Degrees-of-Freedom Region of the MISO Broadcast Channel with General Mixed-CSIT

# Degrees-of-Freedom Region of the MISO Broadcast Channel with General Mixed-CSIT

Jinyuan Chen and Petros Elia 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.
###### 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.

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## I Introduction

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, 2, 3, 4, 5]). In this context of multiuser communications, we here consider the broadcast channel (BC), and specifically focus on the two-user multiple-input single-output (MISO) BC, where a two-antenna transmitter communicates to two single-antenna receivers. In this setting, the channel model takes the form

 y(1)t =hTtxt+z(1)t (1a) y(2)t =gTtxt+z(2)t, (1b)

where for any time instant , represent the channel vectors for user 1 and 2 respectively, where represent unit power AWGN noise, where is the input signal with power constraint , and where in this case, 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 degree-of-freedom (DoF) per user, whereas the complete absence of CSIT causes a substantial degradation to just DoF per user111We remind the reader that for an achievable rate pair , the corresponding DoF pair is given by The corresponding DoF region is then the set of all achievable DoF pairs..

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 , the transmitter knows the delayed channel states () up to time , the work in [6] showed that each user can achieve DoF, providing a clear improvement over the case of no CSIT.

This result was later generalized in [7, 8, 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 the transmitter had estimates of and , with estimation errors

 ~ht=ht−^ht,   ~gt=gt−^gt (2)

having i.i.d. Gaussian entries with power

 12E(∥~ht∥2)=12E(∥~gt∥2)=P−α,

for some non-negative parameter 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 denoting the DoF for the first and second user over the aforementioned two-user BC, the work in [7, 8, 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 , nicely bridging the gap between the case of explored in [6], and the case of (and naturally ) corresponding to perfect CSIT.

### I-a Notation and conventions

Throughout this paper, , , , 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 denotes a diagonal matrix. Logarithms are of base 2. comes from the standard Landau notation, where implies . We also use to denote exponential equality, i.e., we write to denote . Finally, in the spirit of [7, 8, 9] we consider a unit coherence period, as well as perfect knowledge of channel state information at the receivers (perfect CSIR).

## Ii The 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, 8, 9] to consider unequal quality of current CSIT knowledge for and . 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 , the transmitter has estimates of the current and , with estimation errors

 ~ht=ht−^ht,   ~gt=gt−^gt (4)

having i.i.d. Gaussian entries with power

 12E(∥~ht∥2)=P−α1,  12E(∥~gt∥2)=P−α2,

for some non-negative parameters that describe the generally unequal quality of the estimates of the current CSIT for the two users’ links.

We proceed to describe the optimal DoF region of the general mixed-CSIT two-user MISO BC (two-antenna transmitter). The optimal schemes are presented in Section III, parts of the proof of the schemes’ performance are presented in Appendix V, while the outer bound proof is placed in Appendix VI.

### Ii-a DoF region of the MISO BC with generalized mixed-CSIT

Without loss of generality, the rest of this work assumes that

 1≥α1≥α2≥0. (5)
###### Theorem 1

The DoF region of the two-user MISO BC with general mixed-CSIT, is given by

 d1≤1,  d2≤1 (6a) 2d1+d2≤2+α1 (6b) d1+2d2≤2+α2 (6c)

where the region is a polygon which, for has corner points

 {(0,0),(1,0),(1,α1),(2+2α1−α23,2+2α2−α13),(α2,1),(0,1)},

and otherwise has corner points

 {(0,0),(1,0),(1,1+α22),(α2,1),(0,1)}.

The above corner points, and consequently the entire DoF inner bound, will be attained by the schemes to be described later on. The result generalizes the results in [7, 8, 9] as well as the result in [10] which considered the case of (), where one user had perfect CSIT and the other only prior CSIT.

Figure 1 depicts the general DoF region for the case where (case 1) and the case where (case 2).

We proceed to describe the communication schemes.

## Iii Design of communication schemes for the two-user general mixed-CSIT MISO BC

As stated, without loss of generality, we assume that . We describe the three schemes , and that achieve the optimal DoF region (in conjunction with time-division between these same schemes). Specifically scheme achieves (case 1), scheme achieves DoF points (case 1) and (case 2), and scheme achieves (case 1 and case 2). The scheme description is done for , and for rational . The cases where , or , or where are not rational, can be readily handled with minor modifications. We proceed to describe the basic notation and conventions used in our schemes.

The schemes are designed with phases ( varies from scheme to scheme), where the th phase consists of channel uses, . The vectors and will denote the channel vectors seen by the first and second user respectively during timeslot of phase , while and will denote the estimates of these channels at the transmitter during the same time, and , will denote the estimation errors.

Furthermore and will denote the independent information symbols that may be sent during phase-, timeslot-, and which are meant for user 1, while symbols and are meant for user 2. Vectors and are the unit-norm beamformers for and respectively, chosen so that is orthogonal to , and so that is orthogonal to . Furthermore are the randomly chosen unit-norm beamformers for and respectively.

Another notation that will be shared between schemes includes

 ¯c(b)s,t ≜~hTs,tvs,tbs,t+hTs,tv′s,tb′s,t, ¯c(a)s,t ≜~gTs,tus,tas,t+gTs,tu′s,ta′s,t, t=1,⋯,Ts (7)

that denotes the interference seen by user 1 and user 2 respectively, during timeslot of phase . For being the accumulated interference to both users during phase , we will let be a quantized version of , and we will consider the mapping where the total information in is split evenly across symbols transmitted during the next phase. In addition we use to denote the randomly chosen unit-norm beamformer of .

Furthermore, unless stated otherwise,

 xs,t=ws,tcs,tP(c)s+us,tas,tP(a)s+u′s,ta′s,tP(a′)s+vs,tbs,tP(b)s+v′s,tb′s,tP(b′)s (8)

will be the general form of the transmitted vector at timeslot of phase . As noted above under each summand, the average power that is assigned to each symbol, throughout a specific phase, will be denoted as follows:

 P(c)s≜E()|cs,t|2,P(a)s≜E()|as,t|2,P(a′)s≜E()|a′s,t|2P(b)s≜E()|bs,t|2,P(b′)s≜E()|b′s,t|2.

Furthermore each of the above symbols carries a certain amount of information, per timeslot, where this amount may vary across different phases. Specifically we use to mean that, during phase , each symbol carries bits. Similarly we use to describe the prelog factor of the number of bits in respectively, again for phase .

Finally the received signals during phase for the first and second user, are respectively denoted as and , where generally the signals take the following form

 y(1)s,t =hTs,txs,t+z(1)s,t, y(2)s,t =gTs,txs,t+z(2)s,t, t=1,⋯,Ts. (9)

### Iii-a Scheme X1 achieving C=(2+2α1−α23,2+2α2−α13) (case 1)

As stated, scheme has phases, where the phase durations are chosen to be integers such that

 T2 =T1ξ,Ts=Ts−1μ=T1ξμs−2,∀s∈{3,4,⋯,S−1}, TS =TS−1γ=T1ξμS−3γ, (10)

where , , , and where is any constant such that .

#### Iii-A1 Phase 1

During phase 1 ( channel uses), the transmit signal is

 x1,t=u1,ta1,t+u′1,ta′1,t+v1,tb1,t+v′1,tb′1,t, (11)

while the power and rate are set as

 P(a)1≐P,P(a′)1≐P1−α2,P(b)1≐P,P(b′)1≐P1−α1r(a)1=1,r(a′)1=1−α2,r(b)1=1, r(b′)1=1−α1. (12)

The received signals at the two users then take the form

 y(1)1,t =hT1,tu1,ta1,tP+hT1,tu′1,ta′1,tP1−α2+¯c(b)1,t~hT1,tv1,tb1,tP1−α1+hT1,tv′1,tb′1,tP1−α1+z(1)1,tP0, y(2)1,t =¯c(a)1,t~gT1,tu1,ta1,tP1−α2+gT1,tu′1,ta′1,tP1−α2+gT1,tv1,tb1,tP+gT1,tv′1,tb′1,tP1−α1+z(2)1,tP0, (13)

where under each term we noted the order of the summand’s average power.

At this point, and after the end of the first phase, the transmitter can use its knowledge of delayed CSIT to reconstruct (cf.(III)), and quantize each term as

 ¯c(a)1,t=^c(a)1,t+~c(a)1,t,¯c(b)1,t=^c(b)1,t+~c(b)1,t,t=1,2,⋯,T1,

where are the quantized values, and where are the quantization errors. Noting that , we choose a quantization rate that assigns each a total of bits, and each a total of bits, thus allowing for ([11]). At this point the bits representing , are distributed evenly across the set which will be sequentially transmitted during the next phase. This transmission of will help each of the users cancel the interference from the other user, and it will also serve as an extra observation that allows for decoding of all private information of that same user.

#### Iii-A2 Phase 2

During phase 2 ( channel uses), the transmit signal takes the exact form in (8)

 x2,t=w2,tc2,t+u2,ta2,t+u′2,ta′2,t+v2,tb2,t+v′2,tb′2,t (14)

where we set power and rate as

 P(c)2≐P,r(c)2=1−α1−ΔP(a)2≐Pα1+Δ,r(a)2=α1+ΔP(a′)2≐Pα1−α2+Δ,r(a′)2=α1−α2+ΔP(b)2≐Pα1+Δ,r(b)2=α1+ΔP(b′)2≐PΔ, r(b′)2=Δ, (15)

and where we note that satisfies .

The received signals during this phase are given as

 y(1)2,t =hT2,tw2,tc2,tP+hT2,tu2,ta2,tPα1+Δ+hT2,tu′2,ta′2,tPα1−α2+Δ (16) y(2)2,t =gT2,tw2,tc2,tP+~gT2,tu2,ta2,tPα1−α2+Δ+gT2,tu′2,ta′2,tPα1−α2+Δ +gT2,tv2,tb2,tPα1+Δ+gT2,tv′2,tb′2,tPΔ+z(2)2,tP0, (17)

for , where under each term we noted the order of the summand’s average power.

At this point, based on (16),(17), each user decodes by treating the other signals as noise. After decoding and fully reconstructing , user 1 goes back one phase and subtracts from to remove (up to bounded noise) the interference corresponding to . The same user will also use the estimate of as an extra observation which, together with the observation , present the user with a MIMO channel that allows for decoding of both and Similarly user 2, after fully reconstructing , subtracts from , to remove (up to bounded noise) the interference corresponding to , and also uses the estimate of as an extra observation which, together with the observation , allow for decoding of both and Further exposition to the details regarding the achievability of the mentioned rates, can be found in Appendix V.

Consequently after the end of the second phase, the transmitter can use its knowledge of delayed CSIT to reconstruct , and quantize each term to . With , we choose a quantization rate that assigns each a total of bits, and each a total of bits, thus allowing for . Then the bits representing , are split evenly across the set which will be sequentially transmitted in the next phase so that user 1 can eventually decode , and user 2 can decode .

We now proceed with the general description of phase .

#### Iii-A3 Phase s,  3≤s≤S−1

Phase  ( channel uses) is almost identical to phase 2, with one difference being the different relationship between and . The transmit signal takes the same form as in phase 2 (cf. (8),(14)), the rates and powers of the symbols are the same (cf. (15)) and the received signals () take the same form as in (16),(17).

Most of the actions are also the same, where based on (16),(17) (corresponding now to phase ), each user decodes by treating the other signals as noise, and then goes back one phase and reconstructs . As before, user 1 then subtracts from to remove, up to bounded noise, the interference corresponding to . The same user also employs the estimate of as an extra observation which, together with the observation obtained after decoding , allow for decoding of both and . Similar actions are performed by user 2.

As before, after the end of phase , the transmitter can use its knowledge of delayed CSIT to reconstruct , and quantize each term to with the same rate as in phase 2 ( bits for each , and bits for each ). Finally the accumulated bits representing all the quantized values , are distributed evenly across the set which will be sequentially transmitted in the next phase. More details can be found in Appendix V.

#### Iii-A4 Phase S

During the last phase ( channel uses), the transmit signal is

 xS,t=wS,tcS,t+uS,taS,t+vS,tbS,t (18)

where we set power and rate as

 P(c)S≐P,r(c)S=1−α2P(a)S≐Pα2,r(a)S=α2P(b)S≐Pα2,r(b)S=α2. (19)