May 15, 2012 - and provide novel multi-phase broadcast schemes that achieve ... EURECOM, Sophia Antipolis, France (email: {chenji, elia}@eurecom.fr).
Degrees-of-Freedom Region of the MISO Broadcast Channel with General Mixed-CSIT
arXiv:1205.3474v1 [cs.IT] 15 May 2012
Jinyuan Chen and Petros Elia
Abstract—In the setting of the two-user broadcast channel, recent work by Maddah-Ali and Tse has shown that knowledge of prior channel state information at the transmitter (CSIT) can be useful, even in the absence of any knowledge of current CSIT. Very recent work by Kobayashi et al., Yang et al., and Gou and Jafar, extended this to the case where, instead of no current CSIT knowledge, the transmitter has partial knowledge, and where under a symmetry assumption, the quality of this knowledge is identical for the different users’ channels. Motivated by the fact that in multiuser settings, the quality of CSIT feedback may vary across different links, we here generalize the above results to the natural setting where the current CSIT quality varies for different users’ channels. For this setting we derive the optimal degrees-of-freedom (DoF) region, and provide novel multi-phase broadcast schemes that achieve this optimal region. Finally this generalization incorporates and generalizes the corresponding result in Maleki et al. which considered the broadcast channel with one user having perfect CSIT and the other only having prior CSIT.
˜ t = ht − h ˆ t, h
I. I NTRODUCTION In many multiuser wireless communications scenarios, having sufficient CSIT is a crucial ingredient that facilitates improved performance. While being useful, perfect CSIT is also hard and time-consuming to obtain, hence the need for communication schemes that can utilize partial or delayed CSIT knowledge (see [1]–[5]). In this context of multiuser communications, we here consider the broadcast channel (BC), and specifically focus on the two-user multiple-input singleoutput (MISO) BC, where a two-antenna transmitter communicates to two single-antenna receivers. In this setting, the channel model takes the form (1)
= hTt xt + zt
(2)
= gtT xt + zt ,
yt
yt
(1)
(2)
whereas the complete absence of CSIT causes a substantial degradation to just 1/2 DoF per user1 . An interesting scheme that bridges this performance gap by utilizing partial CSIT knowledge, was recently presented in [6] which showed that delayed CSIT knowledge can still be useful in improving the DoF region of the broadcast channel. In the above described two-user MISO BC setting, and under the assumption that at time t, the transmitter knows the delayed channel states (h, g) up to time t − 1, the work in [6] showed that each user can achieve 2/3 DoF, providing a clear improvement over the case of no CSIT. This result was later generalized in [7]–[9] which considered the natural extension where, in addition to the aforementioned perfect knowledge of prior CSIT, the transmitter also had imperfect knowledge of current CSIT; at time t the transmitter ˆ t , gˆt of ht and gt , with estimation errors had estimates h
(1a) (1b)
where for any time instant t, ht , gt ∈ C2×1 represent the (1) (2) channel vectors for user 1 and 2 respectively, where zt , zt represent unit power AWGN where xt is the input signal noise, with power constraint E kxt k2 ≤ P , and where in this case, P also takes the role of the signal-to-noise ratio (SNR). It is well known that in this setting, the presence of full CSIT allows for the optimal 1 degree-of-freedom (DoF) per user, The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. 257616 (CONECT), from the FP7 CELTIC SPECTRA project, and from Agence Nationale de la Recherche project ANR-IMAGENET. J. Chen and P. Elia are with the Mobile Communications Department, EURECOM, Sophia Antipolis, France (email: {chenji, elia}@eurecom.fr) This paper was submitted in part to the Information Theory Workshop (ITW) 2012.
g˜t = gt − gˆt
(2)
having i.i.d. Gaussian entries with power 1 ˜ 2 1 E kht k = E k˜ gt k2 = P −α , 2 2 for some non-negative parameter α that described the quality of the estimate of the current CSIT. In this setting of ‘mixed’ CSIT (perfect prior CSIT and imperfect current CSIT), and for d1 , d2 denoting the DoF for the first and second user over the aforementioned two-user BC, the work in [7]–[9] showed the optimal DoF region to take the form, {d1 ≤ 1; d2 ≤ 1; 2d1 + d2 ≤ 2 + α; 2d2 + d1 ≤ 2 + α} (3) corresponding to a polygon with corner points 2+α {(0, 0), (1, 0), (1, α), ( 2+α nicely 3 , 3 ), (α, 1), (0, 1)}, bridging the gap between the case of α = 0 explored in [6], and the case of α = 1 (and naturally α > 1) corresponding to perfect CSIT. A. Notation and conventions Throughout this paper, (•)−1 , (•)T , (•)H , respectively denote the inverse, transpose, and conjugate transpose of a matrix, while (•)∗ denotes the complex conjugate, and || • || denotes the Euclidean norm. |•| denotes the magnitude of a scalar, and diag(•) denotes a diagonal matrix. Logarithms are of base 2. o(•) comes from the standard Landau notation, where f (x) = . o(g(x)) implies limx→∞ f (x)/g(x) = 0. We also use = to 1 We remind the reader that for an achievable rate pair (R , R ), the 1 2 Ri , i = 1, 2. corresponding DoF pair (d1 , d2 ) is given by di = limP →∞ log P The corresponding DoF region is then the set of all achievable DoF pairs.
. denote exponential equality, i.e., we write f (P ) = P B to log f (P ) = B. Finally, in the spirit of [7]–[9] we denote lim P →∞ log P consider a unit coherence period, as well as perfect knowledge of channel state information at the receivers (perfect CSIR). II. T HE GENERALIZED MIXED -CSIT
BROADCAST
CHANNEL
Motivated by the fact that in multiuser settings, the quality of CSIT feedback may vary across different links, we extend the approach in [7]–[9] to consider unequal quality of current CSIT knowledge for ht and gt . Specifically under the same set of assumptions mentioned above, and in the presence of perfect prior CSIT, we now consider the case where at time t, ˆ t , gˆt of the current ht and gt , the transmitter has estimates h with estimation errors ˜ t = ht − h ˆ t, h
g˜t = gt − gˆt
(4)
Fig. 1. DoF region when 2α1 − α2 < 1 (case 1) and when 2α1 − α2 ≥ 1 2 (case 2). The corner points take the following values: A = (1, 1+α ), B = 2 (α2 , 1), C = ( 2+2α31 −α2 , 2+2α32 −α1 ) and D = (1, α1 ).
having i.i.d. Gaussian entries with power 1 ˜ 2 1 E kht k = P −α1 , E k˜ gt k2 = P −α2 , 0), where one user had perfect CSIT and the other only prior 2 2 for some non-negative parameters α1 , α2 that describe the CSIT. Figure 1 depicts the general DoF region for the case where generally unequal quality of the estimates of the current CSIT 2α 1 − α2 < 1 (case 1) and the case where 2α1 − α2 ≥ 1 for the two users’ links. (case 2). We proceed to describe the optimal DoF region of the We proceed to describe the communication schemes. general mixed-CSIT two-user MISO BC (two-antenna transmitter). The optimal schemes are presented in Section III, III. D ESIGN OF COMMUNICATION SCHEMES FOR THE parts of the proof of the schemes’ performance are presented TWO - USER GENERAL MIXED -CSIT MISO BC in Appendix V, while the outer bound proof is placed in As stated, without loss of generality, we assume that 1 ≥ Appendix VI. α1 ≥ α2 ≥ 0. We describe the three schemes X1 , X2 and X3 that achieve the optimal DoF region (in conjunction with timeA. DoF region of the MISO BC with generalized mixed-CSIT division between these same schemes). Specifically scheme X1 achieves C = ( 2+2α31 −α2 , 2+2α32 −α1 ) (case 1), scheme X2 Without loss of generality, the rest of this work assumes achieves DoF points D = (1, α1 ) (case 1) and A = (1, 1+α2 ) 2 that (case 2), and scheme X3 achieves B = (α2 , 1) (case 1 and 1 ≥ α1 ≥ α2 ≥ 0. (5) case 2). The scheme description is done for 1 > α1 > α2 ≥ 0, and for rational α1 , α2 . The cases where α1 = 1, or α1 = α2 , Theorem 1: The DoF region of the two-user MISO BC with or where α1 , α2 are not rational, can be readily handled with general mixed-CSIT, is given by minor modifications. We proceed to describe the basic notation and conventions used in our schemes. d1 ≤ 1, d2 ≤ 1 (6a) The schemes are designed with S phases (S varies from 2d1 + d2 ≤ 2 + α1 (6b) scheme to scheme), where the sth phase consists of Ts channel d1 + 2d2 ≤ 2 + α2 (6c) uses, s = 1, 2, · · · , S. The vectors hs,t and gs,t will denote the channel vectors seen by the first and second user respectively where the region is a polygon which, for 2α1 − α2 < 1 has during timeslot t of phase s, while h ˆ s,t and gˆs,t will denote corner points the estimates of these channels at the transmitter during the ˜ s,t = hs,t − h ˆ s,t , g˜s,t = gs,t − gˆs,t will 2+2α1 −α2 2+2α2 −α1 same time, and h , ),(α2 ,1),(0,1)}, {(0,0),(1,0),(1,α1 ),( denote the estimation errors. 3 3 ′ Furthermore as,t and as,t will denote the independent and otherwise has corner points information symbols that may be sent during phase-s, timeslot1 + α2 t, and which are meant for user 1, while symbols bs,t and ), (α2 , 1), (0, 1)}. {(0, 0), (1, 0), (1, ′ 2 bs,t are meant for user 2. Vectors us,t and vs,t are the unitThe above corner points, and consequently the entire DoF norm beamformers for as,t and bs,t respectively, chosen so , and so that vs,t is orthogonal inner bound, will be attained by the schemes to be described that us,t is orthogonal ′to gˆs,t ˆ s,t . Furthermore u , v ′ are the randomly chosen unitlater on. The result generalizes the results in [7]–[9] as well as to h s,t s,t ′ ′ the result in [10] which considered the case of (α1 = 1, α2 = norm beamformers for as,t and bs,t respectively.
Another notation that will be shared between schemes includes (b) c¯s,t (a) c¯s,t
˜ vs,t bs,t +h v b , ,h s,t s,t s,t s,t T
′
T
′
T
(b)
c¯1,t
′
(1)
′
, g˜s,tus,t as,t +gs,tus,t as,t , t = 1, · · · , Ts T
The received signals at the two users then take the form
(7)
y1,t
z }| { ′ ′ ˜ T v1,t b1,t +hT v ′ b′ +z (1) , =hT1,tu1,t a1,t +hT1,tu1,t a1,t +h 1,t 1,t 1,t 1,t 1,t | {z } | {z } | {z } | {z } |{z} P 1−α2
P
P 1−α1
P 1−α1
P0
(a) that denotes the interference seen by user 1 and user 2 respecc¯1,t (a) (b) s z }| { tively, during timeslot t of phase s. For {¯ cs,t , c¯s,t }Tt=1 being ′ ′ ′ ′ (2) (2) T T T T the accumulated interference to both users during phase s, we y1,t =g˜1,tu1,t a1,t +g1,tu1,t a1,t +g1,tv1,t b1,t +g1,tv1,t b1,t + z1,t , | {z } | {z } | {z } | {z } |{z} (a) (b) s (a) (b) s , be a quantized version of {¯ cs,t , c¯s,t }Tt=1 will let {ˆ cs,t , cˆs,t }Tt=1 P P0 P 1−α2 P 1−α2 P 1−α1 (13) and we will consider the mapping where the total information Ts+1 (a) (b) Ts in {ˆ cs,t , cˆs,t }t=1 is split evenly across symbols {cs+1,t }t=1 where under each term we noted the order of the summand’s transmitted during the next phase. In addition we use ws+1,t to average power. denote the randomly chosen unit-norm beamformer of cs+1,t . At this point, and after the end of the first phase, the transFurthermore, unless stated otherwise, mitter can use its knowledge of delayed CSIT to reconstruct ′ ′ (a) (b) 1 {¯ c1,t , c¯1,t }Tt=1 (cf.(7)), and quantize each term as xs,t = ws,t cs,t +us,t as,t +us,t a′s,t +vs,t bs,t +vs,t b′s,t |{z} |{z} |{z} |{z} |{z} (a) (a) (a) (b) (b) (b) (c) (b) (a) (b′ ) (a′ ) Ps Ps Ps c¯1,t = cˆ1,t +˜ c1,t , c¯1,t = cˆ1,t +˜ c1,t , t = 1, 2, · · · , T1 , Ps Ps (8) (a) (b) will be the general form of the transmitted vector at timeslot t where cˆ1,t , cˆ1,t are the quantized values, and where . (a) of phase s. As noted above under each summand, the average c˜(a) , c˜(b) are the quantization errors. Noting that E|¯ c1,t |2 = 1,t 1,t . (b) power that is assigned to each symbol, throughout a specific P 1−α2 , E|¯ c1,t |2 = P 1−α1 , we choose a quantization rate phase, will be denoted as follows: (a) that assigns each cˆ1,t a total of (1 − α2 ) log P + o(log P ) ′ (c) (a) (a′ ) 2 2 2 (b) Ps , E|cs,t | , Ps , E|as,t | , Ps , E|as,t | bits, and each cˆ1,t a total of (1 − α1 ) log P + o(log P ) bits, ′ (b) (b′ ) . . (a) (b) Ps , E|bs,t |2 , Ps , E|bs,t |2 . thus allowing for E|˜ c1,t |2 = E|˜ c1,t |2 = 1 ( [11]). At this the T1 (2 − α1 − α2 ) log P + o(log P ) bits representing Furthermore each of the above symbols carries a certain point (a) (b) T1 2 c1,t , cˆ1,t }t=1 , are distributed evenly across the set {c2,t }Tt=1 amount of information, per timeslot, where this amount {ˆ (a) which will be sequentially transmitted during the next phase. may vary across different phases. Specifically we use rs T2 to mean that, during phase s, each symbol as,t , t = This transmission of {c2,t }t=1 will help each of the users (a) 1, · · · , Ts , carries rs log P + o(log P ) bits. Similarly we use cancel the interference from the other user, and it will also ′ ′ serve as an extra observation that allows for decoding of all (a ) (b) (b ) (c) rs , rs , rs , rs to describe the prelog factor of the number ′ ′ private information of that same user. of bits in as,t , bs,t , bs,t , cs,t respectively, again for phase s. 2) Phase 2: During phase 2 (T2 channel uses), the transmit Finally the received signals during phase s for the first and signal takes the exact form in (8) (2) (1) second user, are respectively denoted as ys,t and ys,t , where ′ ′ ′ ′ generally the signals take the following form x2,t = w2,t c2,t + u2,t a2,t + u2,t a2,t + v2,t b2,t + v2,t b2,t (14)
(1)
(1)
ys,t = hTs,t xs,t + zs,t , (2) ys,t
T
= gs,t xs,t +
(2) zs,t ,
t = 1, · · · , Ts .
(9)
A. Scheme X1 achieving C = ( 2+2α31 −α2 , 2+2α32 −α1 ) (case 1) As stated, scheme X1 has S phases, where the phase durations T1 , T2 , · · · , TS are chosen to be integers such that T2 = T1 ξ,
Ts = Ts−1 µ = T1 ξµs−2 , ∀s ∈ {3, 4, · · · , S −1},
TS = TS−1 γ = T1 ξµS−3 γ,
(10)
α1 −α2 +2∆ α1 −α2 +2∆ 1 −α2 , and where ξ = 2−α 1−α1 −∆ , µ = 1−α1 −∆ , γ = 1−α2 1−2α1 +α2 . where ∆ is any constant such that 0 < ∆ < 3 1) Phase 1: During phase 1 (T1 channel uses), the transmit signal is ′
′
′
′
x1,t = u1,t a1,t +u1,ta1,t +v1,t b1,t +v1,t b1,t , while the power and rate are set as (b) . (a′ ) . (a) . P1 = P, P1 = P 1−α2 , P1 = P, (a) (a′ ) (b) r1 = 1, r1 = 1 − α2 , r1 = 1,
(11)
(b′ ) . P1 = P 1−α1 (b′ ) r1 = 1 − α1 . (12)
where we set power and rate as (c) (c) . r2 = 1 − α1 − ∆ P2 = P, (a) (a) . r2 = α1 + ∆ P2 = P α1 +∆ , (a′ ) (a′ ) . P2 = P α1 −α2 +∆ , r2 = α1 − α2 + ∆ (b) (b) . r2 = α1 + ∆ P2 = P α1 +∆ , (b′ ) (b′ ) . r2 = ∆, P2 = P ∆ , (c)
(15)
(c)
and where we note that r2 satisfies T2 r2 = T1 (2−α1 −α2 ). The received signals during this phase are given as (1)
′
′
y2,t = hT2,t w2,t c2,t +hT2,t u2,t a2,t +hT2,t u2,t a2,t {z } | {z } | {z } | P α1 +∆
P
P∆
(2)
P α1 −α2 +∆
˜ T v2,t b2,t +hT v ′ b′ + z (1) , +h 2,t 2,t 2,t 2,t 2,t | {z } | {z } |{z} P∆
′
′
T T T g2,t u2,t a2,t +g2,t u2,t a2,t y2,t = g2,t w2,t c2,t +˜ | {z } | {z } | {z }
P α1 −α2 +∆
P α1 −α2 +∆ (2) g2,t v2,t b2,t + g2,t v2,t b2,t + z2,t , P
+
T
|
(16)
P0
T
{z
P α1 +∆
}
|
′
{z
P∆
′
}
|{z} P0
(17)
Fig. 2.
Received power levels at user 1 (phase 2).
as an extra observation which, together with the observation (1) (b) ys−1,t − hTs−1,t ws−1,t cs−1,t − cˆs−1,t obtained after decoding ′ cs−1,t , allow for decoding of both as−1,t and as−1,t . Similar actions are performed by user 2. As before, after the end of phase s, the transmitter can use (a) (b) s its knowledge of delayed CSIT to reconstruct {¯ cs,t , c¯s,t }Tt=1 , (a) (b) and quantize each term to cˆs,t , cˆs,t with the same rate as in (a) phase 2 ((α1 −α2 +∆) log P +o(log P ) bits for each cˆs,t , and (b) ∆ log P + o(log P ) bits for each cˆs,t ). Finally the accumulated Ts (α1 − α2 + 2∆) log P + o(log P ) bits representing all the (a) (b) s quantized values {ˆ cs,t , cˆs,t }Tt=1 , are distributed evenly across Ts+1 the set {cs+1,t }t=1 which will be sequentially transmitted in the next phase. More details can be found in Appendix V. 2 +2∆ 4) Phase S: During the last phase (TS = TS−1 α1 −α 1−α2 channel uses), the transmit signal is
for t = 1, 2,· · ·,T2 , where under each term we noted the order of the summand’s average power. At this point, based on (16),(17), each user decodes c2,t by 2 treating the other signals as noise. After decoding {c2,t }Tt=1 (a) (b) T1 and fully reconstructing {ˆ c1,t , cˆ1,t , }t=1 , user 1 goes back one (b) (1) phase and subtracts cˆ1,t from y1,t to remove (up to bounded (b) noise) the interference corresponding to c¯1,t . The same user xS,t = wS,t cS,t + uS,t aS,t + vS,t bS,t (18) (a) (a) will also use the estimate cˆ1,t of c¯1,t as an extra observation (1) which, together with the observation y1,t , present the user where we set power and rate as (c) . (c) with a 2 × 2 MIMO channel that allows for decoding of PS = P, rS = 1 − α2 ′ both a1,t and a1,t . Similarly user 2, after fully reconstructing (a) (a) . (19) PS = P α2 , rS = α2 (a) (b) (a) (2) 1 {ˆ c1,t , cˆ1,t , }Tt=1 , subtracts cˆ1,t from y1,t , to remove (up to (b) (b) . α2 PS = P , rS = α2 . (a) bounded noise) the interference corresponding to c¯1,t , and The received signals are (b) (b) also uses the estimate cˆ1,t of c¯1,t as an extra observation (1) ˜ T vS,t bS,t + z (1) , (2) yS,t=hTS,t wS,t cS,t +hTS,t uS,t aS,t +h S,t S,t which, together with the observation y1,t , allow for decoding | | {z } {z } {z } |{z} | ′ P P α2 P α2 −α1 P0 of both b1,t and b1,t . Further exposition to the details regarding (2) (2) T T T the achievability of the mentioned rates, can be found in uS,t aS,t +gS,t vS,t bS,t + zS,t , (20) yS,t=gS,t wS,t cS,t + g˜S,t {z } | {z } | {z } |{z} | Appendix V. P P α2 P0 P0 Consequently after the end of the second phase, the transmitter can use its knowledge of delayed CSIT to reconstruct for t = 1, 2,· · ·, TS . (a) (b) 2 (a) (b) At this point, as before, the power and rate allocation {¯ c2,t , c¯2,t }Tt=1 , and quantize each term to cˆ2,t , cˆ2,t . With of the different symbols allow both users to decode cS,t (b) 2 . (a) 2 . ∆ α1 −α2 +∆ , E|¯ c2,t | = P , we choose a quantizaE|¯ c2,t | = P by treating the other signals as noise. Consequently user 1 (a) tion rate that assigns each cˆ2,t a total of (α1 − α2 + ∆) log P + can remove hT wS,t cS,t from y (1) and decode aS,t , and S,t S,t (b) o(log P ) bits, and each cˆ2,t a total of ∆ log P + o(log P ) similarly user 2 can remove g T w c from y (2) and decode S,t S,t S,t S,t . . (b) (a) c2,t |2 = 1. Then bS,t . Finally each user goes back one phase and reconstructs bits, thus allowing for E|˜ c2,t |2 = E|˜ (a) (b) TS−1 the T2 (α1 − α2 + 2∆) log P + o(log P ) bits representing {ˆ cS−1,t , cˆS−1,t , }t=1 , which allows for decoding of aS−1,t (a) (b) T2 T3 ′ ′ {ˆ c2,t , cˆ2,t }t=1 , are split evenly across the set {c3,t }t=1 which and a at user 1 and of bS−1,t and bS−1,t at user 2, all as S−1,t will be sequentially transmitted in the next phase so that user 1 described for the previous phases (see Appendix V for more ′ 2 can eventually decode {a2,t , a2,t }Tt=1 , and user 2 can decode details). ′ T2 {b2,t , b2,t }t=1 . Table I summarizes the parameters of scheme X1 . The use We now proceed with the general description of phase s. of symbol ⊥ is meant to indicate precoding that is orthogonal 1 −α2 +2∆ 3) Phase s, 3 ≤ s ≤ S −1: Phase s (Ts = Ts−1 α1−α to the channel estimate (rather than random). The table’s last 1 −∆ channel uses) is almost identical to phase 2, with one differ- row indicates the prelog factor of the quantization rate. ence being the different relationship between Ts and Ts−1 . The a) DoF calculation for scheme X1 : We proceed to add up transmit signal takes the same form as in phase 2 (cf. (8),(14)), the total amount of information transmitted during this scheme. ′ (a) (a ) the rates and powers of the symbols are the same (cf. (15)) and , r In accordance to the declared pre-log factors r s s (1) (2) and the received signals ys,t , ys,t (t = 1, · · · , Ts ) take the phase durations (see Table I), we have that same form as in (16),(17). S S−1 X X Most of the actions are also the same, where based on Ti ) Ti (2α1 −α2 +2∆)+TS α2 )/( d1 = (T1 (2−α2 )+ (16),(17) (corresponding now to phase s), each user decodes i=1 i=2 cs,t by treating the other signals as noise, and then goes S−1 X Ts−1 (b) (a) . As back one phase and reconstructs {ˆ cs−1,t , cˆs−1,t , }t=1 (Ti (1−α1 −∆)+Ti (α1 +∆))+TS (1−α2 ) =( (b) (1) i=2 before, user 1 then subtracts cˆs−1,t from ys−1,t to remove, S S−1 (b) X X up to bounded noise, the interference corresponding to c¯s−1,t . Ti ) (21) T )/( +T α + T α − ∆ i S 2 1 1 (a) (a) The same user also employs the estimate cˆs−1,t of c¯s−1,t i=1 i=2
TABLE I S UMMARY OF SCHEME X1 .
Duration r (a) ′ r (a ) (b) r ′ r (b ) (c) r P (a) ⊥ ′ P (a ) (b) P ⊥ ′ P (b ) (c) P Quant.
Phase 1 T1 1 1−α2 1 1−α1 P P 1−α2 P P 1−α1 2−α1−α2
Phase 2 T1 ξ α1 +∆ α1−α2+∆ α1 +∆ ∆ 1−α1 −∆ P α1+∆ P α1−α2+∆ P α1+∆ P∆ P α1−α2+2∆
Ph. s (3≤s≤S−1) T1 ξµs−2 α1 +∆ α1−α2+∆ α1 +∆ ∆ 1−α1 −∆ P α1+∆ P α1−α2+∆ P α1+∆ P∆ P α1−α2+2∆
Phase S T1 ξµS−3 γ α2 α2 1−α2 P α2 P α2 P 0
(22)
where (21) considers the phase durations seen in (10). ConsidPS−3 ering that 0 < µ < 1 (see (10) for case 1), that i=0 µi = 1−µS−2 1−µ , and given an asymptotically high S, we see that d1 = (1 − ∆) + = (1 − ∆) + = (1 − ∆) −
+ ∆ − 1) + T2 µ
T2 1 ξ + T2 ( 1−µ + 1 ξ (α1 + ∆ − 1) 1 1 ξ + 1−µ
S−3
µS−3 (γ −
γ∆
µ 1−µ ))
(23)
1 + α2 − 2α1 − 3∆ 2 + 2α1 − α2 = . 3 3 (24) (b)
′
(b )
Similarly, considering the values for rs , rs , we have that PS−1 T1 (2 − α1 ) + i=2 Ti (α1 + 2∆) + TS α2 d2 = PS i=1 Ti T1 (2−2α1 −2∆)+TS (α2 −α1 −2∆) = α1 +2∆+ PS i=1 Ti T2 S−3 γ(α2 − α1 − 2∆) ξ (2−2α1 −2∆)+T2 µ = α1 +2∆+ µ T2 1 S−3 (γ − 1−µ )) ξ + T2 ( 1−µ + µ which, in the high S limit, gives d2 = α1 + 2∆ +
1 ξ (2
− 2α1 − 2∆) 1 ξ
+
1 1−µ
2(1+α2 −2α1 −3∆) 2+2α2 −α1 = . (25) 3 3 In conclusion, scheme X1 achieves DoF pair C = ( 2+2α31 −α2 , 2+2α32 −α1 ) (case 1). = α1 +2∆+
B. Scheme X2 achieving D = (1, α1 ) (case 1), and A = 2 (1, 1+α 2 ) (case 2) Scheme X2 is designed with S phases, with phase durations T1 , T2 , · · · , TS chosen to be integers such that T2 = T1 τ,
Ts = Ts−1 β = T1 τ β s−2 , ∀s ∈ {3, 4, · · · , S −1},
TS = TS−1 η = T1 τ β S−3 η,
′
(26)
′
x1,t = u1,t a1,t + u1,t a1,t + v1,t b1,t , with power and rate set as (b) . (a′ ) . (a) . P1 = P, P1 = P 1−α2 , P1 = P α1 ′ (a) (a ) (b) r1 = 1, r1 = 1 − α2 , r1 = α1 . The received signals take the form ′ ′ (1) ˜ T v1,t b1,t + z (1) , y1,t = hT1,t u1,t a1,t + hT1,t u1,t a1,t + h 1,t 1,t | {z } | {z } | {z } |{z}
P0
P 1−α2
P
T1 (α1 + ∆ − 1) + TS ∆ , = (1 − ∆) + PS i=1 Ti
T2 ξ (α1
1−α2 1 −α2 1 −α2 , β = α1−α , η = α1−α . where τ = 1−α 1 1 2 The scheme is similar to X1 , but with a different power and rate allocation, and a different input structure since now user 2 only receives a single private information symbol. 1) Phase 1: During phase 1 (T1 channel uses), the transmitter sends
P0
(a) c¯1,t
}| { z ′ ′ (2) (2) T T T u1,t a1,t + g1,t u1,t a1,t + g1,t v1,t b1,t + z1,t . y1,t = g˜1,t | {z } | {z } | {z } |{z} P 1−α2
P α1
P 1−α2
P0
After the end of the first phase, the transmitter reconstructs (a) 1 {¯ c1,t }Tt=1 (cf.(7)), and quantizes each term as (a)
(a)
(a)
c¯1,t = cˆ1,t +˜ c1,t ,
t = 1, 2, · · · , T1 .
. (a) Noting that E|¯ c1,t |2 = P 1−α2 , we choose a quantization rate (a) that assigns each cˆ1,t a total of (1 − α2 ) log P + o(log P ) bits, . (a) thus allowing for E|˜ c1,t |2 = 1. Then the T1 (1 − α2 ) log P + (a) 1 o(log P ) bits representing {ˆ c1,t }Tt=1 are distributed evenly T2 across the set {c2,t }t=1 which will be transmitted in the next 2 phase. As before, transmission of {c2,t }Tt=1 aims to help user 2 cancel out interference, as well as aims to provide user 1 with an extra observation which will allow for decoding of the user’s private information. 2) Phase 2: During phase 2 (T2 channel uses), the transmitter sends ′
′
x2,t = w2,t c2,t + u2,t a2,t + u2,t a2,t + v2,t b2,t with power and rate set as (c) . P2 = P, (a) . P2 = P α1 , (a′ ) . P2 = P α1 −α2 , (b) . P2 = P α1 , (c)
(c)
r2 = 1 − α1 (a) r2 = α1 (a′ ) r2 = α1 − α2 (b) r2 = α1 ,
(28)
(c)
where we note that r2 satisfies T2 r2 = T1 (1 − α2 ). The received signals in this phase are ′ ′ (1) ˜ T v2,tb2,t +z (1) y2,t =hT2,tw2,t c2,t +hT2,tu2,t a2,t +hT2,tu2,t a2,t +h 2,t 2,t | {z } | {z } | {z } | {z } |{z}
P
P α1
P α1 −α2
P0
P0
(29)
(2)
′
(2)
′
T T T T u2,t a2,t +g2,t u2,t a2,t +g2,t v2,tb2,t +z2,t y2,t =g2,t w2,t c2,t + g˜2,t | {z } | {z } | {z } | {z } |{z}
P
P α1 −α2
P α1 −α2
P α1
P0
(30)
for t = 1, 2,· · ·,T2 .
Then, based on (29),(30), each user decodes c2,t by treating resulting in received signals of the form the other signals as noise, and then proceeds to reconstruct (a) 1 (a) (1) {ˆ c1,t }Tt=1 . User 1 combines each cˆ1,t with its corresponding ˜ T vS,t bS,t + z (1) , yS,t=hTS,t wS,t cS,t +hTS,t uS,t aS,t +h S,t S,t (1) | {z } {z } {z } |{z} | | observation y1,t , to introduce T2 independent 2 × 2 MIMO ′ P P α2 P α2 −α1 P0 channels that allow for decoding of all a1,t and a1,t . At the (2) (2) T T T wS,t cS,t + g˜S,t yS,t=gS,t uS,t aS,t +gS,t vS,t bS,t + zS,t , (a) (2) same time, user 2 subtracts cˆ1,t from y1,t to remove (up to | {z } | {z } | {z } |{z} (a) P P α2 P0 P0 bounded noise) the interference corresponding to c¯1,t , which in turn allows for decoding of b1,t . Consequently after the end of the second phase, the trans- (t = 1,· · ·, TS ). As before, both receivers decode cS,t by treating all other mitter can use its knowledge of delayed CSIT to reconstruct . (a) 2 (a) (a) {¯ c2,t }Tt=1 , and quantize each term to cˆ2,t . With E|¯ c2,t |2 = signals as noise. Consequently user 1 removes hTS,t wS,t cS,t (1) (a) T wS,t cS,t P α1 −α2 , we choose a quantization rate that assigns each cˆ2,t from yS,t and decodes aS,t , and user 2 removes gS,t (2) a total of (α1 − α2 ) log P + o(log P ) bits, a choice that allows from yS,t and decodes bS,t . Finally each user goes back one . (a) (a) TS−1 for E|˜ c2,t |2 = 1. Then the T2 (α1 − α2 ) log P + o(log P ) bits phase and reconstructs {ˆ cS−1,t }t=1 , which in turn allows for (a) T2 ′ representing {ˆ c2,t }t=1 , are distributed evenly across the set decoding of aS−1,t and a S−1,t at user 1 and of bS−1,t at user 2, 3 {c3,t }Tt=1 which will be transmitted in the next phase. all as described in the previous phases. The DoF achievability 1 −α2 details follow those of scheme X1 (Appendix V). 3) Phase s, 3 ≤ s ≤ S − 1: Phase s (Ts = Ts−1 α1−α 1 channel uses) is almost identical to phase 2, except for the Table II summarizes the parameters of scheme X2 . The last relationship between Ts and Ts−1 . Specifically the transmit row indicates the prelog factor of the quantization rate. signal takes the same form as in phase 2 TABLE II S UMMARY OF SCHEME X2 .
′
xs,t = ws,t cs,t +us,t as,t +us,t a′s,t +vs,t bs,t , |{z} |{z} |{z} |{z} (c)
Ps
(a)
Ps
(a′ )
Ps
(b)
Ps
the rates and powers of the symbols are the same (cf. (28)), (1) (2) and the received signals ys,t , ys,t (t = 1, · · · , Ts ) take the same form as in (29),(30). The actions are also the same, where based on (29),(30) (corresponding now to phase s), each user decodes cs,t by treating the other signals as noise, and then goes back one phase Ts−1 (a) . As before, user 1 then employs and reconstructs {ˆ cs−1,t }t=1 (a) (a) the estimate cˆs−1,t of c¯s−1,t as an extra observation which, (1) together with the observation ys−1,t − hTs−1,t ws−1,t cs−1,t attained after decoding cs−1,t , allow for decoding of both ′ (a) as−1,t and as−1,t . At the same time, user 2 subtracts cˆs−1,t (2) from ys−1,t to remove (up to bounded noise) the interference (a) corresponding to c¯s−1,t , which allows for decoding of bs−1,t . Again as before, after the end of phase s, the transmitter can (a) s use delayed CSIT to reconstruct {¯ cs,t }Tt=1 , and quantize each (a) term to cˆs,t with the same rate as in phase 2 ((α1 −α2 ) log P + o(log P ) bits per channel use). Finally the total of the Ts (α1 − α2 ) log P + o(log P ) bits representing the quantized values Ts+1 (a) s {ˆ cs,t }Tt=1 is split evenly to the set {cs+1,t }t=1 which will be transmitted in the next phase. 1 −α2 4) Phase S: During the last phase (TS = TS−1 α1−α 2 channel uses), the transmitter sends (31)
. (c) = P, rS = 1 − α2 . α2 (a) = P , rS = α2 . (b) = P α2 , rS = α2 .
Ph.s (3≤s≤S−1) T1 τ β s−2 α1 α1−α2 α1 1−α1 P α1 P α1−α2 P α1 P α1 −α2
(32)
Phase S T1 τ β S−3 η α2 α2 1−α2 P α2 P α2 P 0
(a)
′
(a )
In accordance to the declared pre-log factors rs , rs and phase durations (see Table II), and irrespective of whether α1 , α2 fall under case 1 or case 2, we have that
d1=(T1 (2−α2 )+
S−1 X
S X Ti ) Ti (2α1 −α2 )+TS α2 )/(
i=2 S−1 X
=(T1+T1 (1−α2 )+
i=1
S X (Ti α1 +Ti (α1−α2 ))+TS α2 )/( Ti )
i=2
i=1
S X Ti ) (Ti (1−α1 )+Ti α1 )+TS (1−α2 )+TS α2 )/(
S−1 X i=2
T1 + T2 + T3 + · · · + TS−1 + TS = =1 T1 + T2 + · · · + TS
with power and rates set as (c) PS (a) PS (b) PS
Phase 2 T1 τ α1 α1−α2 α1 1−α1 P α1 P α1−α2 P α1 P α1 −α2
a) DoF calculation for scheme X2 : We proceed to add up the total amount of information transmitted during this scheme.
=(T1+ xS,t = wS,t cS,t + uS,t aS,t + vS,t bS,t
Phase 1 T1 1 1−α2 α1 P P 1−α2 P α1 1−α2
Duration r (a) ′ r (a ) (b) r r (c) P (a) ⊥ ′ P (a ) (b) P ⊥ P (c) Quant.
i=1
(34) (35)
where (34) is due to (26). (b) Regarding the second user and the declared rs , for case 1
(2α1 − α2 < 1) we see that PS−1 Ti α1 + TS α2 TS (α1 − α2 ) = α1 − PS d2 = i=1 PS i=1 Ti i=1 Ti S−3 T1 τ β η(α1 − α2 ) = α1 − PS−3 i T1 + T1 τ i=0 β + T1 τ β S−3 η β S−3 η(α1 − α2 ) = α1 − 1 PS−3 i S−3 η i=0 β + β τ + S−3 β η(α1 − α2 ) = α1 − 1 1−β S−2 + β S−3 η τ + 1−β = α1 −
1 τ
β S−3 η(α1 − α2 ) = α1 , β 1 + 1−β + β S−3 (η − 1−β )
(36) (37)
IV. C ONCLUSIONS (38)
where we have used (26) to get (36), where we have used that 2α1 − α2 < 1 implies β < 1, and where we have considered an asymptotically large S. When 2α1 − α2 > 1 (β > 1), then (37) gives that d2 = α1 − = α1 −
1 τ
β S−3 η(α1 − α2 ) β 1 + 1−β + β S−3 (η − 1−β )
η(α1 1−β+τ β S−3 τ (1−β)
− α2 )
+ (η −
β 1−β )
which, in the high S regime, gives d2 = α1 −
After transmission, both receivers first decode c by treating the other signals as noise, and then user 1 utilizes its knowlˆ gˆ} to reconstruct hT wc and remove it from edge of {h, g, h, (1) y , thus being able to decode a, while after decoding c, user 2 removes g T wc from y (2) , and decodes b. The details for the achievability of r(a) , r(b) , r(c) follow closely the exposition in Appendix V. Consequently the DoF point (d1 = α2 , d2 = 1) can be achieved by associating c to information intended entirely for the second user.
1−2α1 +α2 1+α2 η(α1 − α2 ) = . (39) = α1 + β 2 2 η − 1−β
When 2α1 − α2 = 1 (β = 1), then (37) gives that d2 = 1 −α2 ) α1 − 1η(α which, for large S, gives +S−2+η τ
1 + α2 d2 = α1 = . (40) 2 In conclusion, scheme X2 achieves DoF pair D = (1, α1 ) 2 (case 1), else it achieves A = (1, 1+α 2 ).
The work provided analysis and communication schemes for the setting of the two-user MISO BC with general mixed CSIT. The work can be seen as a natural extension of the result in [10] and of the recent results in [6]–[9], to the case where the CSIT feedback quality varies across different links. V. A PPENDIX - D ETAILS OF
ACHIEVABILITY PROOF
We will here focus on achievability details for scheme X1 . The clarifications of the details carry over easily to the other two schemes. (c) Regarding rs (2 ≤ s ≤ S − 1 - see (15)), we recall that (1) (2) during phase s, both users decode cs,t (from ys,t , ys,t , t = 1, · · · , Ts - see (29),(30) ) by treating all other signals as noise. ˆ i,j , gˆi,j , ∀i, j}, we note Consequently for H , {hi,j , gi,j , h that (2)
(1)
I(cs,t ; ys,t , H) = I(cs,t ; ys,t , H) = (1 − α1 − ∆) log P + o(log P ), to get 1 (1) (2) min{I(cs,t ; ys,t , H), I(cs,t ; ys,t , H)} log P = 1 − α1 − ∆.
rs(c) =
Similarly for the last phase S (see (18),(19),(20)), we note that (1)
(2)
C. Scheme X3 achieving B = (α2 , 1)
I(cS,t ; yS,t , H) = I(cS,t ; yS,t , H) = (1−α2 ) log P +o(log P ),
This is the simplest of all three schemes, and it consists of a single channel use2 (S = 1, T1 = 1) during which the transmitter sends
to get
x = wc + ua + vb, ˆ and where where u is orthogonal to gˆ, v is orthogonal to h, the power and rates are set as . P (c) = P, r(c) = 1 − α1 (a) . α2 (41) P = P , r(a) = α2 (b) (b) . α1 P = P , r = α1 , resulting in received signals of the form T T ˜ T vb + z (1) , y (1) = hT x + z (1) = h wc} + h ua} + |h | {z {z } |{z} | {z
P
y
(2)
T
=g x+z
(2)
P α2
P0
P0 (2)
z . = g wc + g˜ ua + g vb + |{z} | {z } | {z } | {z } T
P
T
P0
T
P α1
P0
2 We will henceforth maintain the same notation as before, but for simplicity we will remove the phase and time index.
(c)
rS =
1 (2) (1) min{I(cS,t ; yS,t , H), I(cS,t ; yS,t , H)} = 1 − α2 . log P (a)
Regarding achievability for r1 ′
(b )
′
(a )
= 1, r1
(b)
= 1 − α2 , r1 =
1 and r1 = 1 − α1 (see (11),(12),(13)), we note that each (c) 2 element in {c2,t }Tt=1 has enough bits (recall that r2 = 1 − (a) (b) T1 α1 − ∆), to match the quantization rate of {ˆ c1,t , cˆ1,t }t=1 that is necessary in order to have a bounded quantization noise. Consequently going back to phase 1, user 1 is presented with T1 linearly independent 2 × 2 equivalent MIMO channels of the form " # " (1) (b) # (1) (b) y1,t −ˆ c1,t z +˜ c hT1,t ′ a1,t = T u1,t u1,t ′ + 1,t (a)1,t (a) g a cˆ1,t −˜ c1,t 1,t 1,t (t = 1, 2, · · · , T1 ), where again we note that the described quantization rate results in a bounded equivalent′ noise, which (a)
then immediately gives that r1
(a )
= 1 and r1
= 1 − α2
are achievable. Similarly for user 2, the presented T1 linearly independent 2 × 2 equivalent MIMO channels " # # " (b) (b) cˆ1,t −˜ c1,t ′ b1,t hT1,t v1,t v1,t ′ + (2) (a) (2) (a) = g T b1,t y1,t −ˆ c1,t z1,t +˜ c1,t 1,t
b) The equivalent degraded compound BC: Towards the equivalent BC, directly from (1a),(1b) we have that (1)
yt
(t = 1, 2, · · · , T1 ), ′allow for decoding at a rate corresponding (b)
(b )
to r1 = 1 and r1
= 1 − α1 .
(a)
Regarding achievability for rs
′
′
(a )
= α1 + ∆, rs
= α1 −
(1)
= hTt xt + zt √ 1 (1) = hTt P Qt √ Q−1 t xt + z t P √ ′ (1) = hTt P Qt xt + zt √ T √ ˜ T vt x2 + z (1) = P h ut x1 + P h t
(2) yt
T
t (2) zt
t
t
t
(43a)
(b) (b ) = gt xt + α2 + ∆, rs = α1 + ∆ and rs = ∆, (2 ≤ s ≤ S − 1 √ ′ (2) - see (8),(14), (15)), we note that during phase s, both users = gtT P Qt xt + zt T √ √ can decode cs,t , and as a result user 1 can remove hs,t ws,t cs,t (2) (43b) = P g˜tT ut x1t + P gtT vt x2t + zt , (1) (2) T from ys,t , and user 2 can remove gs,t ws,t cs,t from ys,t (t = 1, · · · , Ts ). As a result user 1 is presented with Ts linearly where ′ 1 independent 2 × 2 equivalent MIMO channels of the form xt , [x1t x2t ]T , √ Q−1 t xt , P " # " # (1) (b) (1) (b) ys,t − hTs,t ws,t cs,t −ˆ cs,t hT ′ as,t z +˜ c = s,t us,t us,t ′ + s,t (a)s,t where Qt , [ut vt ] ∈ C2×2 is, with probability 1, an T (a) gs,t as,t cˆs,t −˜ cs,t invertible matrix, where ut is chosen to be of unit norm and orthogonal to gˆt , and where vt is chosen to be of unit norm Ts+1 (t = 1, · · · , Ts ). Given that the rate associated to {cs+1,t }t=1 , and orthogonal to h ˆ t . Furthermore each receiver normalizes (a) (b) s matches the quantization rate for {ˆ cs,t , cˆs,t }Tt=1 , allows for to get a bounded variance of the equivalent noise, and in turn for ′ (1) (a) s ′ y decoding of {as,t , as,t }Tt=1 at a rate corresponding to rs = (1) yt = Tt ′ (a ) ht ut α1 + ∆ and rs = α1 − α2 + ∆. Similarly user 2 is presented √ T (1) ˜ vt x2 √ zt Ph with Ts independent 2 × 2 MIMO channels of the form t t 1 P x + = + t " # # hTt ut hTt ut " (b) (b) √ 1 √ hTs,t ′ bs,t cˆs,t −˜ cs,t ′ ′ (1) vs,t vs,t ′ + (2) (a) = P xt + P 1−α1 ht x2t + zt , (44a) (a) = g T (2) T b ws,t cs,t −ˆ cs,t ys,t − gs,t +˜ c z s,t s,t s,t s,t (2) ′ y (2) ′ yt = Tt s (t = 1, · · · , Ts ) at allowing for decoding of {bs,t , bs,t }Tt=1 gt vt ′ √ T (b) (b ) (2) √ 2 rates corresponding to rs = α1 + ∆ and rs = ∆. P g˜t ut x1t zt (a) (b) P x + = + t Regarding achievability for rS = α2 and rS = α2 g T vt gtT vt √ 2 √ t ′ (see (18),(19),(20)), we note that, after decoding cS,t , user 1 ′ (2) = P xt + P 1−α2 gt x1t + zt , (44b) (1) T can remove hS,t wS,t cS,t from yS,t , and user 2 can remove √ (2) (1) T ′ ′ (2) T gS,t wS,t cS,t from yS,t , (t = 1, · · · , TS ). Consequently during where z (1) = zTt , h′ = P αT1 h˜ t vt , z (2) = zTt , g ′ = t t t t h u h u g vt t t t t t this phase, user 1 sees TS linearly independent SISO channels √P α2 g˜T ut √ √ t α1 h α2 g ˜ ˜ . Consequently and have identity P P t t of the form gtT vt ′ ′ ′ (1) covariance matrices, and the average power of ht , gt , zt (1) (1) ˜ T vS,t bS,t +z (1) ′ y˜S,t , yS,t −hTS,t wS,t cS,t = hTS,t uS,t aS,t + h (2) S,t S,t and zt does not scale with P , i.e., in the high-SNR region 0 (a) (t = 1, · · · , TS ) which can be readily shown to support rS = this power is of order P . With the same CSIT knowledge (b) mapped from the original BC, it can be shown (see [9]) that α2 . A similar argument gives achievability for rS = α2 . the DoF region of the equivalent BC in (44a)(44b) matches the DoF region of the original BC in (1a)(1b). VI. A PPENDIX - P ROOF OF O UTER BOUND Towards designing the degraded version of the above equiv′ (1) We here adopt the outer bound approach in [9] to the alent BC, we supply the second user with knowledge of yt , asymmetric case of α1 6= α2 . As in [9], we first linearly and towards designing the compound version of the above convert the original BC in (1a),(1b) to an equivalent BC degraded equivalent BC, we add two extra users (user 3 and (see (43a),(43b)) having the same DoF region as the original 4). In this compound version, the received signals for the first BC (cf. [9]), and we then consider the degraded version two users are as in (44a)(44b), while the received signals of of the equivalent BC in the absence of delayed feedback, the added (virtual) users are given by √ √ which matches in capacity the degraded BC with feedback ′′ ′′ ′′ (1) (1) (45a) yt = P x1t + P 1−α1 ht x2t + zt , (for the memoryless case), and which exceeds the capacity √ √ ′′ ′′ ′′ (2) (2) of the equivalent BC. The final step considers the compound (45b) yt = P x2t + P 1−α2 gt x1t + zt . and degraded version of the equivalent BC without delayed ′′ ′′ feedback, whose DoF region will serve as an outer bound on We here note that by definition, ht and gt are statistically ′′ ′ ′ (1) equivalent to the original ht and gt respectively, and that zt the DoF region of the original BC.
′′
′
(2)
′
(1)
(1)
′′
(1)
+ I(W2 ; y[n] , y[n] , z[n] |H[n] , W1 ) and zt are statistically equivalent to the original zt and ′ (2) zt . Furthermore we note that user 3 is interested in the + no(log P ) + no(n), (51) same message as user 1, while user 4 is interested in the same where the transition to (50) uses the fact that the high SNR message as user 2. Also we recall′ that in the specific degraded ′ 0 α1 (1) (2) which compound BC, user 1 knows yt , user 2 knows yt and variance of z¯t and zt scales as P and P′ (1) respectively, ′′ (1) ′ ′′ ′′ ′′ (1) (1) (2) (1) in turn means that knowledge of {y , y , z , H }nt=1 , t [n] t t yt , user 3 knows yt , and user 4 knows yt and yt . 2 n 1 Finally we remove delayed feedback - a removal known to not implies knowledge of W1 , W2 and of {xt , xt }t=1 , up to affect the capacity of the degraded BC without memory [12]. bounded noise level. Furthermore We now proceed to calculate an outer bound on the DoF region of this degraded compound BC which at least matches nR1=h(W1 ) the DoF of the previous degraded BC and which serves as an ′ ′ (1) ′′ (1) (1) ′′ (1) =I(W1 ; y[n] , y[n] ,z[n] |H[n])+h(W1 |y[n] , y[n] ,z[n] , H[n]) outer bound on the DoF region of the original BC. ′ (1) ′′ (1) c) Outer bound: We consider communication over the =I(W1 ; y[n] , y[n] , z[n] |H[n])+no(logP )+no(n), (52) described equivalent degraded compound BC, letting n be the large number of fading realizations over which communication since again knowledge of {y ′ (1) , y ′′ (1) , z , H }n provides t [n] t=1 t t takes place, and letting R1 , R2 be the rates of the first and for W up to bounded noise level. ′ 1 ˆ t , gˆt }n , y (i) , second user. We also let H[n] , {ht , gt , h t=1 Now combining (51) and (52), gives [n] ′′ ′ ′′ (i) n (i) n (i) ′ {yt }t=1 and y[n] , {yt }t=1 for i = 1, 2. (1) ′′ (1) nR2 = I(W2 ; y[n] , y[n] , z[n] |H[n] , W1 )+no(log P )+no(n) Using Fano’s inequality, we have ′ (1) ′′ (1) ′ (1) , y[n] |H[n] , W1 ) = I(W ; y 2 [n] nR ≤ I(W ; y |H ) + no(n) 1
1
[n]
[n]
′
′
(1)
≤ n logP +no(logP )−h(y[n] |W1 , H[n] )+no(n), (46) as well as ′′
(1)
(1)
≤ n logP +no(logP )−h(y[n] |W1 , H[n] )+no(n), (47) which is added to (46) to give 2nR1 ≤ 2n log P + 2no(log P ) − ′′
(1)
(1) h(y[n] |W1 , H[n] )
+ 2no(n).
" ′ # ′ −1 (1) √ 1 h yt t y¯1 , diag(1, P α1 ) ′′ ′′ (1) 1 ht y ′ ′′ ′′ ′ t (1) (1) zt ht −zt ht √ 1 ′′ ′ P x h −h t t t = √ 2 + √ ′′ (1) ′ (1) zt −zt P xt α 1 P ′′ ′ h −h √ 1 t t z¯ 0 Px = √ t2 + t + 0 zt P xt z[n] ,
{zt }nt=1 .
′′ (1)
′′
ht −zt ′′ ′ ht −ht
(48)
, zt =
t
Consequently
=
′′
(1)
(1)
−
(1) ′′ (1) h(z[n] |y[n] , y[n] , H[n] , W1 , W2 )
|
′′
{z
no(log P ) ′
(1)
′′
(1)
}
≤h(z[n] )
(1)
′′
(1)
≤ h(y[n] , y[n] |W1 , H[n] ) + nα1 log P + no(log P ) + no(n),
which in turn proves the outer bound 2d1 + d2 ≤ 2 + α1 , (49)
t
(54)
as described in (6b). Finally interchanging the roles of the two users and of α1 , α2 , gives d1 + 2d2 ≤ 2 + α2 .
(55)
Naturally the single antenna constraint gives that d1 ≤ 1, d2 ≤ 1.
(1)
= I(W1 , W2 ; y[n] , y[n] , z[n] |H[n] ) +
(1)
2nR1 + nR2 ≤ 2n log P + nα1 log P + no(log P ) + no(n), (53)
nR1 + nR2 = h(W1 , W2 ) ′
′
(1)
which is combined with (48) to give
′′ (1) ′ (1) √ z −z P α1 t h′′ −ht′ , and let
′
ht
′′
′
′
(1) ′′ (1) h(y[n] , y[n] |W1 , H[n] ) ′
′ (1)
(1)
(1) ′′ (1) ≤ h(y[n] , y[n] |H[n] , W1 )+h(z[n])+no(log P )+no(n)
Let
zt
′
= h(y[n] , y[n] |H[n] , W1 )−h(y[n] , y[n] |H[n] , W1 , W2 ) | {z }
′
≤ 2n log P + 2no(log P )
where z¯t =
(1)
+ h(z[n] |y[n] , y[n] , H[n] , W1 ) +no(log P ) + no(n) | {z }
′
− h(y[n] |W1 , H[n] ) + 2no(n)
−
′′
no(log P )
nR1 ≤ I(W1 ; y[n] |H[n] ) + no(n) ′′
(1)
+I(W2 ; z[n] |y[n] , y[n] , H[n] , W1 )+no(log P )+no(n)
R EFERENCES
(1) ′′ (1) h(W1 , W2 |y[n] , y[n] , z[n] , H[n] ) ′
(1) ′′ (1) I(W1 , W2 ; y[n] , y[n] , z[n] |H[n] ) ′
+ no(log P ) + no(n) ′
(1)
′′
(1)
= I(W1 ; y[n] , y[n] , z[n] |H[n] )
(50)
[1] A. Ghasemi, A. S. Motahari, and A. K. Khandani, “On the degrees of freedom of X channel with delayed CSIT,” in Proc. IEEE Int. Symp. Information Theory (ISIT), Jul. 2011. [2] C. S. Vaze and M. K. Varanasi, “The degrees of freedom region of two-user and certain three-user MIMO broadcast channel with delayed CSI,” Dec. 2010, submitted to IEEE Trans. Inform. Theory, available on arXiv:1101.0306v2 [cs.IT].
[3] M. J. Abdoli, A. Ghasemi, and A. K. Khandani, “On the degrees of freedom of three-user MIMO broadcast channel with delayed CSIT,” in Proc. IEEE Int. Symp. Information Theory (ISIT), Jul. 2011. [4] A. Ghasemi, A. S. Motahari, and A. K. Khandani, “Interference alignment for the MIMO interference channel with delayed local CSIT,” Fer. 2011, available on arXiv:1102.5673v1 [cs.IT]. [5] J. Xu, J. G. Andrews, and S. A. Jafar, “Broadcast channels with delayed finite-rate feedback: Predict or observe?” May 2011, submitted to IEEE Trans. on Wireless Communications, available on arXiv:1105.3686v1 [cs.IT]. [6] M. A. Maddah-Ali and D. N. C. Tse, “Completely stale transmitter channel state information is still very useful,” Sep. 2011, submitted to IEEE Trans. Inform. Theory, available on arXiv:1010.1499v2 [cs.IT]. [7] M. Kobayashi, S. Yang, D. Gesbert, and X. Yi, “On the degrees of freedom of time correlated MISO broadcast channel with delayed CSIT,” Feb. 2012, submitted to Proc. IEEE Int. Symp. Information Theory (ISIT) 2012, available on arXiv:1202.1909v1 [cs.IT]. [8] S. Yang, M. Kobayashi, D. Gesbert, and X. Yi, “Degrees of freedom of time correlated MISO broadcast channel with delayed CSIT,” Mar. 2012, submitted to IEEE Trans. Inform. Theory, available on arXiv:1203.2550v1 [cs.IT]. [9] T. Gou and S. Jafar, “Optimal use of current and outdated channel state information - degrees of freedom of the MISO BC with mixed CSIT,” Mar. 2012, submitted to IEEE Communications Letters, available on arXiv:1203.1301v1 [cs.IT]. [10] H. Maleki, S. Jafar, and S. Shama, “Retrospective interference alignment over interference networks,” IEEE Journal of Selected Topics in Signal Processing, vol. 6, no. 3, pp. 228 – 240, Mar. 2012. [11] T. Cover and J. Thomas, Elements of information theory. New York: Wiley-Interscience, 2006. [12] A. E. Gamal, “The feedback capacity of degraded broadcast channels,” IEEE Trans. Inf. Theory, vol. 24, no. 3, pp. 379 – 381, Apr. 1978.
