I Introduction
Multiuser multipleinput multipleoutput (MUMIMO) refers to a system in which a base station (BS) exploits multiple antennas to simultaneously serve many terminals [2][4]. Nonorthogonal multiple access (NOMA) holds great promise in carrying massive connectivity in MUMIMO systems [5]. In [6]
, an artificial intelligence (AI) based cooperative spectrum sensing framework was studied for NOMA to improve the spectral efficiency. In recent years, MUMIMO has attracted significant interest especially for largescale antenna arrays, namely massive MIMO
[7][9]. In massive MIMO, smallscale fading is averaged out and the transmit power of each antenna can be aggressively scaled down, leading to significantly improved spectral and energy efficiencies [10], [11].To enhance the coverage of a MUMIMO, relay has been introduced for the scenarios where direct link between source and destination is weak due to heavy pathloss and shadowing. In [12]
, the capacity of a MUMIMO relay system was studied, while the degree of freedom of the system was analyzed in
[13]. Specific transmission designs for the MIMO relay was optimized in [14]. Further incorporating the idea of massive MIMO, studies [15] and [16] analyzed the performance of a massive MIMO relay system serving multiple users. Then in [17], the analysis was further explored for a multipair twoway amplify and forward (AF) relay massive MIMO system. For multihop communication systems using multiple frequency bands, an algorithm was proposed in [18] for the relay to choose the optimal modulation method and coding rate.Precoding is one of the key techniques for achieving the performance in MUMIMO downlink [19]. Nonlinear precoding such as dirty paper coding (DPC) is known to be capacityachieving [20]. However, it is rather complex to implement in practice even in a conventional smallscale MIMO setup. Alternatively, linear precoders such as zeroforcing (ZF) precoding can be adopted to asymptotically approach the theoretical benchmark performance of the optimal nonlinear precoding [21]. In the aforementioned works, conventional fully digital signal processing techniques are adopted where each antenna requires a dedicated radiofrequency (RF) chain. This is in general highcost especially for the massive MIMO with a large number of antennas. To reduce the hardware and power consumptions, we may resort to constraining the number of RF chains, resulting in a hybrid transceiver architecture [22], [23].
A hybrid precoding scheme, consisting of a digital precoder in baseband and an analog precoder equipped with phase shifters in RF [24], was proven to approach the benchmark performance achieved by conventional fully digital precoding techniques. In [25], a nearoptimal iterative hybrid precoding scheme was proposed based on a lowcost subarray structure. Then, a successive interference cancelation (SIC)based hybrid precoder was further studied in [26]
with reduced computational complexity. A deeplearningenabled hybrid precoder was proposed in
[27] for massive MIMO framework. In [28], spectral efficiency was characterized for a multipair relay network with a hybrid transceiver. This analysis was then extended in [29] for a multipair massive MIMO twoway relay network.In most of the existing studies on massive MIMO networks using hybrid transceivers, i.e., [24][29], the analog processing matrix was designed by extracting the phases of the corresponding singlehop channel. Few works has investigated the cascaded twohop relay channels where the hybrid analog processing matrix is designed according to the equivalent cascaded channel, which can be more practical in applications. In this paper, we study a massive MIMO uplink network assisted with a lowcomplexity analog relay [30][33]. Hybrid transceiver architecture is adopted at the BS with limited RF chains. We investigate the performance of the system, especially revealing the effect of limited RF chains on the achievable rate. The derived result quantitatively characterizes the tradeoff between performance and hardware cost. The main contributions of this paper are summarized as follows.

The phase of each element of the hybrid analog detecting matrix is elementwisely chosen as that of the cascade twohop channel matrix. This operation generates an equivalent random channel matrix that does not follow any typical multivariate distribution as we know. We derive new results on statistics of the phasealigned cascaded twohop Gaussian channel matrices, which is shown essential in conducting the performance analysis of the massive relay network with hybrid processing.

A tight bound for the achievable rate in massive MIMO relay uplink is obtained in closed form. Further for low signaltonoise ratios (SNRs) where energy efficiency matters, the ergodic rate of the th user is asymptotically expressed as where is the number of antennas at BS while and are equivalent SNRs at the relay and BS, respectively. Apparently, there is a decaying factor of on the ergodic rate compared to the fully digital scheme.

Power scaling laws are obtained to reveal the ability of simultaneously reducing the power consumption of the users and relay as tends to infinity. The asymptotic rate remains constant if the transmit powers of users and relay scale down by and (), respectively. Comparison with the fullRFchain structure verifies that the hybrid detection performs rather close to the fully digital ZF detection in massive MIMO.

The optimal power allocation (PA) problem for sum rate maximization of the system is rather intractable due to the nonlinear relationship between the sum rate and PA factors. We impose an auxiliary constraint to the original optimization problem, and transform it into an equivalent one which allows the optimal PA factors to be obtained in closed form through KarushKuhnTucker (KKT) analysis.
The rest of the paper is organized as follows. Section II introduces the system model. New preliminaries are derived in Section III to assist performance analysis. In Section IV, asymptotical achievable rate is derived and power saving scenarios are elaborated. Section V deals with the optimal PA problem for sum rate maximization. Numerical results and conclusions are presented in Section VI and VII, respectively.
Notations: and represent the Hermitian and transpose of a matrix, respectively. and Tr represent the inverse and trace of a square matrix, respectively. takes expectation.
denotes the identity matrix of size
while returns a diagonal matrix containing on the diagonal. denotes the th element of a matrix. andtake the norm of a complex number and a vector, respectively.
returns the phase of a complex value.is the uniform distribution between
and .Ii System Model
In this paper, we investigate a multiuser massive MIMO relaying uplink where active users transmit signals to an antenna BS with the help of an antenna analog AF relay, as illustrated in Fig. 1. Assume that the BS is equipped with a massive antenna array but driven by a far smaller number, , of RF chains.
The channel between users and the relay can be represented as , where is the channel between the th user and the relay and denotes the path loss. The channel between the relay and BS is denoted by whose entries are independent and identically distributed (i.i.d.) as . Note that the number of users, , can be generally an arbitrary value while is fixed in a certain application. If , the system usually schedules a subset with users for simultaneous transmission because the maximum number of independent data streams that can be supported is in theory [34]. If , a direct but effective way is to choose only of the chains for serving the users. Upon these considerations, we therefore conduct the following discussion and analysis by considering without loss of generality. Then in the following, we describe transmission model in Fig. 1 in three steps.
1) users simultaneously transmit signals to the relay. The receive signal at the relay is
(1) 
where is the transmit power of each user, is the transmit signal vector satisfying , and is the Gaussian noise vector satisfying .
2) The relay amplifies and forwards the receive signals to BS. The receive signal at BS is
(2) 
where is an amplification factor to guarantee the power constrain at the relay, and is the Gaussian noise vector satisfying . Denoting by the transmit power at the relay, the amplification factor is
(3) 
3) The hybrid detection at BS can be composed of an analog RF detector, , and a subsequent digital baseband detector, . Specifically, exploits phase shifts to adjust the phases of receive signals while makes adjustments to both signal amplitudes and phases. After the hybrid detection, we have
(4) 
Note that there have already existed amounts of design methods for the hybrid processing matrices. We follow a tractable and effective design philosophy, like in [24], where the analog detector is designed based on the cascade channel from users upto the BS, i.e.,
as a single entity, for several considerations. Firstly, estimating
and is resource consuming and challenging especially at the relay. The channel, , is even harder to obtain since it is a matrix with a very large number of elements in the massive MIMO. Secondly, the setup with an analog relay as in [30][33] is more implementable, especially for the massive MIMO relay network. We do not require individual estimates of and separately, or their statistics. By treating the cascade channel matrix as a single entity, we can apply an uplink channel training for equivalent channel estimation. With this design, we choose by extracting the phases of , i.e.,(5) 
where . While for digital detection, is designed as a commonly used ZF detector according to the equivalent channel . It follows
(6) 
By substituting (1), (2) and (6) into (4), we can rewrite the detected signal as
(7) 
From (7), without loss of generality, the received signaltointerferenceplusnoise ratio (SINR) of user is
(8) 
Then, the ergodic rate of the th user can be expressed as
(9) 
Iii New Preliminaries
Due to the use of hybrid detection, it is necessary to derive the properties of and in (7) for analyzing . The main difficulty relies on the dependence of the phases of and according to the design of in (5). The following theorem and two propositions are new essential results which facilitate our following analysis.
Theorem 1.
For independent random vectors , and , where and are identically distributed as , we have
(10) 
where and .
Proof.
The major difficulty comes from the dependence between and . Considering and , the expectation of in (10) is taken jointly over different , and , which implies
(11) 
Firstly we need to calculate the expectation of for any given averaging over and . However, according to the definition of and , is also a nonlinear function of and , which makes the expectation in (11) over and hard to evaluate in general. Fortunately in the following, we are able to prove that the conditional expectation in (11) given is a value irrelevant to . It implies that we can calculate the expectation conditioned on any realization of , e.g., where is an unit vector whose first element is 1 and others are 0s. By substituting into (11) and dividing the calculation of the expectation into two cases, i.e., and , we then obtain the analytical result of the expectation.
1) Now, we first prove that the above expectation of for any given returns a value that is irrelevant to . Let us focus on the inner conditional expectation taken over and in (11
). Firstly we give the singular value decomposition (SVD) of
as(12) 
where is the singular value of and is an unitary matrix. For a certain , there can be many possible ’s. Specifically, we can construct a by
(13) 
where are a spanned orthogonal basis which makes unitary, and . Using (12), the expectation of conditioned on follows
(14) 
where applies (12), follows from the fact that and are irrelevant to the scaling factor , comes from the fact that the left multiplication of unitary does not change the value of , i.e., for unitary . Then, let us define a new vector as and we have because of unitary and the condition in (b), i.e., . Now we have
(15) 
which implies . These above steps then establish the equality from (b) to (c). In step (d), we simply use definitions of vectors that and . According to (15), we have and . Finally, we obtain (e) due to the fact that and have the same distribution with and , respectively [35, Th. 3.7.10]. From (III), it shows that the conditional expectation for any is irrelevant to and the expectation equals to that with . Substituting (III) in (11), we have
(16) 
2) In the sequel, we deduce the analytical expression of the expectation in (16) by separating calculations for the two cases, i.e., and .
(a) For any , we have
(17) 
where and are the phases of the first element of and , respectively, and applies the independence of and . Concerning the term , we have
(18) 
where represents the first entry of and comes from the fact that the magnitude of follows the Rayleigh distribution with mean , since is distributed as . For the th element of , we have
(19) 
where represents the th element of and is obtained by using the independence of , and . As both and follow , their phases follow . Hence = = 0 and is obtained.
To conclude, can be expressed from (18) and (19) as
(20) 
Similarly, we have
(21) 
Substituting (20) and (21) into (17), the expectation of for any is
(22) 
(b) For any , we have
(23) 
where is obtained by using the property of Wishart matrix [38, Th. 3.2.14] and is derived by using the property of Gamma function that for .
Remark: Theorem 1 does not show the favorable propagation property of massive MIMO as in some existing cases like [11]. In Theorem 1, the dependence of and with and
makes the common way of applying the law of large numbers (LLN) in massive MIMO not applicable. This dependence among the random variables results in the value of
in general nonzero for which differs from the popular observation in traditional massive MIMO.Note that the result in Theorem 1 enables us to further obtain the exact expectation result in closed form in Proposition 1, which will be shown useful later in the performance analysis.
Proposition 1.
Assume two independent channel matrices where and with entries following i.i.d. . Let be the, namely phasealigning, matrix where . The expectation of the diagonal element of equals
(25) 
where
is the variance of the Gaussian entries in
.Proof.
Since the elements of are extracted from the phases of , is obviously dependent of which makes the expectation difficult to evaluate directly. To tackle this difficulty, we derive the value of by first expanding the term in summation as follows and then elementwisely evaluate the terms with the help of Theorem 1. From the definition of , we can equivalently write the th entry of as
(26) 
where is the th row of and is the th column of . Now we have
(27) 
where and . Then,
(28) 
where applies Theorem 1. This completes the proof. ∎
Proposition 2.
Let , and be defined as in the above Proposition 1. Then, we have and
(29) 
where the almost sure convergence, , corresponds to large tending to infinity.
Proof.
Since , we have and
(30) 
where is obtained by defining and as the amplitude and phase of , respectively. Since and are independent complex Gaussian random variables (RVs), from [36, eq. (17)], we obtain that the phase of , i.e., , follows the distribution and is independent of . For , we assume its phase .
Now the uniform distribution of can be proved by showing that any two realizations of
have the same probability of occurrence. From (
30), it show that a given set of values of , say , yields a realization of some , say . Meanwhile for an arbitrary fixed value , a given set of values of yields the realization of . Since is uniformly distributed, it is directly known that the probability of occurrence of is equal to that of . It implies that the occurrence of is the same as that of for any fixed value of .More specifically, for any combination of and , generating a certain phase as in (30), we can add to each . It yeilds
(31) 
In this way, we get another realization of with phase . Due to the uniform distribution of , one combination of and the corresponding combination of share the same probability. Hence, all combinations of for any have the same probability. Then, all for any share the same probability. For any other combinations of and , we can arrive at the same conclusion. We thus safely obtain that , i.e., , is a uniform RV. Equivalently, we arrive at
(32) 
which is the first part of Proposition 2.
Subsequently we consider the asymptotic behavior of , whose th diagonal element is
(33) 
For the th nondiagonal element, according to the LLN, we have
(34) 
where utilizes the independence of and conditioned on any given , and the last equality uses (32). By combining (33) and (34), we complete the proof. ∎
Iv Achievable Rate Analysis
In this section, asymptotic user rate is derived under the assumption of large antenna arrays. Power scaling laws are obtained to reveal the tradeoff between power consumption and hardware cost, while keeping a constant user rate. For notational brevity, we define .
Iva Asymptotic Rate Analysis
From (9), the ergodic achievable sum rate is obtained as
(35) 
Then we focus on characterizing . It appears difficult to evaluate the exact value of even though we have obtained the preliminary results with respect to the complicated and coupling random matrices, like , in (8). Here we resort to characterizing a tight performance bound to under the massive MIMO setup, as given in the following theorem.
Theorem 2.
Under the assumption of large antenna arrays, a lower bound for the ergodic user rate is
(36) 
Proof.
Note that the rate bound in (36) is tight for the massive MIMO setup. It can alternatively be regarded as an accurate approximation of the exact rate, , as proved in [22, Lemma 1]. Moreover, the previous analysis in [12] showed that the effects of MIMO can still be reflected even though an analog AF relay with scalar is utilized. This is also evidenced from our analytical results, e.g., through (36) in the massive MIMO relay network. It indicates that the ergodic sum rate logarithmically increases with respect to the number of antennas at the relay, i.e., , which implies an array gain of .
Let and denote the equivalent SNRs at the relay and BS, respectively. We can rewrite (36) equivalently as
(39) 
where
(40) 
where in (40) is the normalized largescale fading from user to the relay.
From (39), it is observed that logarithmically increases with large . The term in (39) is the product of the equivalent SNRs at the relay and BS. The coefficients before , and , respectively represent the effect of limited RF chains and array gains on the achievable rate.
Moreover, from (40), is associated with . It evaluates the influence of the equivalent SNR at BS on the asymptotic rate. The factor is caused by the hybrid detection and it increases with causing the rate degradation. is associated with . It represents the impact of the equivalent SNR at relay on the rate. The coefficient comes from the amplification factor of the relay and it increases with which also contributes to the rate degradation of each data stream.
More specifically, we then investigate the impact of different equivalent SNRs at the relay and BS on the achievable rate. The following three typical cases are discussed.
1) Case 1: Low SNR Analysis. For and , in (39) can be expressed as follows:
(41) 
In the lowSNR scheme, the achievable rate is a function of a scaled product of equivalent SNRs at relay and BS. The SINR of each user is proportional to . The scaling factor equals where comes from the array gain and is due to the effect of limited RF chains.
2) Case 2: High SNR Analysis. For and , in (39) can approximately be given by
(42) 
In the highSNR scheme, the achievable rate is a nonlinear function of and due to the relatively complex relationship between SINR and and under the relay network with limited RF chains. Assuming the same path loss from users to relay, we have .
3) Case 3: Intuitively, when the SNR at BS is far higher than that at relay, the relay system degenerates into a singlehop one. Specifically for and , we have and . in (39) is approximately given by
(43) 
From (43), is rarely affected by . The achievable rate only depends on the channel parameter from users to relay. Compared to the achievable rate of the singlehop system using pure digital detection as studied in [11], the hybrid processing introduces an SINR reduction by a multiplier factor . As , the ergodic rate achieved by hybrid detection approaches to that achieved by fully digital detection.
4) Case 4: On the other hand, when the SNR at BS is far lower than that at relay, the relay system also degenerates into a singlehop one. Specifically for and , we have and . in (39) is approximately given by
(44) 
IvB Power Scaling Law
In this section, we assume for all , and normalize . We focus on a nondecreasing achievable rate when the transmit power of users and/or relay is reduced as the number of antennas increases, i.e., and for fixed
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