Degrees of Freedom of Full-Duplex Cellular Networks with Reconfigurable Antennas at Base Station

# Degrees of Freedom of Full-Duplex Cellular Networks with Reconfigurable Antennas at Base Station

Minho Yang,  Sang-Woon Jeon,  and Dong Ku Kim,  M. Yang and D. K. Kim are with the School of Electrical and Electronic Engineering, Yonsei University, Seoul, South Korea (e-mail: {navigations, dkkim}@yonsei.ac.kr).S.-W. Jeon is with the Department of Information and Communication Engineering, Andong National University, Andong, South Korea (e-mail: swjeon@anu.ac.kr).
###### Abstract

Full-duplex (FD) cellular networks are considered in which a FD base station (BS) simultaneously supports a set of half-duplex (HD) downlink (DL) users and a set of HD uplink (UL) users. The transmitter and the receiver of the BS are equipped with reconfigurable antennas, each of which can choose its transmit or receive mode from several preset modes. Under the no self-interference assumption arisen from FD operation at the BS, the sum degrees of freedom (DoF) of FD cellular networks is investigated for both no channel state information at the transmit side (CSIT) and partial CSIT. In particular, the sum DoF is completely characterized for no CSIT model and an achievable sum DoF is established for the partial CSIT model, which improves the sum DoF of the conventional HD cellular networks. For both no CSIT and partial CSIT models, the results show that the FD BS with reconfigurable antennas can double the sum DoF even in the presence of user-to-user interference as both the numbers of DL and UL users and preset modes increase. It is further demonstrated that such DoF improvement indeed yields the sum rate improvement at the finite and operational signal-to-noise ratio regime.

Blind interference alignment, degrees of freedom (DoF), full-duplex (FD), interference management, reconfigurable antennas.

## I Introduction

To meet soaring wireless demand with limited spectrum, there has been considerable researches for boosting utilization of wireless resources. Recently, full-duplex (FD) radios have emerged as a potential way of improving spectral efficiency by enabling simultaneous transmission and reception at the same time with the same wireless spectrum. Because of such simultaneous transmission and reception, FD has a potential to double the spectral efficiency compared to the conventional half-duplex (HD) mode such as frequency division duplex (FDD) and time division duplex (TDD). Nonetheless, FD involves the practical issue of suppressing high-powered self-interference arisen from simultaneous transmission and reception [1, 2, 3, 4]. In recent researches, there has been remarkable progress on analog and digital domain self-interference cancellation (SIC) techniques, showing that the point-to-point bidirectional FD system can achieve nearly twice higher throughput than the corresponding HD system, which demonstrates the possibility of implementing FD radios in practice [2, 3, 4].

Unlike the point-to-point bidirectional FD system, we cannot simply argue that the network throughput can be doubled for cellular systems even under the ideal assumption that self-interference is perfectly suppressed. In particular, consider the cellular system in Figure 1 in which a FD base station (BS) simultaneously supports a set of HD downlink (DL) users and a set of HD uplink (UL) users, one of the feasible scenarios of FD radios considering compatibility with legacy HD users in the current communication systems. For such case, a new source of interference from UL users to DL users appears, which does not exist in HD cellular systems where DL and UL traffic is orthogonalized by frequency or time domain. The impact of such user-to-user interference in FD cellular systems has been widely discussed in several researches [5, 6, 7, 8, 9]. They showed that if interference from UL users to DL users is not properly mitigated, the network throughput may be degraded even though self-interference is perfectly suppressed. Therefore, efficient interference management from UL users to DL users is a key challenge to boosting the network throughput of cellular systems by adapting FD operation at BSs [5, 6, 7, 8, 9].

In order to understand fundamental limits of FD radios in cellular networks, there have been several recent researches on characterizing the degrees of freedom (DoF) of FD cellular networks [10, 11, 12, 13]. In particular, a single-cell FD cellular network has been studied in [12, 13], in which a FD BS with perfect self-interference suppression supports both HD DL and UL users as seen in Fig. 1. In [12], the authors characterized the sum DoF of the single-cell FD cellular network assuming that global channel state information (CSI) is available at the BS, i.e., full CSI at the transmit side (CSIT). They showed that FD operation at the BS can double the sum DoF compared to HD operation when both the numbers of DL and UL users become large even in the presence of user-to-user interference, concurrently reported in [13]. However, asymptotic interference alignment (IA) techniques proposed in [12, 13] require perfect CSIT and an arbitrarily large number of time extension to achieve the optimal sum DoF, which is quite challenging in practice due to feedback delay, system overhead and complexity, and etc [14, 15, 16, 17, 18, 19, 20].

Motivated by such advantages of FD radios and reconfigurable antennas, we consider FD cellular networks in which a FD BS equipped with reconfigurable transmit and receive antennas supports HD DL and UL users simultaneously in the same frequency spectrum. For comprehensive understanding on the impact of FD radios and CSI conditions in the context of IA or blind IA using reconfigurable antennas, we consider two different CSI models: For no CSIT case, both the BS and each UL user do not know their CSIT; For the partial CSIT case, the BS only knows its CSIT. For both models, we assume that CSI at the receive side (CSIR) is available. Similar to the previous full CSIT models in [10, 11, 12, 13], the primary aim is to characterize whether the sum DoF can be doubled or not with partial or no CSIT by FD operation at the BS equipped with reconfigurable antennas. The main contributions of this paper are as follows:

• For no CSIT model, we completely characterize the sum DoF of FD cellular networks. We propose a novel blind IA technique, which perfectly aligns user-to-user interference at each DL user while preserving intended signal space at the BS, and establish the converse showing the optimality of the proposed scheme in terms of the sum DoF. The result shows that the sum DoF is asymptotically doubled if both the numbers of UL users and preset modes at the receiver of the BS increase, which is the first result demonstrating the benefit of FD radios on cellular networks under no CSIT.

• For the partial CSIT model, we establish an achievable lower bound on the sum DoF of FD cellular networks, which characterizes the sum DoF for a broad class of network topologies. We propose a novel blind IA technique combined with zero-forcing beamforming based on partial CSIT, which partially aligns user-to-user interference at each DL user while preserving intended signal space at the BS. The result shows that the sum DoF is doubled if there exist two DL and two UL users and two preset modes at the transmitter and the receiver of the BS. For the single-antenna case, our result for the partial CSIT model extends the previous achievability result in [13] to a general antenna configuration assuming different numbers of preset modes at the transmitter and receiver of the BS.

• We further demonstrate that such DoF improvement indeed yields the sum rate improvement at the finite and operational signal-to-noise ratio (SNR) regime, which presents the benefit of blind IA using reconfigurable antennas compared with the previous works [10, 11, 12, 13].

The rest of this paper is organized as follows. In Section II, we introduce the network model and DoF metric considered throughout the paper. In Section III, we state the main results of this paper, the sum DoF of FD cellular networks, and remark several observations possibly deduced from the main results. We present achievability and converse proofs of the main results in Section IV and Section V respectively. We finally conclude in Section VII.

## Ii Problem Formulation

In this section, we introduce FD cellular networks consisting of a FD BS and HD DL and HD UL users and then formally define the sum DoF metric, which will be analyzed throughout the paper.

### Ii-a Notation

For integer numbers and , and denote the quotient and the remainder respectively when dividing by . For integer numbers and , when and when . For matrices and , is the Kronecker product of and . For a matrix , denote the Frobenius norm, transpose, and conjugate transpose of by , , and , respectively. For a set of matrices , denotes the block-diagonal matrix consisting of as the th diagonal block. For natural numbers and , , , and denote the identity matrix, the all-one matrix, and the all-zero matrix respectively. Let be the th column vector of where .

### Ii-B Full-Duplex Cellular Networks

We consider a FD cellular network in which a FD BS simultaneously supports HD DL users and HD UL users. Both the transmitter and receiver of the BS are equipped with reconfigurable antennas. In particular, the transmitter of the BS is equipped with a reconfigurable antenna capable of switching among preset modes at each time and the receiver of the BS is equipped with a reconfigurable antenna capable of switching among preset modes at each time. Notice that (or ) corresponds to the case where the transmitter (or the receiver) of the BS is equipped with a conventional antenna. Each DL and UL user is equipped with a conventional antenna. In this paper, we assume that self-interference within the BS due to FD operation is perfectly suppressed. We will discuss about the impact of imperfect self-interference suppression in Section VI.

We assume block fading in this paper, i.e., each channel coefficient remains the same in a consecutive time slots of coherence time and is drawn independently in the next consecutive time slots of coherence time. The length of the coherence time is assumed to be sufficiently large. Let be the channel from the transmitter of the BS to the th DL user when the BS selects its transmit mode as the th preset mode, where and . Similarly, let be the channel from the th UL user to the receiver of the BS when the BS selects its receive mode as the th preset mode, where and . Let be the channel from the th UL user to the th DL user. All channel coefficients are assumed to be independent and identically distributed (i.i.d.) drawn from a continuous distribution.

Denote the transmit mode and the receive mode of the BS at time by and , respectively. Then the received signal of the th DL user at time is given by

 ydi(t) =hi(α(t))xd(t)+Ku∑j=1gijxuj(t)+zdi(t) (1)

for and the received signal of the BS at time is given by

 yu(t) =Ku∑j=1fj(β(t))xuj(t)+zu(t) (2)

where is the transmit signal of the BS at time , is the transmit signal of the th UL user at time , is the additive noise of the th DL user at time , and is the additive noise of the BS at time . The additive noises are assumed to be i.i.d. drawn from and independent over time. The BS and each UL user should satisfy the average power constraint , i.e., and for all .

For notational convenience, from (1) and (2), we define the length- time-extended input–output relation as

 ydi =Hi(¯α)xd+Ku∑j=1gijxuj+zdi, yu =Ku∑j=1Fj(¯β)xuj+zu (3)

where

 ¯α =[α(1),⋯,α(n)]T, ¯β=[β(1),⋯,β(n)]T, Hi(¯α) =diag(hi(α(1)),⋯,hi(α(n))), Fj(¯β) =diag(fj(β(1)),⋯,fj(β(n))), ydi =[ydi(1),⋯,ydi(n)]T, yu=[yu(1),⋯,yu(n)]T, xd =[xd(1),⋯,xd(n)]T, xui=[xui(1),⋯,xui(n)]T, zdi =[zdi(1),⋯,zdi(n)]T, zu=[zu(1),⋯,zu(n)]T.

For comprehensive understanding on the DoF improvement achievable by reconfigurable antennas at the FD BS, we consider the following two different scenarios for CSI assumption:

• No CSIT model (CSIT is not available):
The BS knows its receive side CSI, ; The th DL user knows its receive side CSI, ; The th UL user does not know any CSI.

• Partial CSIT model (CSIT is only available at the BS):
The BS knows both its transmit and receive side CSI, i.e., and ; The th DL user knows its receive side CSI, ; The th UL user does not know any CSI.

###### Remark 1.

For the considered network, CSIR might not immediately lead to CSIT even if channel reciprocity holds because a FD BS supports HD DL users and HD UL users. That is, a set of DL users and a set of UL users are fixed and separate. Furthermore, the validity of such channel reciprocity will depend on the relative difference between channel coherence time and time difference between UL and DL frames allocated to an user. If the time difference between UL and DL frames allocated to an user is longer than the coherence time, then additional channel feedback from the receive side to the transmit side is required to attain CSIT [32]. Moreover, the RF front-ends of transmit and receive antennas are different and have their own delays and gains, which necessarily cause reciprocity error and impose reciprocity calibration [33]. For the above reasons, we consider both no CSIT and partial CSIT models in this paper.

###### Remark 2.

Notice that, for both no CSIT and partial CSIT models in this paper, each DL user does not require CSI from its UL users. Therefore, CSIR is available by using the conventional UL channel training (for CSI from UL users to the BS) and DL channel training (for CSI from the BS to DL users) without additional channel training from UL to DL users.

### Ii-C Degrees of Freedom

For the network model stated in Section II-B, we define a set of length- block codes and its achievable DoF. Let and be the th DL message and the th UL message respectively, where and . For no CSIT model, a code consists of the following set of encoding and decoding functions:

• Encoding: For , the encoding function of the BS at time is given by

 (xd(t),α(t))=ϕt(Wd1,⋯,WdKd,yu(1),⋯,yu(t−1),{fj(k)}j∈[1:Ku],k∈[1:Mu]).

For , the encoding function of the th UL user () at time is

 xuj(t)=φjt(Wuj).
• Decoding: Upon receiving (i.e., to ), the decoding function of the BS is

 ^Wuj=χj(yu,Wd1,⋯,WdKd,{fj(k)}j∈[1:Ku],k∈[1:Mu]) for j∈[1:Ku].

Upon receiving , the decoding function of the th DL user () is given by

 ^Wdi=ψi(ydi,{hi(k)}k∈[1:Md]).

If there exists a sequence of codes such that and as increases for all and , a rate tuple is said to be achievable. Then the achievable DoF tuple is given by

 (dd1,⋯,ddKd,du1,⋯,duKu)=limP→∞(Rd1logP,⋯,RdKdlogP,Ru1logP,⋯,RuKulogP).

Finally, the sum DoF for no CSIT model is defined as

 dΣ,noCSIT=max(dd1,⋯,ddKd,du1,⋯,duKu)∈D{Kd∑i=1ddi+Ku∑j=1duj}

where denotes the achievable DoF region.

For the partial CSIT model, the encoding and decoding functions of the BS are replaced as

 (xd(t),α(t)) =ϕt(Wd1,⋯,WdKd,yu(1),⋯,yu(t−1),{hi(k)}i∈[1:Kd],k∈[1:Md],{fj(k)}j∈[1:Ku],k∈[1:Mu]), ^Wuj=χj(yu,Wd1,⋯,WdKd,{hi(k)}i∈[1:Kd],k∈[1:Md],{fj(k)}j∈[1:Ku],k∈[1:Mu]),

respectively. Then the sum DoF can be defined in the same manner. Let denote the sum DoF for the partial CSIT model.

For the rest of this paper, we characterize the sum DoF of the FD cellular network under both no CSIT model and the partial CSIT model.

## Iii Main Results

In this section, we state our main results, the sum DoF of the FD cellular network for both no CSIT and partial CSIT models, and provide a numerical example for demonstrating the benefit of FD operation and reconfigurable antennas at the BS.

For no CSIT model, we completely characterize the sum DoF of the FD cellular network in the following theorem.

###### Theorem 1.

For the FD cellular network with no CSIT,

 dΣ,noCSIT =min{max(Kd,Ku),max(1+min(Kd,1)(Lu−1)Lu,1)} (4)

where .

###### Proof:

We refer achievability proof to Section IV-A and converse proof to Section V. ∎

###### Remark 3.

From Theorem 1, is independent of the parameters and if and . That is, for no CSIT case, equipping a reconfigurable antenna at the transmitter of the BS cannot increase the sum DoF and similarly a single DL user is enough to achieve the optimal sum DoF. More importantly, is asymptotically doubled if both and increase. Therefore, for no CSIT case, arbitrarily large numbers of UL users and preset modes at the receiver of the BS are required to double the sum DoF by FD operation at the BS.

For the partial CSIT model, we establish an upper and achievable lower bounds on the sum DoF of the FD cellular network in the following theorem.

###### Theorem 2.

For the FD cellular network with partial CSIT,

 dΣ,pCSIT≤min{2,max(Kd,Ku),max(1+Ku(Kd−1)Kd,1+Kd(Ku−1)Ku)} (5)

and

 dΣ,pCSIT≥min{2,max(Kd,Ku),max(1+Lu(Ld−1)Ld,1+Ld(Lu−1)Lu)} (6)

where and .

###### Proof:

We refer to the converse in [12, Theorem 1] for the proof of the upper bound in (5). In particular, [12] considers the FD BS equipped with conventional multiple transmit and receive antennas (instead of reconfigurable antennas) and assumes that full CSI is available at the BS and each user. The upper bound in (5) is attained from [12, Theorem 1] by assuming a single transmit and receive antenna at the BS. We can easily see that the converse argument in [12, Theorem 1] is applicable to the reconfigurable antenna model in Fig. 2 for the full CSIT case. Hence (5) can be an upper bound on . We refer to Section IV-B for the proof of the achievable lower bound in (6). ∎

###### Corollary 1.

For the FD cellular network with partial CSIT,

 dΣ,pCSIT=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩2if Kd,Ku,Md,Mu≥2,1+Ku−1Kuif Kd=1,Mu≥Ku≥1,1+Kd−1Kdif Ku=1,Md≥Kd≥1. (7)
###### Proof.

By comparing the upper and lower bounds on in Theorem 2, (7) can be straightforwardly obtained. ∎

For the single-antenna case, Theorem 2 and Corollary 1 extend the previous achievability result for the partial CSIT model in [13] to a general antenna configuration assuming different numbers of preset modes at the transmitter and receiver of the BS.

###### Remark 4.

From Theorem 2 and Corollary 1, is asymptotically doubled if both and increase when or both and increase when . Hence, similar to no CSIT case, arbitrarily large numbers of users and preset modes are required to double the sum DoF by FD operation only at the DL or UL side. On the other hand, is doubled if . That is, only two DL and UL users and the FD BS equipped with reconfigurable antennas having two preset modes are enough to double the sum DoF if the BS can attain its downlink CSI. Lastly, unlike no CSIT case in which reconfigurable antennas are only beneficial at the receiver of the BS, reconfigurable antennas are equally beneficial at the transmitter and receiver of the BS for the partial CSIT case.

In summary, from Theorems 1 and 2, the sum DoF is doubled even in the presence of user-to-user interference by FD operation at the BS. Furthermore, reconfigurable antennas can effectively improve the sum DoF under both partial and no CSIT cases. The following example plots the sum DoFs in Theorems 1 and 2 for the symmetric case.

###### Example 1.

For comparison, consider the symmetric case where and . Then, from Theorem 1 and 2,

 dΣ,noCSIT=2−1min(K,M) (8)

and

 min(K,M,2)≤dΣ,pCSIT≤min(K,2). (9)

Fig. 3 plots (8) and (9) with respect to . Obviously, if the BS operates as the conventional HD operation, i.e., serving either DL users or UL users, the sum DoF is limited by one. From (8) and the lower bound in (9), the sum DoF is still one if the FD BS is equipped with conventional non-reconfigurable antennas, i.e., . For the partial CSIT case, is enough to double the sum DoF. On the other hand, arbitrarily large and are required to double the sum DoF in the case of no CSIT.

In Section VI, we further demonstrate that the above sum DoF improvement achievable by FD operation and reconfigurable antennas at the BS yields the sum rate at the finite and operational SNR regime, which presents the benefit of blind IA using reconfigurable antennas compared with the previous works [10, 11, 12, 13].

## Iv Achievability

In this section, we establish the achievability in Theorems 1 and 2 and then present the achievable sum rates of the proposed schemes at the finite SNR regime.

Recall that and . When or , the right-hand sides of (4) and (6) in Theorems 1 and 2 are expressed as

 min{max(Kd,Ku),1}.

In this case, the sum DoF is trivially achievable by single-user transmission (supporting a DL user if and a UL user if ). Thus, we now focus on the achievability proof of Theorems 1 and 2 when .

Let us define the -point inverse discrete Fourier transform (IDFT) matrix as , given by

 Ωn=1√n⎡⎢ ⎢ ⎢ ⎢ ⎢⎣11⋯11ω⋯ωn−1⋮⋮⋱⋮1ωn−1⋯ω(n−1)(n−1)⎤⎥ ⎥ ⎥ ⎥ ⎥⎦

where [34]. In the followng, the IDFT matrix will be used for transmit precoding matrices to exploit the following properties of the IDFT matrix: 1) is an orthonormal matrix, i.e.,

 ΩHnΩn=In; (10)

2) Every submatrix of is of full-rank [35]. In particular, the above properties will be used to prove Lemma 1.

### Iv-a Achievability for Theorem 1 when Kd,Ku≥1

When , the right-hand side of (4) is given by

 2−1Lu.

In the following, we establish the achievability of Theorem 1, showing that the sum DoF of is achievable for no CSIT model. In particular, the BS sends information symbols to only the first DL user and UL users send a single information symbol each to the BS during time slots.

Let be the information symbol vector for the first DL user satisfying that and be the information symbol for the th UL user, , satisfying that . These information symbols will be delivered by symbol extension, i.e., beamforming over time slots. In particular, let be the submatrix consisting of the first through ()th column vectors of and be the th column of . That is, . The BS and the th UL user set their length- time-extended transmit signal vectors as

 xd =W1sd1, xuj=w2suj for j∈[1:Ku], (11)

each of which satisfies the average power constraint , i.e., and for . Here, is used as the transmit precoding matrix for sending and is used as the transmit precoding vector for sending , which is the same for all . During signal transmission, the BS fixes its transmit mode, i.e., for all . During signal reception, on the other hand, the BS sets its receive mode differently at each time, i.e., for all . Denote the above transmit mode vector and receive mode vector by and , respectively.

Then, from (II-B) and (11), the length- time-extended input–output relation is given by

 yd1 =h1(1)W1sd1+w2Ku∑j=1g1jsuj+zd1, (12) yu =Rsu+zu (13)

where and . Here, (13) holds from the fact that .

For decoding its DL message, the first DL user multiplies to , which is represented as

 WH1yd1=h1(1)sd1+WH1zd1 (14)

where the equality holds from (10). Then, the first DL user estimates its information symbols based on (14). Hence, the achievable DoF of the first DL user is

 dd1=1−1Lu.

Now consider decoding of UL messages at the BS. The BS estimates its information symbols based on (13). From the definition of , can be rewritten as

 R=diag(w21,⋯,w2Lu)[F1(¯β1),⋯,FKu(¯β1)]

where for is the th element of and thus almost surely. Therefore, from (13), the achievable sum DoF of the UL users is given by

 Ku∑j=1duj=rank(R)Lu=1.

Consequently, the sum DoF of is achievable for no CSIT model, which completes the achievability proof of Theorem 1.

### Iv-B Achievability for Theorem 2 when Kd,Ku≥1

In this section, we show the achievability proof of Theorem 2 when . For better understanding, we first illustrate the proposed scheme when and then provide the achievability proof for the general case.

#### Iv-B1 Example case

Consider the FD cellular network defined in Section II and assume that . We now show that the transmitter of the BS sends two information symbols to each DL user and each UL user sends two information symbols to the receiver of the BS for four time slots (). As a result, the achievable sum DoF of the proposed scheme is given by two. For intuitive explanation, we skip the power constraint issue and some proof steps in this example case, which will be given in the next subsection.

Let , be the information vectors sent to the first DL user and the second DL user and let , be the information vectors sent by the first UL user and the second UL user. Let be the submatrix consisting of the first and the second columns of and be the submatrix consisting of the third and the fourth columns of . Note that and . We set the transmit mode and the receive mode of the BS for 4 time slots, denoted by and respectively, as and set the DL transmit precoding matrices as

 [U1,U2]=[WH3H1(¯α)WH3H2(¯α)]−1.

Here we skip the proof of the existence of the above inverse matrix, which will be proved in the next subsection. Then, the BS and the th UL user construct their length- time-extended transmit signal vector as

 xd =U1sd1+U2sd2, xuj=W4suj for% j∈[1,2]

From (II-B), the length- time-extended input–output relation is given by

 ydi =Hi(¯α)(U1sd1+U2sd2)+2∑j=1gijW4suj+zdi for j∈[1,2], (15) yu =[F1(¯β)V,F2(¯β)W4][sTu1,sTu2]T+zu (16)

Then, the th DL user estimates its information symbols by multiplying to in (15). From the definition of and ,

 WH3ydi=sdi+WH3zdi for j∈[1,2], (17)

which shows that the th DL user can obtain almost surely. The BS estimates its information symbols from (16), showing that it can obtain and almost surely because is invertible almost surely, which will be proved in the next subsection. Consequently, eight information symbols are delivered for four time slots and thus the achievable sum DoF of the proposed scheme is given by two.

#### Iv-B2 General proof

Note that from the assumption that . In this case, the right-hand side of (6) is given by

 min{2,max(1+Ld(Lu−1)Lu,1+Lu(Ld−1)Ld)}.

In the following, we will show that the sum DoF of is achievable for all integer values satisfying that

 nd ∈[1:Lu], nu ∈[1:Ld], nd+nu ∈[2:LdLu]. (18)

Notice that and satisfy (IV-B2), which result in the sum DoFs of and respectively. Then, the following relation holds:

 dΣ,pCSIT ≥max{min(2,1+Ld(Lu−1)Lu),min(2,1+Lu(Ld−1)Ld)} =min{2,max(1+Ld(Lu−1)Lu,1+Lu(Ld−1)Ld)}. (19)

Therefore, in order to establish the achievablility of Theorem 2, it is enough to show that the sum DoF of is achievable for all integer values satisfying that (IV-B2).

From now on, assume that satisfies (IV-B2). In the proof, the BS sends information symbols to each of DL users (out of DL users) and each of UL users sends information symbols each to the BS for time slots.

Let be the information vector for the th DL user, , satisfying that . Let be the information vector for the th UL user, where , satisfying that . These information symbols will be delivered by symbol extension, i.e., beamforming over time slots. Let be the transmit precoding matrix for sending , where , satisfying that and be the transmit precoding matrix for sending , which is same for all , satisfying that . We will discuss designing of transmit precoding matrices of the BS and the UL users later. The BS and the th UL user set their length- time-extended transmit signal vector as

 xd =Ld∑i=1Uisdi, xuj=Vsuj for j∈[1:Ku], (20)

each of which satisfies the average power constraint , i.e., and for .

During signal transmission and reception, the BS sets its transmit and receive mode differently at each time with cycle of and respectively, i.e., and for . Denote the above transmit mode vector and receive mode vector by and , respectively.

Then, from (II-B) and (20), the length- time-extended input–output relation is given by

 ydi =Hi(¯α2)[U1,⋯,ULd]sd+Ku∑j=1gijVsuj+zdi, yu =[F1(¯β2)V,⋯,FKu(¯β2)V]su+zu (21)

where and .

Now consider designing of the DL transmit precoding matrix for and the UL transmit precoding matrix . Let be the submatrix consisting of the first through th columns of and be the submatrix consisting of the ()th through ()th columns of . Let us define

 P =[(WH3H1(¯α2))T,⋯,(WH3HLd(¯α2))T]T∈CLdnd×LdLu, Q =[F1(¯β2)W4,⋯,FKu(¯β2)W4]∈CLdLu×Kunu. (22)

The following lemma is used for designing the transmit precoding matrices of the BS and the UL users.

###### Lemma 1.

and almost surely.

###### Proof:

We refer to the Appendix for the proof. ∎

Now, we determine the transmit precoding matrices of the BS and the UL users as

 [U1,⋯,ULd]=P†∥P†∥, V=1√nuW4 (23)

where is the right inverse matrix of satisfying that , which exists almost surely from Lemma 1.

For decoding its DL message, the th DL user multiplies to . From (21) and (23),

 WH3ydi=sdi∥P†∥+WH3zdi (24)

where the equality holds from the definition of in (22) and the property of the IDFT matrix in (10). Then, the th DL user estimates its information symbols based on (24). Hence, the achievable sum DoF of the DL users is given by

 Ld∑i=1ddi=ndLu.

Now consider decoding of the UL messages at the BS. From (21) and (23), the received signal of the BS is given by

 yu=Qsu+zu. (25)

Then, the BS estimates its information symbols based on (25), provided that the achievable sum DoF of the UL users is given by

 Ku∑i=1dui=rank(Q)LdLu≥nuLd

where the inequality follows from Lemma 1.

Consequently, the sum DoF of is achievable for all and satisfying , which completes the achievability of Theorem 2.

## V Converse

In this section, we establish the converse of Theorem 1. When