DoF Analysis of the MIMO Broadcast Channel with Alternating/Hybrid CSIT
We consider a -user multiple-input single-output (MISO) broadcast channel (BC) where the channel state information (CSI) of user may be instantaneously perfect (P), delayed (D) or not known (N) at the transmitter with probabilities , and , respectively. In this setting, according to the three possible CSIT for each user, knowledge of the joint CSIT of the users could have at most states. In this paper, given the marginal probabilities of CSIT (i.e., , and ), we derive an outer bound for the DoF region of the -user MISO BC. Subsequently, we tighten this outer bound by taking into account a set of inequalities that capture some of the states of the joint CSIT. One of the consequences of this set of inequalities is that for , it is shown that the DoF region is not completely characterized by the marginal probabilities in contrast to the two-user case. Afterwards, the tightness of these bounds are investigated through the discussion on the achievability. Finally, a two user MIMO BC having CSIT among P and N is considered in which an outer bound for the DoF region is provided and it is shown that in some scenarios it is tight.
In contrast to point to point multiple-input multiple-output (MIMO) communication where the channel state information at the transmitter (CSIT) does not affect the multiplexing gain, in a multiple-input single-output (MISO) broadcast channel (BC), knowledge of CSIT is crucial for interference mitigation and beamforming purposes . However, the assumption of perfect CSIT may not always be true in practice due to channel estimation error and feedback latency. Therefore, the idea of communication under some sort of imperfection in CSIT has gained more attention recently. The so called MAT algorithm was presented in  where it was shown that in terms of the degrees of freedom, even an outdated CSIT can result in significant performance improvement in comparison to the case with no CSIT. Assuming correlation between the feedback information and current channel state (e.g., when the feedback latency is smaller than the coherence time of the channel), the authors in  and  consider the degrees of freedom in a time correlated MISO BC which is shown to be a combination of zero forcing beamforming (ZFBF) and MAT algorithm. Following these works, the general case of mixed CSIT and the -user MISO BC with time correlated delayed CSIT are discussed in  and , respectively. While all these works consider the concept of delayed CSIT in time domain,  and  deal with the DoF region and its achievable schemes in a frequency correlated MISO BC where there is no delayed CSIT but imperfect CSIT across subbands, which is more inline with practical systems as Long Term Evolution (LTE) . In , the synergistic benefits of alternating CSIT over fixed CSIT was presented in a two user MISO BC with two transmit antennas. In  and , the MISO BC with hybrid CSIT (Perfect or Delayed) was considered. The recent work of  investigates the DoF region of the K-user MISO BC with hybrid CSIT and linear encoding at the transmitter.  and  show that the optimal sum DoF is achievable if the CSIT is not too delayed in broadcast channels and interference networks, respectively.
The complete characterization of the general MISO BC with perfect, delayed or unknown CSIT is an open problem. The main aim of this paper is to investigate this problem and provide some answers toward this goal. To this end, our contributions are as follows.
Given the marginal probabilities of CSIT in a -user MISO BC, we derive an outer bound for the DoF region.
A set of inequalities is proposed that captures not only the marginals, but also the joint CSIT distribution. This shows that for the K-user case (), marginal probabilities are not sufficient for characterizing the DoF region.
The tightness of the outer bounds is investigated in certain cases.
Finally, a two-user MIMO BC is considered in which the CSI of a user is either perfect or unknown. An outer bound for the DoF region is provided and it is shown to be tight when the joint CSIT probabilities satisfy a certain relationship.
The paper is organized as follows. In section II the system model and preliminaries are presented. An outer bound is provided in section III based on the marginal probabilities and the proof is given in section IV. Section V provides an outer bound that depends on the joint CSIT probabilities. The tightness of the outerbounds will be discussed in section VI. Section VII investigates a two user MIMO BC with CSIT either perfect or unknown, and section VIII concludes the paper.
Throughout the paper, is equivalent to . and denote the transpose and conjugate transpose, respectively. is the circularly symmetric complex Gaussian distribution with covariance matrix . For a pair of integers , the discrete interval is defined as . , and .
Ii System Model
We consider a MISO BC, in which a base station with antennas sends independent messages to single-antenna users (). In a flat fading scenario, the discrete-time baseband received signal of user at channel use (henceforth, time instant) can be written as
where is the transmitted signal at time instant satisfying the (per codeword) power constraint . and are the additive noise and channel vector of user , respectively, and are also assumed i.i.d. over the time instants and the users. We assume global perfect Channel State Information at Receivers (CSIR).
The rate tuple , in which , is achievable if there exists a coding scheme such that the probability of error in decoding at user can be made arbitrarily small with sufficiently large coding block length. The DoF region is defined as where is the capacity region (i.e., the closure of the set of achievable rate tuples).
The probabilistic model used in this paper for CSIT availability allows the transmitter to have a Perfect (P) instantaneous knowledge of the CSI of a particular user at some time instants, whereas at some other time instants it receives the CSI with Delay (D) and finally, for the remaining time instants the CSI of the user is Not known (N) at the transmitter. The CSIT model can be fixed (i.e., as in the hybrid model), alternating or both (i.e., fixed for a subset of the users and alternating for the remaining subset.) When there is delayed CSIT, we assume that the feedback delay is much larger than the coherence time of the channel making the feedback information completely independent of the current channel state. In this configuration, the joint CSIT of all the users has at most states. For example, in a 3 user MISO BC, they will be with corresponding probabilities and, as an example, the marginal probability of perfect CSIT for user 1 is .
By CSIT pattern we refer to the knowledge of CSIT represented in a space-time matrix where the rows and columns represent users and time slots, respectively. The channel remains fixed within each time slot, while it changes independently from one slot to another. For simplicity, we assume the delayed CSI arrives at the transmitter after one time slot. Figure 1 shows an example of a CSIT pattern, in which the transmitter knows the channels of users 2 and 3 perfectly at time slot 1 and has no information about the channel of user 1. The CSI of user 1 will be known in the next time slot due to feedback delay and is completely independent of the channel in time slot 2.
Finally, a symmetric CSIT pattern means that the marginal probabilities of perfect, delayed and unknown CSIT are the same across the users, i.e. . As an example figure 2 shows a symmetric CSIT pattern for the 3-user MISO BC in which .
Iii An outer bound given the marginals
Theorem 1. Let be an arbitrary permutation of size over the indices , and be a permutation of satisfying111The reason for arranging the users according to the sum of the perfect and delayed CSIT probabilities becomes clear in (28).
Given the marginal probabilities of CSIT for user (which can be any two of and , since ), an outer bound for the DoF region of the -user MISO BC with transmit antennas at the transmitter () is defined by the following sets of inequalities
For the symmetric scenario, the sets of inequalities are simplified as
For , the outer bound boils down to the optimal DoF region in .
Iv Proof of theorem 1
For simplicity, we assume , since it is obvious that each subset of users with cardinality () can be regarded as a -user BC. Also, we assume the identity permutation (i.e., ) while the results could be easily applied to any other arbitrary permutation.
Iv-a Proof of
First, we improve the channel by giving the message and observation of user to users (). Hence, from Fano’s inequality,
where denotes the global CSIR up to time instant , and goes to zero as goes to infinity. By this improvement, channel input and outputs (i.e., the enhanced observations of users) form a Markov chain which results in a physically degraded broadcast channel . Therefore, according to , since feedback does not increase the capacity of physically degraded broadcast channels, we can ignore the delayed CSIT (D) and replace them with No CSIT (N). This is equivalent to having the channel of user perfectly known with probability and not known otherwise. From now on, we ignore the term for simplicity (since later it will be divided by and ) and write
where and we have used the fact that , since with the knowledge of and , the observations can be reconstructed within the noise distortion. Before going further, the following lemma is needed.
Lemma 1. Let be a set of arbitrary random variables and be a sliding window of size over () starting from i.e.,
where defines the modulo operation. Then, the following inequality holds for
where is an arbitrary condition.
We prove the lemma by showing that for every fixed , (10) holds for all using induction. It is obvious that for every , (10) holds for . In other words, . Now, considering that (10) is valid for , we show that it also holds for . Replacing with , we have
where in (11), and we have used the validity of (10) for . In (12), we have used the fact that for . In (13), the chain rule of entropies is used and in (14), the sliding window is written in terms of its elements. Finally, in (15), the fact that conditioning reduces the differential entropy is used. Therefore, since was chosen arbitrarily and (10) is valid for and from its validity for we could show it also holds for , we conclude that (10) holds for all values of and satisfying . ∎
Each term in the summation of (9) can be rewritten as
where , and . (17) is from the application of lemma 1 () and (18) is from the chain rule of entropies. Before going further, the following lemma is needed. This lemma, which is based on , is the key part in the proof.
Lemma 2. In the -user MISO BC defined in (1), for the users (), we have
Based on the four possible states for the joint CSIT of and , we have
Iv-A1 CSIT of m is N or P and CSIT of q is P
A Gaussian input with the conditional covariance matrix of achieves the upper bound, where is a unit vector in the direction orthogonal to (since is known).
Iv-A2 CSIT of m is N and CSIT of q is N
In this case both and are statistically equivalent (i.e., having the same probability density functions, and subsequently, the same entropies.) Therefore,
Iv-A3 CSIT of m is P and CSIT of q is N
This is the second result of Theorem 1 in 222The differential entropy terms in the left hand side of (19) can be written in terms of the expectation of the difference of entropies conditioned on the realizations of A. Since the conditional probability density functions exist and have a bounded peak, the same steps of  as discretization, considering the cannonical form and bounding the cardinality of aligned image set can be applied.. ∎
Iv-B Proof of
We enhance the channel in two ways:
Like the approach in , whenever there is delayed CSIT (), we assume that it is perfect instantaneous CSIT (), but we keep the probability of delayed CSIT. In other words, the CSIT of user is perfect with probability and unknown otherwise.
We give the message of user to users .
By summing (40) over users and writing the mutual information in terms of differential entropies,
By applying the results of lemma 2 to (25), we have
Let be an arbitrary permutation of size on . Applying the same reasoning, we have
(27) results in inequalities all having the same left hand side. Therefore,
V An outer bound capturing the joint CSIT probabilities
In the previous section, an outer bound was provided in terms of the marginal probabilities. In this section, we tighten the outer bound by introducing a set of inequalities that captures the joint CSIT probabilities. We start with simple motivating examples. Consider the pattern shown in figure 3. By Fano’s inequality, we write,
By doing the same for , we have
Finally, the rate of user 3 is written as
where in (34), lemma 2 is applied to the differences resulting in the values written under the braces and in (36), . We have split the observations of users 1 and 2 in terms of the joint CSIT, i.e., and . (36) is due to the fact that there is at least one unknown CSIT (N) in the joint states of user 1 and user 2 (i.e., PN, NP and NN. see rows 1 and 2 of the CSIT pattern shown in figure 3). Therefore, we have the following inequalities for the pattern shown in figure 3
From (38), the sum DoF of the pattern in figure 3 has the upper bound of , while it can be easily verified that for the pattern with PPP in the first slot and NNN in the next two slots, which has the same marginals as in figure 3, the sum DoF is . This simple example confirms that for the K-user MISO BC (), the marginal probabilities are not sufficient in characterizing the DoF region333It is important to emphasize on the difference between the following two statements a) Two CSIT patterns with different marginals can have the same DoF regions. b) Two CSIT patterns with the same marginals can have different DoF regions. The first statement is already known in literature. For example, by comparing the original 2-user MAT (i.e., ) and the scheme DN,ND,NN in , it is concluded that both of them have the sum DoF of 4/3, while having different marginal prbabilities (for the latter, ). However, the set of inequalities proposed in this section addresses the second statement which is a new problem and cannot result from the first statement.. Motivated by this simple example, we can have the following set of inequalities for the 3-user MISO BC with P and N
where a dashed line in the above means that the CSIT of the corresponding user is not important (for example, which is a summation over all the possible states for the CSIT of user 3). By looking at the difference of entropies in (35), it is observed that this difference is of order when there is at least one N in the joint CSIT of users 1 and 2 (i.e., PNN, PNP, NPN, NPP, NNP and NNN) and, therefore, is upperbounded by . This results in the first inequality of (V) and the same reasoning applies to the remaining two inequalities. (V) is a set of inequalities that captures the joint CSIT probabilities and is not only a function of the marginals.
Now consider the pattern shown in figure 4 for the 4-user MISO BC.