\{\psi _{i,M}\}_{i=1}^{n_{\mathrm {t}}}$$
.

     
  3. 3.

    The diagonal matrix 
$$\widehat {\mathbf {F}}_{\mathrm {BB},M}$$
implements the waterfilling-based power loading.

     
  4. 4.

    The permutation matrices P 1,M and P 2,M achieve the optimal reverse pairing of the singular values of the previous precoded transmissions 
$${\mathbf {I}}_{N_{\mathrm {s}}}+\beta \boldsymbol {\varLambda }_{M-1}$$
with the largest N s elements of the set 
$$\{\psi _{1,M},\ldots ,\psi _{n_{\mathrm {t}},M}\}$$
related to the current transmission.

     
  5. 5.

    For the initial transmission, i.e., M = 1, the permutation matrices become an identity matrix, i.e., 
$${\mathbf {P}}_{1,M}={\mathbf {P}}_{2,M}={\mathbf {I}}_{N_{\mathrm {s}}}$$
, and the transmit power is loaded according to 
$$f_{M}^{j}=\sqrt {\frac {1}{\sigma _{k_{j},2M}}\left (c-\frac {1}{\beta \psi _{k_{j},M}}\right )^{+}}$$
, 1 ≤ j ≤ N s.

     

4.4 Progressive Hybrid Combining Design

In the previous section, by heuristically assuming that the fully digital linear MMSE combiner can be perfectly reconstructed by the hybrid RF-baseband counterpart, we were able to abstract the hybrid combining effect on the achievable rate, and decouple the precoding from the combining optimization. In the existing works such as [1], the reconstruction was addressed in terms of the combining structure, and was formulated as a matrix reconstruction problem. In doing this, the fully digital solution needs to be known. However, as discussed in Sect. 3.​2, such knowledge might not be practically obtainable in light of the storage requirement imposed by the high-dimensional received signals across multiple ARQ rounds. Furthermore, since the matrix reconstruction formulation attempts to generate the RF and baseband combiners simultaneously, suboptimal baseband solutions are likely to result. In view of these drawbacks, we consider applying the two-step approach from the previous section to the hybrid combining design in this section.

In hopes of alleviating the storage and computational complexity at the receiver, we first seek to decrease the dimension of the received signals through RF combining, which is carried out independently across different ARQ rounds, and then combine the reduced-dimensional RF-processed signals from all the previous and current retransmissions accessible to the baseband, as illustrated in Fig. 4.1. In particular, given the hybrid RF-baseband precoders 
$$\{{\mathbf {F}}_{\mathrm {RF},m}^{\star },{\mathbf {F}}_{\mathrm {BB},m}^{\star }\}_{m=1}^{M}$$
generated from the procedure developed in Sect. 4.3 and the RF combiners 
$$\{{\mathbf {W}}_{\mathrm {RF},m}^{\star }\}_{m=1}^{M-1}$$
, the idea is to design the hybrid RF-baseband combiner for the current ARQ round, i.e., {W RF,M, W BB,M}, as a good approximation of the fully digital solution in terms of error performance. To this end, we consider the minimization of MSE between the transmitted signal and the aggregate received signals, which leads to the problem formulation as

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle \underset{{\scriptstyle {\mathbf{W}}_{\mathrm{RF},M},{\mathbf{W}}_{\mathrm{BB},M}}}{\min} &\displaystyle \mathbb{E}\left[\left\Vert \mathbf{s}-{\mathbf{W}}_{\mathrm{BB},M}^{\mathrm{H}}\mathbb{W}_{\mathrm{RF},M}^{\mathrm{H}}\tilde{\mathbf{y}}_{M}\right\Vert _{2}^{2}\right] \\ &\displaystyle \mathrm{s.t.} &\displaystyle {\mathbf{W}}_{\mathrm{RF},M}\in\mathcal{W}_{RF},{} \end{array} \end{aligned} $$
(4.6)
where 
$$\mathcal {W}_{RF}$$
denotes the set of feasible RF combiners with constant-modulus elements. It is straightforward to show that the objective in (4.6), denoted as 
$$\mathcal {E}$$
, can be evaluated as

$$\displaystyle \begin{aligned} \mathcal{E} & =1+\sigma_{n}^{2}\mathrm{tr}\left[{\mathbf{W}}_{\mathrm{BB},M}^{\mathrm{H}}\mathbb{W}_{\mathrm{RF},M}^{\mathrm{H}}\left(\beta\mathbb{H}\mathbb{H}^{\mathrm{H}}+{\mathbf{I}}_{MN_{\mathrm{r}}}\right)\mathbb{W}_{\mathrm{RF},M}{\mathbf{W}}_{\mathrm{BB},M}\right] \\ & \quad -2\frac{\sqrt{\rho}}{N_{\mathrm{s}}}\Re\left\{ \mathrm{tr}\left(\mathbb{H}^{\mathrm{H}}\mathbb{W}_{\mathrm{RF},M}{\mathbf{W}}_{\mathrm{BB},M}\right)\right\} .{} \end{aligned} $$
(4.7)
In a similar manner to the hybrid precoding design, the two-step approach to the issue of hybrid combining first chooses an RF combiner from the feasible set 
$$\mathcal {W}_{RF}$$
, say B RF,M, and then conditioned on the RF combiner B RF,M, computes the optimal baseband combiner 
$${\mathbf {W}}_{\mathrm {BB},M}^{\star }({\mathbf {B}}_{\mathrm {RF},M})$$
. This procedure is repeated to generate a pair 
$$\{{\mathbf {B}}_{\mathrm {RF},M},{\mathbf {W}}_{\mathrm {BB},M}^{\star }({\mathbf {B}}_{\mathrm {RF},M})\}$$
for each chosen 
$${\mathbf {B}}_{\mathrm {RF},M}\in \mathcal {W}_{RF}$$
, where the MMSE-achieving pair is treated as the solution to the original problem. By solving the first-order derivative of (4.7), i.e.,

$$\displaystyle \begin{aligned} \frac{\partial\mathcal{E}}{\partial{\mathbf{W}}_{\mathrm{BB},M}^{\mathrm{H}}}=\sigma_{n}^{2}\mathbb{W}_{\mathrm{RF},M}^{\mathrm{H}}\left(\beta\mathbb{H}\mathbb{H}^{\mathrm{H}}+{\mathbf{I}}_{MN_{\mathrm{r}}}\right)\mathbb{W}_{\mathrm{RF},M}{\mathbf{W}}_{\mathrm{BB},M}-\frac{\sqrt{\rho}}{N_{\mathrm{s}}}\mathbb{W}_{\mathrm{RF},M}^{\mathrm{H}}\mathbb{H}=\mathbf{0}, \end{aligned}$$
the baseband combiner 
$${\mathbf {W}}_{\mathrm {BB},M}^{\star }\left ({\mathbf {W}}_{\mathrm {RF},M}\right )$$
, as a function of the RF combiner, can be expressed in a closed form as

$$\displaystyle \begin{aligned} {\mathbf{W}}_{\mathrm{BB},M}^{\star}\left({\mathbf{W}}_{\mathrm{RF},M}\right)=\frac{\sqrt{\rho}}{N_{\mathrm{s}}\sigma_{n}^{2}}\left[\mathbb{W}_{\mathrm{RF},M}^{\mathrm{H}}\left(\beta\mathbb{H}\mathbb{H}^{\mathrm{H}}+{\mathbf{I}}_{MN_{\mathrm{r}}}\right)\mathbb{W}_{\mathrm{RF},M}\right]^{-1}\mathbb{W}_{\mathrm{RF},M}^{\mathrm{H}}\mathbb{H}.{} \end{aligned} $$
(4.8)
To facilitate the implementation of such a two-step procedure, we further reduce the infinite set of constant-modulus RF combiners 
$$\mathcal {W}_{F}$$
to either the finite set of N cl N ray receive array response vectors 
$$\{{\mathbf {a}}_{\mathrm {r}}(\phi _{il}^{\mathrm {r}})\}_{i,l}$$
[1] or the N r columns of the N r-dimensional DFT matrix 
$$\mathcal {D}_{\mathrm {r}}=\left [\frac {1}{\sqrt {N_{\mathrm {r}}}}e^{-\frac {\jmath 2\pi mn}{N_{\mathrm {r}}}}\right ],0\leq m,n\leq N_{\mathrm {r}}-1$$
. We formally summarize the proposed strategy for MMSE hybrid combining for the Mth ARQ round as follows.

Algorithm 2 Two-step progressive hybrid RF-baseband combining design

  1. 1.

    Construct B RF,M and accordingly 
$$\mathbb {B}_{\mathrm {RF},M}\triangleq \mathrm {blkdiag}\{{\mathbf {W}}_{\mathrm {RF},1}^{\star },\ldots ,{\mathbf {W}}_{\mathrm {RF},M-1}^{\star },{\mathbf {B}}_{\mathrm {RF},M}\}$$
by choosing n r vectors from either the set of N cl N ray receive array response vectors 
$$\mathcal {A}_{\mathrm {r}}=\{{\mathbf {a}}_{\mathrm {r}}(\phi _{il}^{\mathrm {r}})\}_{i,l}$$
or the N r columns of the DFT matrix 
$$\mathcal {D}_{\mathrm {r}}$$
.

     
  2. 2.

    For the selected B RF,M, compute the MMSE baseband combiner based on (4.8) and the MSE for the pair 
$$\left \{ {\mathbf {B}}_{\mathrm {RF},M},{\mathbf {W}}_{\mathrm {BB},M}^{\star }\left ({\mathbf {B}}_{\mathrm {RF},M}\right )\right \} $$
based on (4.7).

     
  3. 3.

    Repeat step 2 for each combination of B RF,M from 
$$\mathcal {A}_{\mathrm {r}}=\{{\mathbf {a}}_{\mathrm {r}}(\phi _{il}^{\mathrm {r}})\}_{i,l}$$
or from the columns of 
$$\mathcal {D}_{\mathrm {r}}$$
, and choose the pair 
$$\left \{ {\mathbf {B}}_{\mathrm {RF},M},{\mathbf {W}}_{\mathrm {BB},M}^{\star }\left ({\mathbf {B}}_{\mathrm {RF},M}\right )\right \} $$
from step 2 that minimizes the MSE as the solution to (4.6).

     

4.5 Illustrative Results and Discussions

In this section, we present illustrative results comparing the performance of various precoding and combing methods for massive MIMO systems with packet retransmissions. The channel model is realized assuming N cl = 5 scattering clusters with N ray = 2 rays per cluster. We assume ULAs with directional antennas at the transmitter and omni-directional antennas at the receiver. The antenna elements are critically spaced, i.e., d λ = 0.5. For each cluster, the mean azimuth AoD is assumed to be uniformly distributed over a 60 sector angle, i.e., 
$$\left (0^{\circ },60^{\circ }\right )$$
, whereas the mean azimuth AoA at the receiver is uniformly distributed over 
$$\left (0^{\circ },360^{\circ }\right )$$
. The azimuth AoA and AoD of each ray are Laplacian distributed with angle spread of 7.5. We assess the two proposed schemes of progressive hybrid precoding and combining (PHPC): (1) the columns of the RF precoder and combiner are chosen from the set of array response vectors, as denoted by PHPC-AR; (2) the columns of the RF precoder and combiner are chosen from those of the DFT matrices, as denoted by PHPC-DFT. The performance of PHPC is numerically evaluated in terms of achievable rates and MSE. In the former case, we present for comparison the optimal progressive digital precoding (OPDP) scheme of [7], where an ML receiver was employed. Hence, the rate performance of such a design can be treated as a benchmark for any other precoding and combining scheme. In the latter case, we plot for comparison the OPDP scheme of [9] which addressed the minimization of MSE. The sparse precoding and combining (SPC) approach of [1] serves as another baseline. We note that SPC [1] does not take into account packet retransmissions, and it cannot be straightforwardly extended to systems equipped with packet retransmission. This is because the premise of matrix approximation that the idea of SPC was built upon would no longer hold if the previous retransmission attempts were incorporated. Hence, we adapt SPC [1] to the case of ARQ by designing the sparse precoder and combiner for each ARQ round independently. In doing this, we may gain some insights into when the use of time diversity in the precoding/combining design would become advantageous. In view of the limited scattering in the environment, only a small number of data streams is assumed to be transmitted. All the results presented here are averaged over 1000 random realizations of the massive MIMO channel.

4.5.1 Small M

Figure 4.2 shows a performance comparison of different precoding and combining schemes with n t = n r = 3 for the case of M = 2 (i.e., the occurrence of one additional retransmission) in response to different array dimensions and different numbers of data streams. The following observations can be made. For the single-stream beamforming, i.e., N s = 1, all the schemes exhibit an almost indistinguishable performance. For an increased number of data streams, e.g., N s = 3, the performance of the PHPC-AR method is close to that of the OPDP scheme in the low SNR regime (less than − 20 dB), while suffering a slightly widening performance gap as the SNR increases. With N s = 3, the SPC scheme offers a much poorer performance even at a low SNR (see Fig. 4.2a). This can be explained by the fact that SPC [1] minimizes the distance between the hybrid precoder/combiner and the fully digital optimal counterpart in one attempt without considering waterfilling-based power loading/MMSE combining at baseband. By increasing the antenna array dimensions at the transmitter and receiver, the performance loss for all the precoding and combing schemes relative to the OPDP method is reduced, as shown in Fig. 4.2b. It is noted that for a larger antenna array dimension as in Fig. 4.2b, the DFT-based PHCP method is outperformed by SPC in the regime of SNR above − 20 dB. This can be attributed to the fact that PHPC-DFT only exploits the coarse-grained quantized AoDs/AoAs for RF precoding/combining while the RF precoder/combiner of SPC draws on the fine-grained information of AoDs/AoAs of all the scattering paths. In the regime of low received SNR due to (1) low beamforming gain (enabled by a not-so-large-dimensional antenna array in Fig. 4.2a and/or (2) low transmit power (SNR below − 20 dB in Fig. 4.2b), power loading at baseband has a noticeable effect on the achievable rate. In this case, SPC suffers a significant performance loss from suboptimal power allocation, and is thus outperformed by PHCP-DFT. However, as the received SNR increases (above − 20 dB) coupled with a substantial beamforming gain as in Fig. 4.2b, the benefit of optimal power loading tends to diminish, while the effectiveness of spatial separation between data streams in the beam domain becomes increasingly relevant to the rate performance. In this case, the performance loss caused by the quantized angular information used in the RF precoding/combining of PHCP-DFT cannot be compensated for by the diminishing performance gain from the optimal power loading at baseband. Therefore, SPC delivers a superior rate to PHCP-DFT.
../images/470489_1_En_4_Chapter/470489_1_En_4_Fig2_HTML.png
Fig. 4.2

Achievable rates of various ARQ precoding and combining schemes with n t = n r = 3, M = 2, and angle spread of 7.5°. (a) N t = 32, N r = 8. (b) N t = 128, N r = 32

4.5.2 Large M

Figure 4.3 shows a performance comparison with M = 6. With more rounds of signal retransmission, the performance degradation of the PHPC methods relative to OPDP lessens. Interestingly, by exploiting the increased time diversity, the rate performance of PHPC-AR becomes comparable to that of OPDP even for a smaller array dimension of N t = 32, N r = 8 while SPC still suffers a remarkable performance degradation as the hybrid precoding and combining are optimized independently of the previous transmission attempts. In this case, one has to rely on increased spatial diversity, as provided by an increased array size, to alleviate the performance loss. This is evidenced by comparing the performance of SPC in Fig. 4.3a with that in Fig. 4.3b.
../images/470489_1_En_4_Chapter/470489_1_En_4_Fig3_HTML.png
Fig. 4.3

Achievable rates of various ARQ precoding and combining schemes with n t = n r = 3, M = 6, and angle spread of 7.5°. (a) N t = 32, N r = 8. (b) N t = 64, N r = 16

4.5.3 Increasing Number of RF Chains

We would like to study how the performance gap between OPDP and the PHPC schemes behaves in response to an increased number of RF chains at the transmitter and receiver. Figure 4.4 shows a performance comparison of various methods where we increase the number of RF chains available at the transmitter and receiver from n t = n r = 3 to n t = n r = 5. For the simulation, we consider an antenna configuration of N t = 32, N r = 8 with N s = 3 data streams and M = 2 ARQ rounds. Not surprisingly, we see that the performance loss for all the methods relative to OPDP is reduced since a greater flexibility of generating hybrid precoding/combining solutions is enabled by an increased number of RF chains. In particular, PHPC-AR has a negligible performance degradation compared with the optimal OPDP scheme.
../images/470489_1_En_4_Chapter/470489_1_En_4_Fig4_HTML.png
Fig. 4.4

Achievable rates of various ARQ precoding and combining schemes with different number of RF chains at the transmitter and receiver for N t = 32, N r = 8, M = 2, N s = 3, and angle spread of 7.5°

4.5.4 Impact of Angle Spread

In Fig. 4.5, we illustrate a performance comparison of different precoding and combining methods for a varied degree of scattering typically found in a mmWave propagation environment. It is seen from Fig. 4.5a, b that the hybrid precoding/combining structure is more sensitive to increased angle spread, i.e., richer scattering, than the fully digital implementation. Intuitively, this can be attributed to the fact that the use of a limited number (n t = n r = 2) of array response/DFT beamforming vectors for the RF precoding and combining in the hybrid solutions results in an inevitable loss of channel power in the eigen-domain. Fortunately, the achievable rate of the proposed PHCP-AR remains fairly close to that of the optimal solution OPDP across the varied angle spread of interest. Furthermore, by comparing Fig. 4.5a, b, we see that PHCP-DFT outperforms SPC as the observed number of ARQ rounds increases from M = 2 to M = 6 for the array configuration of N t = 128, N r = 32. Although SPC benefits from the detailed information of AoDs/AoAs in the design of the RF precoder/combiner, it does not leverage the time diversity by ignoring the previous failed packet retransmissions. For a small number of ARQ rounds (M = 2), the advantage provided by the fine-grained AoA/AoD information outweighs the performance loss from being oblivious to the time diversity. However, as the number of ARQ rounds increases (M = 6), the benefit of incorporating the time diversity in the precoding/combining design outweighs the loss of angle information, which leads to the superior performance of PHPC-DFT to that of SPC.
../images/470489_1_En_4_Chapter/470489_1_En_4_Fig5_HTML.png
Fig. 4.5

Achievable rates versus azimuth angle spread at the transmitter and receiver for N s = 2, n t = n r = 2. (a) M = 2. (b) M = 6

4.5.5 Quantization of RF Precoder/Combiner

In the proposed approach of PHCP-AR discussed in Sects. 4.3 and 4.4, we consider choosing the columns of the RF precoder F RF,M and those of the RF combiner W RF,M during the Mth ARQ round from the set of array response vectors 
$$\left \{ {\mathbf {a}}_{\mathrm {t}}\left (\phi _{il}^{\mathrm {t}}\right )\right \} _{i,l}$$
and 
$$\left \{ {\mathbf {a}}_{\mathrm {r}}\left (\phi _{il}^{\mathrm {r}}\right )\right \} _{i,l}$$
, respectively. In practice, such knowledge of the exact AoDs 
$$\left \{ \phi _{il}^{\mathrm {t}}\right \} _{i,l}$$
and AoAs 
$$\left \{ \phi _{il}^{\mathrm {r}}\right \} _{i,l}$$
of all the scattering paths might not always be readily available. For example, some paths might not be spatially resolvable as a result of the finite dimension of the antenna array [10]. However, the AoD/AoA support can be inferred through estimating the mean AoD/AoA and angle spread. Suppose that the azimuth AoDs and AoAs lie within 
$$(\phi _{min}^{\mathrm {t}},\phi _{max}^{\mathrm {t}})$$
and 
$$(\phi _{min}^{\mathrm {r}},\phi _{max}^{\mathrm {r}})$$
, respectively. One potential approach to alleviating the channel estimation overhead is to employ uniform quantization of the azimuth angular space such that the array response vectors for F RF,M and W RF,M during the Mth ARQ round are generated from the set of angles

$$\displaystyle \begin{aligned} \begin{array}{rcl} S_{\phi}^{a} &\displaystyle = &\displaystyle \left\{ \phi_{min}^{a}+\frac{k\left(\phi_{max}^{a}-\phi_{min}^{a}\right)}{2^{N_{\phi}}-1},k=0,\ldots,2^{N_{\phi}}-1\right\} ,a\in\{\mathrm{t},\mathrm{r}\} \end{array} \end{aligned} $$
with N ϕ denoting the number of bits used for the quantization. In Fig. 4.6, we illustrate the impact of choosing the RF precoder/combiner from a quantized angular space at the transmitter/receiver on the achievable rate. The following observations can be made. For a smaller antenna configuration of N t = 32, N r = 8, 4-bit quantization is sufficient for the single-stream beamforming. For the same antenna configuration, increasing N s to three requires an additional bit for quantization to achieve a rate performance close to that of the perfect CSI case with complete angle information.
../images/470489_1_En_4_Chapter/470489_1_En_4_Fig6_HTML.png
Fig. 4.6

Achievable rates versus the number of quantization bits 
$$ \left (N_{\phi } \right )$$
for the azimuth AoD and AoA at the transmitter and receiver with n t = n r = 3, M = 2, and azimuth angle spread of 7.5°. (a) SNR = −15 dB. (b) SNR = 0 dB

For a larger antenna array configuration of N t = 128, N r = 32, even for single-stream beamforming, we would need 6 bits for quantization to deliver a comparable performance with the perfect CSI case. Since there is a trade-off between the number of quantization bits to achieve a better performance and the number of combinations to be searched to generate the RF precoder/combiner, we see that N ϕ = 5 can be a suitable choice.

4.5.6 MSE

In Sect. 4.4, we consider the minimization of MSE as the objective function for the design of the RF and baseband combiners for each ARQ round such that the fully digital linear MMSE combiner can be well approximated. In Fig. 4.7, we evaluate the error performance of various precoding and combining methods in terms of MSE. The OPDP curves in the MSE plots are obtained using the method proposed in [9]. As we can see from Fig. 4.7a, b, the PHPC method with both choices of RF precoders and combiners enjoys an error performance similar to that of OPDP. We also observe that the error performance of SPC is independent of the number of ARQ rounds, i.e., M, which is expected since the hybrid combiners of SPC were derived independently for each ARQ round.
../images/470489_1_En_4_Chapter/470489_1_En_4_Fig7_HTML.png
Fig. 4.7

MSE performance of various ARQ precoding and combining schemes with n t = n r = 3, and angle spread of 7.5°. (a) N s = 1. (b) N s = 3

4.5.7 Complexity

Both PHPC-AR and PHCP-DFT use an exhaustive search to find the desirable RF precoding and combining solutions. In the case of PHPC-AR, the RF precoding and combining involve evaluating 
$$\binom {N_{\mathrm {cl}}N_{\mathrm {ray}}}{n_{\mathrm {t}}}$$
and 
$$\binom {N_{\mathrm {cl}}N_{\mathrm {ray}}}{n_{\mathrm {r}}}$$
combinations from the set of array response vectors 
$$\mathcal {A}_{\mathrm {t}}$$
and 
$$\mathcal {A}_{\mathrm {r}}$$
, respectively. In the case of PHCP-DFT where the feasible sets of the RF precoder and combiner are constrained to the columns of N t-dimensional and N r-dimensional DFT matrices, respectively, the number of possibilities to be evaluated is 
$$\binom {N_{\mathrm {t}}}{n_{\mathrm {t}}}$$
for the RF precoding and 
$$\binom {N_{\mathrm {r}}}{n_{\mathrm {r}}}$$
for the RF combining. If the values of N cl, N ray, N t and N r are very large, finding the solution using PHPC-AR and PHCP-DFT in real-time may not be practical. On the contrary, the quantization approach examined in Sect. 4.5.5 requires the evaluation of 
$$\binom {2^{N_{\phi }}}{n_{\mathrm {t}}}$$
and 
$$\binom {2^{N_{\phi }}}{n_{\mathrm {r}}}$$
combinations for the RF optimization at the transmitter and receiver, respectively, which are independent of the number of antennas and the number of scattering paths in the propagation environment. Using PHCP-AR and PHCP-DFT as benchmarks, one can consider the use of reduced-complexity heuristic search algorithms such as the Tabu search [11] to generate the RF solutions in real time. Once the RF precoder and combiner are found, the closed-form baseband precoder and combiner can be computed with the number of flops on the order of 
$$\mathcal {O}\left (n_{\mathrm {t}}^{3}\right )$$
and 
$$\mathcal {O}\left (M^{3}N_{\mathrm {r}}n_{\mathrm {r}}^{2}\right )$$
, respectively. This is a significant decrease in the computational complexity compared with the optimal solution OPDP, which requires 
$$\mathcal {O}\left (n_{\mathrm {t}}^{3}\right )$$
flops in the derivation of the fully digital precoder and worse-case receiver complexity scaled exponentially with N r. For the case of SPC, it takes 
$$\mathcal {O}\left (n_{\mathrm {t}}^{2}N_{\mathrm {t}}N_{\mathrm {cl}}N_{\mathrm {ray}}\right )$$
flops to generate the hybrid precoding solution and 
$$\mathcal {O}\left (N_{\mathrm {cl}}N_{\mathrm {ray}}N_{\mathrm {r}}^{2}n_{\mathrm {r}}\right )$$
flops to generate the hybrid combining solution. For a more intuitive comparison, we list in Table 4.1 the number of real flops per ARQ round required by the dominant operations of the presented precoding and combining schemes. We focus on the case of N s = 3 data streams and M = 6 ARQ rounds of retransmission for both the small and large antenna configurations.
Table 4.1

Comparison of computational complexity (real flops)

 

Real flops (in million)

Antenna configuration

OPDP

PHPC-AR

PHPC-DFT

SPC

N t = 32, N r = 8

0.43

0.032

0.037

0.084

N t = 128, N r = 32

27.3

0.087

0.43

0.52

4.6 Summary

In this chapter, we considered progressive hybrid RF-baseband precoding and combining to increase the spectral efficiency by exploiting time diversity for massive MIMO with hybrid ARQ-enabled packet retransmissions. By assuming that the fully digital linear MMSE combiner can be perfectly reconstructed by the hybrid RF-baseband combiner, the development of hybrid precoding and combining solutions becomes decoupled. Toward deriving the hybrid precoder/combiner, we developed a two-step strategy for sequential joint RF-baseband optimization. Specifically, for each ARQ round, we chose the columns of the RF precoder/combiner either from the set of transmit/receive array response vectors or from the DFT-based codebooks. Conditioned on the RF precoder/combiner, we analytically derived the optimal baseband precoder/combiner. The optimal baseband precoder for the current retransmission was shown to consist of beamforming directions, which lie in the subspace spanned by the generalized eigenvectors of the effective channel and RF precoders, and power loading that depends on the precoding solutions from the previous and current retransmissions. To minimize the performance loss due to separate precoding/combining optimization, the hybrid combiner was formulated as an approximation of the linear digital combiner in terms of MSE. Illustrative results showed that the proposed progressive hybrid solutions with a limited number of RF chains provide performance improvement via exploiting the knowledge of previous ARQ retransmissions in comparison with the baseline that does not, and deliver a comparable performance with the optimal progressive digital counterpart.