SpaceTime 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 [19] at IEEE 50th Annual Allerton Conference on Communication, Control, and Computing, Monticello, Oct. 2012.
Abstract
This paper characterizes the degrees of freedom (DoF) regions for the multiuser vector broadcast channel with periodic channel state information (CSI) feedback. As a part of the characterization, a new transmission method called spacetime interference alignment is proposed, which exploits both the current and past CSI jointly. Using the proposed alignment technique, an inner bound of the sumDoF 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of the result is that the achievable sumDoF gain is improved significantly when a user sends back both current and outdated CSI compared to the case where the user sends back current CSI only.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 tradeoff between CSI feedback delay and the sumDoF gain is characterized for the multiuser 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.
I Introduction
Channel state information at the transmitter (CSIT) is important for optimizing wireless system performance. In the multipleinputsingleoutput (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 [1][2]. Prior work on the MISO broadcast channel focused on the CSIT uncertainty caused by limited rate feedback [5][9]. 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 In particular, if a transmitter has past CSI that is independent of the current CSI, i.e., outdated CSIT, it was commonly believed that only unity DoF gain could be achieved, which is the same DoF gain in the absence of channel knowledge at the transmitter [10]. Recently, it was shown in [11] that outdated CSIT is helpful to obtain the DoF gains beyond that achieved by timedivisionmultipleaccess (TDMA). The key idea from [11] was to exploit the perfect outdated CSIT as sideinformation so that the transmitter aligns interuser interference between the past and the currently received signals. Motivated by the work in [11], extensions have been developed for other networks [12][15]. The DoF gain was studied for the interference channel and the X channel when only outdated CSI is used at the transmitter in [12]. The DoF was characterized for the twouser MIMO interference channel with delayed CSI according to different transmitandreceiveantenna configurations in [13]. The outdated CSI feedback framework was also considered for the twohop interference channel in [14] and [15], where the DoF gain increased beyond that achieved by the TDMA method.
The common assumption of previous work [11][15] 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.
Ia Two Different Models of Periodic CSI Feedback
Unlike aperiodic CSI feedback where each user opportunistically sends back CSI whenever the channel condition is changed [21], 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 [22] 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 twothirds 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 sumDoF gain in this fastfading 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 onethird of channel coherence time, only delayed CSIT is available during onethird of channel coherence time, but both delayed and current CSIT is available in the remaining time as illustrated in Fig 2(b).
IB Contribution
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 sumDoF of the vector broadcast channel in both periodic CSI feedback models. For the feedback frequency limited model, we provide a characterization of the achievable sumDoF region as a function of CSI feedback frequency . As illustrated in Fig. 3, we show that it is possible to increase the achievable sumDoF 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 (cutset bound) without knowledge of all current CSIT, i.e., . This result is appealing because it was believed that the cutset 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 tradeoff region between CSI feedback delay and the sumDoF gain for the multiuser vector broadcast channel by the proposed STIA algorithm and leveraging results in [11]. From the proposed region, we show that the joint use of outdated and current CSIT improves significantly the sumDoF 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 tradeoff region achieves the cutset bound for the range of . Further, we demonstrate that the proposed tradeoff region is optimal for the threeuser vector broadcast channel using a recently derived converse result [30].
To show these results, we propose a new transmission method called spacetime 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 interuser 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 [11] for using both outdated and current CSI.
In terms of related work, new transmission methods combining MAT in [11] and partial zeroforcing (ZF) using both current and outdated CSI were developed for the twouser vector broadcast channel in [16] and [17]. The main difference lies in the system model. In [16] and [17], 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 twouser vector broadcast channel [29] to obtain the DoF gain, which is more closely related to our work. The limitation of work in [29] is that it considers a simple twouser 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 sumDoF 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 tradeoff 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
Iia 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 inputoutput relationship at the th channel use is given by
(1) 
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.
IiB Periodic CSI Feedback Model
IiB1 Model 1 (CSI feedback frequency limited model)
A feedback cycle ( time slots) consists of two sub durations: a nonfeedback duration ( time slots) and a CSI feedback duration ( time slot). During the nonfeedback 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 nonfeedback 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 delayfree feedback link at regularlyspaced time intervals.
In this model, we define a parameter that measures the normalized feedback speed, i.e., feedback frequency, which is
(2) 
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.
IiB2 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:
(3) 
We refer to the case where as the completely outdated CSI regime as considered in [11]. 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 twothirds of the channel coherence time and outdated CSI for the previous channel blocks.
IiC SumDoF 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 sumDoF characterizing the high SNR behavior of the achievable rate region is defined as
(4) 
Iii SpaceTime Interference Alignment
Before proving our main results, we explain the spacetime 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.
Iiia 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.
IiiA1 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 nonfeedback 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
(5) 
where . Neglecting noise at the receiver, the observation at each receiver is a function of three linear combinations:
(6)  
(7)  
(8) 
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 .
IiiA2 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
(9) 
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
(10) 
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 spacetime interuser interference alignment conditions:
(11) 
and
(12) 
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
(13)  
(14) 
and
(15) 
If we denote and for , at time slot 2 and time slot 3, the received signals at user 1 are given by
(16)  
(17) 
If we denote and for , the received signals at user 2 during time slot 2 and 3 are given by
(18)  
(19) 
Finally, for user 3, if we denote and for , the received signals at time slot 2 and 3 are given by
(20) 
(21) 
IiiA3 Decoding
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:
(22) 
(23) 
After removing the interference signals, the effective channel inputoutput relationship for user 1 during the three time slots is given by
(24) 
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.
IiiB The Relationship between STIA and Wireless Index Coding
In the above example, we showed that the proposed STIA achieves the optimal DoF for the 3user 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 [25] and has been studied in subsequent work [26][28]. 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 sideinformation. 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 sideinformation, 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 sideinformation 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 sideinformation 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 sideinformation, which results in minimizing the number of transmissions; it leads to an increase in the DoF.
IiiC 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 [20] ): 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 sumrate performance loss but does not affect the DoF gain. The sumrate 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 multicarrier 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 feedforward information. This feedforward, however, is a source of overhead in the system. Assuming a multicarrier system, effective channel knowledge at the receiver can be obtained without feedforward 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 multicarrier system, which allows channel coherence in the frequency domain to be exploited. Consider a multicarrier 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
(25) 
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,
(26) 
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 multicarrier systems.
Iv An Inner Bound of SumDoF Region
In this section, we characterize an achievable sumDoF region as a function of feedback frequency for the user MISO broadcast channel. The main result is established in the following theorem.
Theorem 1
For the user MISO broadcast channel with the periodic CSI feedback (Model 1), the achievable sumDoF region is characterized as a function of feedback frequency ,
(27) 
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.
Iva 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 [10]. At the other extreme, when the transmitter has current CSI over all time slots at , it is possible to achieve the cutset bound DoF by the ZF method.
IvB Achievability of Point B
Let us consider , where the nonCSI 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.
IvB1 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
(28) 
where and the noise term was dropped for simplicity because it does not affect the DoF calculation.
IvB2 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
(29) 
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