# MISO Broadcast Channel with Delayed and Evolving CSIT

## Abstract

The work considers the two-user MISO broadcast channel with a gradual and delayed accumulation of channel state information at the transmitter (CSIT), and addresses the question of how much feedback is necessary, and when, in order to achieve a certain degrees-of-freedom (DoF) performance. Motivated by limited-capacity feedback links with delays, that may not immediately convey perfect CSIT, and focusing on the block fading scenario, we consider a gradual accumulation of feedback bits that results in a progressively increasing CSIT quality as time progresses across the coherence period ( channel uses - current CSIT), or at any time after (delayed CSIT).

Specifically, for any set of feedback quality exponents describing the high-SNR rates-of-decay of the mean square error of the current CSIT estimates at time (), given an average , and given perfect delayed CSIT (received at any time ), the work here derives the optimal DoF region to be the polygon with corner points . Aiming to now reduce the overall number of feedback bits, we also prove that the above optimal region holds even with imperfect delayed CSIT for any (delayed-CSIT) quality exponent .

Additionally, motivated by settings where users have different feedback qualities and delays, we prove the above to hold true even when the users’ quality exponents are different but share a common average. The work further proceeds to derive the optimal DoF region in the general asymmetric setting.

The results are supported by novel multi-phase precoding schemes that utilize gradually improving CSIT. The approach here incorporates different settings such as the delayed CSIT setting of Maddah-Ali and Tse (), the imperfect current CSIT setting of Yang et al. and of Gou and Jafar (), the asymmetric setting of Maleki et al., and the not-so-delayed CSIT setting of Lee and Heath ( for some ).

## 1Introduction

### 1.1Channel model

We consider the multiple-input single-output broadcast channel (MISO BC) with an -transmit antenna () transmitter communicating to two receiving users with a single receive antenna each. Within the block fading setting, we consider a coherence period of channel uses, during which the channel remains the same. For and denoting this channel during the th coherence block for the first and second user respectively, and for denoting the transmitted vector during timeslot of this th block, the corresponding received signals at the first and second user take the form

(), where denote the unit power AWGN noise at the receivers. The above transmit vectors accept a power constraint , for some power which also here takes the role of the signal-to-noise ratio (SNR). The fading coefficients are assumed to be independent and identically distributed (i.i.d.) complex Gaussian random variables with zero mean and unit variance, and are assumed to remain fixed during a coherence block, and to change independently from block to block.

### 1.2Delay-and-quality effects of feedback

As in many multiuser wireless communications scenarios, the performance of the broadcast channel depends on the timeliness and quality of channel state information at the transmitter (CSIT). This timeliness and quality though may be reduced by limited-capacity feedback links, which may offer consistently low feedback quality, or may offer good quality feedback which though comes late in the communication process and can thus be used for only a fraction of the communication duration. The corresponding performance degradation, as compared to the case of having perfect feedback without delay, forces the delay-and-quality question of how much feedback is necessary, and when, in order to achieve a certain performance.

These delay-and-quality effects of feedback, naturally fall between the two extreme cases of no CSIT and of full CSIT (immediately available and perfect CSIT), with full CSIT allowing for the optimal degree-of-freedom (DoF) per user (cf., [1])^{1}

A valuable tool towards bridging this gap and further understanding the delay-and-quality effects of feedback, came with [4] showing that arbitrarily delayed feedback can still allow for performance improvement over the no-CSIT case. In a setting that differentiated between current and delayed CSIT - delayed CSIT being that which is available after the channel elapses, i.e., after the end of the coherence period corresponding to the channel described by this delayed feedback, while current CSIT corresponded to feedback received during the channel’s coherence period - the work in [4] showed that perfect delayed CSIT, even without any current CSIT, allows for an improved DoF per user.

Within the same context of delayed vs. current CSIT, the work in [5] introduced feedback quality considerations, and managed to quantify the usefulness of combining perfect delayed CSIT with immediately available imperfect CSIT of a certain quality that remained unchanged throughout the entire coherence period. In this setting the above work showed a further bridging of the gap from to DoF, as a function of this current CSIT quality.

Further progress came with the work in [8] which, in addition to exploring the effects of the quality of current CSIT, also considered the effects of the quality of delayed CSIT, thus allowing for consideration of the possibility that the overall number of feedback bits (corresponding to delayed plus current CSIT) may be reduced. Focusing again on the specific setting where the current CSIT quality remained unchanged for the entirety of the coherence period, this work revealed among other things that imperfect delayed CSIT can achieve the same optimality that was previously attributed to perfect delayed CSIT, thus equivalently showing how the amount of delayed feedback required, is proportional to the amount of current feedback.

A useful generalization of the delayed vs. current CSIT paradigm, came with the work in [10] which deviated from the assumption of having invariant CSIT quality throughout the coherence period, and allowed for the possibility that current CSIT may be available only after some delay, and specifically only after a certain fraction of the coherence period^{2}

The above settings^{3}^{4}

Can a specific accumulation-rate of feedback bits, guarantee a certain target DoF performance?

If we send feedback bits without delay (at ), then send bits at , bits at , and bits at any time , then what performance can be guaranteed?

Can imperfect CSIT allow for the optimal 1 DoF?

Can CSIT with very small delays allow for the optimal 1 DoF?

What is better: less feedback early, or more feedback later?

Given a certain target DoF, what is the tradeoff between feedback delays and feedback quality?

Given imperfect feedback, what feedback delays allow for a certain DoF?

How many feedback bits must be accumulated before the channel changes, in order to achieve a certain performance?

How many (delayed) feedback bits must be gathered after the channel changes in order to achieve the best possible performance?

When is delayed feedback unnecessary?

Under what conditions of feedback asymmetry, do two uneven feedback links behave similarly?

How do the feedback capabilities of one user, affect the other user?

Is a reduction in a user’s feedback quality made worse, for that user, by an increase or a decrease of the other user’s feedback quality?

### 1.3Quantification of evolving CSIT quality

In terms of current CSIT, i.e., in terms of CSIT corresponding to feedback received during the coherence period of the channel in question, we consider the case where at time of the th coherence block, the transmitter has estimates of and respectively, with estimation errors

having i.i.d. Gaussian entries with power

for some non-negative parameter describing the quality of the estimates at any given time during the channel’s coherence period^{5}

In terms of delayed CSIT, and again focusing on the aforementioned channels appearing during the th coherence block, we consider the case where at any time after the end of the th block, the transmitter has delayed estimates with estimation errors

again having i.i.d. Gaussian entries, but this time with power

for some non-negative parameter .

We can now see how the evolving CSIT generalization naturally incorporates different settings such as the perfect-delayed CSIT setting in [4] (), the perfect-delayed and imperfect current CSIT setting in [5] (), the bounded-overall-feedback setting with imperfect current and imperfect delayed CSIT [8] (), as well as the ‘not-so-delayed’ CSIT setting in [10] corresponding to having for some integer .

Furthermore proceeding to the asymmetric setting where the CSIT quality differs from user to user, we consider the case where

for describing the current CSIT quality for user 1 and user 2 respectively, and where

for describing the delayed CSIT exponents for the two users. The asymmetric setting here incorporates the setting in [19] corresponding to having and .

### 1.4Structure of paper

Section ? provides the optimal DoF regions for the different cases of evolving CSIT, with Theorem ? describing the optimal DoF region for the case of having symmetrically evolving current CSIT and perfect delayed CSIT, with Theorem ? considering the same symmetric setting but with imperfect delayed CSIT, with Theorem ? considering the partially symmetric setting where the two users’ quality exponents are different but share a common average , and with Theorem ? describing the optimal DoF region for the general asymmetric setting where the aforementioned averages need not be the same. In addition to the theorems, we also provide corollaries and examples that are meant to offer insight. Section ? is dedicated to presenting the different schemes and their DoF performance, and it applies towards the achievability part of the proof of the aforementioned results. Specifically, after a brief description in Section ? of the notation that is common to all schemes, the subsequent subsections ?, ? and ? describe different schemes that jointly achieve the optimal DoF region in the general asymmetric case, then Section ? describes the scheme for the case of having symmetric or partially symmetric evolving current CSIT and perfect delayed CSIT, and then Section ? describes the scheme for the case of having symmetric or partially symmetric evolving current CSIT and imperfect delayed CSIT. Section ? provides the DoF outer bound for the asymmetric case with perfect delayed CSIT, where this outer bound directly supports Theorem ?, while it also supports Theorem ? after setting , as well as supports Theorem ? and Theorem ? after setting . Appendix Section 6 presents some details from the achievability proofs, some DoF calculations as well as some encoding details, and finally Appendix ? provides brief proofs of the different corollaries.

### 1.5Notation and conventions

Throughout this paper, , and denote the transpose, conjugate transpose and Frobenius norm of a matrix respectively, while denotes a diagonal matrix, 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 . Similarly and denote exponential inequalities. Logarithms are of base . Finally we adhere to the common convention (see [4]) of assuming perfect and global knowledge of channel state information at the receivers (perfect global CSIR), where the receivers know all channel states and all estimates^{6}

## 2DoF region of the MISO BC with evolving CSIT

We proceed with the main results, which we divide in four cases; the case of symmetrically evolving current CSIT with perfect delayed CSIT, of symmetrically evolving current CSIT and imperfect delayed CSIT, the partially symmetric case with perfect and imperfect delayed CSIT, and finally the more general asymmetric case. As stated, the corresponding schemes can be found in Section ?, while the corresponding outer bound proof can be found in Section ?.

### 2.1Symmetrically evolving current CSIT and perfect delayed CSIT

We here consider the case of evolving current CSIT with perfect delayed CSIT, and focus on the case where the two users enjoy the same quality of current CSIT corresponding to the same set of quality exponents (). This statistical symmetry is meant to reflect scenarios where the quality of the feedback links is similar across different users. We also focus for now on the case where delayed CSIT can be considered to be perfect; an assumption that is meant to reflect the ability to eventually, after sufficiently large delay, receive sufficient feedback to allow for perfect CSIT estimates. For notational convenience, we define

to be the average (current) CSIT quality exponent.

This is depicted in Figure 1.

Drawing from the above, the following corollary is partially motivated by the possibility of having imperfect feedback and/or having feedback with delays. The proof is brief and can be found in Appendix ?. The use of the term *symmetric DoF* is meant to correspond to the case where the two users have equal DoF.

The above applies to settings such as that in [10] which considers delays in receiving current CSIT, thus corresponding to having for some , and thus having . The corollary shows that, unlike in the ()-user user case in [10] where the optimal sum DoF is achieved even in the presence of the aforementioned (current feedback) delays, in the two-user case here, any delay or imperfection in the current CSIT, will result in suboptimal DoF performance.

The following examples provides insight.

feedback | feedback | extra bits | |||

to | to | to | delay | bits in | after |

period | |||||

### 2.2Symmetrically evolving current CSIT with imperfect delayed CSIT

We now proceed to the more general case where, in addition to imperfections in the current CSIT, imperfections can be found in delayed CSIT estimates as well (). Having could reflect a limitation in the feedback link quality or a limitation in the total number of (current plus delayed) feedback bits, which in turn results in coarse CSIT, irrespective of how long we wait for this delayed feedback. We recall that delayed feedback is not considered to be evolving, again because such delayed feedback can, without loss of generality, be considered to arrive at any point after the end of the coherence period, and after CSIT has reached its maximum refinement. As before, is the average of the quality exponents.

The following corollaries provide further insight and conclusions that hold in the same DoF context.

The above is direct from the theorem and simply considers that current CSIT estimates can be recalled at a later point in time. It applies towards answering the question of how many (delayed) feedback bits must be gathered after the channel changes in order to achieve the best possible performance, offering insight on understanding when delayed feedback is necessary.

Furthermore we have the following, which gives insight on how many feedback bits to send, and when, in order to achieve a certain performance . The proof is again direct.

In addition, the following corollary describes feedback delays that allow for a given target symmetric DoF in the presence of constraints on current and delayed CSIT qualities. We will be specifically interested in the allowable fractional delay of feedback

i.e., the fraction for which . A constraint on the current quality exponents, is meant to reflect a constraint on the total number of feedback bits sent during the coherence period, while bounding corresponds to having a limited total number of (current plus delayed) feedback bits per coherence period^{7}

The following bounds the quality of current and of delayed CSIT needed to achieve a certain target symmetric DoF .

The proof of this is straightforward; the corresponding quality exponents can be . We proceed with some simple examples.

feedback | extra bits | ||||

to | to | to | delay | after | |

### 2.3Asymmetrically evolving current CSIT

We here consider the asymmetric case where need not be equal to , corresponding to having CSIT quality that evolves differently from user to user. Such asymmetry could reflect feedback links with different capacity or different delays. The approach here seeks to shed light on the question of how the feedback capabilities of one user, affect the other user. The exposition of the results is done for two distinct cases. In the first case, which could be described as a partially symmetric case, we show that the results of the two previous theorems hold even when the two users’ quality exponents are different but share a common average , thus revealing among other things the condition (equal exponent average) under which two uneven feedback links behave similarly. The results are derived based on the design of specific schemes that will be shown to properly utilize this partial asymmetry. In the second case we derive the optimal DoF region in the general asymmetric setting where the averages need not be the same. The subsequent results are supported by the outer bound in Section ?, while the achievability part of Theorem ? is supported by the schemes in Section ? and Section ?, and the achievability part of Theorem ? is supported by the schemes in Section ?, Section ? and Section ?, where these latter schemes are specifically designed to handle asymmetric feedback qualities.

Proceeding to a more general asymmetric case, without loss of generality we assume that

and focus on the practical case where

as well as on the case of perfect delayed CSIT.

Figure 2 depicts the above.

The following corollaries provide further insight and conclusions that hold in the above context of asymmetrically evolving current CSIT and perfect delayed CSIT.

The next corollary provides insight on how a reduction in a user’s feedback quality, is exacerbated by quality asymmetry. The proof is brief and can be found in Appendix ?.

## 3Communication schemes for the MISO BC with evolving CSIT

We proceed to describe precoding schemes that achieve the corresponding DoF corner points, by properly utilizing different combinations of superposition coding, successive cancelation, power allocation, and phase durations. As before, we will consider a channel coherence period of time slots, but clarify that the schemes’ DoF performance does not depend on the channel being temporally independent.

We first present the basic notation and conventions used in our schemes. This preliminary description allows for brevity in the subsequent description of the details of our schemes.

### 3.1Precoding schemes: Basic notation and conventions

The schemes are designed to have phases, with phase () spanning coherence blocks, and where will be separately designed in each scheme. The labels of the blocks in each phase , will constitute a set , where^{8}

The transmitted vector at timeslot of block will typically take the form

where are symbols meant for user 1, for user 2, and are common symbols. Their respective powers are denoted as

and the prelog factors of their corresponding rates are respectively denoted as^{9}

In addition

will denote the interference at user 1 and user 2 respectively, and

will denote the transmitter’s delayed estimates of , while we will use

to denote the quantized versions of and respectively, with denoting the corresponding quantization errors. Furthermore in the setting where we quantize a set of complex numbers, we will use to mean that the corresponding number of quantization bits is .

We proceed to first describe the three schemes for the asymmetric quality setting^{10}

### 3.2Scheme : utilizing asymmetric and evolving CSIT to achieve DoF point for case 1 ()

As stated, scheme is designed to have phases, with phase () spanning blocks, where are integers satisfying

where , , , and where can be any number^{11}

The labels of the blocks in each phase , constitute the set as this was described in .

#### Phase 1

During phase 1 (consisting of blocks ), the transmitter sends

, , with power and rates set as

The received signals at the two users then take the form

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

At this point, and after the end of the first phase, the transmitter uses its perfect knowledge of delayed CSIT to reconstruct perfect delayed estimates (cf. ,), and to quantize them into (cf. ) with

which, given that and , allows for bounded quantization noise power

(see for example [18]). 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 later on) that allows for decoding of all private information of that same user.

#### Phase ,

During phase (consisting of block , ), the transmitted signal takes the exact form in

, , where we set power and rates as

Then the received signals at the two users take the form

Upon reception, based on ,, each user first decodes the common signal by treating the other signals as noise. The details for the achievability of follow closely the exposition of the details of scheme , as these details are shown in Appendix ?. 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 . 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 , again corresponding to the phase (note that ). Specifically user 1 is presented, at this instance, with a equivalent MIMO channel of the form