Optimizing Pilot Overhead for Ultra-Reliable Short-Packet Transmission

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1 Optimizig Pilot Overhead for Ultra-Reliable Short-Packet Trasmissio Mohammadreza Mousaei Departmet of Electrical ad Computer Egieerig Uiversity of Illiois at Chicago, Chicago, Illiois Besma Smida, Seior Member, IEEE Departmet of Electrical ad Computer Egieerig Uiversity of Illiois at Chicago, Chicago, Illiois arxiv: v1 [cs.it] 8 May 2017 Abstract I this paper we optimize the pilot overhead for ultra-reliable short-packet trasmissio ad ivestigate the depedece of this overhead o packet size ad error probability. I particular, we cosider a poit-to-poit commuicatio i which oe sesor seds messages to a cetral ode, or base-statio, over AWGN with Rayleigh fadig chael. We formalize the optimizatio i terms of approximate achievable rates at a give block legth, pilot legth, ad error probability. This leads to more accurate pilot overhead optimizatio. Simulatio results show that it is importat to take ito accout the packet size ad the error probability whe optimizig the pilot overhead. I. INTRODUCTION Wireless etwork research has traditioally focused o icreasig the iformatio rate to meet the demad geerated by huma-operated mobiles [1] [4]. However, all sorts of autoomous machies thigs with commuicatio capabilities will soo eed to be coected as well. The data trasmitted to ad from autoomous machies is very differet from the data to ad from huma-operated mobile devices [5]. The autoomous machies exchage a massive umber of short data bursts at moderate data rates [6] [8] but with striget reliability requiremets. These data bursts may result from idustrial automatio, wireless coordiatio amog vehicles, smart grid cotrol fuctios, or health-moitorig activities [9] [12]. The cetral challege with these ew wireless services is that curret wireless systems are ot properly desiged to support high-reliable short-packet trasmissio. The goal of this research is to icrease packet efficiecy by optimizig the pilot overhead for short packet trasmissio. I practical commuicatio schemes, oe sed packets, each of which has bits dedicated to cotrol overhead, pilots for chael estimatio, ad data payload. I short-packet commuicatios, low packet efficiecy is a cocer: a packet typically carries less tha 40% to 50% of actual data ad the relative proportios allocated to differet portios must be carefully optimized [13]. ad block-fadig model is ivestigated i [26], [27], ad the depedece of the optimum overhead o various system parameters of iterest (e.g., fadig rate, sigal-to-oise ratio is quatified. Optimizatio posed i prior works is predicated o the maximizatio of the ergodic chael capacity. They all assume that the packet error probability ca be made arbitrarily small by choosig the packet legth sufficietly large. This optimizatio, based o large block-legth, is usuitable for short-packet trasmissio. Ideed, we eed a ew aalysis of the achievable rate to assess the performace of shortpacket commuicatio [28]. Ufortuately, the exact value of achievable rate is ukow eve for chael models that are much simpler to aalyze tha the oe ecoutered i wireless commuicatios [13]. Polyaskiy ad al. recetly provided a uified approach to obtai tight bouds o achievable rate by providig lower boud that coicides with a upper boud i [28]. They showed that for various chaels, the datarate varies with packet sizes, desired error probabilities, ad chael dispersios. B. Cotributios The key departure from prior work o pilot overhead optimizatio is that we (a use achievable-rate tight-boud expressios that are more accurate for short-packet trasmissio, (b derive the miimum mea square error for cotiuous fadig as fuctio of the packet size, ad (c ivestigate the depedece of the pilot overhead o various system parameters, e.g. packet size, error probability, fadig rate ad sigal-tooise ratio (SNR. II. PRELIMINARIES We cosider a poit-to-poit commuicatio, i which oe sesor wishes to sed messages to a cetral ode, or basestatio. The sesor set traiig symbols kow to the basestatio, eablig the base-statio to estimate the chael gai. A. Related works Pilot overhead optimizatio The optimizatio of pilot overhead, predicated o the maximizatio of the ergodic chael capacity, has bee largely studied i the literature [14] [25]. I more commo systems, where the pilot symbol power is fixed, the optimizatio is over the umber of pilot symbols. I that case, some explicit results have bee established i both low ad high power regimes. Numerical solutios are derived for geeral power levels. By maximizig a tight lower-boud of the average chael capacity, a closed-form solutio for the average rate of pilot symbol i block-fadig [18] ad i cotiuous fadig with rectagular Doppler spectrum is derived i [20]. More recetly, the optimizatio of the pilot overhead i a uified cotiuous A. Chael Model I this system, we cosider a Rayleigh fadig chael ad additive white Gaussia oise (AWGN. Pilot symbols are periodically iserted i every packet. Let each packet cotai symbols. The trasmissio is divided ito two phases. The traiig phase icludes t symbols ad the data trasmissio phase icludes t symbols. We defie parameter α t / 1. Uder this model, the iput-output relatioship of i th received symbol is give by: y(i x(ih(i+w(i, i 0,1,..., 1 (1 1 α should be greater tha α mi 1/.

2 where x(i is the i th symbol, y(i is the correspodig received symbol, w(i is AWGN with zero-mea. Without loss of geerality, we ormalize the Rayleigh fadig chael ( h 2 1, ad we assume x(i ad w(i have uit mea square. Thus, is the SNR at the receiver. B. Chael Estimatio The t traiig symbols are used to estimate h(i for all i i the data trasmissio phase. We first evaluate the miimum chael estimatio error of the chael vector h : [h(0,...,h(] T, as fuctio of t, which is eeded to subsequetly derive the approximate achievable rate. Let h h ĥ deote the mismatch betwee the true chael vectorhad its estimateĥ : [ĥ(0,...,ĥ(]t. I this paper, we use miimum mea square error (MMSE estimator with two differet fadig models: 1 Block Fadig: The block-fadig model applies to a chael i which several adjacet symbols (referred as a block or packet are affected by the same fadig value ad the fadig values i differet blocks are idepedet ad idetically distributed [29]. For example, this model is applicable to wearable health sesors which are trasferrig short-packet commuicatio (such as body temperature ad heart bit rate to smartphoes. Commuicatio eviromet i such system chages i a low speed so that the chael gai, albeit radom, varies so slowly with time that it ca be assumed as costat alog a block. Usig this fadig model we ca derive MMSE as σ 2 h as [18]: σ 2 h 1. (2 2 Cotiuous Fadig: For wireless commuicatio the cotiuous fadig chael model is more realistic. Ideed, the chael is cotiuously chagig, so the actual chael will deviate progressively from the chael estimate obtaied at the traiig time. The chael estimatio error for cotiuous fadig chaels is caused by oise as well as the temporal variatio of the chael [30]. We ca hece model our chael as a block fadig chael with a additioal oise due to the temporal variatio of the chael. Assumig this model we ca derive σ 2 h σ 2 h 1 +σ2 Doppler, (3 where the additioal chael estimatio error σdoppler 2 is derived, i the appedix, for Rayleigh fadig as ( 2( σdoppler 2 παfd 2 α 2. (4 2 where f D is the Doppler frequecy ormalized to the data rate. C. Ergodic Capacity i Fiite Blocklegth Regime Cosider a source which is modeled as a radom variable equi-probably takig values i the set {1,...,M}. The chael is a oisy commuicatio medium which takes a iput i some alphabet A ad output a symbol i alphabet B. A ecoder maps messages ({1,...,M} ito legth sequeces of chael iput symbolsa ( codewords. Therefore, the ecoder is a fuctiof : {1,...,M} A. A decoder that produces a estimate of origial sigal by observig -sequece of chael outputs is a fuctio g : B {1,...,M}. The goal of commuicatio is to fid a ecoder-decoder pair (code which is capable of commuicatig messages with some fixed probability of error ǫ. Such code is called (, M, ǫ- code. Polyaskiy ad al. proved i [28], [31] that give a fixed block-legth, probability of error ǫ ad fadig chael with SNR ad perfect CSI, the maximum umber of messages M (,ǫ, : max{m, (,M,ǫ code}, ca be tightly approximated by where log 2 (M (,ǫ, C( V(Q 1 (ǫ, (5 C( log 2 (ee 1/ E 1 ( 1, (6 ad E 1 (x 1 t 1 e xt dt is the expoetial itegral. The chael dispersio V( ca be derived as [31]: V( Var [ C( h 2 ] + log2 (e 2 [ ] 1 (1 E 2 h 2 1+ h 2, where Var[.] ad E[.] are variace ad expectatio over distributio of h 2, ad Q 1 is fuctioal iverse of Q-fuctio. The ratio R(,ǫ, : log 2M(,ǫ, is kow as the rate. The maximum achievable rate ca be tightly approximated by [28]: R (,ǫ, : 1 log 2(M (,ǫ, C( (7 V( Q 1 (ǫ. (8 III. PILOT-ASSISTED DETECTION FOR SHORT-PACKET TRANSMISSION I this sectio, we provide a approximatio of the achievable rate of a poit-to-poit commuicatio(rtr (,ǫ, whe traiig symbols ad MMSE estimator are used to extract CSI at the receiver. Cotrary to the assumptio i Sectio. II-C, here h is ot kow to the receiver but estimated usig pilot symbols. Durig the data trasmissio phase, after MMSE estimatio chael ca be rewritte as: y(i x(iĥ(i+ x(i h(i+w(i, (9 where the chael state iformatio, ĥ(i, is perfectly kow at receiver. The mai problem here is the fact that the oise x(i h(i +w(i icludes the chael estimatio error. So the oise is ot ecessarily idepedet from the trasmitted sigal or Gaussia. First, we assume MMSE estimator, the h(i ad ĥ(i are orthogoal. The we cosider Gaussia oise, followig the same approach used i [18]. Usig those assumptios, the chael defied i Eq. (9 became similar to the chael itroduced i Sectio II-C, with SNR eff (1 σ2 h 1+σ 2 h. (10

3 0.11 vs ǫ for block ad cotiuous fadig for ifiite blocklegth (Block Fadig for fiite blocklegth (Block Fadig for ifiite blocklegth (Cotiuous Fadig for fiite blocklegth (Cotiuous Fadig ǫ Fig. 1. Optimal pilot overhead for ifiite ad fiite blocklegth i block ad cotiuous fadig model vs. ǫ with 30 ad SNR 15dB. Fig. 3. Optimal pilot overhead for ifiite ad fiite blocklegth i cotiuous fadig model vs. Blocklegth with SNR 23dB, ǫ 1e-9 ad f D Fig. 2. Optimal pilot overhead for ifiite ad fiite blocklegth i block fadig model vs. Blocklegth with SNR 8dB ad ǫ 1e-9. Fially, we took ito accout the umber of symbols dedicated to pilot, t symbols do t carry data, to approximate the achievable rate as: R Tr (,ǫ, eff (1 αc( eff Q 1 (ǫ IV. NUMERICAL RESULTS (1 αv(eff. (11 I this sectio, we umerically evaluate the optimal pilot overhead for ultra-reliable short-packet trasmissio. These umerical results are derived by solvig the derivative of Eq. (11 w.r.t α equal to zero. For compariso purpose we also optimize the pilot overhead usig the ergodic capacity [26]. These comparisos are show i Fig Our simulatio results prove that our optimizatio approach will result i icrease of aroud 10% i achievable rate. We performed simulatios for a broad rage of variables such as: Probability of Error: We first compare the optimal pilot overhead for differet error probabilities. The differece betwee our optimal pilot overhead values ad those derived usig ergodic capacity icreases with decreasig probability of error. Thus, it is very importat to use the ew formulatio for ultra-reliable commuicatio systems. We have similar results for both block ad cotiuous fadig (Fig. 1. Blocklegth: We umerically evaluated the optimal pilot overhead for differet blocklegth. We cosidered both block ad cotiuous fadig. Note that whe we use ergodic capacity optimizatio, the blocklegth is assumed ifiite but the variace of the chael estimatio error varies with for block fadig. As show i Fig. 2, the differece is higher i small blocklegth. This suggests that our approach is more adequate to short packet trasmissio. Moreover, we ca see i Fig. 3 that the pilot overhead icreases with small blocklegth whe we eed more ad more pilot symbols to compesate the chael estimatio mismatch. SNR ad Normalized Doppler Frequecy: The simulatios, illustrated i Fig. 4 ad Fig. 5, show that the optimal pilot overhead decreases with SNR ad icreases with f D, as expected. I additio, the differece betwee our optimal pilot overhead values ad those derived usig ergodic capacity is greater at low SNR ad high ormalized Doppler frequecy. Optimal Rate: We evaluated the rate at the optimum values of α evaluated i this paper ad optimum alpha for ifiite blocklegth. As illustrated i Fig. 7 our optimizatio approach will result i icrease of roughly 10% i rate with block fadig. Fig. 6 shows eve more sigificat icrease i rate with cotiuous fadig. Simulatios show that our approach always results i a higher rate. Fig. 6 also shows that due to the chael estimatio mismatch i cotiuous fadig, icreasig blocklegth after a certai blocklegth ( 29 i this case results i decreasig rate.

4 Fig. 4. Optimal pilot overhead for ifiite ad fiite blocklegth i block ad cotiuous fadig model vs. SNR with 40 ad ǫ 1e-9. Fig. 6. Acheivable rate usig ifiite ad fiite blocklegth i cotiuous fadig model vs. with SNR 20dB, ǫ 1e-12 ad f D Acheivable Rate vs for block fadig R * Tr (α for ifiite blocklegth opt R * Tr (α for fiite blocklegth opt 0.2 R * Tr ( Fig. 5. Optimal pilot overhead for ifiite ad fiite blocklegth i cotiuous fadig model vs. f D with 10, SNR 16dB ad ǫ 1e-9. V. CONCLUSION The goal of this research is to icrease the packet efficiecy by optimizig the pilot overhead for ultra-reliable short packet trasmissio. We cosidered a poit-to-poit commuicatio i which oe sesor seds messages to a cetral ode, or base-statio, over additive white Gaussia oise with Rayleigh fadig chael. We formalized the optimizatio i terms of approximate achievable rates as fuctio of block legth, pilot legth, ad error probability. Simulatio results proved that it is very importat to take ito accout the packet size ad the error probability whe optimizig the pilot overhead. ACKNOWLEDGMENT Fudig for this research was partially supported by NSF uder Award Number Fig. 7. Achievable rate usig ifiite ad fiite blocklegth i block fadig model vs. with SNR 7dB, ǫ 1e-9 ad f D APPENDIX The mobile trasmit t traiig symbols kow to the mobile ad base-statio, eablig the base-statio to estimate the chael gai. The MMSE chael gai estimator ca be derived as [18] ĥ (1+ x t 2 1 x t y t 1 ( x t 2 x t y t (12 where x t, y t are iput ad output traiig symbol vectors. Note that y t x t h + w t + x t h t, where y t [y(1,y(2,...,y(α], x t [x(1,x(2,...,x(α], w t

5 [w(1,w(2,...,w(α] are output, iput ad oise vectors respectively. Also h t [ h(1, h(2,..., h(α] is the chael mismatch due to the temporal variatio of the chael. Sice x t 2 α, we get ( 1 ĥ [ αh+ wx t + h t x t x t ] ( α h+ ( α + 1 ( wx t h t. (13 Hece the chael estimatio error h h ĥ ca be derived as: h h ĥ ( 1 1 h ( α w t x t h t (14 Fially, we derive the mea square error of the MMSE chael estimatio error as σ 2 h 1 +σ2 Doppler (15 where we derived σdoppler 2 usig the mathematical derivatio proposed i [30] for maximum likelihood (ML chael estimator i Rayleigh fadig chael ( 2( σdoppler 2 παfd 2 α 2, (16 2 where f D is the Doppler frequecy ormalized to the symbol rate (R symbol 1/T s give by T svf c, T s is the symbol c period, v is the mobile velocity, f c is carrier frequecy ad c is the speed of electromagetic wave. REFERENCES [1] M. K. Kiskai, H. Sadjadpour, ad M. 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