On the Synergistic Benefits of Alternating CSIT for the MISO BC††thanks: E-mail: firstname.lastname@example.org, email@example.com, firstname.lastname@example.org, email@example.com. The work of H. V. Poor was supported in part by the Air Force Office of Scientific Research under MURI Grant FA 9550-09-1-0643.
The degrees of freedom (DoF) of the two-user multiple-input single-output (MISO) broadcast channel (BC) are studied under the assumption that the form, of the channel state information at the transmitter (CSIT) for each user’s channel can be either perfect (), delayed () or not available (), i.e., , and therefore the overall CSIT can alternate between the resulting states . The fraction of time associated with CSIT state is denoted by the parameter and it is assumed throughout that , i.e., . Under this assumption of symmetry, the main contribution of this paper is a complete characterization of the DoF region of the two user MISO BC with alternating CSIT. Surprisingly, the DoF region is found to depend only on the marginal probabilities , , which represent the fraction of time that any given user (e.g., user 1) is associated with perfect, delayed, or no CSIT, respectively. As a consequence, the DoF region with all 9 CSIT states, , is the same as the DoF region with only 3 CSIT states , under the same marginal distribution of CSIT states, i.e., . The sum-DoF value can be expressed as , from which one can uniquely identify the minimum required marginal CSIT fractions to achieve any target DoF value as when and when . The results highlight the synergistic benefits of alternating CSIT and the tradeoffs between various forms of CSIT for any given DoF value.
The availability of channel state information at transmitters (CSIT) is a key ingredient for interference management techniques . It affects not only the capacity but also the degrees of freedom (DoF) of wireless networks. Perhaps the simplest setting that exemplifies the critical role of CSIT is the two-user vector broadcast channel, also known as the multiple input single output broadcast channel (MISO BC), in which a transmitter equipped with two antennas sends independent messages to two receivers, each equipped with a single antenna. Degrees of freedom characterizations for the MISO BC are available under a variety of CSIT models, including full (perfect and instantaneous) CSIT , no CSIT [3, 4, 5, 6], delayed CSIT [7, 8], compound CSIT [9, 10, 11], quantized CSIT [12, 13, 14], mixed (perfect delayed and partial instantaneous) CSIT [15, 16, 17], asymmetric CSIT (perfect CSIT for one user, delayed CSIT for the other) [18, 19] and with knowledge of only the channel coherence patterns available to the transmitter [18, 20]. Yet, the understanding of the role of CSIT for the MISO BC is far from complete, even from a DoF perspective, as exemplified by the Lapidoth-Shamai-Wigger conjecture , which is but one of the many open problems along this research avenue.
In this work we focus on an aspect of CSIT that has so far received little direct attention – that it can vary over time. Consider the MISO BC for the case in which perfect CSIT is available for one user and no CSIT is available for the other user. Incidentally, the DoF are unknown for this problem. Now, staying within the assumption of full CSIT for one user and none for the other, suppose we allow the CSIT to vary, in the sense that half the time we have full CSIT for user 1 and none for user 2, and for the remaining half of the time we have full CSIT for user 2 and none for user 1. This is one example of what we call the alternating CSIT setting. In general terms, the defining feature of the alternating CSIT problem is a joint consideration of multiple CSIT states.
We motivate the alternating CSIT setting by addressing three natural questions — 1) is it practical, 2) is it a trivial extension, and 3) is it desirable/beneficial, relative to the more commonly studied non-alternating/fixed CSIT settings?
To answer the first question, we note that alternating CSIT may be already practically unavoidable due to the time varying nature of wireless networks. However, more interestingly, the form of CSIT may also be deliberately varied as a design choice, often with little or no additional overhead. For example, acquiring perfect CSIT for one user and none for the other for half the time and then switching the role of users for the remaining half of the time, carries little or no additional overhead relative to the non-alternating case in which perfect CSIT is acquired for the same user for the entire time while no CSIT is obtained for the other user. Thus, alternating CSIT is as practical as the non-alternating CSIT setting.
The second question relates to the novelty of the alternating CSIT setting with respect to the non-alternating CSIT setting. Is the former just a direct extension of the latter? As we will show in this work, this is not the case. Surprisingly, we find that the lack of a direct relationship between the alternating and non alternating settings works in our favor. Indeed, we are able to solve the alternating CSIT DoF problem in several cases for which the non-alternating case remains open. In particular, this includes the above mentioned case of full CSIT for one user and none for the other. As mentioned previously, for this problem the DoF remain open in the non-alternating CSIT setting. However, we are able to find the DoF for the same problem under the alternating CSIT assumption.
The third question, whether there is a benefit of alternating CSIT relative to non-alternating CSIT, is perhaps the most interesting question. Here, we will show that the constituent fixed-CSIT settings in the alternating CSIT problem are inseparable (for more on separability, see [22, 23, 24]), so that the DoF of the alternating CSIT setting can be strictly larger than a proportionally weighted combination of the DoF values of the constituent fixed-CSIT settings. We call this the synergistic DoF gain of alternating CSIT. As we will show in this work, the benefits of alternating CSIT over non-alternating CSIT can be quite substantial.
Related work: In terms of the constituent fixed-CSIT schemes, this work is related to most prior studies of the MISO BC DoF. While several recent works on mixed CSIT models, such as [15, 16, 17], also jointly consider multiple forms of CSIT, it is noteworthy that these works are fundamentally distinct as in [15, 16, 17], the multiple forms of CSIT are assumed to be simultaneously present in what ultimately amounts to a fixed-CSIT setting, as opposed to the alternating CSIT setting considered in this work. More closely related to our setting, are the recent works in  and  which involve alternating perfect and delayed CSIT models. In particular, the three receiver MISO BC with two transmit antennas is studied in , leading to an interesting observation that the presence of a third user, even with only two transmit antennas, can strictly increase the DoF.
Organization: Our model of MISO broadcast channel with alternating CSIT is described in Section 2. In Section 3, we present the DoF region of the MISO BC under alternating CSIT and highlight several aspects and interpretations of the results. In Section 4, we present constituent encoding schemes which highlight the benefits of alternating CSIT. Achievability of the DoF region with alternating CSIT is presented in Section 5 and the converse is presented in Section 6.
2 System Model
A two user MISO BC is considered, in which a transmitter (denoted as ) equipped with two transmit antennas wishes to send independent messages and , to two receivers (denoted as , and , respectively), and each receiver is equipped with a single antenna. The input-output relationship is given as
where (resp. ) is the channel output at (resp. ) at time , is the channel input which satisfies the power constraint , and are circularly symmetric complex additive white Gaussian noises at receivers and respectively. The channel vectors (to receiver ) and (to receiver ) are independent and identically distributed (i.i.d.) with continuous distributions, and are also i.i.d. over time. The rate pair , with , where is the number of channel uses, is achievable if the probability of decoding error for can be made arbitrarily small for sufficiently large . We are interested in the degrees of freedom region , defined as the set of all achievable pairs with .
While a variety of CSIT models are conceivable, here we identify the two most important characteristics of CSIT as — 1) precision, and 2) delay. Based on these two characteristics we identify three forms of CSIT to be considered in this work.
Perfect CSIT (): Perfect CSIT, or , denotes those instances in which CSIT is available instantaneously and with infinite precision.
Delayed CSIT (): Delayed CSIT, or , denotes those instances in which CSIT is available with infinite precision but only after such delay that it is independent of the current channel state.
No CSIT (): No CSIT, or , denotes those instances in which no CSIT is available. The users’ channels are statistically indistinguishable in this case.
The CSIT state of user 1, , and the CSIT state of user 2, , can each belong to any of these three cases,
giving us a total of 9 CSIT states for the two user MISO BC. Further, let us denote by the fraction of time that the state occurs, so that
We will assume throughout this paper that . Specifically,
This assumption is justified by the inherent symmetry of the problem, e.g., it is easy to see that if DoF were to be optimized subject to a symmetric CSIT cost constraint (the cost for acquiring CSIT state equals the cost of ) then the optimal choice of CSIT states will always satisfy the property . Furthermore, we assume that both the receivers have perfect global channel state information.
Problem Statement: Given the probability mass function (pmf), , the problem is to characterize the degrees-of-freedom region .
3 Main Results and Insights
Starting with the parameters , even if we use the 4 constraints (3)-(6) to eliminate parameters (say, ), we are still left with free parameters (, and a challenging task of characterizing the DoF region which is a function of these remaining parameters, i.e., a mapping from a region in to a region in . While such a problem can easily become intractable or at least extremely cumbersome, it turns out — rather serendipitously — to be not only completely solvable but also surprisingly easy to describe.
3.1 Main Result
We start with the main result, stated in the following theorem.
The DoF region , for the two user MISO BC with alternating CSIT is given by the set of non-negative pairs that satisfy
Note the dependence of the DoF region in Theorem 1 on the 5 remaining parameters , , , , . As remarkable as the simplicity of the DoF region description in Theorem 1 may be, it is possible to simplify it even further, in terms of only two marginal parameters – and . This simplification and associated insights are presented next through a set of remarks.
[Representation in terms of Marginals] The DoF region in Theorem 1 can also be expressed as follows:
where and defined below denote the total fraction of time that perfect and delayed CSIT, respectively, are associated with a user:
Note that these two marginal fractions satisfy
where is the total fraction of time that no CSIT is associated with a user.
[Same-Marginals Property] From Remark 1, we make a surprising observation. Given any alternating CSIT setting considered in this work, i.e., given any , there exists an equivalent alternating CSIT problem, having only three states: PP, DD and NN, with fractions , and as defined above. The two are equivalent in the sense that they have the same DoF regions. Thus, all alternating CSIT settings considered in this work can be reduced to only symmetric CSIT states with the same marginals, without any change in the DoF region. The sum DoF as a function of , where is shown in Figure 1.
This equivalence, which greatly simplifies the representation of the DoF region, remains rather mysterious because we have not found an argument that could establish this equivalence a priori. The equivalence is only evident after Theorem 1 is obtained, which allows us to simplify the statement of the theorem, but does not simplify the proof of the theorem. Nevertheless, the possibility of a general relationship along these lines is intriguing.
we have two very distinct observations. We note that in the region where (22) is true, delayed CSIT is interchangeable with no CSIT, because the DoF depends only on . Here, delay makes CSIT useless. On the other hand, in the region where , delayed CSIT is as good as perfect CSIT.
[Minimum Required CSIT for a DoF value] This tradeoff between marginal and is explicitly illustrated in Fig. 2. The most efficient point, in terms of marginal CSIT required to achieve any given value of DoF, is uniquely identified to be the bottom corner of the left most edge (highlighted corner in Fig. 2) of the corresponding trapezoid. Note that any other feasible CSIT point involves either redundant CSIT or unnecessary “instantaneous” CSIT requirements when delayed CSIT would have sufficed just as well. For example, following are the minimum CSIT requirements for various sum-DoF target values:
In fact, a general expression for the minimum CSIT required to achieve a sum-DoF value is easily evaluated to be
3.2 Synergistic Benefits
As mentioned previously, the most interesting aspects of the alternating CSIT problem are the synergistic DoF gains. Representative examples of this phenomenon are presented next.
Example 1: Consider the non-alternating CSIT setting, , in which perfect CSIT is available for one user and delayed CSIT is available for the other user. It has been shown in  that this setting has DoF. Now, let us make this an alternating CSIT setting. Suppose that half of the time the CSIT is of the form and remaining half of the time, the CSIT is of the form . From the main result stated in Theorem 1, it is easy to see that the optimal DoF value is now increased to . This is an example of a synergistic DoF gain from alternating CSIT. Figure 3 shows the DoF regions corresponding to the three fixed-CSIT states – and ; and the DoF region resulting by permitting alternation between states and in which each state occurs for half of the total communication period. This result also highlights the inseparability of operating over such CSIT states and shows that by jointly coding across these states, thereby collaboratively using the CSIT distributed over time, significant gains in DoF can be achieved.
Example 2: Another interesting example for which alternating CSIT provides provable DoF gains over non-alternating CSIT is the case when states and are present. Individually, the optimal DoF for state is as shown in . For the and states, the optimal DoF value is not known; however an upper bound of can be readily established. In contrast, if alternation is permitted among and , according to , then the optimal DoF value is , which is larger than both and , thereby showing strict synergistic gains made possible by alternating CSIT.
Example 3: As mentioned above, the DoF value is not known individually for fixed-CSIT state . In fact, it is our conjecture that for fixed-CSIT state , the optimal DoF value is only . However, in the alternating CSIT setting, if the states and are present for equal fractions of the time, then is the optimal DoF value.
Example 4: Interestingly enough, the Maddah-Ali and Tse (henceforth referred as MAT) scheme , or rather the alternative version of it presented in , may also be seen as an alternating CSIT scheme that achieves DoF with . Since the DoF of the setting by itself is and the DoF of the setting is , and , the synergistic gains are evident here as well.
We conclude this section by highlighting some of key aspects of the achievability and converse proofs. The converse proofs are inspired by the techniques developed for mixed CSIT configurations in  but also include some novel elements. A simple setting that highlights the novel aspects of the converse proof may be the case in which . For the achievability proof, the main challenge lies in identifying the core constituent schemes. In particular, core constituent schemes achieving DoF values of and by using minimal CSIT under various CSIT states are fundamental to the achievability of the DoF region. These constituent schemes are the topic of the next section.
4 Constituent Schemes
In proving the achievability of the respective DoF regions, we first present so called constituent encoding schemes that form the key building blocks for the achievability of the region stated in Theorem 1. Furthermore, through these constituent encoding schemes, the benefits of alternating CSIT states can be easily appreciated.
4.1 Scheme achieving DoF
Achieving DoF requires no CSIT; and thus any state can be used for this purpose. We denote the scheme achieving DoF as follows:
: uses the state NN and achieves .
4.2 Scheme achieving DoF
The only scheme that achieves DoF corresponds to the state PP, i.e., when the transmitter has perfect CSIT from both receivers. This is achievable via zero-forcing. We denote this scheme as follows:
: uses the state PP and achieves .
4.3 Schemes achieving DoF
The following schemes achieve DoF:
: using DD and achieving .
This is the scheme presented in  and achieves sum DoF of as follows: at , the transmitter sends two symbols intended for receiver ; this step delivers a useful information symbol at receiver and creates side-information at receiver . By a useful information symbol for receiver , we refer to a random linear combination of and . Similarly, at , the transmitter sends two symbols intended for receiver ; delivering a useful symbol at receiver while creating side-information at receiver . Due to delayed CSIT, the transmitter can reconstruct the side-information symbols created at . At , the transmitter sends a linear combination of these side-information symbols. After , each receiver, upon receiving this linear combination, can remove the interference by using its past overheard information. Therefore, DoF is achievable.
: using DD, NN for fractions and achieving .
We show this scheme by a modification of the MAT scheme described next. At , the transmitter sends on the first antenna and on the second antenna. Channel outputs at are as follows: receiver obtains , whereas receiver obtains . Via delayed CSIT from , the transmitter can reconstruct and within noise distortion. At , it transmits to both receivers using one antenna and at , it transmits to both receivers. This scheme also achieves a DoF of . The interesting aspect is that delayed CSIT from both receivers is required only at ; however no CSIT is required from . Thus, by alternation between (DD, NN) for fractions , DoF is achievable. This modification of the original MAT scheme is also mentioned in  and .
: using DN, ND for fractions and achieving .
: using DN, ND, NN for fractions and achieving .
We now present the combined explanation of the schemes and . In the original MAT scheme mentioned for , after , the transmitter requires CSIT only from receiver ; after , the transmitter requires CSIT only from receiver and at , the transmitter requires no CSIT. From this observation, we note that the original assumption of global delayed CSIT can be relaxed to one in which the transmitter can choose to select the available CSIT from a set of three states: state ND–no CSIT from receiver and delayed CSIT from receiver ; state DN–delayed CSIT from receiver and no CSIT from receiver ; and state NN–no CSIT from either of the receivers. If in addition, it is required that these states have to be chosen for an equal fraction (i.e., one-third) of time, then the original MAT scheme applies verbatim and is also the optimal DoF under this alternating CSIT model with a relaxed CSIT assumption. Therefore, the schemes and also achieve a DoF of .
4.4 Schemes achieving DoF
The following schemes achieve DoF:
: using PD, NN for fractions and achieving .
To show the achievability of , we show that it is possible to reliably transmit two symbols to receiver and one symbol to receiver in two channel uses. The CSIT configuration is chosen as PD at and NN at . At , the encoder sends
where the precoding matrix is chosen such that . The outputs at the receivers at are given as
(25) (26) (27) (28)
Due to delayed CSIT, the transmitter has access to . Hence, at , it simply sends
Having access to , along with , the symbols can be decoded at receiver . At receiver , the symbol can be decoded from by canceling out the interference which is received at . The scheme is illustrated in Figure 4.
: using DP, NN for fractions and achieving .
: using PN, NP for fractions and achieving .
To show the achievability of , we show that it is possible to reliably transmit two symbols to receiver and one symbol to receiver in two channel uses. The CSIT configuration is chosen as PN at and NP at . At , the encoder sends
where the precoding matrix is chosen such that . The outputs at receivers at are given as
(32) (33) (34) (35)
At this point, receiver requires cleanly in order to decode . At , the CSIT configuration changes to NP, and the transmitter can send cleanly to receiver ; but at the same time it uses the second antenna to transmit which is intended for receiver .
where the precoding matrix is chosen such that so that
(37) (38) (39) (40)
Having access to , along with , the symbols can be decoded at receiver . At receiver , the symbol can be decoded from by canceling out the interference which is received within noise distortion at . The scheme is illustrated in Figure 5.
: using PN, NP for fractions and achieving .
: using ND, PN for fractions and achieving .
To show the achievability of , we show that it is possible to reliably transmit two symbols to receiver and one symbol to receiver in two channel uses. The CSIT configuration is chosen as ND at and PN at . At , the encoder sends
The outputs at receivers at are given as
At this point, side information is created at receiver , and if receiver can obtain cleanly, then it can decode . Due to delayed CSIT from receiver after , the transmitter can obtain within noise distortion.
At , the CSIT configuration changes to PN, and the transmitter can send cleanly to receiver ; but at the same time it uses the second antenna to transmit which is intended for receiver .
where the precoding matrix is chosen such that so that
(44) (45) (46) (47)
Having access to , along with , the symbols can be decoded at receiver . At receiver , the symbol can be decoded from by canceling out the interference which was received within noise distortion previously at . The scheme is illustrated in Figure 6.
: using DN, NP for fractions and achieving .
4.5 Schemes achieving DoF
The following schemes achieve DoF:
: using PD, DP for fractions and achieving .
: using DP, PD for fractions and achieving .
: using PD, PN, NP for fractions and achieving .
In this scheme, we show that it is possible to reliably transmit three symbols to receiver and two symbols to receiver in a total of three channel uses. The CSIT states are chosen as PD at , PN at , and NP at . At , the encoder sends
where the precoding matrix is chosen to satisfy . The channel outputs are given as
(49) (50) (51) (52)
Due to delayed CSIT, transmitter has access to after . It can reconstruct the interference seen at receiver . Hence, at , it sends
where the precoding matrix is chosen to satisfy . The channel outputs are given as
(54) (55) (56)
The key consequence of this encoding step is that receiver still faces the same interference (up to a known scaling factor) as it encountered at . However, to successfully decode , it still requires this interference cleanly, i.e., it requires .
The transmitter now uses the freedom provided under the alternating CSIT model and switches from CSIT state PN at to the state NP at . Having access to , it sends
where . The outputs are given as
(58) (59) (60) (61)
Having access to , the symbols can be decoded. Finally, upon receiving , receiver successfully decodes . The scheme is illustrated in Figure 7.
Note that this scheme also shows that DoF is achievable as mentioned for schemes and , since the states PD, DP at can always be used as PN, NP states as above by ignoring the respective delayed CSIT components.
: using DP, PN, NP for fractions and achieving .
4.6 Scheme achieving DoF
The following scheme achieves DoF:
: using DD, PN, NP for fractions and achieving .
To this end, we show that it is possible to reliably transmit symbols to receiver , and symbols to receiver in a total of five channel uses. The CSIT configurations are chosen as DD, PN, NP, PN, and NP for and respectively. At , the transmitter sends the following:
so that the channel outputs are
Due to delayed CSIT from both receivers (the state DD at ), the transmitter can reconstruct and (which are the interference components at receivers and respectively).
At , the transmitter sends cleanly to receiver , and uses the second antenna to send :
where . The outputs at are
where is a scaled version of .
At , the transmitter switches the role by alternating to the NP state and sends cleanly to receiver and uses the second antenna to send . We thus have,