Coding Versus ARQ in Fading Channels: How reliable should the PHY be?

1 Coding v ersus ARQ in Fading Channels: Ho w reliabl e should the PHY b e? Peng W u and Nihar Jindal Univ ersity of Minnesota, Minneapolis, MN 55455 Email: { pengwu,nihar } @umn.edu Abstract This paper studies the tradeoff between channel coding and ARQ (automatic repeat request) in Rayleigh block-fading channels. A hea vily coded system correspon ds to a low transmission rate with few ARQ re-transmission s, whereas lighter codin g corr esponds to a high er transmitted rate but more re- transmissions. The optimu m erro r prob ability , where optim um ref ers to the m aximization of the average successful thro ughpu t, is derived an d is sh own to be a decreasin g function of th e average signal-to-noise ratio and of the chann el d iv ersity order . A general con clusion of the work is that th e optimum err or probab ility is quite large (e.g., 10% or larger) fo r re asonable ch annel parameters, a nd th at operating at a very sma ll error probability can lead to a significantly reduced through put. This conclusion holds ev en when a numb er o f practical ARQ consider ations, su ch as d elay co nstraints and ack nowledgement feedback er rors, are taken into accoun t. I . I N T R O D U C T I O N In contemporary wireless co mmunication systems , ARQ (automatic repe at req uest) is gen erally used above the physical layer (PHY) to compensa te for packet errors: incorrectly decoded packets are de- tected by the recei ver , and a negative acknowledgement is sent back to the trans mitter to request a re-transmission. In such an architecture there is a n atural tradeoff between the transmitted rate and ARQ re-transmissions. A h igh trans mitted rate c orresponds to many packet e rrors a nd thus ma ny ARQ re- transmissions, but e ach s ucces sfully rece i ved packet co ntains many information bits. On the other hand, a lo w transmitted rate corresponds to fe w ARQ re-transmissions , but few information bits are contained per pac ket. T hus, a fundamental design c hallenge is determining the transmitted rate that maximizes the rate a t which bits are s ucces sfully delivered. Sinc e the packet e rror p robability is an inc reasing fun ction of the transmitted rate, this is equiv alent to determining the optimal packet error probability , i.e. , the optimal PHY reliability le vel. October 29, 2018 DRAFT 2 W e consider a wireless channel where the transmitter chooses the rate based only on the fading statistics becaus e kn owl edge of the instantaneou s cha nnel conditions is not avail able (e.g., high velocity mobiles in ce llular s ystems). T he transmitted rate-ARQ tradeoff is intere sting in this se tting bec ause the pac ket error probability depen ds on the transmitted rate in a non-tri vial f ashion; on the other hand, this tradeoff is somewhat trivial when instantaneo us chann el s tate information at the transmitter (CSIT) is av a ilable (see Remark 1). W e begin by analyzing an ide alized system, for which we find tha t making the PHY too reliable can lea d to a significant penalty in terms of the a chieved goo dput (long-term av erage succ essful thr oughput ), and that the optimal packet error probability is dec reasing in the average SNR a nd in the fading selectivity experienced by each transmitted codeword. W e also see that for a lar ge lev el of system p arameters, choosing a n error prob ability of 10% leads to ne ar- optimal performance. W e the n c onsider a numbe r of important practical considerations , such as a limit on the nu mber of ARQ re-transmission s a nd u nreliable acknowledgement feed back. Even after taking these issues into account, we find that a relati vely unreliable PHY is still p referred. Beca use of fading, the PHY can be made reliable on ly if the trans mitted rate is significantly reduc ed. Howe ver , this red uction in rate is not made u p for by the corresponding reduc tion in ARQ re-transmissions. A. Pr ior W ork There has bee n some recent work on the joint optimization of packet-lev el erasure-correction c odes (e.g., fountain c odes) and PHY -layer error correction [1]–[4]. The fund amental metric w ith erasure cod es is the product of the transmitted rate a nd the packet s ucces s prob ability , wh ich is the same as in the idealized ARQ s etting studied in Section III. Even in that idealized s etting, our work d if fers in a number of ways. References [1], [3], [4] study multicas t (i.e., multiple receivers) while [2] co nsiders unicast assuming no diversity per transmission, wherea s ou r focus is on the unicast s etting with diversity p er transmission. F urthermore, our analysis provides a ge neral explanation of how the PHY reliability s hould depend on both the diversit y a nd the average SNR . In add ition, we consider a numb er of practical issues specific to ARQ, such a s acknowledgement e rrors (Section IV ), as well as hyb rid-ARQ (Section V). I I . S Y S T E M M O D E L W e con sider a Rayleigh bloc k-fading chann el where the cha nnel remains constan t within ea ch block but change s independ ently from one bloc k to another . The t -th ( t = 1 , 2 , · · · ) received channel symb ol October 29, 2018 DRAFT 3 in the i -th ( i = 1 , 2 , · · · ) fading block y t,i is gi ven by y t,i = √ SNR h i x t,i + z t,i , (1) where h i ∼ C N ( 0 , 1) represen ts the chan nel gain and is i.i.d. across fading blocks, x t,i ∼ C N ( 0 , 1) denotes the Ga ussian input sy mbol c onstrained to ha ve unit av erage po wer , and z t,i ∼ C N (0 , 1) models the additi ve Ga ussian noise assume d to be i.i.d. across cha nnel uses a nd fading blocks. Although we focus on sing le an tenna s ystems a nd Rayleigh fading channel, our model can b e e asily extended to multiple-input a nd multiple-output (MIMO) systems and o ther fading distributi ons as commented up on in Remark 2. Each transmission (i.e., c odewor d) is ass umed to span L fading blocks, and thus L represents the time/frequency selectivity expe rienced by eac h codeword. In a nalyzing ARQ systems , the packet e rror probability is the key qua ntity . If a s trong ch annel c ode (with suitably long blocklength) is used , it is we ll known that the packet error probability is accurately ap proximated by the mutual information outage proba bility [5 ]–[8]. Under this a ssumption (which is examined in Section IV -A), the p acket error probability for transmission at rate R bits/symb ol is gi ven by [9, eq (5.83)]: ε ( SNR , L , R ) = P " 1 L L X i =1 log 2 (1 + SNR | h i | 2 ) ≤ R # . (2) Here we explicitly de note the depe ndence of the error prob ability on the average signal-to-noise ratio SNR , the selectivity order L , and the transmitted rate R . W e are g enerally interested in the relationsh ip between R and ε for particular (fixed) values of SNR and L . When SNR an d L are con stant, R can be in versely computed given s ome ε ; thus, throughout the paper we replace R with R ε wherever the relationship between R and ε needs to b e explicitl y pointed out. The focus of the pape r is on simple ARQ, in which packets rece i ved in error are re-transmitted and decoding is p erformed only on the ba sis of the mo st rece nt transmission . 1 More sp ecifically , whenever the receiv er detec ts that a codeword has be en de coded incorrectly , a NA CK is fed back to the transmitter . On the other ha nd, if the receiver de tects c orrect deco ding an A CK is fed back. U pon reception of an ACK, the trans mitter moves o n to the next packet, whe reas rece ption o f a NA CK triggers re-transmission o f the previous packet. ARQ transforms the s ystem into a variable-rate sche me, and the relevant performance metric is the rate a t which pac kets are success fully re ceiv ed. This quantity is gen erally referred to as the long-term average goodpu t , and is clearly de fined in each of the relev ant sections. And consisten t with the assump tion of no CSIT (and fast fading), we a ssume fading is indepe ndent acros s re-tr ansmiss ions. 1 Hybrid -ARQ, which is a more sophisticated and powerful form of ARQ, is considered in Section V. October 29, 2018 DRAFT 4 I I I . O P T I M A L P H Y R E L I A B I L I T Y I N T H E I D E A L S E T T I N G In this section we in vestigate t he optimal PHY reliabili ty lev el under a number of ide alized assumptions. Although not entirely realistic, this idealized model yields important de sign ins ights. In particular , we make the following key assumptions: • Channel co des that o perate at the mutual information limi t (i.e., packet error prob ability is equal to the mutual information outage p robability). • Perfect error detection at the receiv er . • Unlimited number of ARQ re-transmissions. • Perfect A CK/NA CK feedback. In Section IV we relax the se assu mptions, and find tha t the ins ights from this idea lized se tting g enerally also apply to real s ystems. In order to ch aracterize the long-term goo dput in this idea lized setting. In order to do so, we must quantify the n umber o f transmission a ttempts/ARQ rou nds needed for su ccess ful trans mission of e ach packet. If we use X i to deno te the nu mber of ARQ rounds for the i -th packet, then a total of P J i =1 X i ARQ round s are us ed for transmitting J packets; note tha t the X i ’ s are i.i.d. d ue to the indepen dence of fading a nd nois e across ARQ rounds . Each c odeword is a ssumed to span n cha nnel symbols an d to contain b information b its, correspond ing to a transmitted rate of R = b/n bits/symb ols. The average rate at wh ich bits are succe ssfully deli vered is the ratio of the bits deliv ered to the total n umber of c hannel symbols required. Th e goodput η is the long-term average at which bits are succes sfully delivered, and by taking J → ∞ we get [10]: η = lim J →∞ J b n P J i =1 X i = lim J →∞ b n 1 J P J i =1 X i = R E [ X ] , (3) where X is the random vari able de scribing the ARQ rounds req uired for suc cessful deliv ery o f a pac ket. Becaus e each ARQ round is succ essful with proba bility 1 − ε , with ε define d in (2), and roun ds are independ ent, X is geo metric with parameter 1 − ε an d thus E [ X ] = 1 / (1 − ε ) . Ba sed upon (3), we ha ve η , R ε (1 − ε ) , (4) where the transmitted rate is den oted as R ε to emphasize its dependen ce o n ε . Based on this expression , we ca n immed iately se e the tradeoff be tween the transmitted rate, i.e. the number of bits per packet, and the number of ARQ re-transmiss ions per packet: a lar ge R ε means many bits a re containe d in eac h packet but that many re-transmissions are req uired, whe reas a s mall R ε October 29, 2018 DRAFT 5 correspond s to fewer bits p er packet and fewer re-transmissions . Our objec ti ve is to find the optimal (i.e., goodput maximizing) operating point o n this trade off curve for any gi ven parame ters SNR and L . Becaus e R ε is a function o f ε (for SNR and L fixed), this one-dimens ional optimization can be ph rased in terms of R ε or ε . W e find it most insightful to co nsider ε , which lea ds to the following d efinition: Definition 1: The op timal packet e rror proba bility , whe re optimal refers to goodput maximization with goodput defined in (3), for average sign al-to-noise ratio SNR and per-codew ord selec ti vity orde r L is: ε ⋆ ( SNR , L ) , arg max ε R ε (1 − ε ) . (5) By finding ε ⋆ ( SNR , L ) , we thus determine the optimal PHY reliability level a nd ho w this optimum depend s on chan nel parameters SNR and L , which a re g enerally static over the timesca le o f interest. 2 For L = 1 , a simple calculation shows 3 ε ⋆ ( SNR , 1) = 1 − e (1 − SNR ) / ( SNR · W ( SNR )) , (6) where W ( · ) is the La mbert W function [11]. Unfortunately , for L > 1 it does not seem feasible to find an exac t analytical solution beca use a close d-form expression for the outage probab ility exists only for L = 1 . Howe ver , the o ptimization in (5) can be easily solved nu merically (for arbitrary L ). In a ddition, an accurate approximation to ε ⋆ ( SNR , L ) c an be solved analytically , as w e d etail in the next subs ection. In order to provide a gene ral understand ing o f ε ⋆ , Fig. 1 c ontains a plot o f goodp ut η (nu merically computed) versus outage probab ility ε for L = 2 and L = 5 at SNR = 0 and 10 dB . For each curve, the goodput-maximizing v alue of ε is c ircled. F rom this figure, we make the follo wing observations: • Making the physical layer too reliable or too unreliable yields poor g oodput. • The optimal outage probability decreases with SNR and L . These turn out to be the key behaviors of the coding-ARQ tradeof f, and the rema inder of this section is dev oted to ana lytically explain these behaviors through a Gaussian approximation. Remark 1: Throughp ut the p aper we consider the setting without ch annel state information at the transmitter (CSIT). If the re is CSIT , which gene rally is the case when the fading is slow relativ e to the delay in the chan nel feedback loop, the optimization problem in Definition 1 turns o ut to be tri vial. When CSIT is av a ilable, the cha nnel is essen tially A WGN w ith an instantaneo us SNR that is d etermined 2 Note t hat in this definition we assume all possible code rates are possible; nonetheless, this formulation provides v aluable insight f or systems in which the transmitter must choose from a finite set of code rates. 3 The exp ression for L = 1 is also derived in [2]. Howe ver , authors in [2] only consider L = 1 case rather than L > 1 scenarios, which are further inv estigated in our work. October 29, 2018 DRAFT 6 by the fading realization but is known to the T X. If a capac ity-achieving code with infi nite codeword block-length is u sed in the A WGN c hannel, the relationship b etween e rror a nd rate is a step -function: ε =    0 , if R < log 2  1 + SNR | h | 2  (7a) 1 , if R ≥ log 2  1 + SNR | h | 2  . (7b) Thus, it is op timal to choose a rate very slightly below the instantaneo us capac ity log 2  1 + SNR | h | 2  . For realistic codes with finite b locklength, the ε - R cu rve is no t a s tep function but none theless is very steep. For example, for turbo c odes the waterfall chara cteristic of e rror vs . SNR curves (for fixed rate) translates to a step-function-like error vs. rate cu rve for fixed SNR. T herefore, the trans mitted rate sh ould be chosen close to the bo ttom o f the ste p fun ction. A. Gau ssian A ppr o ximation The p rimary difficulty in fi nding ε ⋆ ( SNR , L ) stems from the f act that the ou tage probability in (2) ca n only be expresse d as an L -dimensional integral, except for the s pecial case L = 1 . T o c ircumvent this problem, we utilize a Gau ssian approximation to the outage probab ility u sed in prior work [12]–[14]. The random variable 1 L P L i =1 log 2  1 + SNR | h i | 2  is a pproximated by a N  µ ( SNR ) , σ 2 ( SNR ) /L  random variable, where µ ( SNR ) and σ 2 ( SNR ) are the mean and the variance of log 2  1 + SNR | h | 2  , respectively: µ ( SNR ) = E | h |  log 2 (1 + SNR | h | 2 )  , (8) σ 2 ( SNR ) = E | h |  log 2 (1 + SNR | h | 2 )  2 − µ 2 ( SNR ) . (9) Closed forms for the se quantities can be foun d in [15], [16]. Bas ed on this approximation we hav e ε ≈ Q √ L σ ( SNR ) ( µ ( SNR ) − R ε ) ! , (10) where Q ( · ) is the tail prob ability of a standa rd normal. Solving this equation for R ε and p lugging into (4) yields the following approximation for the g oodput, which we deno te a s η g : η g =  µ ( SNR ) − Q − 1 ( ε ) σ ( SNR ) √ L  (1 − ε ) , (11) where Q − 1 ( ε ) is the in verse o f the Q function. B. Optimization of Goodp ut Ap pr ox imation The optimization of η g turns out to be more tractable. W e fi rst re write η g as η g = µ ( SNR )  1 − κ · Q − 1 ( ε )  (1 − ε ) , (12) October 29, 2018 DRAFT 7 where the constant κ ∈ (0 , 1) is the µ -normalized standard deviation o f the received mutual information: κ , σ ( SNR ) µ ( SNR ) √ L . (13) W e can ob serve t hat κ decreases in SNR a nd L . W e now define ε ⋆ g as the η g -maximizing outage probability: ε ⋆ g ( SNR , L ) , arg max ε  1 − κ · Q − 1 ( ε )  (1 − ε ) , (14) where we hav e p ulled o ut the c onstant µ ( SNR ) from (12) b ecause it does not affect the max imization. Pr o position 1: The PHY reliability lev el that maximizes the Gaussian approx imated goodput is the unique solution to the following fi xed point equa tion:  Q − 1 ( ε ⋆ g ) − (1 − ε ⋆ g ) ·  Q − 1 ( ε )  ′ | ε = ε ⋆ g  − 1 = κ. (15) Furthermore, ε ⋆ g is increasing in κ . Pr o of: See Appe ndix A. W e immediately see that ε ⋆ g depend s on the c hannel p arameters only through κ . Furthermore, beca use κ is d ecreasing in SNR a nd L , we see that ε ⋆ g decreas es in L (i.e., the channel selecti vity) and SNR . Straightforward a nalysis shows that ε ⋆ g tends to z ero as L inc reases approx imately a s 1 / √ L log L , wh ile ε ⋆ g tends to zero with SNR a pproximately a s 1 / √ log SNR . In Fig. 2, the exact optimal ε ⋆ and the a pproximate-optimal ε ⋆ g are plotted vs. SNR (dB) for L = 2 , 5 , and 10 . Th e Gaussian approx imation is seen to be reasona bly accurate, and most importantly , co rrectly captures behavior with respect to L and SNR . In o rder to gain an intuiti ve understanding of the optimization, in Fig. 3 the succe ss p robability 1 − ε (left) and the g oodput η = R ε (1 − ε ) (right) a re plotted versus the transmitted rate R for SNR = 10 dB. For each L the goodput-maximizing operating point is circled. First con sider the curves for L = 5 . For R u p to a pproximately 1 . 5 bits/symbol the s uccess p robability is nearly on e, i.e ., ε ≈ 0 . As a result, the goodput η is approximately e qual to R for R up to 1 . 5 . Wh en R is increas ed beyond 1 . 5 the suc cess probability begins to decrease non -negligibly but the goodp ut no netheless increase s with R becaus e the increa sed trans mission rate makes u p for the loss in suc cess probability (i.e., for the ARQ re-transmissions). Howe ver , the goodp ut peak s at R = 2 . 3 bec ause beyond this po int the increase in transmission rate n o lon ger makes up for the increased re-tr ansmiss ions; v isually , the optimum rate (for each value of L ) co rresponds to a point beyond which the success probability begins to drop of f sharply with the transmitted rate. T o understand the ef fect of the sele cti vity order L , notice that increasing L leads to a steepen ing of the succe ss probability-rate c urve. This has the ef fect o f moving the goodpu t curve closer to the October 29, 2018 DRAFT 8 transmitted rate, which leads to a larger o ptimum ra te a nd a larger optimum succ ess p robability ( 1 − ε ⋆ ). T o understand why ε ⋆ decreas es with SNR , b ased upon the rewritten version of η g in (12) we s ee that the governing relationship is be tween the success prob ability 1 − ε and the no rmalized, rathe r than abs olute, transmission rate R/µ ( SNR ) . Th erefore, increasing SNR steepens the succe ss probab ility-normalized rate curve (similar to the effect of increas ing L ) and thus leads to a s maller v alue of ε ⋆ . Is is important to notice that the op timum error probabilities in Fig. 2 are quite large, ev en for large selectivity and at high SNR lev els. Th is follows from the e arlier explanation that dec reasing the error probability (and thus the rate) b eyond a certain p oint is inefficient be cause the decrease in ARQ re- transmissions does not ma ke up for the loss in transmission rate. T o undersc ore the importance o f not operating the PHY too reliably , in Fig. 4 goo dput is plotted versus SNR (dB) for L = 2 a nd 10 for the optimum error probab ility η ( ε ⋆ ) as well as for ε = 0 . 1 , 0 . 01 , and 0 . 001 . Choosing ε = 0 . 1 leads to near-optimal performance for both selecti vity values. On the othe r hand, there is a signific ant penalty if ε = 0 . 01 or 0 . 001 when L = 2 ; this pena lty is red uced in the highly selec ti ve cha nnel ( L = 10 ) but is still no n-negligible. Ind eed, the most important insigh t from this analysis is that making the PHY too reliable can lead to a significa nt performance pe nalty; for example, choosing ε = 0 . 001 leads to a power penalty of approximately 10 dB for L = 2 and 2 dB for L = 10 . Remark 2: Pr oposition 1 shows ε ⋆ g is on ly determined by κ , wh ich is comp letely determined by the statistics of the receiv ed mu tual information per packet. Th is implies our results can be e asily extended to dif ferent fading d istrib utions and to MIMO b y a ppropriately modifying µ ( SNR ) a nd σ ( SNR ) . I V . O P T I M A L P H Y R E L I A B I L I T Y I N T H E N O N - I D E A L S E T T I N G While the p revi ous section illustrated the nee d to ope rate the PHY a t a relatively u nreliable level un der a numb er of idealized a ssumptions, a legitimate qu estion is whe ther that c onclusion still holds when the idealizations of Se ction III are removed. T hereby moti vated, in this section we b egin to carefully study the follo wing scenarios one b y one: • Finite codeword bloc k-length. • Imperfect error detection. • Limited number of ARQ roun ds per p acket. • Imperfect A CK/N A CK feedback . As we shall s ee, our ba sic conclusion is upheld even unde r more realistic ass umptions. October 29, 2018 DRAFT 9 A. F inite Codewor d Block-length Although in the previous section we assu med op eration at the mutual information of infinite block length codes, real s ystems must use finite bloc klength codes . I n o rder to determine the effect of finite blocklen gth upon the optimal PHY reliability , we s tudy the mu tual information outag e prob ability in terms of the information sp ectrum , which captures the block error probability for finite blocklength co des. In [17 ], it was s hown that actual codes p erform quite close to the information spec trum-based outage proba bility . By extending the results of [17], [18], the outage probability with bloc klength n (symbo ls) is ε ( n, SNR , L , R ) = P   1 L L X i =1 log  1 + | h i | 2 SNR  + 1 n L X i =1   s | h i | 2 SNR 1 + | h i | 2 SNR · n/L X j =1 ω ij   ≤ R   , (16) where R is the transmitted rate in nats/symbol, a nd ω i,j ’ s a re i.i.d. La place random variables [18 ], each with ze ro mean an d variance two. The first term in the sum is the s tandard infinite bloc klength mutual information expression , wherea s the seco nd term is due to the finite blockle ngth, and in particular ca ptures the e f fect of atyp ical noise realizations. This second term goes to zero as n → ∞ (i.e., atypica l noise does not occur in the infinite blocklen gth limit), but c annot b e ignored for finite n . The sum of i.i.d. Laplace random variables has a Bessel-K distrib ution, which is dif ficult to compute for large n but can b e very acc urately approximated by a Gaussian as verified in [17]. Thus, the mutual information conditioned o n the L channel rea lizations is a pproximated by a Ga ussian random v ariable: N 1 L L X i =1 log  1 + | h i | 2 SNR  , 1 L L X i =1 2 | h i | 2 SNR n (1 + | h i | 2 SNR ) ! (17) (This is diff erent from Section III-A, whe re the Gaussian a pproximation is made with respect to the fading realiza tions). Therefore, we c an a pproximate the outage probability with finite block-length n by av eraging the cu mulati ve dis trib u tion function (CDF) of (17) ov er different ch annel rea lizations: ε ( n, SNR , L , R ) ≈ E | h 1 | ,..., | h L | Q   1 L P L i =1 log  1 + | h i | 2 SNR  − R q 1 L P L i =1 2 | h i | 2 SNR n (1+ | h i | 2 SNR )   . (18) In Fig. 5 , we compa re fin ite a nd infinite blockleng th cod es by plotting suc cess proba bility 1 − ε vs. R ε (bits/symbol) for L = 10 a t SNR = 0 and 10 d B. It is clearly seen that the s teepness of the succes s-rate curve is reduce d by the finite blockleng th; this is a conseq uence of atypical noise realizations. W e ca n now c onsider goodput maximization for a given blockle ngth n : ε ⋆ ( SNR , L, n ) , arg m ax ε R ε (1 − ε ) , (19) where both R ε and ε are computed (nu merically) in the fi nite c odeword bloc k-length regime. October 29, 2018 DRAFT 10 In Fig. 6, the op timal ε vs. SNR (dB) is p lotted for both finite block-leng th cod ing and infinite block- length coding. W e see that the optimal error probability becomes lar ger , a s expected by suc cess-rate curves with redu ced steep ness in Fig. 5. At high SNR, the fi nite bloc k-length coding c urve almost overlaps the infinite block-leng th c oding curve because the unus ual noise term in the mutual information expression is negligible for large values of SNR. A s expected, the optimal reliability level with finite blocklength codes does not dif fer significa ntly from the idealized cas e. B. Non -ideal Err or Detection A critical component of ARQ is error detection, which is generally performed us ing a cyclic redundancy check ( CRC). The s tandard usa ge of CRC corresponds to appending k parity check bits to b − k information bits, yielding a total of b bits that a re then enc oded (by the cha nnel encode r) into n channe l sy mbols. At the receiver , the ch annel d ecoder (which is generally agnostic to CRC) takes the n channel symbols as inputs and produc es an estimate of the b bits, which are in turn passe d to the CRC d ecoder for error detection. A basic analys is in [19] shows that if the c hannel decod er is in e rror (i.e., the b bits input to the c hannel en coder do not match the b dec oded bits), the p robability of an un detected error (i.e., the CRC d ecoder signals correct even though an e rror ha s occurred) is rough ly 2 − k . Th erefore, the overall probability of an undetected error is well approximated by ε · 2 − k . Undetected errors can lead to s ignificant prob lems, who se severity depen ds upon higher ne twork layers (e.g., whethe r or no t an ad ditional layer of error de tection is pe rformed at a higher laye r) and the application. Howev er , a general persp ectiv e is provided b y imposing a cons traint p on the undetec ted error probability , i.e ., ε · 2 − k ≤ p . Base d on this co nstraint, we s ee tha t the constraint can be met by increasing k , wh ich come s at the cost of overhead, o r by reducing the packet error probability ε , wh ich can signific antly reduce go odput (Se ction III). The question mos t relev a nt to this pa per is the follo wing: does the prese nce of a s tringent con straint on un detected error probability moti vate red ucing the PHY packet error prob ability ε ? The relev an t q uantity , along with the undetec ted e rror proba bility , is the rate at which information bits are correctly deli vered, which is: η = b − k n · (1 − ε ) =  R ε − k n  · (1 − ε ) , (20) where R ε − k n is the effecti ve transmitted rate after acc ounting for the parity check overhead. It is the n October 29, 2018 DRAFT 11 relev an t to maximize this rate subject to the c onstraint on un detected error: 4 : ( ε ⋆ , k ⋆ ) , arg max ε,k  R ε − k n  · (1 − ε ) (21) subject to ε · 2 − k ≤ p Although this optimization prob lem (nor the version based on the Gaussian a pproximation) is not analytically tractable, it is ea sy to se e tha t the so lution correspond s to k ⋆ = ⌈− log 2 ( p/ε ⋆ ) ⌉ , where ε ⋆ is roughly the optimum packet error probability as suming perfect error detec tion (i.e. the solution from Section III). In other words, the u ndetected e rror probability constraint should be satisfied by ch oosing k sufficiently large while leaving the PHY transmitted rate nearly untouche d. T o better understan d this, note that reducing k by a bit requires reduc ing ε by a factor of two. The co rresponding reduction in CRC overhead is very s mall (rough ly 1 /n ), while the reduction in the transmitted rate is muc h larger . T hus, if we consider the choices of ε an d k that achieve the cons traint wi th equa lity , i.e., k = − log 2 ( p/ε ) , goodpu t decreas es as ε is d ecrease d be lo w the pa cket error proba bility which is optimal unde r the assumption o f perfect e rror detection. In othe r words, ope rating the PHY at a more reliable p oint is not worth the small reduction in CRC overhead. C. End -to-End De lay Constraint In ce rtain a pplications such as V oice-over -IP (V oIP), there is a limit o n the numb er of re-transmissions per pa cket as well as a con straint on the fraction of packets that a re not succe ssfully deli vered within this limit. If s uch con straints are imposed, it ma y not be clear how aggres si vely ARQ shou ld be utilized. Consider a sys tem where any packet that fails on its d -th attempt is discarde d (i.e., at mos t d − 1 re-transmissions are allowed), but at most a fraction q of pac kets can be discarde d, where q > 0 is a reliability constraint. Under these conditions, the prob ability a pac ket is discarded is ε d , i.e., t he probability of d cons ecutiv e decod ing failures, while the long-term average rate at which p ackets are s uccess fully deliv ered s till is R ε (1 − ε ) . T o understand why the goodput expression is unaffected by the delay limi t, note that the number of succes sfully de li vered p ackets is equal to the number of transmissions in which decoding is succ essful, regardless of which pa ckets are trans mitted in eac h slot. T he delay con straint only af fects which packets are delivered in dif ferent slots, a nd thus does not af fect the g oodput. 5 4 For the sake of compactness, the depen dence of ε ⋆ and k ⋆ upon SNR , L and n is suppressed henceforth, excep t where explicit notation is required. 5 The goodput exp ression can alternativ ely be deriv ed by computing the ave rage number of ARQ rounds per packet (accounting for the limit d ), and then applying the rene wal-re ward theorem [20]. October 29, 2018 DRAFT 12 Since the discarde d packet probability is ε d , the reliability c onstraint requires ε ≤ q 1 /d . W e can thu s consider maximization of goodpu t R ε (1 − ε ) subject to the constraint ε ≤ q 1 /d . Beca use the go odput is observed to be co ncave in ε , only two possibilities exist. If q 1 d is larger than the op timal value of ε for the unco nstrained p roblem, then the op timal value of ε is unaffected by q . In the more interesting and relev ant case where q 1 d is smaller than the optimal unc onstrained ε , then goo dput is maximized by choosing ε equal to the uppe r bo und q 1 d . Thus, a strict delay and reliability constraint forces the PHY to b e more reliable than in the u ncon- strained cas e. Howe ver , amongst all allowed packet error prob abilities, goodput is maximize d by cho osing the la r gest. Thu s, althoug h s trict co nstraints d o n ot allow for very agg ressiv e use of ARQ, none theless ARQ should be utilized to the maximum extent p ossible. D. Noisy ACK/N ACK F eedback W e finally remove the assumption of pe rfect acknowledgemen ts, a nd conside r the realistic sc enario where A CK/NA CK feed back is not pe rfect and where the acknowledgement overhead is factored in. The main is sue confronted here is the joint optimization of the reliability level of the forw ard data channe l and of the rev erse ackn owl edgeme nt (feed back/con trol) ch annel. As intuition sugge sts, reliable communication is possible only if some combination o f the forw ard and reverse reliability le vels is sufficiently large; thu s, it is not clear if op erating the PHY at a relativ ely un reliable level as sug gested in earlier sec tions is approp riate. The eff ects of acknowledgemen t errors c an sometimes be reduced through higher-l ayer mec hanisms (e.g., seq uence number check ), b ut in o rder to she d the most light on the issu e of forward/re verse reliability , we focus on an extreme cas e where ackn owledgement err ors are most ha rmful. In particular , we consider a setting with de lay and reliability constraints a s in Section IV -C, and whe re any NA CK to A CK error lea ds to a packet missing the delay dea dline. W e first de scribe the feedba ck channe l model, and then analyze performance. 1) F eedba ck Channel Mode l: W e assume A CK/N A CK feedback is pe rformed over a Ra yleigh fading channe l us ing a total of f symbols which are d istrib uted on L fb independ ently faded s ubchan nels; here L fb is the diversity order of the feedba ck ch annel, w hich need no t be equa l to L , the forward ch annel div ersity order . Sinc e the feedbac k is b inary , BPSK is use d with the symb ol rep eated on e ach sub-ch annel f /L fb times. For the sa ke o f s implicity , we ass ume that the fee dback c hannel has the same average SNR as the forward c hannel, and that the fading on the feedb ack cha nnel is independen t of the fading o n the forward c hannel. After maximum ratio combining at the receiver , the eff ectiv e SNR is ( f /L fb ) · SNR · P L fb i =1 | h i | 2 , where October 29, 2018 DRAFT 13 h 1 , · · · , h L fb are the feed back ch annel fading coefficients. The resulting probability of error (denoted by ε fb ), a veraged over the fading realizations, is [21 ]: ε fb =  1 − ν 2  L fb · L fb − 1 X j =0  L fb − 1 + j j   1 + ν 2  j , (22) where ν = q ( f /L fb ) · SNR 1+( f /L fb ) · SNR . Clearly , ε fb is decreasing in f and SNR . 6 2) P erforma nce An alysis: In order to analyze p erformance with non-idea l feedback, we mus t first specify the rules by wh ich the transmitter and recei ver operate. The trans mitter takes pre cisely the s ame actions as in Sec tion IV -C: the transmitter immediately moves on to the next packet whe never an ACK is received, and after receiving d − 1 consec uti ve N ACK’ s (for a single packet) it attempts that pa cket one last time but then moves on to the next pac ket regardless of the acknowledgement rece i ved for the last attempt. Of course, the presence of feedback errors means that the receiv ed ac knowledgement does not always match the transmitted acknowledgement. The receiv er also operates in the standard ma nner , but we do a ssume tha t the receiver ca n always determine whether or not the p acket being received is the same as the pac ket rec eiv ed in the p re vious slot, as can be acc omplished by a simple co rrelation; this reasona ble assumption is equiv alen t to the receiver having knowledge o f acknowledgement e rrors. In this setup an A CK → NA CK error causes the transmitter to re-transmit the pre vious packet, ins tead of moving o n to the next packet. The receiver is able to rec ognize tha t an ack nowledgement error h as occurred (thr ough correlation of the current and pr evious received packets), and bec ause it already decoded the pa cket co rrectly it does no t a ttempt to deco de again. Instead, it simply transmits a n A CK once again. Thus, each A CK → NA CK error ha s the relatively benign effect of wasting one ARQ round . On the other hand, N A CK → A CK errors ha ve a considerably more deleterious e f fect because upon reception of an ACK, the transmitter automatically moves on to the next packet. Beca use we are considering a stringent delay co nstraint, we assume that suc h a N ACK → A CK e rror can not be rec overed from a nd thus we cons ider it as a los t packet that is co unted towards the reliability co nstraint. This is, in some sens e, a worst-case ass umption tha t acc entuates the e f fect of N A CK → A CK e rrors; some comme nts related to this po int are put forth a t the en d o f this s ection. T o more clearly illustrate the mode l, the complete ARQ proces s is shown in Fig. 7 for d = 3 . Eac h branch is labeled with the succe ss/failure of the transmiss ion a s well as the ack nowledgement (including errors). Circle nodes r efer to states in which the receiv er has yet to success fully decode the packet, whereas 6 Asymmetric decision regions can be used, in which case 0 → 1 and 1 → 0 errors have unequ al probabilities. Howe ver , this does not significantly affe ct performance and thus is not considered. October 29, 2018 DRAFT 14 triangles refer to states in which the receiv er has decode d correctly . A packet loss occurs if there is a decoding failure followed by a N A CK → A CK e rror in the first two rounds, or if d ecoding fails in all three attempts. All othe r outcome s c orrespond to case s where the rece i ver is ab le to decode the packet in some round, a nd thus succe ssful deliv ery of the packet. In these c ases, however , the numbe r of ARQ round s depend s on the first time at which the rec eiv er can d ecode and wh en the A CK is correctly d eli vered. (If an A CK is no t succe ssfully delivered, it may take u p to d rounds be fore the transmitter mov es on to the next pac ket.) Notice tha t a fter the d -th a ttempt, the transmitter moves on to the next pa cket regardless of what acknowledgement is received; this is due to the delay constraint that the transmitter follows. Based on the figu re and the independe nce of decoding a nd feedback errors across rounds, t he probability that a p acket is los t (i.e., it is not suc cessfully delivered within d rounds) is: ξ d = ε · ε fb + ε 2 (1 − ε fb ) ε fb + · · · + ε d − 1 (1 − ε fb ) d − 2 ε fb + ε d (1 − ε fb ) d − 1 , (23) where t he first d − 1 terms represent decoding failures f ollowed by a N A CK → A CK error ( more specifically , the l -th term c orresponds to l − 1 decoding failures a nd l − 1 correct NA CK transmiss ions, followed by another d ecoding failure and a NA CK → A CK e rror), an d the last term is the proba bility of d decoding failures a nd d − 1 correct N A CK transmissions. If we alternati vely co mpute the success probability , we get the follo wing d if ferent expression for ξ d : ξ d = 1 − d X i =1 (1 − ε ) · ε i − 1 · (1 − ε fb ) i − 1 , (24) where the i -th summand is the prob ability that succ essful forward transmission occurs in the i -th ARQ round. Base d upon (23) an d (24) we see that ξ d is inc reasing in both ε and ε fb . Th us, a desired packet loss probability ξ d can be achieved by different co mbinations of the forward channe l reliability and the feedback channel reliability: a les s re liable forward chan nel requires a more reliable feedback c hannel, and vice versa. As in Section IV -C we impos e a re liability constraint ξ d ≤ q , which by (23) translates to a joint constraint o n ε and ε fb . The relativ ely c omplicated joint cons traint can b e a ccurately app roximated by two muc h s impler constraints. Since we must satisfy ε ≤ q 1 d ev en with p erfect feed back ( ε fb = 0 ), for any ε fb > 0 we also mu st s atisfy ε ≤ q 1 d (this en sures tha t d cons ecutive d ecoding fail ures do not oc cur too freque ntly). Furthermore, by examining (23) it is evident that the first term is dominant in the pac ket loss prob ability express ion. Th us the constraint ξ d ≤ q esse ntially translates to the simplified cons traints ε · ε fb ≤ q and ε ≤ q 1 d . (25) October 29, 2018 DRAFT 15 These s implified cons traints are very accurate for values of ε not too close to q 1 d . On the other ha nd, as ε approaches q 1 d , ε fb must go to zero very rapidly (i.e. much faster than q /ε ) in order for ξ d ≤ q . The first con straint in (25) rev eals a general design principle: the co mbination of the forward and feedback channel mus t be sufficiently reliable. Th is is bec ause ε · ε fb is precisely the probab ility that a packet is lost because the initial transmission is de coded incorrectly and is followed by a NA CK → A CK error . Having established the reliability constraint, we now procee d to maximizing goodpu t while taking acknowledgement errors and ARQ overhead into a ccount. W ith respe ct to the lon g-term average goo dput, by applying the renewal-re ward theorem again we obtain: η = n n + f · R ε (1 − ξ d ) E [ X ] . (26) where random variable X is the numb er of ARQ round s per pa cket, and E [ X ] is de ri ved in Appen dix B. Here, n n + f is the feedb ack overhead pe nalty bec ause each packet spann ing n symbols is follo we d by f symb ols to c on vey the ackn owledgement. W e n ow maximize goo dput with re spect to both the forw ard and fee dback c hannel error probab iliti es: ( ε ⋆ , ε ⋆ fb ) , arg max ε,ε fb n n + f · R ε (1 − ξ d ) E [ X ] (27) subject to ξ d ≤ q noting that ε fb is a d ecreasing func tion of the number of feedbac k symbo ls f , acco rding to (22). Th is optimization is n ot a nalytically tractable, but can be easily solved nu merically a nd can be u nderstood through examination of the dominant relationships. T he overhead factor n / ( n + f ) clea rly depen ds o nly on ε fb (i.e., f ). Although the sec ond term R ε (1 − ξ d ) / E [ X ] d epends on both ε and ε fb , the depend ence upon ε fb is relativ ely minor as long as ε fb is reasonab ly small (i.e. les s than 10% ). Thus , it is reasona ble to cons ider the pe rfect feedback setting, in which case the second term is R ε (1 − ε ) . Therefore, the challenge is b alancing the feedba ck channe l overhead factor n n + f with the efficiency of the forward channe l, ap proximately R ε (1 − ε ) , while sa tisfying the c onstraint in (25). If f is chose n small, the feedback errors must b e compens ated with a very reliable, and thus inefficient, forward chann el; on the other han d, c hoosing f lar ge incu rs a lar ge fee dback overhead penalty but allows for a less reliable, and thus more ef ficient, forward ch annel. In Fig. 8, the jointly op timal ( ε ⋆ , ε ⋆ fb ) are plotted for a conse rv ativ e set of forward ch annel parameters ( L = 3 with SNR = 5 o r 10 dB, and n = 200 d ata symbo ls per pac ket), stringent de lay a nd reliability constraints (up to d = 3 ARQ rounds and a reliability con straint q = 10 − 6 ), and diff erent diversit y orders October 29, 2018 DRAFT 16 ( L fb = 1 , 2 and 5 ) for the feed back chan nel. Also plotted is the curve specifying the ( ε, ε fb ) pairs that achieve the reliability constraint ξ d = q . A s d iscusse d earlier , this c urve h as two distinct regions: for ε < 0 . 008 it is e ssentially the straight line ε · ε fb = q , whe reas ε fb must go to zero very quickly as ε approach es q 1 /d = 10 − 2 . When L fb = 2 , the optimal point c orresponds to the transition between these two regions. Mo ving to the right o f the optimal corresp onds to mak ing the PHY more reliable while making the control c hannel less reliable (i.e., decrea sing ε and f ), but this is subo ptimal bec ause the overhead savings d o no t c ompensa te for the loss incurred by a more reliable PHY . On the o ther han d, moving to the left is subo ptimal be cause only a very mo dest inc rease in ε is allowed, and this inc rease co mes a t a large expens e in terms of control symbols. If L fb = 5 , the optimal point is further to the left becaus e the feedba ck overhead requ ired to achieve a desired e rror rate is reduced . Howe ver , the behavior is quite diff erent if there is no diversity on the feedba ck cha nnel ( L fb = 1 ). W ithou t div ersity , the feedb ack error prob ability de creases extremely slowly with f (at orde r 1 /f ), a nd thus a very large f is required to achieve a rea sonable feedba ck error probability . In this extreme ca se, it is optimal to s acrifice significa nt P HY e f ficiency and choose ε quite a bit s maller than q 1 /d = 10 − 2 . Notice that inc reasing SNR moves the optimal to the left for all values of L fb becaus e a lar ger SNR improves the feedba ck channel reliability while not significantly changing the behavior of the forward c hannel. This b ehavior is further explained in Fig. 9, where goodput η (optimized with respect to ε fb ) is plotted versus forw ard error probability ε for the pa rameters of the p re vious figure, with SNR = 5 dB and L fb = 1 an d 2 here. The figure illustrates the stark contrast with res pect to feedba ck channel d i versity: with diversity (e ven for L fb = 2 ), the good put increases monotonically up to a point quite close to q 1 /d , while without div ersity the g oodput pe aks at a point far below q 1 /d . This is due to the hu ge difference in the fe edback chan nel reliability with an d without div ersity: in order to a chieve ε fb = 10 − 3 , a t SNR = 5 dB without d i versity f = 79 symbols are req uired, w hereas f = 9 suffices for L fb = 2 . T o more clearly understand why the optimal p oint with di versity is so close to q 1 /d , let us con trast two different choice s of ε for L fb = 2 . At the optimal ε = 8 × 10 − 3 , we require ε fb = 6 . 3 × 10 − 5 and thus f = 34 . On the other hand, at the subop timal ε = 10 − 3 we require ε fb = 10 − 3 and thus f = 9 . R educing the forward e rror probability by a factor of 8 reduces the fe edback overhead from 34 234 to 9 209 , but reduc es the transmitted rate by about 50% . The takeaway messa ge o f this analysis is clear: a s long as the feedbac k ch annel has at leas t some div ersity (e.g., throug h frequency or an tennas), stringent post-ARQ reliability constraints should be satisfied by increasing the reliability o f the fe edback c hannel instead o f increa sing the forward c hannel October 29, 2018 DRAFT 17 reliability . This is another conseq uence of the fact that d ecreasing the forward c hannel error probability requires a huge bac kof f in terms of transmitted rate, which in this cas e is not compensated by the correspond ing decreas e in feedba ck overhead. V . H Y B R I D - A R Q While up to now we hav e conside red simple ARQ, c ontemporary wireles s sys tems often utilize mo re powerful hybrid-ARQ (HARQ) tec hniques. When incre mental redun dancy (IR) HARQ, which is the mos t powerful type o f HARQ, is implemented, a N A CK triggers the trans mission of extra parity c heck bits instead of re-transmission of the original pa cket, and the receiver attempts to decode a packet on the basis of all previous trans missions related to that packet. This correspon ds to a ccumulation of mutual information ac ross HARQ rounds, and thus e ssentially match es the transmitted rate to the instantaneo us channe l conditions without requiring CSI at the transmitter [10], [14]. T he foc us of this section is understand ing how the PHY transmitted rate should be c hosen when HARQ is us ed. Unlike simple ARQ, HARQ requ ires the rec ei ver to keep information from previous rounds in memory; partly for this reason, HARQ is ge nerally impleme nted in a two-layered system (e.g., in 4G cellular networks suc h as L TE [22] [23]) in which the HARQ process h as to restart (triggered by a higher-layer simple ARQ re-trans mission) if the n umber of HARQ roun ds reaches a de fined maximum. The precise model we study is d escribed as follows. As before, eac h HARQ transmission (i.e., round) experience s a diversit y order of L . Howe ver , a max imum of M HARQ round s a re allowed per pa cket. If a packet cannot b e decod ed after M HARQ rounds , a post-HARQ ou tage is declared. This triggers a higher-l ayer simple ARQ re-transmission, which restarts the HARQ process for that packet. This two-layered ARQ process c ontinues (indefinitely) until the pac ket is suc cessfully receiv ed at the rece i ver . For the sake of simplicity , we proceed u nder the ideal assumptions discus sed in Se ction III. Note that the case M = 1 rev erts to the simple ARQ model discuss ed in the rest of the p aper . Gi ven this model, the first-HARQ-round outag e proba bility , de noted ε 1 , is exac tly the same as the non-HARQ outage probability with the s ame SNR , di versity order L , and rate R , i.e., ε 1 ( SNR , L, R ) = P " 1 L L X i =1 log 2  1 + SNR | h i | 2  ≤ R # . (28) In this expres sion R is the trans mitted rate during the first HARQ round , which we refer to as the HARQ initial rate R init hereafter . Becaus e IR leads to ac cumulation of mutual information, the numb er of HARQ October 29, 2018 DRAFT 18 rounds neede d to decode a p acket is the smallest integer T ( 1 ≤ T ≤ M ) such that T X i =1   1 L L X j =1 log 2  1 + SNR | h i,j | 2    > R init . (29) Therefore, the post-HARQ outage, d enoted by ε , is: ε ( SNR , L , M , R init ) = P   M X i =1   1 L L X j =1 log 2  1 + SNR | h i,j | 2    ≤ R init   . (30) This is the proba bility that a packet fails to be dec oded after M HARQ rounds, an d thus is the probability that the HARQ p rocess has to b e restarted. Using the ren ew al-re ward theorem as in [10] yields the following expression for the long-term average goodput with HARQ: η = R init (1 − ε ) E [ T ] , (31) where the distributi on of T is determined by (29). Our interest is in finding the initial rate R init that maximizes η . T his optimization is n ot ana lytically tractab le, but we can none theless provide s ome insight. In Fig. 10, good put is plotted versus vs. R init for L = 2 and a ma ximum of M = 2 HARQ round s, as well a s for a system using only simple ARQ (i.e., M = 1 ) with the same L = 2 , at SNR = 5 and 10 dB. W e immediately obs erve that go odput with HARQ is maximized at a co nsiderably high er rate than for the system without HARQ. Although we d o not h av e analytical proof, we conjecture that the goodput-maximizing initial rate with HARQ is always larger than the ma ximizing rate without HARQ (for equal di versity order per round/transmission). In f act, with HARQ the initial rate sh ould be c hosen such that the first-round outag e ε 1 is quite large, and for larger values of M the op timizer actually trends tow ards one. If ε 1 is small, then HARQ is rarely used wh ich means that the rate-matching capability provided by HARQ is not exploited. Howe ver , R init should not be chose n so large such that there is significant probability of post-HARQ outage, b ecaus e this lea ds to a s imple AR Q re-transmission a nd thus forces HARQ to re-start. Th e following theorem provides a n uppe r boun d on the optimal initial rate: Theorem 1: For any SNR , L , a nd M , the optimal initial rate with HARQ is u pper boun ded by 1 / M times the o ptimal transmitted rate for a n on-HARQ system with diversity order M L . Pr o of: The HARQ good put can be rewri tten as η = R init M · (1 − ε ) · M E [ T ] . (32) October 29, 2018 DRAFT 19 Based on (30) we see that the post-HARQ ou tage probability ε is prec isely the same as the outage probability for a n on-HARQ system with diversity order M L and transmitted rate R init / M . The refore, the term ( R init / M )(1 − ε ) in (32) is prec isely the goodput for a no n-HARQ sys tem with d i versity order M L . Base d on (29) we can s ee that the term M / E [ T ] is decrea sing in R init / M , and thus the value of R init / M that max imizes (32) is sma ller than the value that maximize s ( R init / M )(1 − ε ) . Notice that M L is the maximum d i versity experienced by a packet if HARQ is use d, w hereas M L is the precise div ersity order experienced by e ach packet in the refe rence system (in the theorem) without HARQ. Co mbined with our e arlier obse rvati on, we see that the initial rate s hould be chose n large en ough such that HARQ is s uf ficiently utilized, but not so lar ge such that simple ARQ is overly us ed. V I . C O N C L U S I O N In this pap er we ha ve conducted a detailed study of the optimum physical layer reliability when simple ARQ is used to re-transmit incorrectly decode d pac kets. Our find ings show that when a cross - layer perspec ti ve is taken, it is optimal to u se a rather unreliable ph ysical layer (e.g., a pac ket e rror probability of 10% for a wide range of chann el parameters). The fundamental reason for this is tha t making the phys ical layer very reliable requires a very c onservati ve transmitted rate in a fading channe l (without instantaneous channel knowledge a t the transmitter). Our findings are q uite general, in the se nse that the PHY sho uld n ot be operate d reliably e ven in scena rios in which intuition might sug gest PHY -lev el reliability is neces sary . For example, if a smaller packet error mis-detection probability is desired, it is much more efficient to utilize additional e rror detection bits (e.g., CRC) as co mpared to performing additional e rror correction (i.e., ma king the PHY more reliable). A delay constraint imposes a n uppe r bound on the number of ARQ re -transmissions and an upper limit o n the P HY error probab ility , but a n optimized sy stem s hould operate at exactly this lev el and no lower . Finally , when a cknowledgement errors are taken into ac count and high end-to-end reliability is required, such reliability s hould be achieved by design ing a reliable fee dback cha nnel instead of a reliable da ta (PHY) channe l. In a broader context, one important message is that traditional diversity metrics, which c haracterize how q uickly the probability of error can be made very s mall, ma y no longer be appropriate for wireless systems due to the pre sence of ARQ. As se en in [24] in the context of multi-antenna co mmunication, this c hange ca n significa ntly reduce the attractiv eness o f transmit di versity techniqu es that red uce e rror at the exp ense of rate. October 29, 2018 DRAFT 20 A P P E N D I X A P RO O F O F P RO P O S I T I O N 1 W e first prove the strict concavity of η g . For any in vertible function f ( · ) , the follo wing holds [25]:  f − 1 ( a )  ′ = 1 f ′ ( f − 1 ( a )) . (33) By combining this with Q ( x ) = R ∞ x 1 √ 2 π e − t 2 2 dt , we get  Q − 1 ( ε )  ′ = − √ 2 π e ( Q − 1 ( ε )) 2 2 , (34) which is strictly negati ve. Acco rding to this, the seco nd de ri vativ e of η g ( ε ) is: ( η g ( ε )) ′′ = κµ  Q − 1 ( ε )  ′  2 + (1 − ε ) √ 2 π e ( Q − 1 ( ε )) 2 2 Q − 1 ( ε )  . (35) Becaus e κ  Q − 1 ( ε )  ′ < 0 , in order to prove ( η g ( ε )) ′′ < 0 we only n eed to show tha t the expression inside the parenthesis in (35) is strictly positiv e. If we s ubstitute ε = Q ( x ) (here we de fine x = Q − 1 ( ε ) ) , the n we only n eed to prove ( Q ( x ) − 1) e x 2 2 x < q 2 π . No tice wh en x ≥ 0 , the left hand s ide is negati ve (becaus e Q ( x ) ≤ 1 ) and the ine quality h olds. Whe n x < 0 , the left hand side becomes Q ( − x ) e x 2 2 ( − x ) . From [26], Q ( − x ) < 1 √ 2 π ( − x ) e − x 2 2 , so if x < 0 , ( Q ( x ) − 1) e x 2 2 x < 1 √ 2 π ( − x ) e − x 2 2 e x 2 2 ( − x ) = 1 √ 2 π < r 2 π . (36) As a result, the seco nd d eri vati ve of η g ( ε ) is strictly smaller than zero a nd thus η g is s trictly conc av e in ε . Since η g is strictly concave in ε , we reach the fixed point equation in (15) by s etting the first deriv ati ve to ze ro. The concavity of η g implies ( η g ( ε )) ′ is decreasing in ε , and thus from (15) we s ee that ε ⋆ g is increasing in κ . A P P E N D I X B E X P E C T E D A R Q R O U N D S W I T H A C K N O W L E D G E M E N T E R R O R S If the ARQ p rocess terminates after i roun ds ( 1 ≤ i ≤ d − 1 ), the rea sons for tha t can be: • The first i deco ding attempts are uns ucces sful, the first i − 1 N A CKs are received co rrectly , but a N A CK → A CK error happen s in the i -th rou nd, the probability of which is ε i · (1 − ε fb ) i − 1 · ε fb . • The packet is decode d correctly in the j -th round (for 1 ≤ j ≤ i ), but the ACK is not correctly receiv ed un til the i -th roun d. This correspon ds to j − 1 de coding failures with co rrect ackn owledge- ments, followed b y a decoding succ ess an d i − j acknowledgement errors (A CK → N A CK), and the n a correct acknowledgement: P i j =1 ε j − 1 (1 − ε fb ) j (1 − ε ) ε i − j fb . October 29, 2018 DRAFT 21 These ev ents a re a ll exclusiv e, a nd thus we c an sum the above proba bilities. For X = d , we notice that the ARQ process takes the maximum of d rounds if: • There are d decoding f ailures with d − 1 correct N A CKs, the probability of which is ε d − 1 · (1 − ε fb ) d − 1 . • The packet is deco ded correctly in the j -th round (for 1 ≤ j ≤ d − 1 ), but the A CK is n ev er rece i ved correctly . This corresp onds to j − 1 decoding failures with correc t NA CKs, followed by a decod ing succe ss and d − j acknowledgemen t errors (A CK → NA CK): P d − 1 j =1 ε j − 1 (1 − ε fb ) j − 1 (1 − ε ) ε d − j fb . These ev ents a re a gain exclusive. Th erefore, the expected numbe r of rounds is: E [ X ] = d − 1 X i =1 i ·   ε i · (1 − ε fb ) i − 1 · ε fb + i X j =1 ε j − 1 (1 − ε fb ) j (1 − ε ) ε i − j fb   + d ·   ε d − 1 · (1 − ε fb ) d − 1 + d − 1 X j =1 ε j − 1 (1 − ε fb ) j − 1 (1 − ε ) ε d − j fb   . (37) R E F E R E N C E S [1] M. Luby , T . Gasiba, T . Stockhammer , and M. W atson, “Reliable multimedia do wnload deliv ery in cellular broadcast networks, ” IEEE T rans. Broadc asting , vol. 53, no. 1 Part 2, pp. 235–246, 2007. [2] C . Berger , S. Zhou, Y . W en, P . W i llett, and K. Pattipati, “Optimizing joint erasure-and error-correction coding for wireless packe t t ransmissions, ” IEEE Tr ansactions on W ireless Communications , vol. 7, no. 11 Part 2, pp. 4586–4 595, 2008. [3] T . A. Courtade and R. D. W esel, “A cross-layer perspectiv e on rateless coding for wireless channels, ” Pro c. of IEEE Int’l Conf. in Commun. (ICC ’09) , pp. 1–6, Jun. 2009. [4] X. Chen, V . S ubramanian, and D. J. 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October 29, 2018 DRAFT 23 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 η (bits/symbol) ε L=2 L=5 SNR = 10 dB SNR = 5 dB Fig. 1. Gooput η (bits/symbol) vs. P HY outage probability ε for L = 2 , 5 , SNR = 10 dB −10 −5 0 5 10 15 20 25 30 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 SNR (dB) ε ⋆ Exact Gaussian Approximation L=10 L=5 L=2 Fig. 2. Optimal ε vs. SNR (dB) for L = 2 , 5 , 10 October 29, 2018 DRAFT 24 0 1 2 3 4 5 6 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 R ε (bits/symbol) Success probability 1− ε L=5 L=20 L= ∞ (a) 1 − ε vs. R ε (bits/symbol) 0 1 2 3 4 5 6 7 0 0.5 1 1.5 2 2.5 3 R ε (bits/symbol) Goodput (bits/symbol) L=5 L=20 L= ∞ (b) η (bits/symbol) vs. R ε (bits/symbol) Fig. 3. Success probability 1 − ε and η ( bits/symbol) vs. R ε (bits/symbol) for SNR = 10 dB −10 −5 0 5 10 15 20 25 30 0 1 2 3 4 5 6 7 SNR (dB) η (bits/symbol) ε =0.001 ε =0.01 ε =0.1 optimal (a) L = 2 −10 −5 0 5 10 15 20 25 30 0 1 2 3 4 5 6 7 8 SNR (dB) η (bits/symbol) ε =0.001 ε =0.01 ε =0.1 optimal (b) L = 10 Fig. 4. η (bit s/symbol) vs. SNR (dB), for ε = 0 . 001 , 0 . 01 , 0 . 1 , and ε ⋆ October 29, 2018 DRAFT 25 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 R ε (bits/symbol) 1 − ε n=50,Gaussian n=200,Gaussian n=inf,Exact L=10 0 dB 10 dB Fig. 5. Success probability 1 − ε vs. t ransmitted rate R ε (bits/symbol) for n = 50 , 200 , ∞ , L = 10 at SNR = 0 and 10 dB −5 0 5 10 15 20 25 30 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 SNR (dB) optmal ε n= ∞ ,Exact n=200,Gaussian n=500,Gaussian L=2 L=10 L=5 Fig. 6. Optimal ε vs. SNR (dB) for L = 2 , 5 , 10 and n = 200 , 500 and ∞ October 29, 2018 DRAFT 26 Fig. 7. The ARQ process with non-ideal feedback with an end-to-end delay constraint d = 3 . 10 −6 10 −5 10 −4 10 −3 1 2 3 4 5 6 7 8 9 x 10 −3 ε ⋆ fb ε ⋆ 10 dB 5 dB L fb =2 L fb =2 L fb =5 L fb =5 L fb =1 n=200, L=3 d=3, q=10 −6 ξ d =q curve L fb =1 Fig. 8. ( ε ⋆ , ε ⋆ fb ) with L fb = 1 , 2 and 5 in Rayleigh fading f eedback channel for n = 200 , d = 3 , q = 10 − 6 , and L = 3 at SNR = 5 and 10 dB. T he curve specifying the ( ε, ε fb ) pairs that achiev e t he reli ability constraint ξ d = q is also plotted. October 29, 2018 DRAFT 27 0 0.002 0.004 0.006 0.008 0.01 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 ε goodput (bits/symbol) L fb =2 L fb =1 d=3, q=10 −6 n=200, L=3 SNR=5 dB q 1/d Fig. 9. Goodput η (bits/symbol) vs. PHY outage probability ε with L fb = 1 and 2 in Rayleigh fading feedback channel for SNR = 5 dB, n = 200 , L = 3 , d = 3 and q = 10 − 6 . 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 initial rate (bits/symbol) goodput (bits/symbol) no HARQ,M=1,L=2 HARQ,M=2,L=2 5 dB 10 dB Fig. 10. Goo dput (bits/symbol) vs. i nitial rate (bits/symbol) with HARQ for M = 2 and L = 2 and without HARQ for M = 1 and L = 2 at SNR = 5 , 10 dB. October 29, 2018 DRAFT