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Achievable DoF Regions of Three-User MIMO Broadcast Channel with Delayed CSIT

For the two-user multiple-input multiple-output (MIMO) broadcast channel with delayed channel state information at transmitter (CSIT) and arbitrary antenna configurations, all the degrees-of-freedom (DoF) regions were obtained. However, for the three-user MIMO broadcast channel with delayed CSIT and arbitrary antenna configurations, the DoF region of order-2 messages is still unclear and only a partial achievable DoF region of order-1 messages was obtained. In this paper, for the three-user MIMO broadcast channel with delayed CSIT and arbitrary antenna configurations, we first design the achievable scheme for each antenna configuration case, and then introduce a structuring method for analyzing the inherent achievable DoF region of the proposed scheme. As a result, the DoF region of order-2 messages is precisely characterized and the achievable DoF region of order-1 messages is completely expressed.


I. INTRODUCTION
The multiple-input multiple-output (MIMO) broadcast channel has one transmitter and multiple receivers, all of them are equipped with multiple antennas. In the MIMO broadcast channel, the transmitter intends to send private messages or common messages to receivers. Specifically, the messages intended by k receivers are denoted as order-k messages. As a fundamental metric of MIMO broadcast channel, the degrees-of-freedom (DoF) denotes the maximal number of independent channels that a wireless network can afford, which is also an first-order approximation of channel capacity in high signal-to-noise ratio (SNR) regime. Usually, the DoF is derived based on the consensus of DoF converse and the achievable DoF. The DoF converse is an information-theoretical upper/outer limit, while the achievable DoF is a lower/inner limit.
The achievable DoF represents the number of achievable independent channels, which is also arXiv:2001.05134v1 [cs.IT] 15 Jan 2020 a first-order approximation of data rate in high SNR regime. Moreover, the value of achievable DoF is also embodied in the corresponding achievable schemes for practical use. The DoF region of the MIMO broadcast channel can be obtained in [1]- [3] when the channel state information at transmitter (CSIT) is instantaneously obtained. However, if the wireless channel is fast timevarying, obtaining instantaneous CSIT needs a high feedback frequency and a rapid feedback link, which are difficult to meet. To solve that problem, utilizing delayed CSIT can alleviate the stringent requirements of feedback frequency and rapidity [4]- [18].
Related works on DoF of MIMO broadcast channel with delayed CSIT were given in [4]- [9]. As a seminal work, Maddah-Ali and Tse first investigated the DoF region of order-k messages for K-user multiple-input and single-output (MISO) in [4], where the transmitter has M antennas and each receiver has one antenna. They obtained the DoF region of order-k messages for M ≥ K −k +1 antenna configurations. In particular, for private messages, i.e., order-1 messages, the DoF region was derived for M ≥ K antenna configurations. Thereafter, for two-user MIMO broadcast channel, even though the antenna configurations are arbitrary, the DoF region was characterized in [5], [6]. For three-user MIMO broadcast channel with order-1 messages, if each receivers has N antennas symmetrically, the sum-DoF as a simpler metric than DoF region was obtained in [7] except for N < M < 2N antenna configurations. Recently, a higher achievable sum-DoF than that in [7] was proposed in [8] based on a holistic transmission design for N < M < 2.5N antenna configurations. Apart from symmetry antenna configurations, since the DoF region of order-3 messages is trivial, finding the DoF regions of order-2 messages and order-1 messages in three-user MIMO broadcast channel with arbitrary antenna configurations is a challenging task, and only a partially achievable DoF region of order-1 messages was obtained [9], due to the difficulty in achievable scheme design and expressing the achievable DoF regions.
To sum up, for three-user MIMO broadcast channel with delayed CSIT and arbitrary antenna configurations, except for the trivial order-3 message case, the DoF region of order-2 messages and DoF region of order-1 messages remain unclear.
In this paper, for the DoF region of order-1 messages, we obtain a complete achievable DoF region of order-1 messages. For the DoF region of order-2 messages, we obtain the precise DoF region of order-2 messages. We further show that the achievable DoF region of order-1 messages can be the DoF region if some antenna configuration conditions are satisfied. To achieve these, we design the achievable schemes of order-1 messages for all antenna configurations except for the existing cases in [9], and, for a case of antenna configurations, we generalize our holistic achievable scheme in [8] with symmetry antenna configurations to the one with arbitrary antenna configurations. In particular, we analyze the inherent achievable DoF regions of the proposed achievable schemes via a structuring method. Through this structuring method, we unravel the identity of a set of inequalities for feasible transmission phases and a set of inequalities for achievable DoF region. As a comparison, the proposed achievable DoF region of order-1 messages is compared with the existing DoF outer region in some representative antenna configurations, where the performance gap between them can be small if an antenna configuration condition holds.
Notations: The scalar, vector and matrix is denoted by h, h, and H, respectively. (•) T and (•) H denote transpose and conjugate-transpose, respectively. R n + denotes the n-tuples non-negative real numbers. The h or H is comprised of partial rows of h or H. The convex hull of set S is denoted by Conv(S). The convex hull of a finite set is the set of all convex combinations of its points. The block-diagonal matrix with blocks A and B is denoted by

A. Three-user MIMO Broadcast Channel
The three-user (N 1 , N 2 , N 3 , M ) MIMO broadcast channel with arbitrary antennas at transmitter and three receivers is depicted in Fig. 1, where the transmitter is equipped with M antennas and receivers 1, 2 and 3 are equipped with N 1 , N 2 and N 3 antennas, respectively. Without loss of generality, we assume that N 1 ≤ N 2 ≤ N 3 . At the time slot (TS) i, i ≥ 1, the channel state information from the transmitter to the receiver j, j = 1, 2, 3, is denoted by the matrix H j [i] ∈ C N ×M , whose elements are i.i.d. across space and time, and drawn from a continuous distribution. The received signal in the TS i at receiver j is expressed as where the input signal at transmitter is x[i] and additive white Gaussian noise (AWGN) at receiver j is n j ∼ CN (0, σ 2 ). Given a maximal average transmit power P , x[i] is subject to an average . . . . . .

B. Delayed CSIT
At each TS, each receiver estimates its channel state information matrix and returns it to the transmitter. We assume that the feedback channel from the receivers to transmitter is subject to a delay, whose value is not smaller than one TS. Therefore, only delayed CSIT, i.e., H j [i − τ ], τ ∈ {1, 2, · · · }, is available at the transmitter, which is different from the current one. For the channel state information at receiver (CSIR), the CSI feedback can be overheard by other receivers without a delay. Hence, CSIR can be instantaneously obtained.

C. DoF Region
Firstly, the order-2 messages rate tuple (R 12 (P ), R 23 (P ), R 13 (P )) is achievable, if there exists a code such that the probability of decoding error approaches zero when the number of channel uses goes to infinity. Then, the channel capacity region of order-2 message C 2 (P ) is the region of all achievable rate tuple of order-2 messages satisfying the average power constraint. Finally, the DoF region is the first-order term of channel capacity when transmit power goes to infinity, which can accurately approximate channel capacity region in high signal-to-noise ratio regime.
DoF region of order-2 messages is defined as follows: Firstly, the order-1 message rate tuple (R 1 (P ), R 2 (P ), R 3 (P )) is achievable, if there exists a code such that the probability of decoding error approaches zero when the number of channel uses goes to infinity. Then, the channel capacity region C 1 (P ) is the region of all achievable rate tuple satisfying the average power constraint. Finally, the DoF region of order-2 messages is defined as follows:

III. DOF REGION OF ORDER-2 MESSAGES
Theorem 1: The DoF region of order-2 messages is given by Proof: The DoF outer region, expressed by equation (5), is given in sub-section A. The achievable DoF region, expressed via Lemmas 1-4, is presented in sub-section B. It can be verified that the DoF outer region is the same as the achievable DoF region. Hence, the DoF region can be expressed by equation (5).

A. DoF Outer region
The idea of presenting the DoF outer region follows that in [4], [5]. First, we create a physically degraded broadcast channel by providing the output of receiver j to receiver j + 1, · · · , 3.
Then, according to [19]- [21], the feedback will not change the capacity region of the physically degraded broadcast channel, and thus the DoF region of order-2 messages with delayed CSIT is equal to the DoF region of order-2 messages with no CSIT. Finally, using Theorem 2 in [22] for the DoF region of order-2 messages with no CSIT and permuting all possible receiver indexes, we can obtain the DoF outer region, which is the same expression as equation (5).

B. Achievable DoF Region
To show the achievability, we divide the general antenna configurations into four specific cases, and then design an achievable scheme for each case.
C. Case 1: M ≤ N 2 Lemma 1: If M ≤ N 2 , the achievable DoF region is given by Proof: To achieve corner points of the above region, we can send min{N 1 , M } order-2 messages for receivers 1 and 2, M order-2 messages for receivers 2 and 3, and min{N 1 , M } order-2 messages for receivers 1 and 3, in each TS, respectively. The entire region can be attained through time-sharing of the schemes used in achieving the corner points.
D. Case 2: N 2 < M ≤ N 3 Lemma 2: If N 2 < M ≤ N 3 , the achievable DoF region of order-2 messages is given by Proof: We first design a two-phase achievable scheme, and then show that the inherent achievable DoF region, which is the same as that in Lemma 2.
Phase-I spans T 12 + T 23 + T 13 TSs, where order-2 messages are transmitted via the strategy given in Appendix A with B 1 = M and B 2 = B 3 = min{M, N 1 + N 2 } antennas. After transmission of order-2 messages, receivers 1 and 2 cannot decode their desired messages, due to the lack of equations. In order to decode x ab , receiver 1 needs additional T 12 (M − N 1 ) equations, and receiver 2 needs additional T 12 (M − N 2 ) equations. In order to decode x bc , receiver 2 needs additional T 23 (min{M, N 1 + N 2 } − N 2 ) equations. In order to decode x ac , receiver 1 needs Phase-II spans T TSs, where, based on CSIT of Phase-I, the order-3 message will be generated at transmitter for assisting receivers 1 and 2 to decode desired order-2 messages. To generate order-3 messages via the strategy given in Appendix A, in particular, y bc are set as building blocks. In this phase, all order-3 messages are transmitted using N 3 transmit antennas.
The feasible transmission time of order-3 messages and order-2 messages is given as follows: In Phase-II, we provide T 23 (min{M, N 1 + N 2 } − N 2 ) + T 12 (M − N 2 ) equations for receiver 2, equations for receiver 1, so that the transmitted order-2 messages in Phase-I can be decoded at receivers 1 and 2. Therefore, T, T 12 , T 23 , and T 13 should satisfy the following inequalities so that receivers 1 and 2 can obtain their lacking equations in order to decode desired order-2 messages, respectively.
We next derive the inherent achievable DoF region of the above proposed scheme by structuring method. For brevity, we denote β T 12 + T 23 + T 13 + T . Adding (β − T )N 1 and (β − T )N 2 at both sides of inequalities (8a) and (8b), respectively, we have Then, dividing both sides of inequalities (9a) and (9b) by βN 1 and βN 2 , respectively, we have The achievable DoF tuple is expressed as Therefore, we can re-write the inequalities (10a) and (10b) through the expression of achievable DoF tuple given in equation (11) as follows, where we have d 12 , d 23 , d 13 ≥ 0.
This shows that the proposed achievable DoF region in equation (7) can be achieved.
the achievable DoF region of order-2 messages is given by Proof: We first design a two-phase achievable scheme, and then show that the inherent achievable DoF region, which is the same as that in Lemma 3.
Phase-I spans T 12 + T 23 + T 13 TSs, where order-2 messages are transmitted via the strategy given Phase-II spans T TSs, where, based on CSIT of Phase-I, the order-3 message will be generated at transmitter for helping receivers 1 and 2 decode desired order-2 messages. To generate order-3 messages via the strategy given in Appendix A, in particular, y bc 1 ∈ C T 23 (M −N 2 ) , y ac 2 ∈ C T 13 (M −N 1 ) , and y ab 3 ∈ C T 12 (M −N 1 ) are set as building blocks. In this phase, all order-3 messages are transmitted using N 3 transmit antennas.
The feasible transmission time of order-2 messages and order-3 messages is given as follows: In Phase-II, we provide (T 12 + T 13 )(M − N 1 ) equations for receiver 1, and (T 12 equations for receiver 2, and (T 13 +T 23 )(M −N 3 ) equations for receiver 3. Therefore, T, T 12 , T 23 , and T 13 should satisfy the following inequalities so that receivers 1, 2 and 3 can obtain their lacking equations, respectively.
This shows that the proposed achievable DoF region in equation (13) can be achieved.
Proof: We first design a two-phase achievable scheme, and then show that the inherent achievable DoF region, which is the same as that in Lemma 4.
Phase-I spans T 12 +T 23 +T 13 TSs, where order-2 messages are transmitted via the strategy given in Appendix A with B 1 = min{M, N 1 +N 3 } and B 2 = B 3 = N 1 +N 2 antennas. After transmission of order-2 messages, receivers 1, 2 and 3 cannot decode their desired messages, due to the lack of equations. In order to decode x ab , receiver 1 needs additional T 12 (min{M, equations, and receiver 2 needs additional T 12 (min{M, to decode x bc , receiver 2 needs additional T 23 N 1 equations, and receiver 3 needs additional In order to decode x ac , receiver 1 needs additional T 13 N 2 equations, and receiver 3 needs additional Phase-II spans T 12 + T 23 + T 13 TSs, where, based on CSIT of Phase-I, the order-3 message will be generated at transmitter for assisting receivers 1 and 2 to decode desired order-2 messages. To generate order-3 messages via the strategy given in Appendix A, in particular, y bc 1 ∈ C T 23 N 1 , y ac 2 ∈ C T 13 N 2 , and y ab 3 ∈ C T 12 (min{M,N 1 +N 3 }−N 1 ) are set as building blocks. In this phase, all order-3 messages are transmitted using N 3 transmit antennas.
The feasible transmission time of order-2 messages and order-3 messages is given as follows: In 3. Therefore, T, T 12 , T 23 , and T 13 should satisfy the following inequalities so that receivers 1, 2 and 3 can obtain their lacking equations, respectively.
This shows that the proposed achievable DoF region in equation (19) can be obtained.

IV. ACHIEVABLE DOF REGION OF ORDER-1 MESSAGES
In this section, we present the achievable DoF region of order-1 messages by dividing the general antenna configurations into four specific cases, and show their achievable DoF region, respectively. The following critical condition of antenna configurations is repeatedly used in characterizing the achievable DoF region of order-1 messages: As a remark, one can see that N 1 + N 2 < N 3 leads to invalidation of the above condition.
A. Case 1: In this case, an achievable DoF region of order-1 messages has been characterized in [9], which can achieve the DoF outer region if an antenna configuration condition holds.
B. Case 2: Theorem 2: The proposed achievable DoF region is given by where regions D 1 , D 2 , and D 3 are given by 2. An illustration of achievable DoF region for Case 2.
and, if condition ( * ) holds, the corner point P 0 is the intersection of the following planes: Otherwise, the corner point P 0 is given by the intersection of following planes: (29a), (29b), and Remark: The proposed achievable DoF region in Theorem 2 is illustrated in Fig. 2.
Proof: Regions D 1 , D 2 , and D 3 are proposed based on setting one coordinate of DoF outer region in [5] to zero. Since D 1 , D 2 , and D 3 belong to regions of two-user broadcast channel with delayed CSIT, they can be obtained using the scheme in [5]. It remains to show that corner point P 0 can be achieved by the following three-phase achievable scheme: Phase-I spans T 1 + T 2 + T 3 TSs, where order-1 messages are transmitted by coded transmission given in Appendix B with A 1 = N 1 + N 2 and A 2 = A 3 = M antennas. After the coded transmission of order-1 messages, receivers 1, 2 and 3 cannot decode their desired messages, due to lack of equations. In order to decode x 1 a and x 2 a , receiver 1 needs N 1 + N 2 equations and M equations, respectively. In order to decode x 1 b and x 2 b , receiver 2 needs N 1 + N 2 equations and M equations, respectively. In order to decode x 1 c and x 2 c , receiver 3 needs M equations and M equations, respectively.
Phase-II spans T 12 + T 23 + T 13 TSs, where, based on CSIT of Phase-I, order-2 messages x ab ∈ C (N 1 +N 2 )T 1 , x bc ∈ C M T 2 and x ac ∈ C M T 3 are generated via the strategy given in Appendix B, to provide lacking equations in order to decode order-1 messages. The proposed achievable scheme of order-2 messages is used to deliver the generated order-2 messages. Therefore, the transmission time for order-2 messages should satisfy the following equivalence relationship: Phase-III spans T TSs, where, to facilitate the decoding of transmitted order-2 messages, based on CSIT of Phase-II, order-3 messages y abc ∈ C T N 3 are generated via the strategy given in Appendix A. In this phase, all order-3 messages are transmitted using N 3 transmit antennas.
The transmission time of Phase-III should be assigned to the value that all receivers can acquire their lacking equations with the same amount of TSs, namely, Equivalently, it can be re-written as the following linear system: If condition ( * ) holds, a non-negative solution of above linear system can be given by We next show that the above solution is equivalent to the corner point given in Theorem 2. For brevity, we denote α T 1 + T 2 + T 3 + T 12 + T 23 + T 13 + T . Adding (α − T )N 1 , (α − T )N 2 and (α − T )N 3 at the both sides of equations (33a), (33b) and (33c), respectively, we have Replacing T 12 , T 23 , and T 13 with T 1 , T 2 , and T 3 using equations (31b), (31c) and (31c), and dividing both sides with αN 1 , αN 2 , and αN 3 , respectively, we have Since the achievable DoF tuple is expressed as we have By re-writing equations (36a), (36b) and (36c) through equations (37) and (38), the corner point P 0 can be expressed as an intersection of equations (29a), (29b) and (29c).
If condition ( * ) does not hold, we cannot obtain a non-negative solution of equations (33a), (33b) and (33c), because of T 12 < 0 for the solution. Therefore, we shift to solve for achieving a corner point. A solution is given by Proof: According to [5], a DoF outer region is given by C. Case 3: Theorem 3: The proposed achievable DoF region is given by where regions D 1 , D 2 , D 3 are given by and, if condition ( * ) holds, the corner point P 0 is given by the intersection of following planes: Otherwise, the corner point P 0 is given by the intersection of following planes: (46a), (46b), and Proof: Regions D 1 , D 2 , and D 3 are proposed based on setting one coordinate of the DoF outer region in [5] to zero. Since D 1 , D 2 , and D 3 belong to regions of two-user broadcast channel with delayed CSIT, they can be achieved using the scheme in [5]. We next show that the corner point P 0 can be obtained by the following scheme with three transmission phases: Phase-I spans T 1 + T 2 + T 3 TSs, where order-1 messages are transmitted by coded transmission given in Appendix B with A 1 = N 1 + N 2 , A 2 = M , and A 3 = N 1 + N 3 antennas. After the coded transmission of order-1 messages, receivers 1, 2, and 3 cannot decode their desired messages, due to the lack of equations. In order to decode x 1 a and x 2 a , receiver 1 needs N 1 + N 2 equations and N 1 + N 3 equations, respectively. In order to decode x 1 b and x 2 b , receiver 2 needs N 1 + N 2 equations and M equations, respectively. In order to decode x 1 c and x 2 c , receiver 3 needs M equations and N 1 + N 3 equations, respectively.
Phase-II spans T 12 + T 23 + T 13 TSs, where, based on CSIT of Phase-I, order-2 messages x ab ∈ C (N 1 +N 2 )T 1 , x bc ∈ C M T 2 and x ac ∈ C (N 1 +N 3 )T 3 are generated via the strategy given in Appendix B, to provide lacking equations in order to decode order-1 messages. The proposed achievable scheme of order-2 messages is used to deliver the generated order-2 messages. Therefore, the transmission time for order-2 messages should satisfy the following equivalence relationship: Phase-III spans T TSs, where, to facilitate the decoding of transmitted order-2 messages, based on CSIT of Phase-II, order-3 messages y abc ∈ C T N 3 are generated via the strategy given in Appendix A. In this phase, all order-3 messages are transmitted using N 3 transmit antennas.
The transmission time of Phase-III should be assigned to the value that all receivers can acquire their lacking equations with the same amount of TSs. That is, we have Equivalently, it can be re-written as the following linear system: If condition ( * ) holds, a non-negative solution of above linear system can be given by We next show that the above solution is equivalent to the corner point P 0 in Theorem 3. For brevity, we denote α T 1 + T 2 + T 3 + T 12 + T 23 + T 13 + T . Adding (α − T )N 1 , (α − T )N 2 and (α − T )N 3 at the both sides of equations (50a), (50b) and (50c) respectively, we have Replacing T 12 , T 23 , and T 13 with T 1 , T 2 , and T 3 using equations (48b), (48c) and (48c), and dividing both sides with αN 1 , αN 2 , and αN 3 , respectively, we have Since the achievable DoF tuple is expressed as we have By re-writing equations (53a), (53b) and (53c) through equations (54), (55a) and (55b), the corner point can be expressed as an intersection of equations (46a), (46b) and (46c). If condition ( * ) holds with equality, we have T 1 = 0, which implies that d 1 + d 2 + d 3 = 0, due to equation (55a). Therefore, if M = N 1 + N 2 and condition ( * ) holds with equality, it can be confirmed that the whole DoF outer region is achievable.
If condition ( * ) does not hold, we cannot obtain a non-negative solution of equations (50a), (50b) and (50c), because of T 12 < 0 for the solution. Therefore, we shift to solve 20 for achieving a corner point. A solution is given by Due to the relationship revealed by equations (55a) and (55b), the corner point can be equivalently expressed as an intersection of equations (46a), (46b) and d 1 + d 2 − d 3 = 0.
Corollary 2: If condition ( * ) holds with equality and M = N 1 + N 3 , the achievable DoF region given in Theorem 3 is the DoF region.
Proof: According to [5], a DoF outer region is given by where regions D 1 , D 2 , and D 3 are given by and the corner point P 0 is given by the intersection of following planes: Meanwhile, the corner point P 0 should satisfy Remark: If the antenna configuration is symmetrical, i.e., N 1 = N 2 = N 3 = N , the corner point In fact, Theorem 4 is a generalization of the method proposed in our conference paper [8].
Proof: Regions D 1 , D 2 , and D 3 are proposed based on setting one coordinate of the DoF outer region in [5] to zero. Since D 1 , D 2 , and D 3 belong to regions of two-user broadcast channel with delayed CSIT, they can be achieved using the scheme in [5]. We next show that the corner point P 0 can be achieved by the following scheme with three transmission phases: Phase-I spans T 1 + T 2 + T 3 TSs, where order-1 messages are transmitted by coded transmission given in Appendix B with A 1 = A 2 = A 3 = M antennas. After the coded transmission of order-1 messages, receivers 1, 2 and 3 cannot decode their desired messages, due to lack of equations.
In order to decode x 1 a and x 2 a , receiver 1 needs M equations and M equations, respectively. In order to decode x 1 b and x 2 b , receiver 2 needs M equations and M equations, respectively. In order to decode x 1 c and x 2 c , receiver 3 needs M equations and M equations, respectively.
Phase-II spans T 12 + T 23 + T 13 TSs, where, based on CSIT of Phase-I, order-2 messages x ab ∈ C (N 1 +N 2 )T 1 , x bc ∈ C (N 2 +N 3 )T 2 and x ac ∈ C (N 1 +N 3 )T 3 are generated via the strategy given in Appendix B, to provide lacking equations in order to decode order-1 messages. Apart from the above generated order-2 messages, additional order-2 messages and order-3 messages are generated for providing more lacking equations, which can be explained as follows: The additional order-2 messages are given by The order-3 messages are given by The proposed achievable scheme of order-2 messages is used to deliver the generated order-2 messages. Therefore, the transmission time for order-2 messages should satisfy the following equivalence relationship: Phase-III spans T TSs, where, to facilitate the decoding of transmitted order-2 messages, based on CSIT of Phase-II, order-3 messages y abc ∈ C T N 3 are generated via the strategy given in Appendix A. In this phase, all order-3 messages are transmitted using N 3 transmit antennas.
The transmission time of Phase-III should be assigned to the value that all receivers can acquire their lacking equations with the same amount of TSs. Thus, we have Equivalently, it can be re-written as the following linear system: If the linear system established by equations (67) and (69) has a non-negative solution, we next show that the above solution is equivalent to the corner point P 0 in Theorem 4. For brevity, we at both sides of equations (69a), (69b) and (69c), respectively, we have Replacing T 12 , T 23 , and T 13 with T 1 , T 2 , and T 3 using equations (67b), (67c) and (67c) and dividing both sides of equations (70a), (70b) and (70c) with αN 1 , αN 2 , and αN 3 , respectively, we have Since the achievable DoF tuple is expressed as we have By re-writing equations (71a), (71b) and (53c) through equations (72), (73b), (73c) and (73c), the corner point P 0 can be expressed as an intersection of equations (63a), (63b) and (63c). Due to equations (73b), (73c) and (73c), the linear system established by equations (67) and (69) has a non-negative solution is equivalent to conditions (64a), (64b) and (64c) hold.

E. Compared with DoF Outer Region
In this section, we compare the order-1 message DoF performance of the proposed achievable DoF region with the DoF outer region in [5]. Numerically, we calculate the achievable sum-DoF from the proposed achievable DoF region and the DoF upper bound from the DoF outer region in [5], respectively. The DoF upper bound is defined as maximizing d 1 + d 2 + d 3 over the DoF outer region and the achievable sum-DoF is defined as maximizing d 1 + d 2 + d 3 over the achievable DoF region. Clearly, the optimality is attained when the corner point is strictly positive.
In Case 2, Fig. 3 depicts that, if condition ( * ) holds, the proposed achievable sum-DoF is extremely close to the DoF upper bound, which implies the satisfactory performance of proposed schemes. If the condition ( * ) is not satisfied, the DoF gap between the achievable sum-DoF and DoF upper bound becomes increasingly larger. This is because, in this special case, the DoF upper bound is related to N 3 , but the proposed achievable sum-DoF is independent of it. In Case 3, Fig. 3 shows that, if condition ( * ) holds, the proposed achievable sum-DoF is close to the DoF upper bound, however, the difference is greater than that of Case 2, and the gap will also grow larger as N 3 increases. In Case 4, which is irrelevant to condition ( * ), Fig. 3 depicts that the gap between DoF upper bound and achievable sum-DoF reduces as N 3 increases. To sum up, Fig. 3 shows that the condition ( * ) is critical because whether or not it holds significantly affects the performance of proposed design. We can also infer from Fig. 3 that, in Case 4, the larger is the difference between M and N 2 + N 3 , the greater is the performance gap.

V. CONCLUSION & FUTURE RESEARCH
We proposed achievable DoF regions of order-2 and order-1 messages and presented a DoF outer region of order-2 messages for three-user MIMO broadcast channel with delayed CSIT.
Our results show that the DoF region of order-2 messages can be exactly attained. We also obtained the completely achievable DoF of order-1 messages for the first time. In the future, research effort can be devoted to derive the complete DoF region of order-1 messages for threeuser MIMO broadcast channel with delayed CSIT. In addition, based on the results of two-user and three-user broadcast channels with delayed CSIT, we envision that the DoF region of order-k messages for (k + 1)-user MIMO broadcast channel with arbitrary antenna configurations and delayed CSIT can be characterized using the methods presented in this paper.

A. General Achievable Scheme of Order-2 Messages
The general achievable scheme has two phases, where in Phase-I, order-2 messages are transmitted and in Phase-II, order-3 messages that assists the transmitted order-2 messages decode are sent. Generally, the number of used antennas and transmission time are undetermined.