On Fading Broadcast Channels with Partial Channel State Information at the Transmitter

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1 On Fading Broadcast Channels with Partial Channel State Information at the Transmitter Ravi Tandon 1, ohammad Ali addah-ali, Antonia Tulino, H. Vincent Poor 1, and Shlomo Shamai 3 1 Dept. of Electrical Engineering, Princeton University, Princeton, NJ, USA. Bell Laboratories (Alcatel-Lucent), Holmdel, NJ, USA. 3 Dept. of Electrical Engineering, Technion, Israel Institute of Technology, Haifa, Israel Abstract The two user multiple-input multiple-output (IO) broadcast channel is studied in which the channel from one of the receivers is known instantaneously and perfectly at the transmitter, whereas the channel from the other receiver is known in a delayed manner. The degrees of freedom (DoF) region of the two-user (,,) IO broadcast channel under this model is completely characterized. The scheme illustrates the joint utilization of current and past knowledge of the channel state information at the transmitter. I. INTRODUCTION A typical assumption while studying fading broadcast and interference networks is the instantaneous availability of the global channel state information at the transmitter (CSIT). However, in practical scenarios, this assumption is not well justified since the channels are usually estimated at the receivers, and then fed back to the transmitter. It is therefore of utmost importance to understand the fundamental performance limits of fading networks under the assumption of delayed CSIT. To this end, we consider the two-user multiple-input multiple-output (IO) broadcast channel (BC) with fast fading, in which the channels to the receivers change identically and independently over time. As a starting motivation, consider the two-user multipleinput single-output (ISO) broadcast channel, with two transmit antennas and two single antenna receivers. Under the perfect CSIT assumption, two degrees of freedom (DoF) can be achieved via zero-forcing beamforming [1]. However, under the no CSIT assumption, the optimal sum DoF collapses to one [], [3]. Rather surprisingly, addah-ali and Tse showed in [4] that under the delayed CSIT assumption, the optimal sum DoF is 4/3, i.e., completely outdated CSIT can provide DoF gains. The interesting aspect of this result is the utilization of side information at the receivers via feedback. All of these problems consider situations that do not distinguish the relative form of CSIT from the point of view of the receivers, i.e., CSIT is either perfect or delayed or absent from both of the receivers. The research of H. V. Poor and R. Tandon was supported by the Air Force Office of Scientific Research URI Grant FA The work of S. Shamai was supported by the Israel Science Foundation (ISF), and the Philipson Fund for Electrical Power. {rtandon, poor}@princeton.edu, {mohammadali.maddahali,a-tulino}@alcatel-lucent.com, sshlomo@ee.technion.ac.il. Recently, an interesting model has been investigated for the two-user ISO BC [5], [6], in which the channel to each receiver has two components: one component is available instantaneously at the transmitter, whereas the other component is available after unit delay. Depending on the relative amount of perfect versus delayed CSIT, which is quantified by a parameter [0, 1], it is shown that the optimal sum DoF for this model is (4+ )/3. When =1, which corresponds to the case of full instantaneous CSIT, one obtains DoF, whereas in the other extreme when =0, the result of 4/3 with fully delayed CSIT is recovered. Another feasible scenario could be one in which the channel to one of the receivers (say receiver 1) is available perfectly at the transmitter; whereas the channel to receiver is available in a delayed manner. This situation can arise if receiver 1 is operating in a controlled environment and the delay in the feedback is insignificant; whereas the feedback channel from receiver suffers significant delay. We call this model as one with partially perfect CSIT. In particular, we consider the case in which there are -transmit antennas, and two receivers, with and receive antennas respectively. The partial CSIT assumption is as follows: the channel to receiver 1 is available instantaneously and perfectly; the channel to receiver is available in a delayed manner. The main contribution of this work is the characterization of the DoF region of the (,, )-IO BC with partially perfect CSIT. As a comparison to 4/3 (optimal sum DoF for (, 1, 1)- ISO BC under delayed CSIT), we show that under the partial CSIT assumption, the optimal sum DoF increases to 3/. The main idea behind the scheme is the joint utilization of two resources: a) the instantaneous knowledge of the channel to receiver 1 via zero-forcing beamforming; and b) the delayed channel knowledge of receiver to use side-information at receiver. We also remark here that the case of (, 1, 1)-ISO BC has also been treated in [7], where the optimal sum DoF of 3/ is reported. II. TWO-USER IO BC WITH PARTIALLY PERFECT CSI We consider the two-user (,, )-IO broadcast channel with fast fading, in which the channel outputs at the /1/$ IEEE 1004

2 H 1 (t) Rx 1 H 1 (t) (W 1,W ) Fig. 1. Tx (t 1) H H (t) Rx The IO broadcast channel with partially perfect CSI. receivers are given as follows: Y 1 (t) =H 1 (t)x(t)+z 1 (t) Y (t) =H (t)x(t)+z (t), where X(t) is the transmitted signal; H 1 (t) C N1 denotes the channel between the transmitter and receiver 1; H (t) C N denotes the channel between the transmitter and receiver ; and Z j (t) CN(0,I Nj ), for j = 1,, is the additive noise at receiver j. The power constraints at the transmitter is given as E X(t) P, for 8 t. It is assumed that the components [H j ] nj,m are statistically equivalent for all m {1,...,}, and n j {1,...,N j }, for j =1,. We focus on the IO-BC model under the following assumptions (see Figure 1): H 1 (t) is available instantaneously at the transmitter; H (t) is available at the transmitter in a delayed manner. Furthermore, it is assumed that both receivers have global channel state information (CSI). We denote the transmitter by Tx and the receivers by Rx 1 and Rx. A coding scheme with block length T for the IO BC with partial CSIT consists of a sequence of encoding functions X(t) =f t W 1,W, H t 1, H t 1, defined for t =1,...,T, and two decoding functions Ŵ j = g T j (Y T j, H T 1, H T ), for j =1,. A rate pair (R 1 (P ),R (P )) is achievable if there exists a sequence of coding schemes such that P(W j 6= Ŵj)! 0 as T!1for j =1,. The capacity region C(P ) is defined as the set of all achievable rate pairs (R 1 (P ),R (P )). We define the DoF region as follows: n D = (,d ) d j 0, and 9(R 1 (P ),R (P )) C(P ) Ŵ 1 Ŵ R j (P ) o s.t. d j = lim P!1 log (P ),j =1,. We state our main result in the following theorem: Theorem 1: The DoF region of the (,, )-IO broadcast channel with perfect CSI from receiver 1 and delayed CSI from receiver is given by min(, ) (1) min(, + ) + d 1. () min(, ) To put this result into perspective, we also recall the DoF regions with perfect CSI of both receivers and with delayed CSI [4], [8] from both receivers. DoF region with perfect CSI from both receivers: min(, ) (3) d min(, ) (4) + d min(, + ). (5) DoF region with delayed CSI from both receivers: min(, + ) + d 1 min(, ) (6) min(, ) + d 1. min(, + ) (7) As a comparison with the result in [8] for delayed CSI from both receivers, we note from Theorem 1 that the constraint in (7) becomes redundant and the single user constraint min(, ) becomes active. In the next section, we first highlight the coding scheme through an example, which shows that zero-forcing pre-coding along with utilization of side-information are jointly required to achieve the optimal DoF region. III. (, 1, 1)-ISO BC WITH PARTIALLY PERFECT CSI To illustrate the main idea behind the coding scheme, we first focus on the (, 1, 1)-ISO BC. We assume that instantaneous CSI corresponding to receiver 1 is available at the encoder and the CSI corresponding to receiver is available in a delayed manner. For this example, the DoF region in Theorem 1 simplifies to 1 (8) + d 1. (9) From Figure (), it is clear that to show the achievability of the DoF region above, we need to show that the pair (,d )=(1, 1/) is achievable under the assumption of partial perfect CSI. The remaining region is achievable by time-sharing between this point and the corner points (1, 0) and (0, 1). Henceforth, we focus on achieving the pair (,d )=(1, 1/). A. Achieving (1, 1/) for (, 1, 1)-ISO BC The scheme works over two channel uses, in which we will show reliable transmission of two symbols (u 1,u ) to Rx 1 and one information symbol v to Rx. In the first channel use, knowing the channel to Rx 1, the transmitter can zeroforce the interference caused by the v symbols at Rx 1. At 1005

3 d d 1 ( 3, 3 ) Partially = Partially (1, 1 ) (, ) No CSI 1 Fig.. DoF regions for the (, 1, 1) ISO broadcast channel. Fig. 3. DoF regions for t =1, the transmitter sends X(1) = u + Bv. (10) The output at Rx 1 (ignoring noise) is given as Y 1 (1) = H 1 (1)X(1) (11) = H 1 (1) + H u 1 (1)Bv (1) = H 1 (1) (13) u, L(u 1,u ), (14) where (13) follows from the fact that the 1 vector B is chosen such that H 1 (1)B =0, i.e., in the first channel use, the transmitter uses zero-forcing so as to not cause interference at receiver 1 by using its perfect knowledge of H 1 (1). The output at Rx (ignoring noise) is given by Y (1) = H (1)X(1) (15) = H (1) + H u (1)Bv (16) = I(u 1,u )+H (1)Bv. (17) After t =1, having access to delayed CSI, i.e., H (1), the transmitter can reconstruct the interference caused at Rx, i.e., I(u 1,u ). At t =, it sends this reconstructed interference (which is a scalar for this example) on one of the antennas: I(,u X() = ). (18) After t =, both Rx 1 and Rx have access to I(u 1,u ). Using L(u 1,u ) and I(u 1,u ), receiver 1 can reliably decode the two information symbols (u 1,u ). Having I(u 1,u ), receiver can cancel this interference from Y (1) and decode v reliably via channel inversion. Hence, the pair (1, 1/) is achievable. IV. CODING SCHEE: ARBITRARY (,, ) We present encoding schemes depending on the relative values of (,, ). We first focus on cases in which. A. For this case, the DoF regions for perfect, delayed and partially perfect settings are the same and are given by + d. This region is trivially achievable by time-sharing between the points (,d )=(,0) and (0,) respectively. B. < In this case, the DoF region with partially perfect CSI is the same as that with perfect CSI, and is given by, + d (19) whereas the region with delayed CSI is given by + d 1. (0) Figure 3 shows the typical DoF regions when <. To show the achievability of the region with partially perfect CSI, we need to show the achievability of the pair (, ). This is feasible by zero-forcing the interference at Rx 1 alone, and without any CSI (not even delayed) from Rx. This is because Rx has enough antennas ( ) to decode all information symbols. C. ( + ) The (achievable) DoF region with partially perfect CSI is The DoF region with delayed CSI is (1) + d 1. () + d 1, + d 1. (3)

4 d ( + ) Partially N1, ) v-variables and interference variables (each of which is a linear combination of u-symbols). Having access to delayed CSI from Rx, {H (t)} N1 t=1, the transmitter can reconstruct {H (t)s(t)u} N1 t=1 ; hence it can reconstruct these interference symbols. It then creates ( ) linearly independent combinations of these interference symbols. This is feasible since ( + ). We denote these linear combinations of the interference variables by I =[i 1,...,i N1( )] T. (9) In phase, which is of duration ( sends ), the transmitter X(t) = S(t)I + B(t)v, (30) Fig. 4. DoF regions for ( + ) Figure 4 shows the typical DoF regions for this case. To show the achievability of the region with partially perfect CSI, we need to show the achievability of the pair (,d )=, N1. (4) In order to show this, we show that over channel uses, it is possible to transmit symbols to Rx 1 and ( ) symbols to Rx. Let us denote by u =[u 1,...,u N1 ] T (5) the vector of information symbols intended for Rx 1, and by v =[v 1,...,v N( )] T (6) the vector of information symbols intended for Rx. The scheme operates over two phases. Phase 1 is of duration, whereas phase is of duration ( ). During phase 1, the transmitter sends X(t) = S(t)u + B(t)v (7) where S(t) (chosen as full rank) is of size, and B(t) is of size ( ). The matrix B(t) is selected to zero-force the interference caused by v-symbols at Rx 1, i.e., the matrix B(t) is selected to satisfy H 1 (t)b(t) =[0] N1 ( ), for t =1,...,. (8) The channel outputs in this phase are Y 1 (t) =H 1 (t)s(t)u Y (t) =H (t)s(t)u + H (t)b(t)v, for t =1,...,. Therefore, at the end of this phase, Rx 1 has N1 equations in u-variables. Therefore, to decode the information symbol vector u, Rx 1 requires additional ( ) linearly independent equations in u-variables. On the other hand, Rx has equations in ( where S(t) (chosen as full rank) is of size ( ), and the matrix B(t) is selected to satisfy H 1 (t)b(t) =[0] N1 ( The channel outputs for this phase are Y 1 (t) =H 1 (t) S(t)I ), for t = +1,...,. (31) Y (t) =H (t) S(t)I + H (t)b(t)v, for t = +1,...,. Thus, at the end of phase, Rx 1 has a total of N1 + ( ) = equations in u-variables, which can be decoded successfully. At Rx, there are a total of + ( )= equations in ( ) v-variables, and interference variables. Thus, it can successfully decode the v information symbols. D. ( + ) The encoding scheme for this case follows in a similar manner as in the previous case and is therefore omitted. The difference comes only in phase, where the transmitter can directly use the interference symbols (suffered at Rx to be used for subsequent transmission). Remark 1: We now emphasize the key features of the encoding scheme for the case in which ( + ). The first feature is the use of zero-forcing in the first phase to nullify interference caused at Rx 1. Furthermore, depending on the relative values of (,, ), zero-forcing is also required in general during phase. To understand this carefully, if we revisit the encoding scheme for the (, 1, 1)- ISO BC, for which we showed the achievability of (1, 1/), phase does not require zero-forcing. This is because in phase 1, enough linear combinations of useful symbols are available at Rx (even though these symbols are corrupted by interference). Hence, depending on? ( ), we may or may not require zero-forcing during phase. To be precise, for the cases in which ( ), only interference forwarding suffices which simplifies the encoding scheme, and which is indeed the case for the (, 1, 1)-ISO BC

5 V. CONVERSE PROOF The bound min(, ) is trivial, hence we focus on showing that min(, + ) + d 1. (3) min(, ) To this end, let us consider a matrix { e H(t)} T t=1, where eh(t) C min(,n1+n) (33) span( e H(t)) = span H 1 (t), H (t) T, (34) and, { H(t)} e T t=1, {H 1 (t)} T t=1, and {H (t)} T t=1 are mutually independent. Now, consider the following artificial channel output: ey (t) = e H(t)X(t)+ Z(t), (35) where Z(t) CN(0,I min(,n1+)), and is independent of all other random variables in the model. In addition, we assume that { H(t)} e T t=1 is available at both receivers, but not available at the transmitter. Consider the original broadcast channel (OBC): Channel outputs: Y 1 at Rx 1, Y at Rx. CSI at Tx: H 1 instantaneous, H delayed. CSI at Rx 1, Rx : H 1, H instantaneous. For simplicity, we have dropped the index (t) in the rest of this section. We next enhance OBC to create a physically degraded broadcast channel (DBC): Channel outputs: (Y 1,Y ) at Rx 1, Y at Rx. CSI at Tx: H 1 instantaneous, H delayed. CSI at Rx 1, Rx : H 1, H instantaneous. Using the fact that feedback does not increase the capacity region of a physically degraded broadcast channel [9], we can drop the assumption of delayed CSI of H without changing the capacity region of DBC. We denote the new channel as DBC : Channel outputs: (Y 1,Y ) at Rx 1, Y at Rx. CSI at Tx: H 1 instantaneous, H -unknown. CSI at Rx 1, Rx : H 1, H instantaneous. We further enhance DBC by giving the artificial output ey (as defined in (35)) to receiver 1. We denote the resulting channel as DBC : Channel outputs: (Y 1,Y, e Y ) at Rx 1, Y at Rx. CSI at Tx: H 1 instantaneous, ( e H, H )-unknown. CSI at Rx 1, Rx : H 1, H, e H1 instantaneous. We now focus on the channel DBC, and observe that from (34), by construction, the row-span of e H equals the row-span of (H 1, H ). Therefore, (Y 1,Y ) can be reconstructed from e Y and (H 1, H, e H) within noise-distortion. Thus, from the above argument, the DoF region of DBC is the same as that of DBC : Channel outputs: e Y at Rx 1, Y at Rx. CSI at Tx: H 1 instantaneous, ( e H, H )-unknown. CSI at Rx 1, Rx : H 1, H, e H instantaneous. In this final channel DBC, the channel gains of (Y, e Y ) are unknown at the transmitter. oreover we have a broadcast channel with min(, + ) antennas at Rx 1 and antennas at Rx. oreover, for the two-user (,, )- IO broadcast channel with no CSI at the transmitter, the degrees of freedom region is given as [], [3] min(, ) + d 1. (36) min(, ) Using (36) for DBC, we arrive at the desired bound: min(, + ) + d 1. (37) min(, ) This completes the proof for the converse. VI. CONCLUSIONS The DoF region of the two-user IO BC has been characterized under the assumption of instantaneous CSIT from one receiver and delayed CSIT from the other receiver. We are also investigating extensions of this work to K-user ISO broadcast channels. In particular, for the K-user ISO BC, there can be perfect CSIT from k receivers and delayed CSIT from (K k) receivers, i.e., this formulation leads to a total of (K + 1) different CSIT configurations, out of which only two configurations are fully understood, corresponding to either perfect CSIT from every receiver (k = K) or delayed CSIT from every receiver (k =0) [4]. VII. ACKNOWLEDGEENT The authors would like to express our sincere gratitude to Dr. Syed A. Jafar for helpful comments which led to strengthening of the results in this paper. REFERENCES [1] G. Caire and S. Shamai. On the achievable throughput of a multiantenna Gaussian broadcast channel. IEEE Trans. Inform. Theory, 49(7): , Jul [] C. Huang, S. A. Jafar, S. Shamai, and S. Viswanath. On degrees of freedom region of IO networks without channel state information at transmitters. IEEE Trans. Inform. Theory, 58(): , Feb. 01. [3] C. S. Vaze and. K. Varanasi. The degrees of freedom regions of IO broadcast, interference, and cognitive radio channels with no CSIT, [arxiv: ]. Submitted to IEEE Trans. Inform. Theory, Sep [4]. A. addah-ali and D. Tse. Completely stale transmitter channel state information is still very useful. IEEE Trans. Trans. Theory, 58(7): , Jul. 01. [5] S. Yang,. Kobayashi, D. Gesbert, and X. Yi. Degrees of freedom of time correlated ISO broadcast channel with delayed CSIT, [arxiv: ]. Submitted to IEEE Trans. Inform. Theory, ar. 01. [6] T. Guo and S. A. Jafar. Optimal use of current and outdated channel state information degrees of freedom of the ISO BC with mixed CSIT, [arxiv: ]. ar. 01. [7] H. aleki, S. A. Jafar, and S. Shamai. Retrospective interference alignment over interference networks. IEEE Journal of Selected Topics in Signal Processing, 6(3):8 40, Jun. 01. [8] C. S. Vaze and. K. Varanasi. The degrees of freedom regions of twouser and certain three-user IO broadcast channels with delayed CSIT [arxiv: v]. Submitted to IEEE Trans. Inform. Theory, Dec [9] A. El Gamal. The feedback capacity of degraded broadcast channels. IEEE Trans. Inform. Theory, 4(3): , ay