Space-Time Interference Alignment and Degrees of Freedom Regions for the MISO Broadcast Channel with Periodic CSI Feedback ††thanks: N. Lee and R. Heath were funded in part by the Army Research Labs, Grant W911NF1010420. This paper was presented in part  at IEEE 50th Annual Allerton Conference on Communication, Control, and Computing, Monticello, Oct. 2012.
This paper characterizes the degrees of freedom (DoF) regions for the multi-user vector broadcast channel with periodic channel state information (CSI) feedback. As a part of the characterization, a new transmission method called space-time interference alignment is proposed, which exploits both the current and past CSI jointly. Using the proposed alignment technique, an inner bound of the sum-DoF region is characterized as a function of a normalized CSI feedback frequency, which measures CSI feedback speed compared to the speed of user’s channel variations. One consequence of the result is that the achievable sum-DoF gain is improved significantly when a user sends back both current and outdated CSI compared to the case where the user sends back current CSI only. Then, a trade-off between CSI feedback delay and the sum-DoF gain is characterized for the multi-user vector broadcast channel in terms of a normalized CSI feedback delay that measures CSI obsoleteness compared to channel coherence time. A crucial insight is that it is possible to achieve the optimal DoF gain if the feedback delay is less than a derived fraction of the channel coherence time. This precisely characterizes the intuition that a small delay should be negligible.
Channel state information at the transmitter (CSIT) is important for optimizing wireless system performance. In the multiple-input-single-output (MISO) broadcast channel, CSIT allows the transmitter to send multiple data symbols to different receivers simultaneously without creating mutual interference by using interference suppression techniques -. Prior work on the MISO broadcast channel focused on the CSIT uncertainty caused by limited rate feedback -. This prior work showed no degrees of freedom (DoF) are lost compared to the perfect CSIT case when the CSI feedback rate per user linearly increases with signal to noise ratio (SNR) in dB scale. Meanwhile, there is little work on the effect of CSIT uncertainty due to feedback delay. In particular, if a transmitter has past CSI that is independent of the current CSI, i.e., outdated CSIT, it was commonly believed that only unity DoF gain could be achieved, which is the same DoF gain in the absence of channel knowledge at the transmitter . Recently, it was shown in  that outdated CSIT is helpful to obtain the DoF gains beyond that achieved by time-division-multiple-access (TDMA). The key idea from  was to exploit the perfect outdated CSIT as side-information so that the transmitter aligns inter-user interference between the past and the currently received signals. Motivated by the work in , extensions have been developed for other networks -. The DoF gain was studied for the interference channel and the X channel when only outdated CSI is used at the transmitter in . The DoF was characterized for the two-user MIMO interference channel with delayed CSI according to different transmit-and-receive-antenna configurations in . The outdated CSI feedback framework was also considered for the two-hop interference channel in  and , where the DoF gain increased beyond that achieved by the TDMA method.
The common assumption of previous work - is that the transmitter has delayed CSI only. Depending on a feedback protocol and relative feedback delay compared to channel variations, however, it may be possible for the transmitter to have knowledge of outdated as well as current CSI. Therefore, one of the fundamental questions is how much gain can the system provide if the transmitter can have both current and outdated CSI? An answer for this question is addressed in the periodic CSI feedback framework in this paper.
I-a Two Different Models of Periodic CSI Feedback
Unlike aperiodic CSI feedback where each user opportunistically sends back CSI whenever the channel condition is changed , periodic CSI feedback allows the transmitter to receive CSI periodically from mobile users through dedicated feedback channels. This is suitable for current cellular standards such as 3GPP LTE  because it is implementable using a control signal with lower overhead. In such a periodic CSI feedback protocol, two different models can be considered depending on the dominant factor that causes delayed CSIT.
Feedback frequency limited model (Model 1): Consider a block fading channel where the channel values are constant for the channel coherence time and the channels change independently between blocks. Further, assume that feedback delay is much less than the channel coherence time . In such a channel, consider the case where a transmitter cannot track all channel changes of users because each user has a limited CSI feedback opportunity given by the periodic feedback protocol. In other words, the CSI feedback frequency is not high enough to continually track the user’s channel variations. Consider the example illustrated in Fig. 1-(a). If a user is able to use the feedback channel over the feedback duration among the overall feedback period , i.e., , the transmitter may have current CSI for two-thirds of the channel variations, assuming that the CSI feedback delay is much smaller than the channel coherence time. Instead of sending back currently estimated CSI only, however, the transmitter can acquire more channel knowledge, provided that each user sends back a set of CSI including outdated and current CSI during the feedback durations. For example, each user may send back one outdated and two current channel state estimates to the transmitter during feedback durations as shown in Fig. 1-(b). In this model, CSIT is outdated due to the lower feedback frequency rather than the feedback link delay. Hence, it is possible to consider an equivalent CSI feedback model in a fast fading channel without considering feedback delay to focus on the effect of CSI feedback frequency as shown in Fig. 2-(a). The equivalence comes from the fact that both models allow for the transmitter to use two outdated and one current observation of CSI periodically. In this model, we establish how the CSI feedback frequency affects the sum-DoF gain in this fast-fading channel configuration.
Feedback link delay limited model (Model 2): We also consider a different CSI feedback scenario in which the feedback link delay is the dominant factor that causes outdated CSIT. Consider a block fading channel where the transmitter can track all channel variations for each user, assuming that the feedback interval is equal to channel coherence time for simplicity. Further, the feedback delay is not negligible compared to the channel coherence time. In this scenario, outdated CSIT is no longer affected by the feedback frequency because the transmitter can track the channel variations over all channel blocks. Rather, the relative difference between the CSI feedback delay and the channel coherence time causes the transmitter have the delayed CSI during a fraction of channel coherence time. For instance, when the feedback delay is one-third of channel coherence time, only delayed CSIT is available during one-third of channel coherence time, but both delayed and current CSIT is available in the remaining time as illustrated in Fig 2-(b).
In this paper, we consider the MISO broadcast channel where a transmitter with antennas supports users with a single antenna. The main contribution of this paper is to show that joint use of outdated and current CSIT improves significantly the sum-DoF of the vector broadcast channel in both periodic CSI feedback models. For the feedback frequency limited model, we provide a characterization of the achievable sum-DoF region as a function of CSI feedback frequency . As illustrated in Fig. 3, we show that it is possible to increase the achievable sum-DoF region substantially from the additional outdated CSI feedback over all the range compared to the case where each user sends back current CSI only at the feedback instant. One notable point is that when , it is possible to achieve the optimal DoF gain (cut-set bound) without knowledge of all current CSIT, i.e., . This result is appealing because it was believed that the cut-set bound can be achieved when the transmitter only has current CSIT over all channel uses. Our result, however, reveals the fact that current CSIT over all channel uses is not necessarily required, and joint use of outdated and current CSIT allows the optimal DoF gain to be achieved.
For the feedback link delay limited model, we also characterize a trade-off region between CSI feedback delay and the sum-DoF gain for the multi-user vector broadcast channel by the proposed STIA algorithm and leveraging results in . From the proposed region, we show that the joint use of outdated and current CSIT improves significantly the sum-DoF gain compared to that achieved by the separate exploitation of outdated and current CSIT. One consequence of the derived region is that not too delayed CSIT achieves the optimal DoF for the system. This is because the proposed trade-off region achieves the cut-set bound for the range of . Further, we demonstrate that the proposed trade-off region is optimal for the three-user vector broadcast channel using a recently derived converse result .
To show these results, we propose a new transmission method called space-time interference alignment (STIA). The key idea of STIA is to design the transmit beamforming matrix by using both outdated and current CSI jointly so that multiple inter-user interference signals between the past observed and the currently observed are aligned in both space and time domains. Therefore, STIA is a generalized version of the transmission method developed in  for using both outdated and current CSI.
In terms of related work, new transmission methods combining MAT in  and partial zero-forcing (ZF) using both current and outdated CSI were developed for the two-user vector broadcast channel in  and . The main difference lies in the system model. In  and , the imperfect current CSI estimated from the temporal channel correlations is used in the transmission algorithms. Meanwhile, our transmission algorithm exploits perfect current CSI. The problem of exploiting perfect outdated and current CSIT has been studied for the two-user vector broadcast channel  to obtain the DoF gain, which is more closely related to our work. The limitation of work in  is that it considers a simple two-user vector broadcast channel.
This paper is organized as follows. In Section II, the system model and periodic CSI feedback models are described. The key concept of the proposed transmission method is explained by giving an illustrative example in Section III. In Section IV, the achievable sum-DoF region for the -user MISO broadcast channel with the feedback frequency limited model (Model 1) is characterized. In Section V, a CSI feedback delay and DoF gain trade-off is studied with the feedback link delay limited model (Model 2). In Section VI, the proposed algorithm is compared with the prior work. The paper is concluded in Section VII.
Throughout this paper, transpose, conjugate transpose, inverse, and trace of a vector are represented by , , and , respectively. In addition, , , and a complex value, a positive integer value and a complex gaussian random variable with zero mean and unit variance.
Ii System Model
Ii-a Singal Model
Consider a -user MISO broadcast channel where a transmitter with multiple antennas sends independent messages to a receiver with a single antenna. The input-output relationship at the -th channel use is given by
where denotes the signal sent by the transmitter, represents the channel vector from the transmitter to user ; and for . We assume that the transmit power at the transmitter satisfies an average constraint . We assume that all the entries of channel vector are drawn from a continuous distribution and the absolute value of all the channel coefficients is bounded between a nonzero minimum value and a finite maximum value to avoid the degenerate cases where channel values are equal to zero or infinity. Each receiver has a perfect estimate of its CSI, i.e. has perfect CSIR.
Ii-B Periodic CSI Feedback Model
Ii-B1 Model 1 (CSI feedback frequency limited model)
A feedback cycle ( time slots) consists of two sub durations: a non-feedback duration ( time slots) and a CSI feedback duration ( time slot). During the non-feedback duration, each user cannot send back CSI to the transmitter at all. Starting with the time slot, each user sends it back to the transmitter until time slots. In the first time slot in the feedback duration, the user sends back both current CSI and the estimated CSI obtained from the previous non-feedback durations. From time slot to , only current CSI is provided to the transmitter . For example, user sends back outdated CSI and one current CSI, through the feedback link at time slot. From time slot to , the only current CSI is sent back to the transmitter . Since the channel values are independently changed at every time slot, the transmitter is able to obtain outdated or obsolete CSI and current CSI within a feedback cycle , where . The CSI is provided through a noiseless and delay-free feedback link at regularly-spaced time intervals.
In this model, we define a parameter that measures the normalized feedback speed, i.e., feedback frequency, which is
For example, if , then the transmitter obtains instantaneous CSI over all time slots because the feedback is frequent enough to track all channel variations, i.e., the perfect CSIT case. Alternatively, if , then the transmitter cannot obtain any CSI from the receivers, i.e., no CSIT case.
Ii-B2 Model 2 (CSI feedback link delay limited model)
We consider an ideal block fading channel where the channel values are constant for the channel coherence time and change independently between blocks. Each user sends backs CSI to the transmitter every time slots periodically where denotes channel coherence time. If we assume that the feedback delay time is less than the channel coherence time, i.e., , then the transmitter learns the CSI time slot after it was sent by the user. For example, if a user sends back CSI at time slot , the transmitter has CSI at time slot in our model.
Let us define a parameter for the ratio between the CSI feedback delay and the channel coherence time. We call this the normalized CSI feedback delay:
We refer to the case where as the completely outdated CSI regime as considered in . In this case, only completely outdated CSI is available at the transmitter. We refer to the case where as the current CSI point. Since there is no CSI feedback delay, the transmitter can use the current CSI in each slot. As depicted in Fig. 1-(b), if , the transmitter is able to exploit instantaneous CSI over two-thirds of the channel coherence time and outdated CSI for the previous channel blocks.
Ii-C Sum-DoF Region with Periodic CSI Feedabck
Since the achievable data rate of the users depends on the parameters or and SNR, it can be expressed as a function of and SNR. Using this notion, for codewords spanning channel uses, a rate of user is achievable if the probability of error for the message approaches zero as . The sum-DoF characterizing the high SNR behavior of the achievable rate region is defined as
Iii Space-Time Interference Alignment
Before proving our main results, we explain the space-time interference alignment (STIA) signaling structure. The periodic CSI feedback protocol motivates us to develop the STIA because it exploits current and outdated CSI jointly. In this section, we focus on the special case of , , and to provide an intuition why joint use of both CSI is useful. We generalize this construction to a more general set of parameters in the sequel.
Iii-a Precoding and Decoding Structure
In this section, we will show that of DoF (outer bound) is achieved for the -user vector broadcast channel when ( and ) under the assumptions of Model 1 as depicted Fig. 2-(a). The proposed STIA algorithm consists of two phases. The first phase involves sending multiple symbols per user to create an interference pattern without CSI knowledge at the transmitter. The second phase involves sending linear combinations of the same symbols to recreate the same interference patterns using both current and outdated CSI. Using the fact that each user receives the same interference patterns during the two phases, it cancels the interference signals to obtain the desired symbols in the decoding procedure.
Iii-A1 Phase One (Obtain the Interference Pattern)
This phase consists of one time slot. During phase one, the transmitter does not have any CSI knowledge according to Model I because each user cannot feedback CSI during the non-feedback duration, i.e., . In time slot 1, the transmitter sends six independent symbols where and intended for user 1, and intended for user 2, and and intended for user 3 without precoding as
where . Neglecting noise at the receiver, the observation at each receiver is a function of three linear combinations:
where denotes a linear combination seen by user for the transmitted symbols for user at the -th time slot:
In summary, during phase one, each receiver obtains a linear combination of one desired equation and two interference equations for .
Iii-A2 Phase Two (Generate the Same Interference Pattern)
The second phase uses two time slots, i.e. . In this phase, the transmitter has knowledge of both current and outdated CSI thanks to feedback. For example, during time slot 2, the transmitter has outdated CSI for the first time slot and current CSI for the second time slot. Using this information, in time slot 2 and time slot 3, the transmitter sends simultaneously two symbols for the dedicated users by using linear beamforming as
where denotes the beamforming matrix used for carrying the same symbol vector at time slot , where and . The main idea for designing precoding matrix is to allow all the receivers observe the same linear combination for interference signals received during time slot 1 by exploiting current and outdated CSI. For instance, user 2 and user 3 received the interference signals in the form of and . Therefore, to deliver the same linear combination for the undesired symbols to user 2 and user 3 repeatedly during , the transmitter constructs the beamforming matrix carrying symbols, and to satisfy
Similarly, to ensure the interfering users receive the same linear combination of the undesired symbols, which is linearly dependent (aligned) with the previously overheard equation during time slot 1, the beamforming matrices carrying data symbols for user 2 and user 3 are constructed to satisfy the space-time inter-user interference alignment conditions:
Since we assume that the channel coefficients are drawn from a continuous distribution, matrix inversion is guaranteed with high probability. Therefore, it is possible to construct transmit beamforming matrices , and as
If we denote and for , at time slot 2 and time slot 3, the received signals at user 1 are given by
If we denote and for , the received signals at user 2 during time slot 2 and 3 are given by
Finally, for user 3, if we denote and for , the received signals at time slot 2 and 3 are given by
To explain the decoding process, let us consider the received signals at user 1. In time slot 1, user 1 acquired knowledge of the interference signals in the form of and . From phase two, user 1 received the same interference signals and at time slot 2 and 3 as shown in (16) and (17). Therefore, to decode the desired signals, the observation made during phase two are subtracted from the observation in phase one, resulting in the equations:
After removing the interference signals, the effective channel input-output relationship for user 1 during the three time slots is given by
Since precoding matrix for was designed independently of channel , the elements of the effective channel vector observed at time slot 2 and 3, i.e., and are also statistically independent random variables. This implies that the three channel vectors, , , and are linearly independent. Therefore, with a probability of one because all channel elements are drawn from a continuous distribution. As a result, user 1 decodes two desired symbols within three time slots. In the same way, user 2 and user 3 are able to retrieve a linear combination of their desired symbols by removing the interference signals and can use the same decoding method. Since the transmitter has delivered two independent symbols for its intended user in three channel uses, a total DoF are achieved.
Iii-B The Relationship between STIA and Wireless Index Coding
In the above example, we showed that the proposed STIA achieves the optimal DoF for the 3-user vector broadcast channel without using current CSI over all channel time slots. We can interpret STIA from a wireless index coding perspective. The index coding problem was introduced in  and has been studied in subsequent work -. Suppose that a transmitter has a set of information messages for multiple receivers and each receiver wishes to receive a subset of while knowing some another subset of as side-information. The index coding problem is to design the best encoding strategy at the transmitter, which minimizes the number of transmissions while ensuring that all receivers can obtain the desired messages. To illustrate, consider the case where a transmitter with a single antenna serves two single antenna users. Suppose each user has the message for the other user as side-information, i.e., user 1 has and user 2 has . In this case, an index coding method is to send the linear sum of two messages to both receivers. Since user 1 and 2 receive and ignoring the noise and assuming the flat fading channel, each receiver can extract the desired message by using side-information as and . STIA mimics this index coding algorithm in a more general sense by using current and outdated CSI jointly. This is because, during phase one, each user acquires side-information in a form of a linear combination of all transmitted data symbols where the linear coefficients are created by a wireless channel. During the second phase, the transmitter constructs the precoding matrices by using outdated and current CSI so that each user can exploit the received interference equations in the first phase as side-information, which results in minimizing the number of transmissions; it leads to an increase in the DoF.
Iii-C Extensions of STIA Algorithm
We proposed the STIA algorithm as a way to take advantage of current and past CSI in the vector broadcast channel. Now, we make some remarks about how the STIA algorithm can be modified and extended to deal with more complex networks and practical aspects.
Remark 1 (Extension into interference networks  ): The proposed algorithm is directly applicable to the -user MISO interference channel where each transmitter has antennas. The reason is that the beamforming matrices used in the second phase are independently generated without knowledge of the other user’s data symbols. In other words, data sharing among the different transmitters is not required to apply the STIA in the -user MISO interference channel. For example, the of DoF gain is also achievable for the -user MISO interference channel with the periodic feedback model when .
Remark 2 (Transmission power constraint): Since it is assumed that the transmitter sends a signal with large enough power in the DoF analysis, the beamforming solutions containing matrix inversion do not violate the transmit power constraint. In practice, however, the proposed STIA algorithm needs to be modified so that the power constraint is satisfied. This modification may incur the sum-rate performance loss but does not affect the DoF gain. The sum-rate maximization or mean square error minimization problems of the STIA algorithm given transmit power constraint can be investigated in future work.
Remark 3 (Effective channel estimation in multi-carrier systems): In the STIA algorithm, each receiver requires the effective channel during phase two. For example, user has to know and when it decodes the desired signal as shown in (24). This effective channel knowledge may be acquired at the receiver if the transmitter sends this information as data to the user by spending other downlink channel resources, i.e., a feed-forward information. This feed-forward, however, is a source of overhead in the system. Assuming a multi-carrier system, effective channel knowledge at the receiver can be obtained without feed-forward processing. The main idea for the effective channel estimation at the receiver is to use a demodulation reference signal and an interference cancellation technique.
Let us explain this by giving an example that uses a multi-carrier system, which allows channel coherence in the frequency domain to be exploited. Consider a multi-carrier system where the channel values are the same across channel coherence band , i.e., for and . When the transmitter sends the training signal (demodulation reference signal) by the same precoding method used for data signal as through subcarrier where and . The received signal at time slot and subcarrier is given by
where the last equality comes from the fact that for by the STIA algorithm. Note that each receiver has knowledge of training signal and the channel at time slot 1 . Hence, user 1 can extract out the desired equation for the channel estimation at subcarrier as where . As an example, the receiver could estimate the effective channel by solving the equation,
Since the size of effective channel is , if the transmitter uses an independent training signal over subcarriers among total subcarriers, it is estimated reliably, if the noise effect is ignored. Therefore, the STIA algorithm can be implemented without forwarding the effective channel to the users. Further, the overhead required for channel estimation is the same as that required for a multiuser MISO system. This implies that the STIA algorithm does not require additional overhead for learning the effective channel in multi-carrier systems.
Iv An Inner Bound of Sum-DoF Region
In this section, we characterize an achievable sum-DoF region as a function of feedback frequency for the -user MISO broadcast channel. The main result is established in the following theorem.
For the -user MISO broadcast channel with the periodic CSI feedback (Model 1), the achievable sum-DoF region is characterized as a function of feedback frequency ,
where and .
We prove Theorem 1 by four different transmission methods, each of which achieves four different corner points of the region as shown in Fig. 3. Since time sharing can be used to achieve the lines connecting each point, we focus on characterizing the conner points.
Iv-a Achievablity of Point A and D
Achievability for the corner point A when is shown by TDMA. Also, achievability for the point D when is proven by ZF. When receivers do not send back CSI to the transmitter, i.e., , then the TDMA method is used to achieve . At the other extreme, when the transmitter has current CSI over all time slots at , it is possible to achieve the cut-set bound DoF by the ZF method.
Iv-B Achievability of Point B
Let us consider , where the non-CSI feedback duration has time slots and the CSI feedback duration has time slots. For this case, we show that DoF are achievable by STIA using time slots in the first phase and time slots in the second phase.
Iv-B1 Phase One
The first phase has time slots. Since the transmitter does not have channel knowledge in this phase, it sends independent data symbols using spatial multiplexing on antennas without using any precoding so that each user receives one linear equation for desired symbols while overhearing other transmissions, each of which contains undesired information symbols. During time slot , the transmitter sends the desired symbols for user , through antennas. Hence, the received signal at user in this phase is given by
where and the noise term was dropped for simplicity because it does not affect the DoF calculation.
Iv-B2 Phase Two
The second phase spans time slots. In this phase, the transmitter has CSI for the current and past time periods because each user sends back both outdated and current CSI from the time slots in phase one.
The main objective of phase two is for the transmitter to provide each user with additional observations that can be used to build additional linearly independent equations in the desired symbols. Since each user has one observation from phase one containing its symbols, if it obtains additional linearly independent equations from this phase, then desired symbols can be decoded at each user. To deliver desired symbols to each user through phase two, the transmitted signal at time slot is given by
where . Notice that the transmitter repeatedly sends the same symbol vector regardless of time index , but using a different beamforming matrix , which varies according to time index. When the transmitter sends the signal in (29) at time slot , the received signal at user is given by