Imperfect Delayed CSIT can be as Useful as Perfect Delayed CSIT: DoF Analysis and Constructions for the BC

# Imperfect Delayed CSIT can be as Useful as Perfect Delayed CSIT: DoF Analysis and Constructions for the BC

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)
###### Abstract

In the setting of the two-user broadcast channel, where a two-antenna transmitter communicates information to two single-antenna receivers, recent work by Maddah-Ali and Tse has shown that perfect knowledge of delayed channel state information at the transmitter (perfect delayed CSIT) can be useful, even in the absence of any knowledge of current CSIT. Similar benefits of perfect delayed CSIT were revealed in recent work by Kobayashi et al., Yang et al., and Gou and Jafar, which extended the above to the case of perfect delayed CSIT and imperfect current CSIT.

The work here considers the general problem of communicating, over the aforementioned broadcast channel, with imperfect delayed and imperfect current CSIT, and reveals that even substantially degraded and imperfect delayed-CSIT is in fact sufficient to achieve the aforementioned gains previously associated to perfect delayed CSIT. The work proposes novel multi-phase broadcasting schemes that properly utilize knowledge of imperfect delayed and imperfect current CSIT, to match in many cases the optimal degrees-of-freedom (DoF) region achieved with perfect delayed CSIT. In addition to the theoretical limits and explicitly constructed precoders, the work applies towards gaining practical insight as to when it is worth improving CSIT quality.

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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 imperfect and delayed CSIT knowledge ([1, 2, 3, 4, 5, 6]). 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 information 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 instance , vectors represent the transmitter-to-user 1 and transmitter-to-user 2 channels respectively, where represent unit power AWGN noise at the two receivers, where is the input signal with power constraint , and where in this case, also takes the role of the signal-to-noise ratio (SNR).

With CSIT often being imperfect and delayed, we here explore the effects of the quality of current CSIT corresponding to how well the transmitter knows at time , as well as the effects of the quality of delayed CSIT, corresponding to how well the transmitter knows the same , at time for some positive . Naturally, reduced CSIT quality relates to limitations in the capacity and reliability of the feedback channel. The distinction between the quality of current and delayed CSIT, is meant to reflect the increased challenge of quickly attaining high quality CSIT.

### I-a Related work

Corresponding to CSIT quality, it is well known that in the two-user BC setting of interest, the presence of perfect 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 utilizing partial CSIT knowledge, was recently presented in [1] by Maddah-Ali and Tse, 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 perfectly knows the delayed channel states () up to time (perfect delayed, no current CSIT), the work in [1] 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, 10, 11] which considered the natural extension where, in addition to perfect delayed CSIT, the transmitter also had partial knowledge of current CSIT.

### I-B Notation and conventions

Throughout this paper, , , respectively denote the transpose and conjugate transpose of a matrix, while denotes the Euclidean norm, and denotes the magnitude of a scalar. comes from the standard Landau notation, where implies . We also use to denote exponential equality, i.e., we write to denote . Logarithms are of base .

Finally adhering to the convention followed in [1, 7, 10], we consider a unit coherence period222Simple interleaving arguments can show that, in the absence of delay constraints, the association of current CSIT with a single coherence period, introduces no loss of generality., as well as perfect and global knowledge of channel state information at the receivers (perfect global CSIR, [1, 7, 10, 9]) where the receivers know all channel states and all estimates.

### I-C Structure of paper

After recalling the quantification of CSIT quality, Section II bounds the DoF region of the described two-user MISO broadcast channel for the general case of having imperfect current and imperfect delayed CSIT of different quality. In many cases, these bounds are identified to be tight, and to in fact match the optimal performance associated to perfect delayed CSIT. Section III presents the novel multi-phase precoding schemes that apply for different cases of CSIT quality. The performance of these schemes is derived in the same section, with some of the proof details placed in the Appendix.

## Ii MISO BC with imperfect delayed CSIT and imperfect current CSIT

### Ii-a Quantification of CSIT quality

In terms of current CSIT, we consider the case where at time , the transmitter has estimates of and respectively, 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 describing the quality of the estimates. In this setting, an increasing implies an improved CSIT quality, with implying very little current CSIT knowledge, and with implying perfect CSIT.

In terms of delayed CSIT for channels that appear at time , we consider the case where, beginning at time , the transmitter has delayed estimates of , and does so with estimation errors

 ¨ht=ht−ˇht,   ¨gt=gt−ˇgt (3)

having i.i.d. Gaussian entries with power

 12E(∥¨ht∥2)=12E(∥¨gt∥2)=P−β,

for some non-negative parameter describing the quality of the estimates.

###### Remark II.1

We here note that without loss of generality, we can restrict our attention to the range (cf.[12]), as well as to the case where since having would be equivalent to having simply because current CSIT estimates can be recalled at a later time. As a result, we will henceforth consider that where corresponds the case of perfect delayed CSIT, and where corresponds to the case of perfect CSIT.

Fig. 1 recalls different DoF regions corresponding to imperfect current CSIT (), but perfect delayed CSIT () ([9, 10, 7, 11]).

### Ii-B DoF region of the MISO BC with imperfect delayed and imperfect current CSIT

We proceed with the main result, the proof of which, together with the description of the associated precoding schemes, will be given in Section III.

###### Theorem 1

For the two-user MISO BC with imperfect delayed CSIT, imperfect current CSIT , and for , the DoF region

 d1≤1,   d2≤1 (1+β′′−2α)d1+(1−β′′)d2≤(1+β′′)(1−α) (1−β′′)d1+(1+β′′−2α)d2≤(1+β′′)(1−α)

is achievable and takes the form of a polygon with corner points

 {(0,0),(0,1),(α,1),(1+β′′2,1+β′′2),(1,α),(1,0)}.

Furthermore when , the region is optimal and it is described by

 d1≤1,d2≤1 2d1+d2≤2+α d1+2d2≤2+α

corresponding to the polygon

 {(0,0),(0,1),(α,1),(2+α3,2+α3),(1,α),(1,0)}

matching the optimal DoF region previously associated to .

The above reveals that, whether with imperfect or no current CSIT, imperfect delayed CSIT can in some cases match the optimal performance associated to perfect delayed CSIT. The following corollaries provide further insight, and make the connection to previous work. The corollaries apply to the same setting as the theorem.

###### Corollary 1

In terms of DoF, having is equivalent to having perfect delayed CSIT. Specifically the optimal region from [9, 10] corresponding to , can in fact be achieved for any , and the optimal region from [1] corresponding to , can in fact be achieved whenever .

Building on the above, we also have the following.

###### Corollary 2

Whenever the desired DoF pair lies within the pentagon , there is no need for any delayed CSIT, and suffices.

This is the case for example, for the optimal , which can be achieved with imperfect current and no delayed CSIT. Consequently whenever the desired DoF pair lies within the aforementioned pentagon, or whenever , then there is no need for improving the quality of delayed CSIT. Otherwise, the DoF penalty due to a reduced , can be at most , which is no bigger than .

Fig. 2 depicts different DoF regions spanning the general setting of imperfect delayed and imperfect current CSIT.

## Iii Multi-phase precoding schemes for the two-user MISO BC with imperfect delayed and imperfect current CSIT

We proceed to describe the two precoding schemes that achieve the corresponding corner DoF points, by properly utilizing different combinations of superposition coding, successive cancelation, power allocation, and phase durations.

As stated, without loss of generality, we assume that . The scheme description is done for , and for rational . The cases where , , , or where are not rational, can be readily handled with minor modifications. We first proceed to describe the basic notation and conventions used in our schemes. This preliminary description allows for brevity in the subsequent description of the details of the schemes.

The schemes are designed with phases ( varies from scheme to scheme), where the th phase () consists of channel uses. At this point, and to more clearly reflect the division of time into phases, we will switch to a double time index where, for example, the vectors and will now denote the channel vectors, during timeslot of phase . Similarly, in terms of current CSIT (cf. (2)), and will respectively denote the transmitter’s estimates of channels and , and , will denote the corresponding estimation errors. We recall that the estimates and become known to the transmitter at time , i.e., they become known instantly. In terms of delayed CSIT (cf. (3)), and will be the estimates of and , where these estimates become known to the transmitter with unit delay (at time ), and are stored and recalled thereafter. Finally , will denote the estimation errors corresponding to delayed CSIT.

Furthermore and will denote the independent information symbols that are precoded and sent during phase , timeslot , and which are meant for user 1, while symbols and are meant for user 2. In addition, will denote the common information symbol generally meant for both users.

The transmitted vector at timeslot of phase will, in most cases, take the form

 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, (4)

where vectors are the unit-norm beamformers for respectively. In our schemes, vectors and will be chosen to be orthogonal to and respectively, with chosen pseudo-randomly (and assumed to be known by all nodes). Corresponding to the transmitted vector in (4), and as noted 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, regarding the amount of information, per time slot, carried by each of the above symbols, we will use to mean that, during phase , each symbol carries bits, and similarly we will use to describe the prelog factor of the number of bits in respectively, again for phase .

 ι(1)s,t ≜hTs,t(vs,tbs,t+v′s,tb′s,t), ι(2)s,t ≜gTs,t(us,tas,t+u′s,ta′s,t), t=1,⋯,Ts (5)

to denote the new interference experienced by user 1 and user 2 respectively, during timeslot of phase , and we will use

 ˇι(1)s,t ≜ˇhTs,t(vs,tbs,t+v′s,tb′s,t), ˇι(2)s,t ≜ˇgTs,t(us,tas,t+u′s,ta′s,t),t=1,⋯,Ts, (6)

to denote transmitter’s (delayed) estimates of at time . To clarify, we mean that the transmitter creates, at time , the estimates of the actual interference experienced during time , by using the delayed CSIT estimates obtained at time .

For being the accumulated delayed estimates of all the interference terms during phase , we will let be the quantized delayed estimates which are obtained by properly quantizing , at a quantization rate that will be described later on. Based on the information in , new symbols are then created, where these new symbols are created to evenly share the total information in (i.e., the information in is evenly split among the elements in ), and where these new common symbols will be sequentially transmitted during the next phase.

Finally the received signals and at the first and second user during phase , take the 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. (7)

We now proceed with the details of the first scheme.

### Iii-a Scheme X1 achieving C1=1+β′′2 (0≤α≤β≤1)

For this scheme, the phase durations are chosen to be integers generated to form a geometric progression where

 Ts =Ts−1ξ=T1ξs−1,∀s∈{2,3,⋯,S−1}, TS =TS−1ζ=T1ξS−2ζ, (8)

and where , . The progression can be made to consist of integers since , and by extension , are rational numbers. For this scheme, is asked to be large.

#### Iii-A1 Phase 1

During phase 1 ( channel uses), the transmitter sends

 x1,t=w1,tc1,t+u1,ta1,t+u′1,ta′1,t+v1,tb1,t+v′1,tb′1,t, (9)

with power and rate set as

 P(c)1≐P,P(a)1≐P(b)1≐Pβ,P(a′)1≐P(b′)1≐Pβ−αr(c)1=1−β,r(a)1=r(b)1=β,r(a′)1=r(b′)1=β−α. (10)

The received signals take the form

 y(1)1,t =hT1,tw1,tc1,tP+hT1,tu1,ta1,tPβ+hT1,tu′1,ta′1,tPβ−α +ˇι(1)1,tˇhT1,t(v1,tb1,t+v′1,tb′1,t)Pβ−α+¨hT1,t(v1,tb1,t+v′1,tb′1,t)P0+z(1)1,tP0, (11) y(2)1,t =gT1,tw1,tc1,tP+gT1,tv1,tb1,tPβ+gT1,tv′1,tb′1,tPβ−α +ˇι(2)1,tˇgT1,t(u1,ta1,t+u′1,ta′1,t)Pβ−α+¨gT1,t(u1,ta1,t+u′1,ta′1,t)P0+z(2)1,tP0, (12)

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

 E()|ˇι(1)1,t|2 =E()|ˇhT1,tv1,tb1,t|2+E()|ˇhT1,tv′1,tb′1,t|2 =E()|(~hT1,t−¨hT1,t)v1,tb1,t|2+E()|ˇhT1,tv′1,tb′1,t|2≐Pβ−α, E()|ˇι(2)1,t|2 =E()|(~gT1,t−¨gT1,t)u1,ta1,t|2+E()|ˇgT1,tu′1,ta′1,t|2≐Pβ−α, (13)

and

 E()|ι(1)1,t−ˇι(1)1,t|2 =E()|¨hT1,t(v1,tb1,t+v′1,tb′1,t)|2≐P0, E()|ι(2)1,t−ˇι(2)1,t|2 =E()|¨gT1,t(u1,ta1,t+u′1,ta′1,t)|2≐P0. (14)

Fig. 4 provides a graphical illustration of the received power levels at user 1 and user 2 during phase 1 of scheme .

At this point, based on the received signals in (11),(12), each user decodes by treating the other signals as noise. The details regarding the achievability of can be found in the Appendix. After decoding , user 1 removes from , while user 2 removes from . Then, at the end of the first phase, the transmitter uses its partial knowledge of delayed CSIT to reconstruct (cf.(III)), and to quantize each term as

 ¯ˇι(2)1,t=ˇι(2)1,t−~ι(2)1,t,¯ˇι(1)1,t=ˇι(1)1,t−~ι(1)1,t,t=1,2,⋯,T1, (15)

where are the quantized delayed estimates of the interference terms, and where are the corresponding quantization errors. Noting that (cf. (III-A1),(III-A1)), we choose a quantization rate that assigns each a total of bits, and each a total of bits, thus allowing for (see for example [13]). At this point, the bits representing , are distributed evenly across the set of newly constructed symbols which will be sequentially transmitted during the next (second) phase. This transmission of in the next phase, will help each of the users cancel the dominant part of the interference from the other user, and it will also serve as an extra observation (which will in turn enable the creation of a corresponding MIMO channel - see (16) later on) that allows for decoding of all private information of that same user.

#### Iii-A2 Phase s,  2≤s≤S−1

Phase  ( channel uses) is similar to phase 1, with the transmit signal taking the same form as in phase 1 (cf. (4),(9)), and so do the rates and powers of the symbols (cf. (10)), as well as the received signals () (cf. (11),(12)).

At the receivers (see (11),(12), corresponding now to phase ), each user decodes by treating the other signals as noise. After decoding , user 1 removes from , and user 2 removes from .

At this point, each user goes back one phase and reconstructs, using its knowledge of , the quantized delayed estimates of all the interference accumulated during the previous phase (cf.(III),(15)). 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 , allow for decoding of both and . Specifically user 1, using its knowledge of , and , is presented, at this instance, with a equivalent MIMO channel of the form

 (16)

where is the equivalent noise that will be seen to be properly bounded. As will be argued further in the Appendix, the above MIMO channel allows for decoding of and .

Similar actions are performed by user 2 which uses knowledge of and to decode both and (see the Appendix for more details on the achievability of the mentioned rates).

As before, after the end of phase , the transmitter uses its imperfect knowledge of delayed CSIT to reconstruct , and quantize each term to with the same rate as in phase 1 ( bits for each , and bits for each ). Finally the accumulated bits representing all the quantized values , are distributed evenly across the set , the elements of which will be sequentially transmitted in the next phase (phase ).

#### Iii-A3 Phase S

During the last phase ( channel uses), the transmitter sends

 xS,t=wS,tcS,t+uS,taS,t+vS,tbS,t (17)

with power and rates set as

 P(c)S≐P,r(c)S=1−αP(a)S≐Pα,r(a)S=αP(b)S≐Pα,r(b)S=α, (18)

resulting in received signals of the form

 y(1)S,t =hTS,twS,tcS,tP+hTS,tuS,taS,tPα+~hTS,tvS,tbS,tP0+z(1)S,tP0, (19) y(2)S,t =gTS,twS,tcS,tP+~gTS,tuS,taS,tP0+gTS,tvS,tbS,t