On the Degrees of Freedom of the Finite State Compound MISO Broadcast Channel

Transcription

1 On the Degrees of Freedom of the Finite State Compound MISO Broadcast Channel Invited Paper Chenwei Wang, Tiangao Gou, Syed A. Jafar Electrical Engineering and Computer Science University of California, Irvine, Irvine, CA, USA {chenweiw, tgou, Abstract We explore the degrees of freedom (DoF) of the finite state compound multiple-input single-output (MISO) broadcast channel (BC) in this paper. In prior work, Weingarten et. al. have found inner and outer bounds on the DoF with users. The bounds have a different character. While the inner bound collapses to unity as the number of states increases, the outer bound does not diminish with the increasing number of states beyond a threshold value. It has been conectured that the outer bound is loose and the inner bound represents the actual DoF. In the complex setting (all signals, noise, and channel coefficients are complex variables) we solve a few cases to find that the outer bound and not the inner bound of Weingarten et. al. is tight. In other words, the DoF results are robust to channel uncertainty at the transmitter. I. INTRODUCTION It is well known that the MIMO point to point channel and the MIMO multiple access channel do not lose any DoF due to the lack of channel state information at the transmitters (CSIT). Evidently, this is because the combination of oint processing of all received signals and perfect channel state information at the receiver (CSIR) is able to compensate for the lack of CSIT. However, for most other MIMO networks the DoF are not believed to be robust to channel uncertainty at the transmiter. Consider, for example, the MIMO broadcast channel with M antennas at the transmitter and N, N antennas at the two receivers. With perfect channel knowledge this channel has a total of min(m, N +N ) DoF, which is the same as with perfect cooperation at the receivers. However, in the ergodic time-varying i.i.d. Rayleigh fading case for example, it is known that with no CSIT, the MIMO broadcast channel loses DoF to the extent that time-division between users is optimal for all points in the DoF region. For most networks where the antennas are not disproportionately located on the receivers such as networks of single antenna nodes the DoF penalty due to the total lack of CSIT can be quite severe. For example, under the i.i.d. fading assumption (independent identically distributed across all dimensions), any distributed network of single antenna nodes has only DoF in the absence of CSIT. This is because all received signals are statistically equivalent and therefore any receiver can decode all the messages. Since a receiver with only antenna can decode all messages, the sum DoF cannot be more than. This work was supported by NSF grants and CCF and by ONR YIP grant N The DoF loss due to lack of CSIT is very significant for larger networks. While perfect CSIT is an overly optimistic assumption, the complete lack of CSIT is overly pessimistic. The collapse of DoF in the total absence of CSIT, while sobering, is not a comprehensive argument against the potential benefits of interference alignment in particular or structured coding approaches in general. Hence the need to investigate the behavior of DoF under partial channel knowledge. Two kinds of approaches have been followed in this regard. The first approach investigates how the quality of CSIT should improve as SNR increases, in order to retain the same DoF as possible with perfect CSIT 5. The second approach seeks the impact on DoF of a fixed amount of channel uncertainty that is independent of SNR. References 6, 7 take this approach for the two user MISO BC. Weingarten et. al. 7 investigates a finite state compound channel setting where a specific channel state is drawn (unknown to the transmitter) from a finite set of allowed states and the chosen state is held fixed throughout the duration of communication. As large as this set may be, its finite cardinality restricts the channel uncertainty at the transmitter to a space of zero measure. For example, with M = antennas at the transmitter, the best outer bound on the DoF in is equal to 4. The achievability of 4 DoF is shown when each user s channel can be in one of two states. However, as the number of possible channel states for either user (or both users) increases, Weingarten et. al. also conecture that the DoF in their setting should collapse to. Our main contribution in this paper is to settle the conecture in the negative. The central concept involved in this work is the idea of interference alignment, i.e. structuring signals in such a way that undesired signals cast overlapping shadows where they are not desired while they remain distinguishable where they are desired. We present the system model in the next section. The main results are presented as theorems in Section III along with the main ideas needed for the proofs. The detailed proofs are presented in Section IV and the conclusions are summarized in Section V. II. SYSTEM MODEL A compound MISO broadcast channel consists of a transmitter with M > antennas and K single antenna receivers. The channel vector h k associated with user k is drawn from a

2 Fig.. User Compound Broadcast Channel with J = J = set J k with finite cardinality J k. To avoid degenerate cases, we assume the channel coefficients are drawn from a continuous distribution. Thus, almost surely the channel states are generic, e.g., the coefficients are algebraically independent. Once the channel is drawn, it remains unchanged during the entire transmission. While the transmitter is unaware of the specific channel state realization, the receivers are assumed to have perfect channel knowledge. The transmitter sends independent messages W k with rates R k to receiver k =,,..., K, respectively. A rate tuple (R, R,..., R K ) is achievable if each receiver is able to decode its message with arbitrary small error probability regardless of state (realization) of the channel. The received signal of user k corresponding to channel state index is given by y k (n) = h k x(n) + z k (n) k =,..., K =,..., J k () h k = h k,..., hk M is a M channel vector between the transmitter and receiver k under state where {,..., J k. x = x (n),..., x M (n) T is an M transmitted complex vector at time n and satisfies the average power constraint E( x ) P. z k represents independent identically distributed (i.i.d.) zero mean unit variance circularly symmetric complex Gaussian noise. The total number of degrees of freedom d is defined as R + + R K d = lim P log P A two user compound broadcast channel with M =, J = J = J = is shown in Fig.. Remark: The compound broadcast channel is equivalent to a broadcast channel with common messages. This can be seen by considering different states as different users. Now instead of a K user compound broadcast channel, we have a J + + J K user broadcast channel with K common messages, one for each group k, k =,..., K, with J k users. III. COMPOUND MISO BROADCAST CHANNEL The degrees of freedom of the complex compound MISO BC are studied by Weingarten, Shamai and Kramer in 7. The exact DoF are found for some cases and conectures are made for more general scenarios. The achievability of the conectured DoF is established in 7. We start with the first conecture, re-stated here in the terminology of this paper. () Fig.. user compound MIMO BC with J =, J = and J =, J = A. J =, J = J M Conecture : (Weingarten et. al. 7) Consider a complex compound BC with K = users, M antennas at the transmitter, and J =, J = J M possible generic states for users, respectively. Then the total number of DoF is + M J, The basic idea of the achievability provided in 7) is timedivision multiplexing. To disprove this conecture, we provide a specific counter example in the following theorem. Theorem : For the complex compound MISO BC with K = users, M = antennas at the transmitter and J =, J = J = generic channel states for users, respectively, the exact number of total DoF =, Since + M J = 4 < in this case, Conecture is disproved by Theorem. Interestingly, Theorem indicates that the total number of DoF does not decrease even as the number of possible channel states for user increases from to. The reason we are able to achieve more than the conectured DoF in this case, can be attributed to interference alignment schemes inspired by recently developed insights on asymmetric complex signaling 9. Asymmetric complex signaling involves reducing the complex setting to a real setting by treating a complex number as a two dimensional vector with real elements. In this setting, the scenario of Theorem can be seen as a transmitter with 4 antennas, while each receiver has two antennas. Very importantly, this mapping to the real setting introduces some structure in the channel as complex channel coefficients are translated to quaternionic matrix forms. Because of this structure the proof of Theorem involves some subtleties that may distract from the main concept behind the interference alignment. For simplicity of exposition, we defer the fine details of the proof to Section IV and highlight the main concepts in this section. Specifically, in the following explanation we ignore the special structure of the channel matrices and treat them as generic MIMO channels between a 4-antenna transmitter and -antenna receivers. As mentioned before, the user compound MIMO BC is equivalent to a MIMO BC with two common messages, one for each of the two groups. Group k =,, consists of J k users corresponding to J k states of user k in the compound BC, and the users in the same group need to decode the same message (see Fig. ). Since there is only one receiver in group

3 Fig.. compound broadcast channel with 4 antennas at transmitter, antennas at receiver, and J =, J = J =, we omit the index and replace with for simplicity. In addition, we use H, H to denote the channel matrix from the transmitter to user (group ) and receiver in group, respectively. Consider first, an alternate achievable scheme for the case of J =, J =. We let the two receivers in group use arbitrarily picked combining column vectors v and v, respectively, along which they require interference free reception in order to achieve DoF for their desired message. In order to protect these group receivers, the transmitter sends user s message along the directions orthogonal to v T H and v T H. Since the transmitter has 4 antennas i.e., a 4 dimensional transmit signal space it is able to find a two dimensional subspace that is orthogonal to the protected dimensions of user. This allows real streams to be sent to receiver that do not interfere with the chosen directions of any of the receivers in group. On the other hand user s message will be sent along the null space of H. Since this is a rank matrix, it also has a two dimensional null space along which user s message can be transmitted. However, because each of group receivers have chosen only one receive dimension, only one interference free stream is sent. Thus, a total of interference free streams are delivered ( streams to user and stream to every receiver of group ). Since these are real signals, the total DoF achieved is. Now consider the case J =. Suppose the third user in group chooses a combining vector v for its interferencefree desired signal dimension. Now in order to protect the chosen dimensions of group receivers, the signals for user should be transmitted orthogonal to the three 4 vectors v T H, vt H and v T H. Without alignment these three vectors will span a three dimensional space that needs to be protected from user s signal, leaving only dimension at the transmitter to send user s message. However, we wish to allow user to still access interference-free dimensions. To achieve this goal, we need to make these three vectors span only a two dimensional vector subspace as seen by the transmitter, i.e. the three vectors should be linearly dependent (see Fig. ). Equivalently, three column vectors H T v, HT v, HT v should be linearly dependent. Since H T, H T and H T are three 4 generic matrices, the column spaces of any two of the three matrices only have null intersection, In other words, any two of H T v, HT v and H T v cannot be aligned along the same direction. Therefore, we align the vector H T v in the space spanned by H T v and H T v. Mathematically, we have H T v span(h T v H T v ) () The above alignment is the key to achieving more than the conectured DoF in this case. Whereas most interferencealignment schemes, 4, 8, 9 align interfering dimensions in a one-to-one fashion, this is impossible in this case as pointed above. What is needed instead, is an alignment of an interfering dimension within the space spanned by several others. As it turns out, this problem bears a striking resemblance to the interference alignment scheme used in for the MIMO interference channel where the interference vectors from one interferer are aligned in the space spanned by the interference vectors of other interferers. The remaining details of the proof including the impact of channel structure in this case are presented in Section IV. Finally, note that the converse is trivial here because DoF are already shown to be optimal for J = and DoF cannot increase with increasing channel uncertainty. B. J = J = J M The second case captures the setting where the channel states of both users are unknown to the transmitter. Conecture : (Weingarten et. al. 7) Consider a complex compound BC with K = users, M antennas at the transmitter, and J = J = J M possible generic states for users J, respectively. Then the total number of DoF is J M+, For the case that J = M, this conecture is shown to be tight in 7. Note that the collapse of DoF to unity as J increases, also evident here, is already implied by Conecture. The following theorem disproves Conecture through another counter example. Theorem : For the complex compound MISO BC with K = users, M = antennas at the transmitter and J = J = J = generic channel states for each user, the exact number of total DoF = 4, J Since J M+ = 6 5 < 4, Conecture is disproved by Theorem. Once again, Theorem indicates that the total number of DoF does not decrease as the number of possible channel states J for each user increases from to. The proof of Theorem is based on asymmetric complex signaling over multiple channel uses and is shown in. Except for the detailed nuances required to deal with the channel structure imposed by channel extensions, the essence of the proof follows from the same interference alignment ideas outlined in the previous section for Theorem. Theorem and present only specific counter examples to disprove Conectures and. In both cases the DoF are shown to remain unchanged as the number of possible states for one or both users is increased by one. From these results it is

4 still not clear what happens as the number of states continues to increase, due to the limitations of the linear interference alignment approach. If the channel is in the real setting, i.e. where all channel coefficients, signals and noise terms are real variables, we show in that DoF of the compound broadcast channel are still robust. The achievability is the lattice alignment schemes, which were first shown for the K user interference channel, 0. Our results in are extended to the complex setting in 4. IV. PROOF OF THEOREM Proof: The converse is shown in 7. The achievable scheme is interference alignment with asymmetric signaling. Consider the received signal at user k under state in a single time slot. y k = h k x + z k (4) By viewing complex variables as two dimensional vectors, the received signal can be written as Re{x Re{y k Im{y k = H k Im{x Re{z k {{ k Re{x + k Im{z k (5) {{ k 4 Im{x {{ y k : z k k : k H k = {{ x:4 Re{h k Im{hk Re{hk Im{hk Im{h k Re{hk Im{hk Re{hk Thus we convert the original complex MISO BC to a 4 real MIMO BC with a special structure in the channel matrices. On this new real channel, therefore, we need to show the achievability of a total of = DoF. Due to J = in this network, we omit the state index of user and replace with to denote the state of user. The transmitter sends independent data streams x and x to user along with beamforming vectors u and u, respectively. In addition, it sends data stream to user with beamforming vector u. Mathematically, we have x = u x +u x = u u x x (6) U x (7) x = u x (8) Thus the transmit vector is x = x + x, and the received signal vectors of two users are y = H x + z (9) y = H x + z (0) At state of user, we use a combining vector v to get one interference free dimension. Thus the signal vectors of user and user under state after linear combination can be represented as r y = H U x + H u x + z () r = v T y =vt H u x +v T H U x +v T z () In order to decode the desired signals without interference at both users, we ust need to zero force the second item (interference item) of the two equations above. Thus, our goal is to design U, u, v, =,, such that following equations are satisfied. { U T H T v = 0, =,, u T H T = 0 () To satisfy the first condition, the dimension of the column space of H T v H T v H T v cannot be larger than. This is because U T is a 4 matrix, which has a dimensional null space. Since the column spaces of H T and H T only have null intersection, the matrix H T v H T v has rank Therefore, we align H T v into the space spanned by column vectors of H T v H T v. To achieve this aim, we first generate v randomly, then let v H v = H T H T T v (4) Therefore, we can find linearly independent beamforming vectors of U for user that are orthogonal to the column vectors of H T v H T v and vector u for user which is orthogonal to the column vectors of H T such that both users are free of interference. What remains to be shown is that at any receiver, the desired signal vectors after linear combination are linearly independent among themselves. First, consider the desired signal after linear combination at user under state which is given by v T H u x. We only need to show u T H T v 0. Note that u can be arbitrarily chosen as any vector orthogonal to the column vectors of H T. Thus, u T H T v that H T v = 0 implies lies in the column space of H T. This, however, cannot be true since H T, H T are 4 matrices generated i.i.d, the column spaces of H T, H T only have null intersection Second, we consider the desired signal of user which is given by H U x. To separate two data streams carried by x, H U or equivalently U T H T should be a full rank matrix. Recall that U is chosen such that U T H T = 0. Thus, to show H T does v H T v lie in the null space of U T, we only need to prove the following matrix has full rank. H T H T v H T v (5) Since h, h, h are three vectors generated i.i.d., we are able to find two non-zero real coefficients β, β such that h T = β h T + β h T (6)

5 Considering the mapping from the complex channel to a real channel, we can see that the matrix H can also be linearly represented by H, H with the same coefficients. H T = β H T + β H T (7) Since H T H T has full rank almost surely, substituting (7) into (5) and multiplying H T H T to the left hand side of (5) do not change the rank of (5). Therefore, we ust need to prove the following matrix has full rank almost surely. β I v O β I O v (8) Recall that v v = H H T H T T v = B v B v (9) where B, B are both 4 4 full rank matrices with the form Re{bm Im{b B m = m I Im{b m Re{b m, m =, b = det( h T h T )/ det( h T h T ) b = det( h T h T )/ det( h T h T ) (0) where indicates the Kronecker product operation. The channels h are generated i.i.d., thus B is not a scaling version of B Scaling the row vectors and column vectors does not change the rank of a matrix, therefore, we only need to show that the following matrix has full rank I B v O I O B v () Let λ be a linear combination vector and λ, λ be two linear combination scalar coefficients. If the matrix () has full rank, the following equations should have only zero solutions. I B v O I O B v λ λ λ Equivalently we can rewrite it as, { B v λ = B v λ B v λ = λ = 0 () () The first equation implies that B v and B v are along the same direction. However, this is not true since B is not a scalar version of B Thus, the only solution to () is λ = λ = 0 and λ = 0. Therefore, all the column vectors of () are linearly independent In other words, () is a full rank matrix Overall, a total of + = DoF can be achievable almost surely. V. CONCLUSION In this work we settle Weingarten et. al. conectures for the finite state compound MISO BC in the negative, thereby showing that in the finite state compound channel setting, the DoF results based on structured codes are robust to channel uncertainty at the transmitters. In the current line of work, the most important issue that remains unresolved is the robustness of DoF characterizations to compound networks with infinite states or a continuum of states. In this regard, the conecture of Lapidoth et. al. 6 is most relevant, as is the recent work on the DoF of the two user MIMO interference channel. The overarching observation is that the best outer bounds known so far are not able to distinguish between channel uncertainty at the transmitters over a finite set of states or over a continuum of states. On one hand, if indeed the DoF collapse to unity with channel uncertainty over a continuous (non-zero measure) channel space, then better outer bounds are needed that can distinguish this setting from the finite state compound setting. On the other hand, if the DoF are indeed resilient to channel uncertainty over a continuum, then a much finer understanding of statistical interference alignment is needed. In either case, settling this issue will have a profound impact on our understanding of both the capacity limits of wireless networks as well as the robustness of these limits. REFERENCES S. Jafar and M. Fakhereddin, Degrees of Freedom for the MIMO interference channel, IEEE Trans. on Information Theory, vol. 5, No. 7, July C. Huang, S. Jafar, S. Shamai and S. Vishwanath, On Degrees of Freedom Region of MIMO Networks without CSIT, Proceedings of ITA workshop at UC San Diego, 009 (Full paper available on Arxiv) V. Cadambe and S. Jafar, Interference alignment and the degrees of freedom of the K user interference channel, IEEE Trans. on Information Theory, vol. 54, pp , Aug V. Cadambe and S. Jafar, Interference Alignment and the Degrees of Freedom of Wireless X Networks, IEEE Trans. on Information Theory, vol. 55, No. 9, Sep G. Caire, N. Jindal and S. Shamai, On the Required Accuracy of Transmitter Channel State Information in Multiple Antenna Broadcast Channels, IEEE ACSSC 007, pp. 87 9, Nov A. Lapidoth, S. Shamai and M. Wigger, On the capacity of Fading MIMO broadcast channels with imperfect transmitter side-information, arxiv:cs/ v, May H. Weingarten, S. Shamai and G. Kramer, On the compound MIMO broadcast channel, Information Theory Workshop, Jan S. Jafar, S. Shamai, Degrees of Freedom Region for the MIMO X Channel, IEEE Trans. on Information Theory, vol. 54, No., pp. 5-70, Jan V. Cadambe, S. Jafar and C. Wang, Interference Alignment with Asymmetric Complex Signaling - Settling the Host-Madsen-Nosratinia Conecture, e-print, arxiv: , April A.S. Motahari, S. O. Gharan and A. K. Khandani, Real Interference Alignment with Real Numbers, arxiv: , Aug R. Etkin and E. Ordentlich, On the Degrees-of-Freedom of the K-User Gaussian Interference Channel, arxiv: , Jan T. Gou, S. Jafar, Degrees of Freedom of the K User M N MIMO Interference Channel, arxiv: , Sept T. Gou, S. Jafar and C. Wang, On the Degrees of Freedom of Finite State Compound Wireless Networks - Settling a Conecture by Weingarten et. al, arxiv: , Sept M.A. Maddah-Ali, On the Degrees of Freedom of the Compound MIMO Broadcast Channels with Finite States, arxiv: v, Oct. 009.