Robust Rate-Adaptive Wireless Communication Using ACK/NAK-Feedback

1 Rob ust Rate-Adapti v e W ire less Communication Using A CK/N AK-Feedback C. Emre K oksal and Philip Schniter Abstract T o com bat the d etrimental effects of the variability in wire less ch annels, we consider cro ss-layer rate adaptation based on limited feedback. In particular , based on limited feedback in the form of link- layer ackn owledgements (A CK) an d negative ackno wledg ements (N AK), we maximize th e ph ysical-layer transmission rate subject to an upper bound on the expected packet error rate. W e take a robust approach in th at we d o n ot assume any par ticular pr ior distribution on the channe l state. W e first analyze the fundam ental limitation s o f s uch systems and der iv e an u pper boun d on th e ach iev able ra te for signa ling schemes based on un coded QAM and random Gaussian ensembles. W e show th at, for cha nnel estimation based on binary A CK/N AK feed back, it may be prefera ble to use a separa te training sequence a t hig h error rates, rather than to explo it lo w-err or-rate data p ackets themselves. W e also develop an ad aptive recursive estimator, which is provably asymp totically optim al and asympto tically efficient. Index T erms — adaptive modulatio n, rate adap tation, auto matic repeat request, cross-layer strategies. I . I N T RO D U C T I O N Channel variation is a principal feature of wireless communication. On one ha nd, ch annel variation poses a hindranc e to reliable co mmunication, in that cha nnel fading can make the received signal-to-noise ratio (SNR) arbitrarily low at any given time instant, making reliable co mmunication virtually imposs ible. On the other hand, c hanne l variation pos es an o pportunity , in that a channel-state-aware transmitter can communicate reliably at high rates during channel quality peaks. The k ey to taming and e xploiting chann el The authors are with the Dept. of Electrical and Computer Engineering at The Ohio State Univ ersity , Columbus, OH 43210. P lease direct all correspond ence to P rof. C. Emre K oksal, Dept. ECE , 2015 Neil A ve., Columbu s OH 43210, phone 614.688.43 69, fax 614.292.7596 , and e-mail koksa l@ece.osu.edu. Philip Schniter can be reached at the same address and fax, phone 614.247.6488, and e-mail schniter@ece.osu.edu. Nov ember 2, 201 8 DRAFT 2 variati on therefore lies in the judiciou s u se of transmitter ch annel state information (CSI). While accurate r eceive r CSI is relati vely easy to maintain, accurate transmitter CSI is o ften difficult to maintain due to limited feedback res ources. W e partition l imited feedback sc hemes (see [1] f or an overview) into tw o classes: those based on channel-state feedback a nd tho se based on er r or-r ate feedback . In limited chan nel-state feedback sche mes (e.g., [2]–[5]), the ch annel-state e stimate computed by the receiver is quantized 1 and then fed bac k to the transmitter . In limited error-rate feedba ck sche mes (e.g., [6]–[17]), a quantized error-rate estimate is fed back to the transmitter , from which it can infer CSI r elative to the previously emp loyed transmission rate. For example, with Au tomatic Repea t reQues t (ARQ) [18], a n egati ve ac knowledgement (N AK) o f packet reception s ugges ts that the cha nnel quality w as below that needed for reliable c ommunication at the previously employed transmission rate, wh ereas a positiv e ac knowledgement (A CK) of pa cket reception sugges ts the opposite. Although ACK/ NAK feedbac k c an be employed for the es timation of transmitter CSI, its primary role is that of maintaining a d esired packet error rate at the link layer throug h controlled pa cket re-transmission (see, e.g., [18]). In fact, since the packet a cknowledgement is a standard provision of most practical link layers, we reason that—for the purpose of ch annel-state estimation—it comes at ess entially no cost to the phy sical layer , unlike traditional channel-state feedback sch emes, which require the dedication of rev erse-cha nnel bandwidth b eyond that required for pa cket ack nowledgements. In this se nse, A CK/N AK- based transmitter-CSI sche mes require even less total feed back ban dwidth tha n “on e-bit” chan nel-state feedback schemes (e.g., [19], [ 20]), giv en that systems employing “one-bit” chan nel-state feedbac k include A CK/N AK a s w ell, for the pu rpose of ARQ. W ith the above motiv ation, we focus on the exclusive u se of limited error-r ate feedback for the maintenance o f transmitter CSI, from which trans mission rate and /or power resources are sub seque ntly adapted. While examples of this strategy can b e found in a number of previous works (e.g., [6]–[17]), there are limitations in how it has be en a pplied. For examp le, in [6]–[10 ], the a daptation algorithms are designed heuristically , ba sed o n practical experiences gained for a spe cific application in a spe cific operating e n vironment. In [11]–[17], on the other hand , transmiss ion rates an d/or p owers are chose n carefully to ma ximize a certain performance metric. T o a chieve this o bjectiv e, a Baye sian approa ch is 1 In s ome cases, the receiv er uses its channel estimate to calculate discrete transmitter rate and/or po wer param eters, and then feeds back those parameters directly . Since these transmitter parameters can be put in one-to-one corresponden ce with some quantized channel-state estimate, we consider such schemes to be equiv alent to channel-state feedback schemes. Nov ember 2, 2018 DRAFT 3 taken, i.e., a model is as sumed for the c hanne l variations an d an as sociated o ptimization problem is solved ba sed on this model. T y pically , the channe l is as sumed to v ary according to a finite-state Markov model [11 ], [12 ], [14 ]–[16] or a Gaus s-Markov process [17]. The sho rtcoming of a mode l-based approa ch is that, it may n ot be pos sible to ass ign a ccurate priors over a wide ran ge o f cha nnel operating co nditions. Consider , for example, that ch annel variations sp an a wide rang e of time sc ales, from bits to thousan ds of packets. For ins tance, relative movement of the transmitter-r ece i ver pair ma y caus e variations a t relativ ely long time scale s, since a very large numbe r of packets can be transmitted during the time it takes for the stations to move far eno ugh to cau se s ignificant chang e in the channe l. On the other h and, c o-chann el interference ca n ch ange signific antly from one pac ket transmiss ion to anothe r . Finally , the multipath nature of the propag ation me dium can cause fast and/or slow fading in the chan nel, depen ding on the relati ve movement of the sc atterers. In this pap er , w e ta ke a rob ust Bayesian [21] approa ch to ra te-adaptation from limited error -rate feedback , wh ere “robust Ba yesian” refers to the fact that we treat the cha nnel state as a ran dom q uantity without as suming a ny pa rticular p rior distributi on on it. In pa rticular , we first de ri ve c onditions on the “quality” of CSI needed for a mo del-independ ent A CK/N AK-based rate adaptation system to ma ximize data rate while keeping the packet error probability belo w a specified threshold. Based on these conditions, we d eri ve funda mental bounds on the rate ac hiev able under a given error probability c onstraint. Finally , we de sign an A CK/N AK-feedback-bas ed non -Bayesian cha nnel-state estimator with prov able asymptotic optimality . Our findings are illustrated through both unc oded QAM and random Gaussian signaling. W e emp hasize that the packet-le vel retransmissions orchestrated by link-layer ARQ would be performed on top of the A CK/N AK-based rate-control that we study . In fact, s ince our phy sical-layer optimization criterion (i.e., maximization o f transmission rate su bject to a giv en target pac ket e rror probability) is by n ature d ecoup led from the functioning of highe r layers, we d o n ot explicitly cons ider ARQ in our analysis. In other words, from the p erspective of our phy sical lay er , the link-layer ARQ mec hanism merely specifies the c ontents o f the packets that are to b e transmitted. The remainde r of the pap er is o r ganize d a s follows. In Sec tion II, we detail the syste m model and provide a mathema tical statemen t o f the problem. In Section III, we de ri ve conditions for succ essful rate adaptation with impe rfect CSI, and in Section IV , we e valuate boun ds on the ac hiev able rates with A CK/N AK fee dback. In Sec tion V, we develop an recursi ve cha nnel es timator b ased on s uch feedback , and in Section VI we con clude. Nov ember 2, 2018 DRAFT 4 P S f r a g r e p l a c e m e n t s rate controller encoder f orward channel decoder f eedback re verse channel data H t , γ t F t − 1 R t X t Y t ˆ X t ε ( γ t , R t ) F t Fig. 1. The rate adaptation system. I I . S Y S T E M M O D E L A. S ystem Componen ts Figure 1 de picts our model of the phys ical-layer a daptiv e co mmunication system. At e ach discrete packet index t , the transmitter transmits a p acket X t = [ X t, 1 , . . . , X t,n ] containing a fixed number , n , of symbols { X t,k } n k =1 , wh ich are e ncode d at a rate o f R t bits/symbol, chose n by the rate controller from the set of possible rates R . W e assu me that the transmit power is cons tant an d n ormalize a ll power lev els such that the energy per symbo l is E  | X t,k | 2  = 1 . For this p acket, the corresp onding cha nnel ou tputs are Y t,k = H t X t,k + W t,k , k = 1 , . . . , n, (1) for c omplex-valued c hannel g ain H t and additiv e white circularly symme tric complex Gau ssian noise W t,k with two-sided power spec tral de nsity N o . Some co mmon mo dels for H t include Ray leigh-, Rician- and Nakag ami-fading (se e e.g., [22]). Howe ver , we will no t assu me any specific statistical model for H t and we will make only weak a ssumptions on the d istrib ution of H t in the sequel. The qua ntity γ t = | H t | 2 / N o can be interpreted as the t th packet’ s channel SNR . Sinc e each symb ol has unit en ergy , γ t is also the r eceive d SNR for pac ket t . Thus, we will simply refer to γ t as the SNR. Due to lac k of power a daptation, γ t is an exog enous qua ntity over which the s ystem has no control. W e assume that, for all t , γ t takes on values from s ome prior distrib ution p ( · ) ∈ P , where P is a set of distrib utions with finite mean a nd variance. Howe ver , we make no further as sumptions on set P . W e do not ev en ass ume kn owledge of this set by the transmitter or the rec eiv er . W e as sume that the recei ver has access to perfect CSI and uses a maximum lik elihood deco der to decode the rec eiv ed packet. Let ˆ X t denote the d ecode d estimate of packet X t based on rec eiv ed pac ket Y t = [ Y t, 1 , . . . , Y t,n ] , and the corresp onding probab ility of d ecoding error be ε ( γ t , R t ) = Pr  ˆ X t 6 = X t | γ t , R t  . Note that ε ( · , · ) depend s on the p acket size n and the coding/modu lation sche mes, wh ich are ass umed Nov ember 2, 2018 DRAFT 5 to be kn own at the de coder . For now , we as sume only that the coding/modulation scheme s are su ch that ε ( γ t , R t ) is a conv ex, continuou s, and increa sing function of R t and a c on vex, continuous, and decreas ing function of γ t . Later , we detail the behavior of our propo sed s chemes for the specific case s of u ncoded QAM and random Ga ussian signaling. Based o n the rece i ved packet Y t and the dec oded packet ˆ X t , the dec oder gene rates a feedback packet F t which is co mmunicated to the transmitter through a reverse c hannel. Ass uming that the receiver is capable of perfect e rror detection, we take F t to be a binary A CK/N AK (i.e., F t = 0 for A CK an d F t = 1 for NAK), so that Pr  F t = f | γ t , R t  =      ε ( γ t , R t ) , f = 1 1 − ε ( γ t , R t ) , f = 0 . (2) W e assume that the rev erse channe l is error- free but introduces a delay of a s ingle 2 packet interval. Thus, the “information” a vailable to the transmitter when choosing rate R t is I t = [ F 1 , F 2 , . . . , F t − 1 , R 1 , R 2 , . . . , R t − 1 ] . W e fin d it c on venient to explicitly include the previous rates { R τ } τ < t in the information vector I t becaus e the A CK/N AK fee dback F τ characterizes chan nel quality relative to the transmission rate R τ . Note that the controller c hoose s the trans mission rate a t time t solely b ased on the information vec tor I t , which is av ailable at the rec eiv er as well. W e a ssume that the rece i ver is also aware of the controller’ s rate allocation strategy , s o tha t it can comp ute the current and previous values of R t . Finally , we ass ume in the sequel that the S NR is c onstant over e ach block o f T ≫ 1 packets, a nd tha t it change s indep endently from block to block , i.e ., that the channel is “ block fading. ” In the s equel, we focus (without los s o f g enerality) on the first bloc k, for which t ∈ { 1 , . . . , T } , and omit the t -depen dence on the SNR, writing γ t as “ γ . ” In a ddition, we use p ( γ | I t ) to d enote the posterior SNR distrib ution, which can be a ssociated with the prior distribution p ( γ ) through the co nditional mass func tion P ( F t | γ , R t ) giv en in (2). Fu rthermore, we deno te the set of possible posterior probability distributions u sing P ( I t ) . B. Ide al Rate S election W e de fine the ideal p -hypothes ized con tr oller as the o ne that, at time t , ba sed on the h ypothesiz ed pos - terior p ( γ | I t ) , jointly optimizes the transmission rates ( R t , . . . , R T ) to max imize the sum-rate P T τ = 1 R τ subject to a co nstraint on expected error probability . In doing s o, we allow any packet to be declared a pr obe packet , wh ich is exempt from the expected-error-probability co nstraint but contributes nothing to 2 It is straightforward to generalize al l of our results to a general delay of d > 1 packet intervals. While the generalization does not alter the fundamental nature of our results, it requires a more complex notation, which we avoid for clari ty . Nov ember 2, 2018 DRAFT 6 sum rate. Probe pa ckets are us ed exclusively to le arn about the SNR γ , in the hope of more efficient allocation of future data packets . In particular , the ideal controller choose s r ates acc ording to the foll owing constrained optimization problem: max ( D t ,...,D T ) ∈{ 0 , 1 } T − t +1 , ( R t ,...,R T ) ∈R T − t +1 T X τ = t D τ R τ (3) subject to D τ E p [ ε ( γ , R τ ) | I t ] 6 e − α for a ll τ = t, . . . , T . (4) Here, D τ ∈ { 0 , 1 } indicates whe ther the τ th packet is a da ta pa cket ( D τ = 1 ) o r a prob e packet ( D τ = 0 ), and α > 0 is an app lication-depende nt quality-of-service (QoS) parameter . Note that the expectation E p [ · ] in (4) is taken over the c onditional distrib ution p ( γ | I t ) . W ith A CK/N AK feedback, recall that I t = [ F 1 , F 2 , . . . , F t − 1 , R 1 , R 2 , . . . , R t − 1 ] . Thus, the choice of R t aff ects n ot only the c ontrib ution to the sum-rate but also the “q uality” of the co nditional SNR distrib ution p ( γ | I τ ) a t times τ ≥ t + 1 . As these future SNR estimates g et worse, the controller is force d to cho ose more con servati ve (i.e., lower) rates in order to satisfy the expec ted error-r ate c onstraint. (W e justify this statement in the sequel.) Th us, the se lection of R t has both short-term a nd long-term cons equenc es, which ma y be in co nflict. Conse quently , the so lution to the idea l rate adaptation p roblem (3,4) under A CK/N AK feedback is a cons trained partially observable Markov dec ision process (POMDP) [23]. For practical horizons T , it is c omputationally impractical to implement this P OMDP , as n ow d escribed. Firstly , n otice tha t the s tate of the c hannel is continuous . Even if the c hanne l state was discretized (at the expen se of some los s in pe rformance), the required memo ry to impleme nt the optimal sch eme would grow exponentially with the horizon T . Ind eed, this P OMDP lies in the sp ace of P SP A CE-complete problems, i.e., it req uires bo th comp lexity and memory that g row exponentially with the horizon T [24]. Next, c onsider the (ge nie-aided) cas e of perfect CSI, i.e., I t = γ for a ll t . Whe n the c hanne l is k nown, there is no ne ed for probe pac kets, and thus the optimal so lution choo ses D τ = 1 ∀ τ . Furthermore, since the rate choice does n ot affect the quality of the SNR estimate, the idea l rate assignme nt problem decoup les, so that the best c hoice for R t becomes R perf-CSI t ( γ ) , arg max R t ∈R R t s.t. ε ( γ , R t ) 6 e − α . (5) Indeed, with pe rfect CSI, cons traint (4) is ac ti ve for all t = 1 , . . . , T , since ε ( γ , R t ) is a conv ex increasing function o f R t and the objective function is linear in R t . Notice that, in this case , ideal rate selection is greedy a nd R perf-CSI t ( γ ) is in variant 3 to time t . 3 This inv ariance holds as long as ε ( · , · ) is t -in v ariant, i.e., the coding/modu lation scheme does not change with time. Nov ember 2, 2018 DRAFT 7 P S f r a g r e p l a c e m e n t s I T p +1 ˆ γ ( I T p +1 ) { R T p +1 , . . . , R T } channel estimator rate allocator Fig. 2. The controller decomposed into two componen ts: a channel estimator and a rate allocator . C. Prac tical Rate Selection In practice, we have neither the exact pos terior p ( γ | I t ) , nor the perfect CSI. Thu s, we c onsider a practical (non-ideal) app roach, motiv ated b y technique s from the field o f a daptive control [25], which deviates from the ideal app roach in two principal ways: 1) the probe packet locations are set at the first T p packets in eac h T - block, and 2) the c ontroller is s plit into two componen ts: a channel e stimator , which produces an SNR estimate ˆ γ ( I T p +1 ) based o n the prob e-packet feedba ck I T p +1 , a nd a rate alloca tor , which a ssigns the data packet rate base d on ˆ γ ( I T p +1 ) . (See Fig. 2.) As before, the rate allocator choos es the data-packet rates ( R T p +1 , . . . , R T ) in order to maximize s um-rate under an expected-error-probability constraint. In particular , at each time t ∈ { T p + 1 , . . . , T } , the rate R t is chosen via: max ( R t ,...,R T ) ∈R T − t +1 T X τ = t R τ (6) subject to E p  ε ( γ , R τ ) | ˆ γ ( I T p +1 )  6 e − α (7) for a ll τ = t , . . . , T , where the expectation in (7) is taken over some posterior d istrib ution p ( γ | ˆ γ ( I T p +1 )) . Let u s denote ˆ γ T p +1 , ˆ γ ( I T p +1 ) and the s et of pos sible p osterior distributions with P ( ˆ γ T p +1 ) , which, in turn, is de cided by the particular c hoice o f the e stimator ˆ γ ( · ) . While related, the co nstraints (4) an d (7) have an important dif ference: the information co ntained by I t in (4) is su mmarized by the poss ibly inc omplete sta tistic ˆ γ ( I T p +1 ) in (7). Cons equen tly , sa tisfaction of (7) do es not necess arily gu arantee s atisfaction of (4) or vice versa. Due to the fact that the probing period is limited to the first T p packets, { R T p +1 , . . . , R T } does n ot aff ect the quality of future SNR estimates, the rate assignmen t prob lem (6)-(7) de couples , an d the value of R t satisfying (6)-(7) redu ces to R ∗ t , arg max R t ∈R R t s.t. E p  ε ( γ , R t ) | ˆ γ ( I T p +1 )  6 e − α . (8) Nov ember 2, 2018 DRAFT 8 Moreover , (8) implies that R ∗ t is i n variant to time t . Note that the decoupling that occurs here is r eminisce nt of the decou pling that occurred with ideal rate selection (3)-(4) un der perfect channe l s tate information, i.e., (5). In the n ext section, we shall se e tha t the choice of estimator plays a key role in the overall performanc e of the practical rate adapta tion s cheme. Recall tha t the es timator determines p ( γ | ˆ γ ( I T p +1 )) , which determines the expected e rror proba bility constraint. Under certain s cenarios , we shall s ee that a solution to (8) do es not exist, i.e., that no rates within R satisfy the expected error probability constraint. La ter , in Section V, we dev elop a non-Bay esian es timator in and s how that, with that estimator , the set P ( ˆ γ T p +1 ) will contain merely the class of Gauss ian distrib utions, as ymptotically as T p → ∞ , for any se t, P , of prior d istrib utions with finite mean and v ariance for the S NR. I I I . R A T E A DA P TA T I O N W I T H I M P E R F E C T C S I Before studying the practical rate allocator (8), we first co nsider a pa rticular “ naiv e” d ata-rate a llocator , in orde r to draw intuition on how estimation errors aff ect syste m performance . Giv en SNR estimate ˆ γ , generated from a pa rticular u nbiased e stimator , the na i ve allocator as signs the data rate R nai ve t ( ˆ γ ) , arg max R t ∈R R t s.t. ε ( ˆ γ , R t ) 6 e − α (9) for all t = T p + 1 , . . . , T . Due to the lack of expectation in the error-probability cons traint o f (9), the naiv e rates may violate the d esired expec ted-error -probability cons traint in (8). This follows from the fact that, when the po sterior distributi on p ( γ t | ˆ γ t ) is non-a tomic (i.e., σ 2 γ | ˆ γ > 0 ), Jense n’ s inequality 4 implies that E p [ ε ( γ , R t ) | ˆ γ ] > ε ( ˆ γ , R t ) ∀ R t . (10) Therefore, to ensure the expected -error-probability constraint in (8), the practical allocator must “back- off ” the rate relativ e to R nai ve t ( ˆ γ ) . T o do s o, it choose s R ∗ t ( ˆ γ ) 6 R nai ve t ( ˆ γ ) , whe re equality occurs if and only if the e stimation e rror N , γ − ˆ γ is zero-valued (with probability o ne). When the estimator is perfect (i.e., ˆ γ = γ ), we n ote that the naiv e ra te coincides with the idea l rate under perfect CSI (i.e., R nai ve t ( ˆ γ ) = R perf-CSI t ( γ ) | γ = ˆ γ ). In this ca se, R nai ve t acts as an up per bound on the ideal R t under A CK/N AK fee dback , as spec ified by (3)-(4). Acco rdingly , we make the follo wing two definitions. 4 For unbiased ˆ γ , (10) immediately follows fr om Jensen’ s inequality . For biased ˆ γ , (10) still holds but requires some effort to deriv e. W e skip these details since our focus is on unbiased ˆ γ . Nov ember 2, 2018 DRAFT 9 Definition 1: Th e rate pena lty asso ciated w ith estimator ˆ γ is the smallest δ (in bits/symbol) t hat satisfies E p  ε ( γ , R nai ve t ( ˆ γ ) − δ ) | ˆ γ  6 e − α . (11) Definition 2: Th e p ower pe nalty ass ociated with es timator ˆ γ is the sma llest sc ale factor µ that sa tisfies E p  ε ( µγ , R nai ve t ( ˆ γ )) | ˆ γ  6 e − α . (12) Next, we analyze two different s cenarios for the de scribed rate a daptation system. In the first sc enario, the n symbols in the packet are ass umed to b e unco ded QAM symb ols, while in the se cond sce nario, the n symbols are a Gauss ian random code d en semble. W ithin the se cond s cenario, we foc us on the high-SNR and low-SNR case s s eparately . For bo th s cenarios, we u se the ana lysis presented next, in Sec. III-A. A. Ga ussian Appr oximation of the Estimation Err or Under the pos terior distribution p ( γ | ˆ γ ) , let the estimation error N = γ − ˆ γ have the dis trib ution q ( N | ˆ γ ) = p ( N + ˆ γ | ˆ γ ) . L et g N | ˆ γ ( r ) and Λ N | ˆ γ ( r ) d enote the moment generating function and the semi- in variant log moment gen erating function [26 ] of N giv en ˆ γ , respec ti vely . W e assu me that the re exists some r max > 0 suc h that Λ N | ˆ γ ( r ) < ∞ for all | r | < r max . It is we ll known [26] that Λ N | ˆ γ (0) = 0 , Λ ′ N | ˆ γ (0) = E q [ N | ˆ γ ] , a nd Λ ′′ N | ˆ γ (0) = σ 2 N | ˆ γ . Then, for any | r | < r max , E q [exp( r N ) | ˆ γ ] = g N | ˆ γ ( r ) = exp  Λ N | ˆ γ ( r )  (13) = exp  E q [ N | ˆ γ ] r + 1 2 Λ ′′ N | ˆ γ ( r ′ ) r 2  (14) for s ome r ′ between 0 a nd r (having the same sign as r ), where (14) follows from T aylor’ s theorem. Furthermore, applying T aylor’ s theorem to the third-order expansion, we ge t g N | ˆ γ ( r ) = exp  E q [ N | ˆ γ ] r + 1 2 σ 2 N | ˆ γ r 2 + 1 6 Λ ′′′ N | ˆ γ ( r ′′ ) r 3  (15) for s ome r ′′ between 0 and r . In many cases , the first two terms of the expa nsion (15) lea d to insightful expres sions to illustrate the impact of the first- an d sec ond-order statistics of “cha nnel variabili ty . ” Th is will be referred to as the Gaussian appr oximation , since, when N | ˆ γ is Gaussian, the cu mulants of h igher o rder than the variance vanish. Further , for an unbiased e stimator , E q [ N | ˆ γ ] = 0 . In this case, the Gauss ian approx imation yields the simple secon d-order approximation: Λ N | ˆ γ ( r ) ≈ 1 2 σ 2 N | ˆ γ r 2 . (16) Nov ember 2, 2018 DRAFT 10 Regardless of the p osterior distribution p ( N | ˆ γ ) , the ap proximation (16) is asymptotically ac curate for the non-Bay esian es timator proposed in Sec tion V, which is asymp totically un biased and asy mptotically normal, as will b e prov ed. B. R ate Adaptation with Un coded QAM Here, we study the sce nario in whic h the n symbo ls { X t,k } n k =1 of packet t are uncod ed and selected from a QAM constellation of size M t . Since the constellation size is c onstant over the pa cket, the rate equals R t = log 2 M t bits/symbol. The follo wing is a tight 5 approximation [2, p. 289 ] on the sy mbol e rr or rate as sociated with minimum-distance decision making [27, p . 280]: ε k ( γ , R t ) ≈ 0 . 2 exp  − 3 2 γ 2 R t − 1  . (17) The associated pa cket e rror rate is ε ( γ , R t ) = 1 − (1 − ε k ( γ , R t )) n , (18) since ε k ( γ , R t ) remains constant for all k , a s γ and R t remain constant over the pac ket. Since we can write (1 − ε k ( γ , R t )) n 6 1 − nε k ( γ , R t ) + 1 2 n ( n − 1) ε 2 k ( γ , R t ) , (19) it follows that ε ( γ , R t ) > n 2 ε k ( γ , R t ) for all ( γ , R t ) su ch that ε k ( γ , R t ) < 1 n − 1 . Similarly , (18) implies that ε ( γ , R t ) < 1 − (1 − 1 n − 1 ) n for the same ( γ , R t ) . This latter bou nd is an increa sing function o f n , and, for n ≫ 1 , it a pproximately equa ls 1 − e − 1 , wh ich is muc h highe r than typ ical error rate s. W e a ssume that n is lar ge en ough and the p ossible o utcomes of ( γ , R t ) are such that ε ( γ , R t ) > n 2 ε k ( γ , R t ) for all t with prob ability close to 1 . W e further elaborate on this next, a fter we derive a sufficient co ndition for the error c onstraint to b e me t. T o me et the expe cted-error- probab ility cons traint (8), it is necess ary that n 2 E p [ ε k ( γ , R t ) | ˆ γ ] ≈ n 2 E p  0 . 2 exp  − 3 2 γ 2 R t − 1      ˆ γ  (20) = n 2 E q  0 . 2 exp  − 3 2 ˆ γ + N 2 R t − 1      ˆ γ  6 e − α . (21) 5 The bound holds within approximately 1 dB from the true v alue for a wide range of SNRs [2 , p. 289]. Nov ember 2, 2018 DRAFT 11 Using the unbias ed Gaussian app roximation (16), condition (21) can be rewri tten as follo ws, after taking the natural log o f b oth s ides: − 3 2 ˆ γ 2 R t − 1 + σ 2 N | ˆ γ 2  3 2 1 2 R t − 1  2 6 − α − ln 0 . 1 n. (22) For the existence of a fea sible rate R t , the s olution se t for Inequa lity (22) must be non-empty , for which it is necess ary that ˆ γ 2 σ 2 N | ˆ γ > 2( α + ln 0 . 1 n ) . (23) Condition (23) implies that ˆ γ 2 /σ 2 N | ˆ γ , the effective SNR of estimator ˆ γ , must b e at lea st 2( α + ln 0 . 1 n ) to gu arantee an expected error rate of e − α . Us ing similar steps, 6 a s ufficient cond ition ˆ γ 2 /σ 2 N | ˆ γ > 2( α + ln 0 . 2 n ) can also be de ri ved, illustrating the tightness of (23). W e will in vestigate the difficulty of achieving this condition in the next section. Gi ven that (23 ) is s atisfied, one c an so lve (22) to find the upper b ound R ∗ t 6 ¯ R ∗ t ( ˆ γ , σ 2 N | ˆ γ ) , where ¯ R ∗ t ( ˆ γ , σ 2 N | ˆ γ ) , log 2 1 + ˆ γ · 3 2 σ 2 N | ˆ γ ˆ γ 2 1 − s 1 − 2( α + ln 0 . 1 n ) σ 2 N | ˆ γ ˆ γ 2 ! − 1   . (24) Fig. 3(a) plots the uppe r b ound (24) as a function of the es timator’ s eff ective SNR ˆ γ 2 /σ 2 N | ˆ γ for ˆ γ ∈ { 13 , 20 , 25 } d B, a d esired pa cket error rate of e − α = 10 − 3 , and a pa cket size of n = 500 symbols. The naiv e rate alloca tion R nai ve t ( ˆ γ ) = log 2  1 + ˆ γ · 3 2 1 α + ln 0 . 1 n  (25) (deri ved from (21) with N = 0 ) is als o shown on the sa me plot. The re quired effecti ve SNR ˆ γ 2 /σ 2 N | ˆ γ , as imposed by (23), is 21 . 6 here. Fig. 3(a) shows that R nai ve t ( ˆ γ ) < 2 bits/symbo l for ˆ γ 6 13 dB. Sinc e 2 bits/symbol is the minimum possible rate for uncode d QAM, we conc lude that it is impos sible to meet the tar get packet-error ra te of 10 − 3 when ˆ γ 6 13 dB, ev en with pe rfect CSI. By d efinition, the rate pena lty is the sma llest δ that satisfie s δ = R nai ve t ( ˆ γ ) − R ∗ t ( ˆ γ , σ 2 N | ˆ γ ) . Thus, a n upper bound o n δ is giv en b y ¯ δ ( ˆ γ , σ 2 N | ˆ γ ) , R nai ve t ( ˆ γ ) − ¯ R ∗ t ( ˆ γ , σ 2 N | ˆ γ ) . (26) 6 From (18 ) and the fact that ( 1 − ǫ t,k ) n > 1 − nǫ t,k , we have ε ( γ , R t ) 6 nε k ( γ , R t ) for al l ( t, k ) with probability 1 . Conseque ntly , for satisfaction of (8), it is sufficient that n E p [ ε k ( γ , R t ) | ˆ γ ] 6 e − α . Replicating (21)-(23), we obtain t he suf ficiency condition. Nov ember 2, 2018 DRAFT 12 40 60 80 100 0 1 2 3 4 5 6 rate (bits/symbol) (a) P S f r a g r e p l a c e m e n t s ˆ γ = 25 dB ˆ γ = 20 dB ˆ γ = 13 dB ¯ R ∗ t R naive t ˆ γ 2 /σ 2 N | ˆ γ 40 60 80 100 0 0.5 1 1.5 2 2.5 3 3.5 (b) power penalty (dB) P S f r a g r e p l a c e m e n t s ˆ γ 2 /σ 2 N | ˆ γ Fig. 3. For QAM signaling, (a) rates ¯ R ∗ t and R naive t versus estimator’ s effecti ve SNR ˆ γ 2 /σ 2 N | ˆ γ , and (b) po wer penalty lo wer bound µ versus estimator’ s effecti ve SNR ˆ γ 2 /σ 2 N | ˆ γ . From Fig. 3(a), we c an see that ¯ δ ( ˆ γ , σ 2 N | ˆ γ ) de pends on the effecti ve SNR ˆ γ 2 /σ 2 N | ˆ γ : it is significa nt when the eff ective SNR is ne ar the minimum value established by (23), but s hrinks as ˆ γ 2 /σ 2 N | ˆ γ gets lar ge. In addition, ¯ δ ( ˆ γ , σ 2 N | ˆ γ ) gro ws in proportion to ˆ γ . By defin ition, the power pen alty is the s mallest µ that s atisfies R ∗ t ( ˆ γ ) = R nai ve t ( ˆ γ /µ ) . T hus, a lower bound µ ( ˆ γ , σ 2 N | ˆ γ ) on the power pena lty can be found by solving ¯ R ∗ t ( ˆ γ , σ 2 N | ˆ γ ) = R nai ve t ( ˆ γ /µ ) for µ . The power pen alty lower bou nd µ ( ˆ γ , σ 2 N | ˆ γ ) is p lotted in Fig. 3(b) as a function of ef fective SNR ˆ γ 2 /σ 2 N | ˆ γ for the s ame expected pa cket-error rate, 10 − 3 , an d pac ket s ize, n = 500 , as in Fig. 3(a). T he power penalty is seen to be as high as 3 dB when the effecti ve SNR is n ear the minimum value es tablished by (23), but shrinks as ˆ γ 2 /σ 2 N | ˆ γ gets lar ge. C. Ra te Adaptation with Ra ndom Gau ssian E nsembles Next, we stud y the random c oding [28], [29] sc enario in which the cod ew ords a re se lected from a Gaussian e nsemble. Le t R max be the maximum rate in R . Then the Gau ssian ens emble cons ists of 2 nR max possible pac kets, wh ere each symbol, X t,k , of packet t is chos en independe ntly from a N ( 0 , 1) distrib ution. 7 (W e u se unit variance h ere b ecaus e earlier we as sumed E  | X t,k | 2  = 1 .) At time t , say that 7 W e use real-value d symbols, i nstead of comple x-valued symbols, f or si mplicity . Consequently , the data rates will be represented in units of bits per real-symbol. For fair comparison with uncode d QAM, one should simply double these data rates. Nov ember 2, 2018 DRAFT 13 transmission rate R t ∈ R is c hosen. Then o ne packet from a s ize- 2 nR t subset of the initially gen erated set o f 2 nR max packets is chose n arbitrarily for transmission. The receiver is assumed to know the su bsets of possible packets correspon ding to eac h ad missible rate R t ∈ R . Based on its observation of the t th packet, the recei ver finds the mos t likely p acket within the subs et o f 2 nR t possible packets. No te that, unlike the u ncode d QAM sc enario, where each s ymbol is dec oded separately , here the entire p acket is deco ded a s a unit. An up per boun d for the asso ciated decoding error p robability is (e .g., [28]) ε ( γ , R t ) 6 exp  nρ  R t ln 2 − 1 2 ln  1 + γ 1 + ρ  , (27) where ρ ∈ [0 , 1] is the u nion bo und parameter . One can minimize (27) over ρ ∈ [0 , 1] to find the tightest bound, if so desired. T o satisfy the expected -error -probability constraint (8 ), it suffices that there exists a ρ ∈ [0 , 1] for which E p  exp  nρ  R t ln 2 − 1 2 ln  1 + γ 1 + ρ      ˆ γ  6 e − α . (28) 1) Low-SNR Regime: Wh en Pr  γ ≪ 1 | ˆ γ  ≈ 1 , we can write ln  1 + γ 1 + ρ  ≈ γ 1 + ρ = ˆ γ + N 1 + ρ . (29) For an u nbiased estimator , E q  ˆ γ + N 1+ ρ   ˆ γ  = ˆ γ 1+ ρ and var q  ˆ γ + N 1+ ρ   ˆ γ  = σ 2 N | ˆ γ (1+ ρ ) 2 . Thus , us ing the Ga ussian approximation (16), the co nstraint (28) is satisfied if there exists a ρ ∈ [0 , 1] for which α 6 − n ρ  R t ln 2 − ˆ γ 2(1 + ρ ) + 1 8 nρ (1 + ρ ) 2 σ 2 N | ˆ γ  , (30) or , e quiv alently , for which R t 6 1 ln 2  − α nρ + ˆ γ 2(1 + ρ ) − 1 8 nρ (1 + ρ ) 2 σ 2 N | ˆ γ  . (31) Thus, if the re exists some ρ ∈ [0 , 1] for which the right s ide o f (31) is pos iti ve, then any R t below it is feasible. For this to be p ossible, we need 2 α (1 + ρ ) 2 − ˆ γ nρ (1 + ρ ) + 1 4 ( nρ ) 2 σ 2 N | ˆ γ 6 0 Nov ember 2, 2018 DRAFT 14 for s ome ρ ∈ [0 , 1] , which leads to the following nec essa ry c ondition 8 for the e stimator: ˆ γ 2 σ 2 N | ˆ γ > 2 α. (32) One can then find an upper bound on R t ∈ R satisfying (28) a s follo ws: ¯ R ∗ t ( ˆ γ , σ 2 N | ˆ γ ) = max ρ ∈ [0 , 1] 1 ln 2  − α nρ + ˆ γ 2(1 + ρ ) − 1 8 nρ (1 + ρ ) 2 σ 2 N | ˆ γ  . (33) Like wise, o ne can deduce from (27) and (29) that the naiv e rate is R nai ve t ( ˆ γ ) = max ρ ∈ [0 , 1] 1 ln 2  − α nρ + ˆ γ 2(1 + ρ )  . (34) The rate upp er bou nd ¯ R ∗ t is plotted in Fig. 4(a) as a function of the e stimator’ s e f fective SNR ˆ γ 2 /σ 2 N | ˆ γ for ˆ γ ∈ {− 3 , − 8 , − 12 } dB, a des ired packet error rate of e − α = 10 − 3 , an d a packet size o f n = 500 symbols. The rate R nai ve t from (34) is also sh own on the same plot. E very point on the rate curves was computed using the optimal v alue of ρ ∈ [0 , 1] , found numerically . W e n ote that, with the se parameters, (32) implies that ˆ γ 2 /σ 2 N | ˆ γ must be at least 13 . 8 . F igure 4(a) also s hows that the rate penalty ¯ δ ( ˆ γ , σ 2 N | ˆ γ ) = R nai ve t ( ˆ γ ) − ¯ R ∗ t ( ˆ γ , σ 2 N | ˆ γ ) is significant when ˆ γ 2 /σ 2 N | ˆ γ is near the lower boun d e stablished by (32), but that the rate pen alty s hrinks as ˆ γ 2 /σ 2 N | ˆ γ increases . For the sa me tar get pa cket e rror rate ( 10 − 3 ) and pa cket size ( n = 500 ), Fig. 4(b) plots ¯ R ∗ t versus ˆ γ for e stimator eff ective SNR ˆ γ 2 /σ 2 N | ˆ γ ∈ { 60 , 100 } . In the sa me figure, R nai ve t and the “na i ve” Shann on limit (i.e., er godic capacity) 1 2 log 2 (1 + ˆ γ ) b its/real-symbol are sh own. By c omparing the naiv e Shannon limit with R nai ve t , one can obse rve tha t, in the low-SNR regime, the power penalty o f Gaussia n sign aling scheme can be significant, espe cially at small values of ˆ γ . From the same plot, one can obse rve that the additional power pe nalty d ue to imperfect SNR e stimation, µ ( ˆ γ ) , is quite small: less than 0 . 5 d B when ˆ γ 2 /σ 2 N | ˆ γ = 100 and less tha n 1 dB when ˆ γ 2 /σ 2 N | ˆ γ = 60 . 2) High-SNR Regime: Whe n P r  γ ≫ 1 | ˆ γ  ≈ 1 , we c an write ln  1 + γ 1 + ρ  ≈ ln  γ 1 + ρ  = ln  ˆ γ + N 1 + ρ  = ln  ˆ γ 1 + ρ  + ln  1 + N ˆ γ  . (35) 8 Note that condition (32) is not exactly analogous to condition (23). Condition (32) is necessary for a non-empty solution set to exist f or inequality (31), whereas (23) i s necessary for the existence of a f easible rate that satisfies the expec ted-error bound. In order to deriv e an analogous necessary condition, one can use a sphere-packing (SP) bound for t he Gaussian channel (see, e. g., [30]). W ith the SP lo wer bound, our findings would be qualitatively similar , but the deriv ation would be extremely tedious. For this reason, we assume that the upper bound is a good approximation for the actual error rate. Nov ember 2, 2018 DRAFT 15 40 60 80 100 120 10 −4 10 −3 10 −2 10 −1 10 0 rate (bits/real−symbol) (a) P S f r a g r e p l a c e m e n t s ˆ γ = − 3 dB ˆ γ = − 8 dB ˆ γ = − 12 dB ¯ R ∗ t R naive t ˆ γ 2 /σ 2 N | ˆ γ −12 −10 −8 −6 −4 10 −4 10 −3 10 −2 10 −1 10 0 rate (bits/real−symbol) (b) P S f r a g r e p l a c e m e n t s R naive t ¯ R ∗ t at ˆ γ 2 /σ 2 N | ˆ γ = 60 ¯ R ∗ t at ˆ γ 2 /σ 2 N | ˆ γ = 100 naive Shannon limit ˆ γ (dB) Fig. 4. For Gaussian si gnaling at low SNR, rates ¯ R ∗ t and R naive t versus (a) estimator’ s effecti ve SNR ˆ γ 2 /σ 2 N | ˆ γ and (b) estimated SNR ˆ γ . Thus, for an unbiase d e stimator , E q  ln γ 1+ ρ   ˆ γ  ≈ ln ˆ γ 1+ ρ and var q  ln γ 1+ ρ   ˆ γ  ≈ σ 2 N | ˆ γ ˆ γ 2 . Similar to the low SNR s cenario, w e can u se the Gau ssian a pproximation (16) to claim that (28) is satisfie d if there exists a ρ ∈ [0 , 1] for which α > − n ρ R t ln 2 − 1 2 ln  ˆ γ 1 + ρ  + 1 8 nρ ˆ γ 2 /σ 2 N | ˆ γ ! , (36) or , e quiv alently , R t 6 1 ln 2 − α nρ + 1 2 ln  ˆ γ 1 + ρ  − 1 8 nρ ˆ γ 2 /σ 2 N | ˆ γ ! . (37) Hence, if there exists some ρ ∈ [0 , 1] for whic h the right s ide of (37) is positiv e, then any R t below it is feasible. In the high-SNR regime, we have ˆ γ ≫ 1 w ith high probability , a nd thus there almos t a lw ays exists some ρ ∈ [0 , 1] for which a feas ible R t > 0 exists. One c an deduc e from this observation that, a principal diff erenc e betwee n the high-SNR a nd low- SNR regimes is that, in the high-SNR regime, the expected error proba bility co nstraint is sa tisfied muc h mo re easily , with nearly any SNR estimator . One can then fi nd an up per bound on R t ∈ R sa tisfying (28) as follows: ¯ R ∗ t ( ˆ γ , σ 2 N | ˆ γ ) = max ρ ∈ [0 , 1] 1 ln 2 − α nρ + 1 2 ln  ˆ γ 1 + ρ  − 1 8 nρ ˆ γ 2 /σ 2 N | ˆ γ ! . (38) Nov ember 2, 2018 DRAFT 16 Like wise, o ne can deduce from (27) and (35) that the naiv e rate is R nai ve t ( ˆ γ ) = max ρ ∈ [0 , 1] 1 ln 2  − α nρ + 1 2 ln  ˆ γ 1 + ρ  . (39) 40 60 80 100 120 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 rate (bits/real−symbol) (a) P S f r a g r e p l a c e m e n t s ˆ γ = 25 dB ˆ γ = 20 dB ˆ γ = 13 dB ¯ R ∗ t R naive t ˆ γ 2 /σ 2 N | ˆ γ 12 14 16 18 20 22 24 26 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 rate (bits/real−symbol) (b) P S f r a g r e p l a c e m e n t s R naive t ¯ R ∗ t at ˆ γ 2 /σ 2 N | ˆ γ = 60 ¯ R ∗ t at ˆ γ 2 /σ 2 N | ˆ γ = 20 naive Shannon limit ˆ γ (dB) Fig. 5. For Gaussian si gnaling at high SNR, rates ¯ R ∗ t and R naive t versus (a) estimator’ s effecti ve S NR ˆ γ 2 /σ 2 N | ˆ γ and (b) estimated SNR ˆ γ . The rate upp er bou nd ¯ R ∗ t ( ˆ γ , σ 2 N | ˆ γ ) is plotted in Fig. 5(a) as a func tion of the estimator’ s effecti ve SNR ˆ γ 2 /σ 2 N | ˆ γ for ˆ γ ∈ { 13 , 20 , 25 } dB, a desired pa cket error ra te of e − α = 10 − 3 , an d a pac ket size of n = 500 symbols. T he rate R nai ve t ( ˆ γ ) from (39) is also shown on the same plot. Every point on the rate c urves was computed using the optimal value of ρ ∈ [0 , 1] , fou nd numerically . W e emp hasize that the rate s plotted in Fig. 5(a) are expresse d in b its per r eal-symb ol , a nd thus s hould be dou bled for fair comparison with the QAM rates presen ted in Fig. 3(a). For Gau ssian signa ling, if w e c ompare the high-SNR results in Figs. 5(a)-5(b) to the low-SNR results in Figs . 4(a)-4(b), we can see that the n ormalized rate penalty ¯ δ / ¯ R ∗ t is muc h s maller in the high-SNR regime. For instance , at ˆ γ 2 /σ 2 N | ˆ γ = 20 , ¯ δ is no mo re than 0 . 5 bits/symbol a nd ¯ δ / ¯ R ∗ t is less than 25% for all three values of ˆ γ . Th is de crease in rate pe nalty is exp ected, since, in the high -SNR regime, the rate scales roug hly with the log of the SNR. For the sa me tar get packet error rate ( 10 − 3 ) and pa cket siz e ( n = 500 ) , Fig. 5(b) plots ¯ R ∗ t ( ˆ γ , σ 2 N | ˆ γ ) versus ˆ γ for e stimator eff ective SNR ˆ γ 2 /σ 2 N | ˆ γ ∈ { 60 , 100 } . In the same fi gure, R nai ve t ( ˆ γ ) and the naive Shannon limit 1 2 log 2 (1 + ˆ γ ) are sh own. There we obs erve that, in the high-SNR regime, the power penalty for Gaus sian signaling is con stant with ˆ γ , a nd no more than 1 . 5 dB. The additional p ower p enalty due to impe rfect SNR estimation, µ ( ˆ γ , σ 2 N | ˆ γ ) , is approximately 1 dB when ˆ γ 2 /σ 2 N | ˆ γ = 60 and approximately Nov ember 2, 2018 DRAFT 17 2 . 5 dB when ˆ γ 2 /σ 2 N | ˆ γ = 20 . I V . F U N DA M E N TA L L I M I TA T I O N S O F AC K / NA K - B A S E D R AT E A DA P T A T I O N In the previous section, we s tudied the performance of the rate ada ptation sys tem for a gene ric unbiased estimator . W e an alyzed the feasible rates with particular c oding/modulation scheme s a s a func tion of the “quality” of the es timation provided by the estimator , for which the relev ant metric was the estimator’ s eff ective SNR ˆ γ 2 /σ 2 N | ˆ γ . Note tha t w e assume d no knowledge of the prior SNR distribution p ( γ ) . In this se ction, we view the SNR of the c urrent bloc k, γ , as an unknown parame ter , 9 and pose the estimation o f γ as a non -Bayesian pa rameter es timation problem. W e first in vestigate the fundame ntal limitations of SNR estimators that are bas ed on packet-lev el A CK/N AK feedb ack, e.g., ˆ γ = ˆ γ ( I T p +1 ) . Using that analysis, we s how that it is dif ficult to make g ood SNR e stimates while simultaneous ly keeping packet-error -rate low . This latter prope rty moti vates SNR-estimation via probe p ackets tha t come without error -rate constraints (in contrast to d ata pa ckets, which are e rror -rate constrained) as assumed in Sec . II. Finally , we discus s o ptimization o f the probing period T p , and we deriv e an upper bound on the optimal sum rate R ∗ sum . A. F undamen tal Limitations of ACK/N AK-Based SNR Estimation Consider the SNR e stimator ˆ γ ( I T p +1 ) , based o n the T p A CK/N AKs in I T p +1 = [ F 1 , F 2 , . . . , F T p , R 1 , R 2 , . . . , R T p ] , where R t denotes the rate and F t denotes the A CK/N AK feed back for pa cket t . In the sequel, we abbreviate ˆ γ ( I T p +1 ) by ˆ γ . Re call that R t and F t are connec ted throug h the pac ket error proba bility ε ( γ , R t ) , as s pecified in (2). Theorem 1: For true SNR γ and any unbiased estimator ˆ γ based on T p A CK/N AKs, the estimation error variance, σ 2 N | ˆ γ , var  γ − ˆ γ | ˆ γ  , is lower b ounded by : σ 2 N | ˆ γ >   T p X t =1 ( ε ′ ( γ , R t )) 2 ε ( γ , R t ) [1 − ε ( γ , R t )]   − 1 , (40) where ε ( γ , R t ) is c ontinuously dif ferentiable in γ and ε ′ ( γ , R t ) , ∂ ∂ γ ε ( γ , R t ) . 9 W e assume that γ is a random variable, taking on an independent value for each block, but that the distribution of γ i s unkno wn to the transmitter . Nov ember 2, 2018 DRAFT 18 Pr oof: Given γ and the rates R 1 , . . . , R T p , the fee dback F 1 , . . . , F T p satisfies Pr  F 1 = f 1 , . . . , F T p = f T p | γ , R 1 , . . . , R T p  = T p Y t =1 Pr  F t = f t | γ , R t  . (41) Then V ( γ , R t , f t ) , ∂ ∂ γ ln Pr  F t = f t | γ , R t  = ∂ ∂ γ ln  [ ε ( γ , R t )] f t [1 − ε ( γ , R t )] 1 − f t  = ε ′ ( γ , R t ) 1 − ε ( γ , R t )  f t ε ( γ , R t ) − 1  . (42) The Fisher information [31] as sociated with F t is: Φ( γ , R t ) = var ( V ( γ , R t , f t ) | γ , R t ) = ( ε ′ ( γ , R t )) 2 ε ( γ , R t ) [1 − ε ( γ , R t )] , (43) and the c umulativ e Fisher information is P T p t =1 Φ( γ , R t ) . Theo rem 1 follows since the Cramer-Rao lower bound (CRLB) for un biased estimators is the reciprocal of the F isher information [31]. B. L ower Bounds on the Requ ir ed Probing P e riod T p In Sec. III, we deri ved lower bounds ( 23) and (32) on the v alue of ˆ γ 2 /σ 2 N | ˆ γ (i.e., the estimator’ s ef fecti ve SNR) required to facilitate the us e o f d ata trans mission via un coded QAM signaling and randomly coded Gaussian signaling, resp ectiv ely . In this section, we translate those lo wer bounds (on required ˆ γ 2 /σ 2 N | ˆ γ ) into lower bounds on req uired prob e-duration T p , recognizing tha t the quality of SNR es timates (and thus ˆ γ 2 /σ 2 N | ˆ γ ) increases with T p . From these boun ds, we sha ll se e that the required value of T p depend s heavily on t he probe error rate, and in pa rticular that the required v alue of T p grows very large as the probe error rate decrea ses. This motiv ates the optimization of probe error ra te, which requires the decou pling of probe e rror rate from da ta e rror rate (since the latter is usually constrained b y the a pplication). In this sec tion, we a ssume that both the modulation/coding sche me and the rate is fixed over the p robe interval, i.e., that R t = R p for t ∈ { 1 , . . . , T p } . In this ca se, the CRLB (40) reduces to σ 2 N | ˆ γ > 1 T p ε ( γ , R p ) [1 − ε ( γ , R p )] ( ε ′ ( γ , R p )) 2 , (44) which is in versely proportional to T p . Nov ember 2, 2018 DRAFT 19 Recall that, to make unc oded QAM signaling feas ible, condition (23 ) must be satisfied, and to make random Gaus sian sign aling feasible in the low-SNR regime, condition (32) must be sa tisfied. Tho ugh (23) and (32 ) are expressed in terms of the es timator’ s effecti ve SNR, w e c an rewri te them as σ 2 N | ˆ γ 6 1 2 ˆ γ 2 / ( α + ln 0 . 1 n ) a nd σ 2 N | ˆ γ 6 1 2 ˆ γ 2 /α , respec ti vely , and a pply the CRLB (44) to arri ve (see Append ix A) at the following. For unco ded QAM, we need T p > T min p , where T min p = 2( α + ln 0 . 1 n ) ε ( γ , R p ) (1 − ε ( γ , R p ))  (1 − ε ( γ , R p )) − 1 /n − 1  2 ln 2  5  1 − (1 − ε ( γ , R p )) 1 /n  ( n ˆ γ /γ ) 2 , (45) and for ran dom G aussian signaling in the low-SNR regime, we need T p > T min p , where T min p = 8 α  1 − ε ( γ , R p )  1 + ρ ∗ + γ  2 ε ( γ , R p )( nρ ∗ ˆ γ ) 2 (46) and where ρ ∗ is the union bound pa rameter c orresponding to the tightest error bound (27), which itself depend s on γ , R p , and n . 10 −4 10 −2 10 0 10 0 10 1 10 2 10 3 10 4 (a) 10 −4 10 −2 10 0 10 0 10 1 10 2 10 3 10 4 (b) P S f r a g r e p l a c e m e n t s ˆ γ = − 3 dB ˆ γ = − 7 dB ˆ γ = − 10 dB ε ( γ , R p ) ε ( γ , R p ) T min p T min p Fig. 6. Lowe r bound on required probing duration T min p versus probe packet-error rate ε ( γ , R p ) for (a) uncoded QAM and (b) random Gaussian signaling. Figures 6(a)-(b) plot T min p as a func tion of the p robe error rate ε ( γ , R p ) for uncoded QAM signa ling and ran dom Ga ussian s ignaling, res pectiv ely . For the p lots, we as sume ˆ γ ≈ γ , w hich eliminates the depend ence of T min p on ˆ γ and γ in the QA M case; for the Gaus sian case , we s how T p for the values ˆ γ ∈ {− 3 , − 7 , − 10 } dB. As in our p revious plots, we a ssumed n = 500 an d e − α = 10 − 3 . T he key observation to make from these plots is that the numbe r of p robe packets inc reases quickly as ε ( γ , R p ) shrinks. In fact, the plots sugges t that T p is roughly propo rtional to 1 /ε ( γ , R p ) . T his in verse relationship Nov ember 2, 2018 DRAFT 20 is somewhat intuitiv e beca use, giv en a prob e packet-error rate of ε ( γ , R p ) , one must wait for 1 /ε ( γ , R p ) packets (on a verage) to see a single N AK. Re call, h owe ver , that Fig. 6 shows o nly a lower bound T min p on the probe du ration required for c ommunication with positiv e rate; the o ptimal value of T p is expected to be even larger . The main conclus ion to draw from this s ection is that, to keep the probing period sma ll, one must allow relati vely high probe error rate ε ( γ , R p ) . For sys tems which estimate SNR using only A CK/N AK feedback from da ta packets, this implies tha t if the data error rate e − α is small, the n the nu mber of packets required to get a decent S NR estimate will be large. Su ch systems would on ly be suitable for channe ls that are very slowly fading. C. An Up per B ound on the Optimal Su m-Rate Recall that, in our practical rate ad aptation sys tem, the data p acket rates { R t } R T t = T p +1 are chosen ba sed on the SNR estimated using A CK/N AKs from probe packets with rates { R t } T p t =1 . T o complete the system design, we must choose the rates { R t } T t =1 as well as the probe duration T p . In doing so , we aim to maximize the sum da ta rate R sum = P T t = T p +1 R t while satisfying the expected error- probab ility constraint in (8). Intuiti vely , we k now tha t increas ing T p improves the SNR e stimate which, in turn, allows a higher data rate (since less rate “ba ck-off ” is nee ded to satisfy the e rror constraint). On the o ther hand, for a fixed block length T , the number of d ata pa ckets, T − T p , s hrinks a s T p increases . Therefore, the choice of T p in v olves a tradeoff b etween these two objectiv es. In this se ction, we discuss the ch oice of { T p , R 1 , . . . , R T } a nd deri ve an upper bound on the su m rate R sum that le verages the rate bounds from Sec. III and the CRLB from Sec. IV -A. In Sec . II-C, we recognized that the da ta-rate assign ment problem deco uples in such a way that the optimal data rates { R ∗ t } T t = T p +1 become indepe ndent of time t . Th us, in the seq uel, we focus on choos ing a single data rate R d , whose optimal value will be deno ted by R ∗ d . The s ystem design problem then reduces to the following su m-rate maximization: R ∗ sum , max T p 6 T , ( R 1 ,...,R T p ,R d ) ∈R T p +1 ( T − T p ) R d (47) s.t. E p  ε ( γ , R d ) | ˆ γ ( I T p )  6 e − α . As argued in Sec. III, the optimal da ta rate R ∗ d increases monotonically with the quality of the SNR estimate, i.e., with the in verse of the estimator variance 1 /σ 2 N | ˆ γ . Thus , the optimal p robe p arameters { T p , R 1 , . . . , R T p } are those that minimize σ 2 N | ˆ γ . From the CRLB in Theo rem 1, we know tha t σ 2 N | ˆ γ > Nov ember 2, 2018 DRAFT 21 σ 2 N | ˆ γ ( γ ) , where σ 2 N | ˆ γ ( γ ) , min ( R 1 ,...,R T p ) ∈R T p   T p X t =1 [ ε ′ ( γ , R t )] 2 ε ( γ , R t )[1 − ε ( γ , R t )]   − 1 (48) =   T p X t =1 max R t ∈R [ ε ′ ( γ , R t )] 2 ε ( γ , R t )[1 − ε ( γ , R t )]   − 1 . (49) Thus, if γ was provided by a g enie, and if the SNR e stimator was efficient (i.e., C RLB achieving), then (49) suggests to s et the p robe rate at R genie p ( γ ) = arg max R t ∈R [ ε ′ ( γ , R t )] 2 ε ( γ , R t )[1 − ε ( γ , R t )] , (50) which is in variant to bo th time t and probe duration T p . This y ields σ 2 N | ˆ γ ( γ ) = 1 T p ε ( γ , R genie p ( γ ))[1 − ε ( γ , R genie p ( γ ))] [ ε ′ ( γ , R genie p ( γ ))] 2 . (51) Using the genie-aided p robe rate R genie p ( γ ) , we ca n uppe r bo und the op timal s um rate (47) by R genie sum , max T p 6 T , R d ∈R ( T − T p ) R d (52) s.t. E p h ε ( γ , R d ) | ˆ γ ( R genie p ( γ ) , T p ) i 6 e − α , where we explicitly denote the d epend ence of the estimate ˆ γ on both T p and R genie p ( γ ) . Next, rec all that we e stablished, in Sec. III, upper bounds on the largest da ta rate that s atisfies an expected error c onstraint of the type in ( ?? ). In particular , (24) gave an upper bou nd for uncoded QAM signaling, and (33) and (38) gave upper bo unds for Gauss ian sign aling in the low-SNR an d high-SNR regimes, respec ti vely . Thes e data-rate upper bound s, ¯ R ∗ d ( ˆ γ , σ 2 N | ˆ γ ) , can be ap plied to ( ?? ) to bound the optimal sum rate as R ∗ sum 6 ¯ R ∗ sum , whe re ¯ R ∗ sum , max T p 6 T ( T − T p ) ¯ R ∗ d ( ˆ γ , σ 2 N | ˆ γ ) , (53) and where ˆ γ an d σ 2 N | ˆ γ are depend ent on both T p and R p ( γ ) . S ince ¯ R ∗ d ( ˆ γ , σ 2 N | ˆ γ ) increases monotonically in 1 /σ 2 N | ˆ γ , we can uppe r bo und ¯ R ∗ d using the lower bo und on σ 2 N | ˆ γ established in (51). This yields ¯ R ∗ sum 6 R max sum for R max sum , max T p 6 T ( T − T p ) ¯ R ∗ d ( ˆ γ , σ 2 N | ˆ γ ( γ )) . (54) Figures 7(a) and 7(b) plot the normalized su m-rate bound 1 T R max sum as a func tion of the estimated SNR ˆ γ for uncode d QAM and Gauss ian e nsembles , respectively , at T = 5 and T = 50 . As before, we use target error rate 10 − 3 and packet size n = 500 . For the genie-aided probe rate R genie p ( γ ) use d to Nov ember 2, 2018 DRAFT 22 15 20 25 30 0 2 4 6 8 10 Rate (bits/symbol) (a) naive Shannon limit P S f r a g r e p l a c e m e n t s R naive t 1 T R max sum , T = 5 1 T R max sum , T = 50 ˆ γ (dB) −10 −8 −6 −4 −2 0 0 0.1 0.2 0.3 0.4 0.5 Rate (bits/real−symbol) (b) naive Shannon limit P S f r a g r e p l a c e m e n t s R naive t 1 T R max sum , T = 5 1 T R max sum , T = 50 ˆ γ (dB) Fig. 7. Normalized sum-rate bound 1 T R max sum as a function of SNR ˆ γ for (a) QAM and (b) Gaussian signaling in the low-SNR regime. calculate σ 2 N | ˆ γ ( γ ) , we a ssumed that γ ≈ ˆ γ . Th e fig ures also show R nai ve d and the naive Sh annon limit 1 2 log 2 (1 + ˆ γ ) , for comp arison. Note that the difference b etween the naiv e rate R nai ve d and the u pper b ound 1 T R max sum increases significantly as T de crease s. T his is due to the fact that, as T dec reases, it is too costly to allocate a long p robing interval, implying that the qua lity of SNR estimates de creases , so that more rate ba ck-off is required. Note a lso that the difference b etween the naive rate a nd the uppe r b ound increases as the SNR inc reases. This implies that the lack o f perfect CS I b ecomes more costly a s the SNR increases . V . A N A S Y M P T OT I C A L L Y O P T I M A L S N R E S T I M A T O R The qua lity of SNR es timates based on A CK/N AKs from a probe interval is strongly dep enden t on both the probe rates { R t } T p t =1 and the p robe interval T p . For the su m-rate upper bound de ri ved in Sec. IV -C, the probe rate R genie p ( γ ) in (50) was selec ted in a genie -aided manner , assuming knowledge of the true SNR γ . Clearly , γ is no t known in prac tice. In this sec tion, we develop a practical SNR estimator that, d uring the probing interval t ∈ { 1 , . . . , T p } , r ecursive ly u pdates the p robe rate R t and ˆ γ t (i.e., the time- t estimate 10 of γ ) u sing the latest fee dback pair { F t − 1 , R t − 1 } . W e sh ow that the probe rate a daptation is a symptotically optimal , in that R t con ver ges 10 W e emphasize that ˆ γ t is t he time- t estimate of t he time-in va riant SNR γ , and should not be confused wit h the time-varying SNR γ t that was briefly used in Sec. II before the time-inv ariance assumption was introduced. Nov ember 2, 2018 DRAFT 23 to R genie p ( γ ) for any initial probe rate R 1 . Moreover , we sh ow that our SNR e stimator is asy mptotically efficient and asymptotically normal , i.e., that the correspo nding estimation error N t , ˆ γ t − γ con ver ges to a z ero-mean Gauss ian ran dom vari able w hose variance is ide ntical to the CRLB ac hieved with the genie-aided probe rate R genie p ( γ ) . T he normality of the error helps to justify the Gauss ian approx imation used to de ri ve the rate bound s (45) and (46) for the u ncoded QAM and Gaus sian ca ses, respectiv ely . The SNR Es timator: 1) At time t = 1 , choose a n arbitrary rate R 1 ∈ R an d a n arbitrary estimate ˆ γ 1 . 2) At each time t = 2 , . . . , T p , update the e stimate a s ˆ γ t = ˆ γ t − 1 + F t − 1 − ε ( ˆ γ t − 1 , R t − 1 ) ( t − 1) ε ′ ( ˆ γ t − 1 , R t − 1 ) , (55) and choose the rate R t as: R t = argmax R ∈R Φ( ˆ γ t , R ) , (56) where Φ( · , · ) is the Fishe r information as defined in (43). W e p rove the following for ou r estimator . Theorem 2: For both uncoded QAM and Gauss ian ens embles, as T p → ∞ , p T p  ˆ γ T p − γ  d → N T p ∼ N  0 , Φ − 1 ( γ , R genie p ( γ ))  . (57) Pr oof: See Appe ndix B. Theorem 2 implies that our es timator (55) is a symptotically efficient a nd consisten t. Moreover , without any prior information o n γ , rate alloca tion (56) gu arantees the p erformance achieved with the ge nie-aided probe rate R genie p ( γ ) . Next, we s imulate the e stimator . Instea d of the t − 1 on the den ominator , we use ( t − 1) β for various values of β ∈ (0 , 1] . 0 100 200 300 400 500 0 1 2 3 4 5 6 time (packets) Rate (bits/symbol) γ =15 dB γ =20 dB γ =10 dB (a) R t vs. t 0 100 200 300 400 500 0 5 10 15 20 25 time (packets) SNR estimate (dB) γ =10 dB γ =15 dB γ =20 dB (b) ˆ γ t vs. t for β = 0 . 5 0 500 1000 1500 2000 0 5 10 15 20 25 time (packets) SNR estimate (dB) γ =20 dB γ =10 dB γ =15 dB (c) ˆ γ t vs. t for β = 1 Fig. 8. Example trajectories of the recursiv e SNR estimator when uncoded QAM is used. Nov ember 2, 2018 DRAFT 24 In Fig. 8, a single realization of the es timator and the correspond ing assigne d rate are illustrated for dif ferent values of γ , over a block o f T p = 500 prob e packets of size n = 500 symbols. The value of γ and the a symptotic rate R genie p ( γ ) are also s hown on the as sociated graphs. The initial points for the e stimator are ˆ γ 1 = 3 d B, R 1 = 1 b it/symbol, an d the s et of poss ible rates are R = { 1 , 2 , . . . , 10 } in bits/complex-symbol, i.e ., the possible co nstellation sizes a re integer p owers of 2 . For β = 0 . 5 , one can obse rve that the o ptimal rate is reac hed with approx imately 2 0 probe pa ckets for all values of SN R. Once that point is reached , the e stimation error vari anc e dec ays fairly s lowl y due to the low de cay rate β = 0 . 5 . W ith a h igher β , it takes longer to approac h the vicinity o f γ , from the initial v alue ˆ γ 1 , but the estimation error variance is lo wer once in stea dy state. This obse rvati on is illustrated in Fig. 8 (c), where β = 1 and the probing block size is T p = 2000 packets. In the rea lization co rresponding to γ = 20 dB, the “stea dy state” is yet to be reached a fter 2000 p ackets. On the other hand, the amplitude of the fluctuations around the fin al point decay muc h faster , as o ne can observe in the realization c orrespond ing to γ = 10 dB. Dif ferent choice s for β and the ass ociated tradeoffs in volv ed in stochastic a pproximation algorithms are s tudied in [32 ]. 0 100 200 300 400 500 0 1 2 3 4 time (packets) Rate (bits/1D symbol) γ =20 dB γ =3 dB γ =10 dB (a) R t vs. t 0 100 200 300 400 500 0 5 10 15 20 25 time (packets) SNR estimate (dB) γ =20 dB γ =10 dB γ =3 dB (b) ˆ γ t vs. t for β = 0 . 5 0 500 1000 1500 2000 0 5 10 15 20 25 time (packets) SNR estimate (dB) γ =20 dB γ =10 dB γ =3 dB (c) ˆ γ t vs. t for β = 1 Fig. 9. Example trajectories of the recursiv e SNR estimator when Gaussian signaling is used. W e illustrate our estimator resp onse for Gau ssian ense mbles in Fig. 9 . As the s et of rates R , we p icked 100 po ints, e qually spa ced betwee n 0 and 5 bits/real-symbo l. The initial SNR es timate, ˆ γ 1 = 0 dB, was much smaller than the initial one in the QAM simulations, but the initial rate, R 1 = 0 . 5 bits/complex- symbol, was identical to the one in the QAM simulations. He re, we analyze SNR realizations γ = 3 , 10 and 20 dB. W ith Gaus sian ensemble s, the con vergence speed is s lightly lower than tha t with QA M. While the co n ver gence is almost immediate for γ = 3 dB, it takes 30 -40 pa ckets for 10 dB a nd 130-14 0 packets for γ = 20 dB. Th is d if ference is mainly due to the d if ference in the distance s between the initial an d final p oints. On the other h and, due to the large s ize of the s et of poss ible rates (unlike QAM, where Nov ember 2, 2018 DRAFT 25 only a few discrete points are p ossible), there exists some R t ∈ R that is very close to the g enie-aided probe rate R genie p ( γ ) . Cons equen tly , the estimation e rror variance deca ys much faster o nce R t comes n ear the vicinity of R genie p ( γ ) . W e also illustrate the estimator with β = 1 in F ig. 9(c) and one can notice the slow con ver genc e, similar to the QAM simulations. V I . C O N C L U S I O N In this pap er , we studied rate adaptation b ased on A CK/N AK feedbac k. In p articular , we s tudied methods that maximize data rate subject to a cons traint on expec ted packet-error prob ability , assuming that the transmitter ha s no knowledge of the SNR distrib ution. Bec ause op timal ra te allocation was identified as a POMDP , which is impractica l to implement, we focus ed on a s uboptimal framew ork where a c hanne l es timate is ca lculated base d on previous fee dback and a rate is chos en based on this channe l estimate. T o aid the initial rate a llocation, we allowed the use of T p probe p ackets at the start of each da ta block. First we con sidered a so-called “ naive” rate a llocator that maximizes rate s ubject to a constraint on instantaneous packet-error probability , calculated from a given u nbiased estimate ˆ γ of the true SNR γ . Due to the inevitable error in SNR estimation, we a r gue d that one must either back-off the naiv e rate, or c orrespond ingly increa se the SNR, to mee t the stricter expec ted error probability co nstraint. Based on a Gauss ian approximation of the estimation error N = γ − ˆ γ , w e d eri ved conditions on the “effecti ve estimator SNR” ˆ γ 2 /σ 2 N | ˆ γ that are nec essa ry for the existence of a feasible transmission rate, as well as an upper bound on the transmission rate when this nec essa ry con dition is satisfied. T his latter analysis was c arried out for both uncode d QAM sign aling a nd random Gaussia n signaling (the latter in both the low- SNR an d high-SNR regimes). Next, we considered unbiase d S NR estimation via A CK/N AK feedback . 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A P P E N D I X A D E R I V AT I O N O F T min p F O R U N C O D E D Q A M A N D G AU S S I A N S I G NA L I N G In this section, we deriv e (45) and (46). For brevity , we write ε , ε p ( γ , R p ) and ε ′ , ε ′ p ( γ , R p ) . Rec all that, from (23) and (44), we have for , unc oded QAM, T min p = ε (1 − ε ) ( ε ′ ) 2 2( α + ln 0 . 1 n ) ˆ γ 2 (58) where, from (18), ε ′ = ∂ ∂ γ  1 −  1 − 0 . 2 exp  − 1 . 5 γ 2 R p − 1  | {z } (1 − ε ) 1 /n  n  (59) = n (1 − ε ) n − 1 n 0 . 2 exp  − 1 . 5 γ 2 R p − 1  | {z } 1 − (1 − ε ) 1 /n  − 1 . 5 2 R p − 1  (60) = (1 − ε )  (1 − ε ) − 1 /n − 1   − 1 . 5 γ 2 R p − 1  | {z } ln(5(1 − (1 − ε ) 1 /n )) n γ . (61) Thus T min p = ε (1 − ε )( ε ′ 1 − ε ) 2 2( α + ln 0 . 1 n ) ˆ γ 2 (62) = ε (1 − ε )  (1 − ε ) − 1 /n − 1  2 ln 2  5  1 − (1 − ε ) 1 /n  × 2( α + ln 0 . 1 n ) ( n ˆ γ /γ ) 2 . (63) From (32) a nd (44), we have for , Gaussian signaling in the low-SNR regime, T min p = ε (1 − ε ) ( ε ′ ) 2 2 α ˆ γ 2 (64) Nov ember 2, 2018 DRAFT 28 where, from (27), ε ′ = ∂ ∂ γ exp  nρ ∗  R p ln 2 − 1 2 ln  1 + γ 1 + ρ ∗  (65) = ε − nρ ∗ 2 1 (1 + γ 1+ ρ ∗ ) 1 1 + ρ ∗ = ε − nρ ∗ 2(1 + ρ ∗ + γ ) . (66) Thus T min p = (1 − ε ) ε ( ε ′ ε ) 2 2 α ˆ γ 2 = 8 α (1 − ε ) ε (1 + ρ ∗ + γ ) 2 ( nρ ∗ ˆ γ ) 2 . (67) A P P E N D I X B P RO O F O F T H E O R E M 2 W e will directly apply Theorem 2 .1 [33, p. 223]. The ne cessa ry conditions for asy mptotic normality and asymptotic ef ficiency to hold in our system a re: 1) The expectation, E [ F t ] , of o bservation F t must exist and must be b ounded : E [ F t ] = ε ( γ , R t ) exists and is clearly bounded by 1 for all t . 2) The partial de ri vati ve    ∂ E [ F t ] ∂ γ    must be jointly continuous (in γ an d R t ) a nd bounded . For both QAM (17) an d Gaussian (27) signals , | ∂ E [ F t ] ∂ γ | = | ∂ ε ( γ ,R t ) ∂ γ | is c ontinuous and boun ded for γ > 0 and R t > 0 . 3) The vari ance var ( F t ) of ob servation F t must be continuous in γ a nd R t . For both QAM and Gauss ian signaling, v ar ( F t ) = ε ( γ , R t )(1 − ε ( γ , R t )) is continuous and boun ded for γ > 0 and R t > 0 . 4) Fisher information Φ ( γ , R t ) must be continuous , positiv e and for ea ch γ , it must have a unique maximum in R t . For both QAM and Ga ussian s ignaling, the Fisher information Φ( γ , R t ) as gi ven in (43) is co ntin- uous and positi ve for γ > 0 an d R t > 0 . Moreover , it has a unique maximum R t = R genie p ( γ ) for each γ > 0 , since Φ( γ , R t ) is a s trictly concave and continuous function of R t . 5) For some b > 2 , E  | F t | b  must be bounded f or all poss ible values o f γ a nd associated rate R genie p ( γ ) . Since F t ∈ { 0 , 1 } , we know that E  | F t | b  is bounded for all b > 2 and for all values of ( γ , R genie p ( γ )) . Nov ember 2, 2018 DRAFT 29 Furthermore, the asymptotic e f ficiency [33, p. 186,22 4] of the estimator is Φ( γ , R t ) ·  ∂ ∂ γ E [ F t ]  2 var ( F t )        R t = R genie p ( γ ) = Φ( γ , R genie p ( γ )) ·  ε ′ ( γ , R genie p ( γ ))  2 ε ( γ , R genie p ( γ ))(1 − ε ( γ , R genie p ( γ ))) = 1 . The asymptotic optimality , i.e., T p σ 2 N T p | ˆ γ T p → h Φ( γ , R genie p ( γ )) i − 1 as T p → ∞ follows as a conseq uence of Theorem 2 .1 [33, p. 223]. Nov ember 2, 2018 DRAFT