MISO Broadcast Channel with Hybrid CSIT: Beyond Two Users 1footnote 11footnote 1This work has been presented in part in [1, 2]. The research of S. Lashgari and A. S. Avestimehr is supported by NSF Grants CAREER 0953117, CCF-1161720, NETS-1161904, and ONR award N000141310094; and the research of R. Tandon is supported by the NSF Grant CCF 14-22090.

MISO Broadcast Channel with Hybrid CSIT: Beyond Two Users 111This work has been presented in part in [1, 2]. The research of S. Lashgari and A. S. Avestimehr is supported by NSF Grants CAREER 0953117, CCF-1161720, NETS-1161904, and ONR award N000141310094; and the research of R. Tandon is supported by the NSF Grant CCF 14-22090.

Sina Lashgari School of Electrical and Computer Engineering, Cornell University, Ithaca, NY Ravi Tandon Discovery Analytics Center and Department of Computer Science, Virginia Tech, Blacksburg, VA Salman Avestimehr Department of Electrical Engineering, University of Southern California, Los Angeles, CA
Abstract

We study the impact of heterogeneity of channel-state-information available at the transmitters (CSIT) on the capacity of broadcast channels with a multiple-antenna transmitter and single-antenna receivers (MISO BC). In particular, we consider the -user MISO BC, where the CSIT with respect to each receiver can be either instantaneous/perfect, delayed, or not available; and we study the impact of this heterogeneity of CSIT on the degrees-of-freedom (DoF) of such network. We first focus on the -user MISO BC; and we completely characterize the DoF region for all possible heterogeneous CSIT configurations, assuming linear encoding strategies at the transmitters. The result shows that the state-of-the-art achievable schemes in the literature are indeed sum-DoF optimal, when restricted to linear encoding schemes. To prove the result, we develop a novel bound, called Interference Decomposition Bound, which provides a lower bound on the interference dimension at a receiver which supplies delayed CSIT based on the average dimension of constituents of that interference, thereby decomposing the interference into its individual components. Furthermore, we extend our outer bound on the DoF region to the general -user MISO BC, and demonstrate that it leads to an approximate characterization of linear sum-DoF to within an additive gap of for a broad range of CSIT configurations. Moreover, for the special case where only one receiver supplies delayed CSIT, we completely characterize the linear sum-DoF.

I Introduction

Channel state information at the transmitters (CSIT) plays a crucial role in the design and operation of multi-user wireless networks. Timely and accurate knowledge of the channels can potentially help the transmitters mitigate the interference that they cause at the unintended receivers, therefore enabling them to increase the communication rate to their intended receivers. The common procedure for obtaining channel state information (CSI) is to send training symbols (or pilots) at the transmitters, and then estimate the channels at the receivers and feed the estimates back to the transmitters. As a result of this feedback mechanism, CSI may not always be perfect and instantaneous. For instance, CSIT may be outdated due to the fast fading nature of the channels or slow feedback mechanism, it can be noisy (imperfect), or not available at all. Therefore, one can expect that in a large network there would be various types of CSI available at the transmitters with respect to different receivers. This results in communication scenarios with heterogeneous or hybrid CSIT.

As a result, there has been a growing interest in studying the impact of CSIT on the capacity of wireless networks, especially the broadcast channel. In particular, it was shown in [3] that even when the transmitter(s) only have access to delayed CSIT, there is significant potential for degrees-of-freedom (DoF) gain. They studied the problem of -user multiple-input single-output broadcast channel (MISO BC) with delayed CSIT, and showed that for such network . This work was followed by several other works which studied other network configurations under the assumption of delayed CSIT, including interference channel [4, 5, 6, 7], X-channel [8, 9], multi-hop networks [10], and other variations of delayed CSIT [11].

Most of these prior works assume that the entire network state information is obtained with delay. However, in a large network, one can expect various types of CSIT available at the transmitters with respect to different receivers. As a result, there have also been several works on studying the impact of heterogeneous (or hybrid) CSIT on the capacity of wireless networks, where the CSIT with respect to each receiver can now be either instantaneous/perfect (), delayed (), or not available () [12, 13, 14, 15, 16, 17, 18]. However, studying networks under the assumption of heterogeneous CSIT becomes quite challenging, to the extent that only the DoF for -user MISO BC is characterized [19, 15]; and beyond the 2-user network configuration even the DoF is unknown and the problem remains widely open.

To make progress on the MISO BC beyond 2 users, in this paper we focus on characterizing the degrees of freedom when restricted to linear schemes (also called LDoF). Our motivation to focus on the linear degrees of freedom is based on recent progress made in [9, 20], where the concept of LDoF was introduced; and it was shown that for 2-user X-channel with delayed CSIT, LDoF can be characterized, while the information theoretic DoF remains open. Linear schemes are also of significant practical interests due to their low complexity; and in fact, the majority of DoF-optimal schemes developed so far for networks with delayed CSIT are linear.

We consider the problem of MISO BC with hybrid CSIT, with a multiple-antenna transmitter and single-antenna receivers (), and study its linear degrees of freedom. The channels are time-varying, and the CSIT provided by each receiver is either instantaneous (), delayed (), or none (). We first study the case of , and fully characterize the LDoF for all possible hybrid CSIT configurations. The result is obtained by developing a general outer bound on the LDoF region, and a matching achievable scheme for each of the CSIT configurations.

The outer bound, which is the main contribution of this paper, is based on three main ingredients. The first ingredient is a novel lemma, called Interference Decomposition Bound. It essentially lower bounds the interference dimension at a receiver with delayed CSIT by the average dimension of its constituents, thereby decomposing the interference into its individual components. As a result of Interference Decomposition Bound, we can then focus on analyzing the dimension of constituents of interference at receivers which supply delayed CSIT, in order to derive an upper bound on LDoF. Proof of Interference Decomposition Bound is based on temporal analysis of dimensions of transmit signals at different receivers, leading to necessary conditions on the increments of such dimensions using the delayed CSIT constraint.

The second main ingredient of the converse proof is MIMO Rank Ratio Inequality for Broadcast Channel, which provides a lower bound on the dimension of interference components at receivers supplying delayed CSIT. In particular, the bound states that if the transmitter employs linear precoding schemes, the dimension of each interference component at a single-antenna receiver supplying delayed CSIT is at least half of the dimension of the corresponding signal at any other single-antenna receiver. This inequality can be viewed as a variation of the Rank Ratio Inequality proved in [9], which shows that if two distributed single-antenna transmitters employ linear strategies, the dimensions of received linear subspace at a single-antenna receiver supplying delayed CSIT is at least of the dimension of the same signal at any other single-antenna receiver. Note that the key difference between the two lemmas lies in the assumption of distributed antennas in Rank Ratio Inequality, which changes the proof techniques required to establish the inequality.

Finally, the third ingredient of the converse, called Least Alignment Lemma, provides a lower bound on the dimension of interference components at receivers supplying no CSIT. In particular, the bound states that once the transmitter(s) in a network has no CSIT of a certain receiver, the least amount of alignment will occur at that receiver, meaning that transmit signals will occupy the maximal signal dimensions at that receiver. As a result, Least Alignment Lemma implies that the dimension of interference caused at a receiver which supplies no CSIT by the message intended for another receiver is at least equal to the dimension of the message itself. Using the three main ingredients we develop a converse proof which characterizes the linear DoF region for all possible hybrid CSIT configurations of the 3-user MISO BC.

We next extend the key proof ingredients of the converse for 3-user MISO BC to the general -user setting. In particular, we extend the Interference Decomposition Bound to the -user setting to provide lower bound on the dimension of any interfering signal in an arbitrary receiver supplying delayed CSIT. In addition, we present a generalized version of MIMO Rank Ratio Inequality for BC, which provides a lower bound on the dimension of joint received signals at any arbitrary subset of receivers supplying delayed CSIT. Additionally, we extend the Least Alignment Lemma and show that under linear schemes, for arbitrary transmit signals the dimension of received signal at a receiver supplying no CSIT cannot be less than any other receiver.

By extending the converse tools to the general -user setting, we provide a new outer bound on the linear DoF region of the general -user MISO BC with arbitrary hybrid CSIT configuration. We demonstrate that our new outer bound leads to an approximate linear sum-DoF characterization to within an additive gap of for networks with more number of receivers supplying instantaneous CSIT than delayed CSIT; and the approximation gap decays exponentially with the increase in number of receivers supplying instantaneous CSIT. Furthermore, by using the outer bound and providing a new multi-phase achievable scheme, we present the exact characterization of linear sum-DoF for networks in which only one receiver supplies delayed CSIT.

Notation. We use small letters (e.g. ) for scalars, arrowed letters (e.g. ) for vectors, capital letters (e.g. ) for matrices, and calligraphic font (e.g. ) for sets. We also use bold letters (e.g. ) for random entities, and non-bold letters for deterministic values (e.g., realizations of random variables).

Ii System Model

We consider the Gaussian -user multiple-input single-output broadcast channel (MISO BC) as depicted in Figure  1. It consists of a transmitter with antennas, and single-antenna receivers, , where . The transmitter has a separate message for each of the receivers.


Fig. 1: Network configuration for -user MISO BC.

Consider communication over time slots. The received signal at () at time is given by

(1)

where is the transmit signal vector at time ; denotes the channel coefficients of the channel from Tx to ; and denotes the additive white noise which is distributed as . The elements of the channel coefficients vector are i.i.d, drawn from a continuous distribution and also i.i.d across time and users. denotes the set of all channel vectors at time . In addition, we denote by the set of all channel coefficients from time 1 to , i.e.,

(2)

We denote the vector of transmit signals in a block of length by , where is the result of concatenation of transmit signal vectors . We assume Tx obeys an average power constraint, .

We focus on scenarios in which channel state information available at the transmitter (CSIT) with respect to different receivers can be instantaneous (), delayed (), or none (). We refer to these scenarios as fixed hybrid scenarios, or hybrid in short. In particular, CSIT with respect to , , is denoted by , as defined in [15]. In this notation, indicates that Tx has access to instantaneous CSIT with respect to ; i.e., at time , Tx has access to . Similarly, indicates delayed CSIT with respect to ; i.e., at time , Tx has access to . Finally, indicates no CSIT, which means the channel to is not known to the Tx at all. We assume that the type of CSIT for each receiver is fixed and does not alternate over time (nevertheless, channels are time-varying). Therefore, there are different fixed hybrid scenarios. As an example, we use to denote the 3-user MISO BC where the first receiver provides instantaneous CSIT, while the other two provide delayed CSIT.

Definition 1.

We denote the set of indices of users in states by , respectively. In addition, for an ordered set we denote by the ordered set obtained by a permutation of the elements of , where we denote the elements of the new ordered set by .

Note that according to Definition 1, and . Based on the above description of channel state information, the channel outcomes available to Tx at time are denoted by the following set:

(3)

We restrict ourselves to linear coding strategies as defined in  [9], in which degrees-of-freedom (DoF) represents the dimension of the linear subspace of transmitted signals. More specifically, consider a communication scheme with block length , in which the Tx wishes to deliver a vector of information symbols to (). Each information symbol is a random variable with variance . These information symbols are then modulated with precoding matrices at times . Note that the precoding matrix depends only upon the outcome of due to the hybrid CSIT constraint:

(4)

Based on this linear precoding, Tx will then send at time . We can rewrite as following.

(5)

where denotes the vertical concatenation of matrices and (i.e.,

We denote by the overall precoding matrix of Tx for , such that the rows of constitute . In addition, we denote the precoding function used by Tx by .

Based on the above setting, the received signal at () after the time steps of the communication will be

(6)

where is the block diagonal channel coefficients matrix where the channel coefficients of timeslot (i.e. ) are in the row , and in the columns of , and the rest of the elements of are zero.222For , we define ; therefore, for instance, .

Now, consider the decoding of at (i.e., decoding the information symbols for ). The corresponding interference subspace at will be

where is the matrix formed by row concatenation of matrices for , and of a matrix corresponds to the sub-space that is spanned by its columns. Let denote the orthogonal subspace of . Then, in the regime of asymptotically high transmit powers (i.e., ignoring the noise), the decodability of information symbols at corresponds to the constraint that the image of on has dimension :

(7)

which can be shown by simple linear algebra to be equivalent to the following:

(8)

Based on this setting, we now define the linear degrees-of-freedom of the -user MISO broadcast channel with hybrid CSIT.

Definition 2.

-tuple degrees-of-freedom are linearly achievable if there exists a sequence such that for each and the corresponding choice of , satisfy the decodability condition of (8) with probability 1; i.e., for all ,

(9)

and

(10)

We also define the linear degrees-of-freedom region as the closure of the set of all linearly achievable -tuples . Furthermore, the linear sum-degrees-of-freedom () is defined as follows:

(11)

In what follows we first focus on the case of , and completely characterize the for 3-user MISO BC with hybrid CSIT. We then extend our bounds and present new outer bounds on the of the general -user MISO BC with hybrid CSIT.

Iii 3-user MISO Broadcast Channel with Hybrid CSIT

In this section we focus on 3-user MISO broadcast channel with hybrid CSIT. In particular, we first state the complete characterization of for all hybrid CSIT configurations; and then, we present the proof based on 3 key lemmas.

Theorem 1.

Given a hybrid CSIT configuration, i.e., a partition of users into disjoint sets and as defined in Definition 1, the is characterized as follows:

(12)

The and the corresponding for different CSIT configurations are summarized in Table I.

Note that although there are different CSIT configurations for 3-user MISO BC, many of them are permutations of one another, e.g. . As a result, there are only distinct CSIT configurations which are presented in Table I.

CSIT States Linear Degrees of Freedom Region ()
PPP 3
PPD
PPN
PDD
PDN
DDD
DDN
PNN, DNN, NNN
TABLE I: and for all possible configurations of hybrid CSIT for 3-user MISO BC.
Remark 1.

The bound in Theorem 1 strictly improves the state-of-the-art bounds, and also leads to complete characterization of for . For instance, for (i.e. supplying instantaneous CSIT, while supply delayed CSIT) the prior results suggest that [17, 21], while by Theorem 1, is indeed equal to . Similarly, for the case of , the prior results [17, 21] imply that , while by Theorem 1,

Remark 2.

Theorem 1 implies that the state-of-the-art achievable schemes presented in [18] for and are both optimal from the perspective of .

Remark 3.

It is worth noting that in any CSIT configuration which involves receivers with state N, the inequalities that constitute the LDoF region have coefficient 1 for the degrees-of-freedom of receivers with state N. In other words, receivers that supply no CSIT do not contribute to the , and unless all receivers have state N, removing the no CSIT receivers from the network will not decrease the .

In the remainder of this section we prove Theorem 1. To this aim, we first present the converse proof in Section III-A, and then discuss the achievability in Section III-B.

Iii-a Proof of Converse for 3-User MISO Broadcast Channel with Hybrid CSIT

We first provide the three main ingredients that are key in proving the converse for 3-user MISO broadcast channel with hybrid CSIT. We then show how those main ingredients are used to prove the converse for two representative CSIT configurations (i.e. and ). The proof of converse for other CSIT configurations can be found in Appendix A. The first two ingredients of the converse proof deal with lower bounding received signal dimension at a receiver which supplies delayed CSIT, while the third ingredient captures the impact of no CSIT.

The first key ingredient is Interference Decomposition Bound, which essentially provides a lower bound on the interference dimension at a receiver supplying delayed CSIT, based on the constituents of that interference, as well as the received signal dimension at other receivers. It is stated below; and its proof is provided in Appendix B.

Lemma 1.

(Interference Decomposition Bound) Consider , and a fixed linear coding strategy , with corresponding precoding matrices as defined in (4). If (i.e., if supplies delayed CSIT),

(13)
Remark 4.

The R.H.S. of Interference Decomposition Bound represents the dimension of interference caused at , which supplies delayed CSIT, by the messages intended for . On the other hand, the third term on the L.H.S. (i.e. ) is the dimension of the remaining interference at after removing the contribution of the message of ; and the first two terms (i.e. ) can be shown by (9) and sub-modularity of rank (stated in Lemma 4) to equal , which is the dimension of message of . Hence, Interference Decomposition Bound provides an inequality which connects the dimension of interference at a receiver to the average dimension of its constituents. Note that statement of Lemma 1 does not assume any specific CSIT with respect to any receiver except .

The second main ingredient, called MIMO Rank Ratio Inequality for BC, provides a lower bound on the dimension of received signal at a receiver supplying delayed CSIT. It is stated below; and its proof is provided in Appendix D.

Lemma 2.

(MIMO Rank Ratio Inequality for BC) Consider , and a linear coding strategy , with corresponding as defined in (4). If (i.e., if supplies delayed CSIT), then, for each beamforming matrix , where , and each , we have

(14)

where denotes the column concatenation of matrices and .

Remark 5.

Lemma 2 implies that for any transmit signal , the corresponding received signal dimension at a receiver with delayed CSIT is at least half of the corresponding received signal dimension at any other receiver. Note that statement of Lemma 2 does not assume any specific CSIT with respect to any receiver except .

The third main ingredient of converse, Least Alignment Lemma, demonstrates that when using linear schemes, once the transmitter has no CSIT with respect to a certain receiver, the least amount of alignment will occur at that receiver, meaning that transmit signals will occupy the maximal signal dimensions at that receiver. The lemma is stated below; and its proof is provided in Appendix E.

Lemma 3.

(Least Alignment Lemma) Consider , and a linear coding strategy , with corresponding as defined in (4). For let denote the row concatenation of the precoding matrices , where . If (i.e., if supplies no CSIT),

Remark 6.

Note that the statement of Lemma 3 does not assume any specific CSIT with respect to any receiver except .

Remark 7.

Lemma 3 can be seen as a variation of the corresponding result in the context of secrecy problems in [22, 23]. Moreover, as shown in [19], Least Alignment Lemma also holds for non-linear schemes; and for this extension the reader is referred to [19].

We now prove the converse for two representative CSIT configurations and , highlighting the applications of the above three lemmas. Converse proofs for other CSIT configurations can be found in Appendix A.

Iii-A1 Proof of Converse for

According to Table I, it is sufficient to show that and ; since the other two inequalities (i.e. , and ) can be proven similarly using symmetry. Moreover, the bound follows directly from the existing state-of-the-art arguments used in [17, 3]. Henceforth, we focus on proving .

Suppose degrees-of-freedom are linearly achievable. Hence, by Definition 2 there exists a sequence such that for each and the corresponding choice of , satisfy the conditions in (9) and (10). Therefore, in order to prove , it is sufficient to show that

(15)

Note that since in the configuration receiver 3 supplies delayed CSIT, we can invoke Lemma 1, which states that:

(16)

We now further bound each side of the above inequality. We first upper bound the left-hand-side of the above inequality:

(17)

On the other hand, for the right-hand-side of (16) we have

(18)

Hence, by considering (16)-(18), we obtain

(19)

which proves (15), and therefore, completes the converse proof for .

Remark 8.

Note that in order to prove for , we did not rely on any specific CSIT assumption with respect to . Therefore, the bound also holds for the case of . Moreover, note that by symmetry one can conclude that also holds for . Hence, since according to Table I, and constitute the LDoF region for , the above derivations suffice in proving the converse for the CSIT configuration as well.

Iii-A2 Proof of Converse for

According to Table I, it is sufficient to show that and . Suppose degrees-of-freedom are linearly achievable. Hence, by Definition 2 there exists a sequence such that for each and the corresponding choice of , satisfy the conditions in (9) and (10). Therefore, in order to prove and , it is sufficient to show that

(20)

and

(21)

We have,

where (a) follows from the sub-modularity of rank of matrices (see Lemma 4 stated below); and (b) follows from Lemma 2 applied to as the receiver which supplies delayed CSIT. Therefore, the proof of (20) is complete. We now prove (21).

which completes the proof of (21). We now state the sub-modularity of rank of matrices (see  [24] for more details).

Lemma 4.

(Sub-modularity of rank) Consider a matrix . Let , denote the sub-matrix of created by those columns in which have their indices in . Then, for any ,

Note that a similar statement is true for sub-modularity of rank with respect to the rows of a matrix, instead of the columns as stated in Lemma 4.


Fig. 2: LDoF Region for .

Iii-B Proof of Achievability for Theorem 1

The regions described in Theorem 1 result in polytopes in ; and therefore, the LDoF regions can be completely described via their extreme points. Many of such extreme points can be trivially achieved (e.g. the point for ); therefore, we only focus on the non-trivial extreme points and provide reference for each of them in Table II.

CSIT States Non-trivial extreme points of the LDoF region and reference to the achievable scheme
PPD
PDD
PDN
DDD
DDN
TABLE II: Achievability results for extreme points of different configurations of hybrid CSIT for 3-user MISO BC

The only non-trivial extreme point that has not yet been achieved in the literature according to Table II belongs to , and is . The LDoF region suggested by Theorem 1 for is shown in Fig. 2. Therefore, we only prove the achievability of for . The scheme is illustrated in Fig. 3. We will show how to deliver 3 symbols to , 2 symbols to , and 2 symbols to over 4 time slots in order to achieve