Performance Analysis of Adaptive Coded Modulation with Antenna Diversity and Feedback Delay

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1 Performace Aalysis of Adaptive Coded Modulatio with Atea Diversity ad Feedback Delay Kjell J. Hole, Herik Holm, ad Geir E. Øie Abstract A geeral adaptive codig scheme for spectrally efficiet trasmissio o flat fadig chaels was itroduced by the authors i a earlier paper [2]. A istace of the codig scheme utilizes a set of multidimesioal trellis codes desiged for additive white Gaussia oise chaels of differet qualities. A feedback chael betwee the decoder ad ecoder makes it possible for the ecoder to switch adaptively betwee these codes based o chael state iformatio fed back from the decoder. I this paper, the adaptive codig scheme is employed i a mobile wireless commuicatio system cosistig of a statioary trasmitter with oe atea, a wireless Rayleigh fadig chael, ad a mobile termial with oe or more receive ateas. The bit-error-rate at the output of the decoder is determied for various termial speeds, time delays i the feedback chael, ad umber of receive ateas. The obtaied results idicate that the proposed adaptive codig scheme is well suited for commuicatios over mobile wireless chaels with carrier frequecies i the high MHz rage, delay spread up to 25 s, ad termial mobility up to pedestria speed. The authors are with the Departmet of Telecommuicatios, Norwegia Uiversity of Sciece ad Techology, O. S. Bragstads plass 2B, N-7491 Trodheim, Norway. K. J. Hole is also with the Departmet of Iformatics, Uiversity of Berge, HiB, N-52 Berge, Norway Kjell.Hole@ii.uib.o, {Herik.Holm,Geir.Oie@tele.tu.o. I. Itroductio May authors have studied adaptive coded modulatio for wireless commuicatios see [1] ad the refereces therei. I a earlier paper [2], we cosidered a geeral adaptive codig scheme for sigle-user chaels with frequecy-flat slowly varyig multipath fadig. A particular istace of this codig scheme utilizes a set of multidimesioal trellis codes desiged for additive white Gaussia oise AWGN chaels of differet qualities. A feedback chael makes it possible for the ecoder to switch adaptively betwee these codes based o chael state iformatio CSI fed back from the decoder, thus resultig i a overall scheme with high spectral efficiecy. The output bit-error-rate BER of a adaptive codig scheme may icrease with growig time delay i the feedback chael ad/or icreasig termial speed [3]. Sice ay implemeted feedback chael has ozero feedback delay, ad sice it is ecessary to allow for mobile termials, it is importat to determie the BER degradatio of the proposed adaptive codig scheme i [2]. Alouii ad Goldsmith [4] have determied the BER degradatio for ucoded adaptive modulatio. I this paper, we exted their techique to determie the BER degradatio of ay istace of the proposed adaptive codig scheme. We first itroduce, i Sectio II, a mobile wireless chael with Rayleigh fadig, where the mobile termial has multiple receive ateas whose sigals are combied usig the maximal ratio combiig MRC method [5, Ch. 5]. For ay istace of the adaptive codig scheme ad ay umber of receive ateas, Sectio III the shows how to determie the BER degradatio associated with a ozero feedback delay ad a ozero termial speed. As a example, Sectio

2 IV evaluates a a specific adaptive ecoder ad decoder codec utilizig a set of four-dimesioal trellis codes. A coclusio is draw i Sectio V. II. System model ad codig scheme The system model cosists of a statioary trasmitter/receiver trasceiver, a wireless frequecy-flat fadig chael, ad a mobile trasceiver, or termial. It is assumed that the distace betwee the statioary trasceiver ad the mobile termial is ot more tha a few hudred meters. We will oly cosider the flow of user iformatio o the dowlik. Hece, i our model the feedback chael or uplik from the termial to the receiver will oly be used for CSI. The statioary trasceiver has oe trasmit atea, while the mobile termial has H 1 receive ateas. Each of the H atea braches is modeled as a Rayleigh fadig chael with ideal coheret detectio. It is assumed that the the brach sigals are statistically idepedet. Deotig the trasmitted complex basebad sigal at time idex t {, 1, 2... by xt, the received sigal at atea h {1, 2,..., H ca the be writte as y h t α h t xt + h t. Here, the statioary ad ergodic fadig evelope α h t is a real-valued radom variable with a Rayleigh distributio, ad h t is complex-valued AWGN with statistically idepedet real ad imagiary compoets. The total oesided power spectral desity of the AWGN is deoted N [W/Hz] ad the oe-sided chael badwidth is deoted B [Hz]. Let S [W] deote the costat average trasmit power. The istataeous received carrier-to-oise ratio CNR o atea brach h at time idex t is the h t α2 h t S, h 1, 2,..., H, N B with expectatio E [ h t] h ΩS/N B where Ω E [ α 2 h t] is assumed idepedet of h. Thus, h is also equal for all h. The mobile termial implemets a MRC combier to process the H received brach sigals [5, p. 316]. Sice the brach sigals are statistically idepedet, the istataeous CNR at the output of the H-brach MRC combier is give by H h1 h. 1 If we deote E [], the h /H, ad the gamma probability desity fuctio pdf of the istataeous CNR at the output of the MRC combier may be writte as [5, Eq ] p H H H 1 H 1! exp H,. 1 It is coveiet to view the combiatio of the H atea braches ad the MRC combier as a sigle chael. The istataeous CNR at the output of this chael determies the chael state at a give time. We assume that the mobile termial has perfect kowledge of. The rage [, of possible CNR values is divided ito N + 1 ooverlappig itervals or fadig regios. At ay give time the CNR will fall i oe of these fadig regios, ad the associated CSI, i.e. the regio idex {, 1,..., N, is set to the statioary receiver via the feedback chael, which is assumed to be error free. Assume that [, 1 i fadig regio, [, +1 i regio {1, 2,..., N 1, ad [ N, i regio N. Also, assume that the BER must ever exceed a target maximum BER. Whe [, +1 we use a multidimesioal trellis code, deoted code {1, 2,..., N, desiged to achieve a BER BER o a AWGN chael of CNR. For < 1, i.e. i fadig regio, the chael coditios are so bad that o iformatio is trasmitted, ad we have a outage durig which the iformatio flow is buffered. Let 4 M 1 < M 2 < < M N deote the umber of symbols i N quadrature amplitude modulatio QAM costellatios of growig size, ad let code be based o the costellatio with symbols. For some small fixed L {1, 2,..., the ecoder for code accepts L log 2 1 iformatio bits at each time idex k L t {, L, 2L, ad geerates L log 2 coded bits. The coded bits specify L modulatio symbols i the th QAM costellatio. These symbols are trasmitted at time idices 1 We suppress the time depedece from ow o for otatioal simplicity. 2

3 k, k+1,, k+l 1. The L two-dimesioal symbols ca be viewed as oe 2L-dimesioal symbol, ad for this reaso the code is said to be a 2L-dimesioal trellis code. I practice, the N codes are chose such that they may be ecoded ad decoded by the same codec [2]. To determie the values of the fadig regio boudaries or thresholds, we eed to determie the BER performace of each code. Whe code is operatig o a AWGN chael of CNR, the BER-CNR relatioship for varyig may be approximated by the expressio b BER a exp, 2 where a > ad b > are costats which deped oly o the weight distributio of the code [2]. These costats ca be foud for ay give code by least-squares curve fittig of data from AWGN chael simulatios to 2. The fittig must be doe separately for each code i the set. Plots of BER foud i the literature idicate that the approximatio i 2 is accurate for ay CNR resultig i BER < 1 1 see Fig. 1 for a example. Ufortuately, for the miimum value, the approximatio reduces to BER a, ad sice a ca be larger tha oe, 2 may be of little use for low CNRs. Whe we oly wat to approximate the BER at moderate-to-high CNRs, as was doe i [2], this is ot a problem. However, we eed to approximate the BER for ay CNR i this paper, ad we will therefore use the followig BER expressio for code { a exp b BER, 1 2, < 3 Here, the boudary l2a b is the smallest CNR such that the BER is o larger tha.5. The boudary was obtaied by assumig equality i 2, settig BER.5, ad solvig for. For a true BER betwee 1 1 ad.5 the expoetial expressio i 3 teds to produce a larger value tha the true BER, assumig that the coded commuicatio system maages to maitai sychroizatio. I practice, it is difficult to maitai sychroizatio for a very high BER, ad the approximatio BER.5 may therefore be close to the true BER of a real system. If the coded system should exhibit a BER >.5 for a very low CNR, the all decoded iformatio bits may be flipped to achieve a BER <.5. Hece,.5 is a reasoable upper boud o the BER. Assumig a target BER such that > ad settig BER equal to BER i 3, the thresholds are give by [2] K /b, 1, 2,..., N N+1 4 where K l BER /a. The probability that falls i fadig regio, P P < +1, is give by [4, Eq. 1] where Γ P H, H Γυ, µ Γ H 1! µ H, H+1 5 t υ 1 e t dt 6 is the complemetary icomplete gamma fuctio [6, Eq ]. Sice H is a iteger i 5, the fuctio may be calculated usig [6, Eq ]. III. BER degradatio The BER degradatio due to ozero feedback delay ad ozero termial speed is determied i this sectio. It is assumed that the commuicatio system utilizes a set of N trellis codes with kow parameters a ad b. Let the total feedback delay, τ [s] be the time betwee the momet the mobile termial acquires a set of L modulatio symbols ad the momet the statioary trasmitter activates a ew code. The total feedback delay is determied by the sum of three delays: i the processig time eeded by the termial to estimate the istataeous CNR ad to determie i which fadig regio the CNR falls, ii the time eeded to feed back the regio idex to the 3

4 trasmitter, ad iii the processig time eeded by the trasmitter to activate code. I a real system, the processig delay i depeds o the techique used to estimate the istataeous received CNR, whereas the processig delay iii depeds o the ecoder complexity. Sice the distace betwee the statioary trasceiver ad the mobile termial is assumed to be o more tha a few hudred meters, the trasmissio delay ii is maily determied by the commuicatio protocols. The size of the sigal costellatio at time idex t is a fuctio of the istataeous received CNR, but the costellatio is used at time t+τ whe has chaged to τ. Cosequetly, while the CNR falls i some fadig regio, i.e. < +1, the CNR τ may fall outside this regio. Substitutig τ for i 3, we ca write the BER as a fuctio of τ for a give : BER τ τ { a exp bτ, τ 1 2, τ < 7 The average BER for i fadig regio is ow give by BER τ +1 { BER τ τ p τ τ d τ p d, 8 where p is give by 1. Furthermore, p τ τ is the pdf of τ coditioed o [4] p τ τ H 1/2 H τ 1 ρ ρ 2H ρτ I H 1 exp 1 ρ Hρ + τ 1 ρ. 9 The fuctio I H 1 i 9 is the H 1th-order modified Bessel fuctio of the first kid [7, Ch. 9]. The pdf also cotais the chael power correlatio coefficiet ρ at lag τ. It is show i Appedix A that ρ is give by the square of the zeroth-order Bessel fuctio of the first kid [7, Ch. 9], ρ J 2 2πf D τ, 1 for ay umber of receive ateas. Here, f D v/λ [Hz] is the maximum Doppler frequecy shift defied by the termial speed v [m/s] ad the wavelegth λ [m] of the carrier. Usig 7, the average BER i fadig regio give by 8 ca be rewritte as the differece betwee two double itegrals, where ad I1 def { I2 def { BER τ I1 I2, +1 a exp b τ p τ τ d τ p d [ a exp b τ 1 ] p τ τ d τ 2 p d. 12 The double itegral I2 is zero for, i.e., whe parameters a ad b result i good BER approximatios for ay τ. Hece, I2 may be viewed as a correctio term eeded whe a ad b are oly useful for τ >. It is show i Appedix B that I1 where ad a H 1! H β H + From Appedix C, we have H ΓH, β ΓH, β +1 ω H 13 Hρ b H + 1 ρb 14 ω H + b. 15 I2 Sa, b S 1 2, 16 4

5 for Sa, b def 1 ρ H a H 1! ρ j [ H j + H 1! j! b j 1 ρ + H [ ] b H ic H + j, + 1 ρ [ Γ H + j, Γ H + j, H 1 ρ ] j+h H +1 1 ρ 17 ], where ic υ, µ µ t υ 1 e t dt 18 is the icomplete gamma fuctio [6, Eq ]. Sice H + j is a iteger i 17, the fuctio may be calculated usig [6, Eq ]. The average BER over all N codes, deoted by BER τ, is equal to the expected umber of iformatio bits i error per modulatio symbol divided by the expected umber of trasmitted iformatio bits per modulatio symbol, BER τ N 1 i BER τ N 1 i P N 1 i [I1 I2] N 1 i. 19 P Here, i log 2 1/L is the umber of iformatio bits per modulatio symbol ad P is defied by 5. I practice, the double itegral I2 ca oly be approximated sice the sum i 17 must be termiated after a fiite umber of terms. Sice each term i the sum is positive, the termiatio causes the expressio i 19 to become a upper boud o the BER. The tightess of the boud improves as the umber of terms is icreased. We will use the te first terms i the sum of 17 i the ext sectio. IV. Evaluatio of example codec A adaptive codec with eight 4-dimesioal trellis codes was described i [2]. The idividual codes Figure 1: The boxes are BER estimates geerated by software simulatio ad the curves are estimates obtaied from 3. The labels idicate the umber of symbols i the QAM sigal costellatios utilized by the 4-dimesioal trellis codes. BER performaces o a AWGN chael were simulated for various CNRs. The obtaied BER poits represeted by boxes are show i Fig. 1. Curve fittig with the least squares method was used to obtai the parameters a ad b listed i Table 1. The correspodig BER approximatios 3 are plotted i Fig. 1. The expressio i 4 was used to determie the tabulated thresholds 2 rouded to oe decimal digit for target BER 1 4. Usig the thresholds, settig L 2, ad h 2 db, the base-1 logarithm of the average BER 19 is plotted as a fuctio of the correlatio coefficiet ρ i Fig. 2 for H {1, 2, 4 receive ateas. We observe that because the thresholds are chose accordig to 4, the istataeous BER is smaller tha the target BER for < < +1 ad ρ close to oe. As a result, the average BER will be below BER for large ρ see Fig The thresholds i Table 1 are larger tha the thresholds i [2, Table I] because we have reduced the target BER from 1 3 to 1 4. Furthermore, the path memory legth of the Viterbi decoder was set to 9 i [2] while a path memory legth of 16 was used i this paper. 5

6 a b [db] Table 1: Parameters a ad b for example codec ad calculated thresholds [db] for target BER 1 4. H mi. ρ τ max [ms] τ max /T [symb.] , , ,76 Table 2: Miimum correlatio coefficiet ρ eeded to achieve BER 1 4 for average atea brach CNR h 2 db ad differet umber H of receive ateas. Maximum tolerable delay τ max [ms] ad umber of modulatio symbols trasmitted durig τ max for carrier frequecy 19 MHz, badwidth 4 khz, ad termial speed v 1 m/s. Let τ max deote the maximum total delay, or maximum tolerable delay, for a give target BER. The expressio 1 for ρ ca be used to determie the maximum tolerable delay τ max for differet Doppler shifts f D ad targets BER. The miimum values of ρ rouded to three decimal digits eeded to achieve BER τ BER 1 4 are listed i Table 2 for H {1, 2, 4. If we let the carrier frequecy be f 19 MHz ad use the value c m/s for the speed of light, the the wavelegth of the carrier frequecy is λ c/f 3/19.16 m. A mobile termial with pedestria speed v 1 m/s the has Doppler shift f D v/λ 19/ Hz. The correspodig maximum tolerable delays τ max rouded to oe decimal digit are listed i Table 2. Figure 2: Base-1 logarithm of average BER for correlatio coefficiet.8 ρ < 1, target BER 1 4, ad average atea brach CNR h 2 db. To see that the fadig is early costat over may hudred modulatio symbols for commuicatios at pedestria speed, we calculate the umber of symbols trasmitted durig the maximum tolerable delay τ max. We first eed to determie a badwidth B for which it is reasoable to assume that the fadig is frequecy flat. The rms delay spread, σ d [s], measures how much a sigal compoet may be delayed durig trasmissio [8, Sec ]. The reciprocal of the delay spread provides a measure of the width of the bad of frequecies which are similarly affected by the chael respose. The chael is therefore approximately frequecy flat if the badwidth B 1/σ d. At 19 MHz, the multipath delay spread is up to σ d 25 s for a cordless phoe i idoor ad outdoor eviromets [9]. Hece, we may assume that a chael with badwidth at least up to B 4 khz has frequecy flat fadig. The time eeded to trasmit oe symbol at the Nyquist sigalig rate is T 1/B 2.5 µs, resultig i τ max /T symbols beig trasmitted durig the maximum tolerable delay. Usig the rouded values of τ max i Table 2, we obtai the τ max /T values listed i the rightmost colum of Table 6

7 2 for termial speed v 1 m/s ad H {1, 2, 4. V. Coclusio It has bee show see Fig. 2 that the BER performace may degrade cosiderably as ρ decreases, which for a give carrier frequecy correspods to icreasig the termial speed. However, the degradatio ca be mitigated by the use of MRC atea diversity. Still, our results idicate that adaptive coded modulatio may be best suited for systems with moderate mobility requiremets, with termials movig at pedestria speed. Appedix A Calculatio of ρ I this appedix we show that the chael power correlatio coefficiet ρ is give by the expressio i 1. The istataeous received CNR o the chael may be expressed as α 2 K where α 2 is the chael power gai ad K S/N B. Sice E[] H h1 E[ h] HKΩ, we have E[α 2 ] HΩ. Assume that α 2 is the power gai at some time t ad let ατ 2 be the power gai at time t + τ for τ >. The correlatio coefficiet ρ betwee α 2 ad ατ 2 is the give by ρ covα2, ατ 2 σα 2 σ 2 2 α 2 τ E[α2 α 2 τ ] E[α 2 ] E[α 2 τ ] σ α 2 σ α 2 τ. 2 The chael gai α 2 is gamma distributed [8, p. 48]. Hece, assumig that the chael power gais α 2 ad α 2 τ have the same expectatios ad stadard deviatios, we have E[α 2 ] E[α 2 τ ] HΩ 2 ad σ α 2 σ α 2 τ E[α 2 ] 2 /H HΩ 2 i 2. To calculate E[α 2 α 2 τ ], we first compare two differet expressios for the istataeous received CNR. Whe the commuicatio chael is viewed as a Rayleigh fadig chael with a H-brach MRC combier, the H h1 h K H h1 α2 h where α2 h is the power gai o the Hth atea brach. Sice we also have α 2 K, it follows that α 2 H h1 α2 h, ad we ca write E[α 2 α 2 τ ] E [ H h1 α 2 h H i1 α 2 i,τ ] H H E[αh 2 αh,τ 2 ] + E[αh 2 αi,τ 2 ]. 21 h1 h1 i h Furthermore, because the sigals o differet atea braches h i are statistically idepedet, the covariace covα 2 h, α 2 i,τ E[α 2 h α 2 i,τ ] E[α 2 h] E[α 2 i,τ ], or equivaletly, E[α 2 h α2 i,τ ] Ω2. The expressio i 21 is the equal to E[α 2 α 2 τ ] H E[α 2 h α 2 h,τ ] + HH 1 Ω 2, ad the correlatio coefficiet i 2 reduces to ρ E[α2 h α2 h,τ ] Ω2 Ω Observe that 22 is idepedet of the umber of receive ateas H. I fact, 22 defies the correlatio coefficiet for a Rayleigh fadig chael without MRC. It is show i [8, Eq. 2.68] that the umerator i 22 is equal to Ω 2 J 2 2πf D τ, ad as a result, ρ is give by the expressio i 1. Appedix B Evaluatio of I1 I the followig we calculate the double itegral i 11. For the ier itegral, BER i 3 is fixed sice the CNR is fixed. It follows from 3 that b lber /a. Usig this expressio for ad settig D lber /a, the ier itegral i 11 is equal to I1, def a exp Itroducig the costat x H 1/2 H τ 1 ρ ρ Hρ + τ 1 ρ I H 1 2H ρτ 1 ρ D τ d τ. ρh ρh + 1 ρd 23 7

8 ad makig the substitutio H z 1 ρ + D τ, the itegral 23 ca be writte as H H I1, a H + 1 ρd ρd H exp 24 H + 1 ρd z H 1/2 e z x I H 1 2 xz dz. x The value of the itegral i 24 is equal to Q H x, where Q H, is the geeralized Marcum Q-fuctio of order H [7, Eq ]. Sice Q H x, Q 1 x, 1 for all x, we have H H I1, a H + 1 ρd ρd H exp. 25 H + 1 ρd for The double itegral i 11 ca ow be writte as I1 F F Fξ ξ I1, p d. To calculate Fξ, we first observe that BER i 3 is o loger a costat sice varies. Usig the coectio D lber /a b /, it follows from 1 ad 25 that H H a H H Fξ H 1! H + 1 ρb ξ H 1 exp β d where β is defied by 14. Substitutig t β ad observig that H H H H 1 1 H + 1 ρb β ω for ω defied by 15, we get H a H ΓH, β ξ Fξ H 1! ω H 27 where Γ, is give by 6. The expressio for I1 i 13 is ow obtaied from 26 ad 27. Appedix C Evaluatio of I2 We shall calculate the double itegral I2 defied by 12. We first split the double itegral i two to obtai I { a exp b τ p τ τ d τ p d 28 { 2 1 p τ τ d τ p d.29 The secod itegral 29 is a special case of the first itegral 28 with a 2 1 ad b. Hece, we oly eed to cosider the first itegral. The pdf p τ τ defied by 9 cotais the H 1th-order modified Bessel fuctio of the first kid defied by [7, Eq. 9.28] ν 1 I ν z 2 z 1 2 z 2j j + ν! j! j for ν a iteger. Usig this defiitio, the ier itegral i 28 is equal to [ ] H H a exp Hρ b 1 ρ + H 1 ρ [ 1 H 2 ρ j + H 1! j! 1 ρ{b j 1 ρ + H [ ] b H ic H + j, + 1 ρ where ic, is defied by 18. The outer itegral i 28 is the equal to 17. Sice the double itegral i 29 is a special case of the double itegral i 28, the correctio term I2 is ow give by 16. Refereces [1] K. J. Hole ad G. E. Øie, Spectral efficiecy of adaptive coded modulatio i urba microcellular etworks, IEEE Tras. Veh. Techol., vol. 5, pp. 1 18, Ja. 21. ] j 8

9 [2] K. J. Hole, H. Holm, ad G. E. Øie, Adaptive multidimesioal coded modulatio over flat fadig chaels, IEEE J. Select. Areas Commu., vol. 18, pp , July 2. [3] D. L. Goeckel, Adaptive codig for time-varyig chaels usig outdated fadig estimates, IEEE Tras. Commu., vol. 47, pp , Jue [4] M.-S. Alouii ad A. J. Goldsmith, Adaptive M-QAM modulatio over Nakagami fadig chaels, Proc. 6th Commuicatios Theory Mii-Coferece CTMC VI i cojuctio with IEEE Global Commuicatios Coferece GLOBECOM 97 Phoeix, Arizoa, Nov. 1997, pp [5] W. C. Jakes, Editor, Microwave Mobile Commuicatios. Piscataway, NJ: IEEE Press, secod ed., [6] I. S. Gradshtey ad I. M. Ryzhik, Table of Itegrals, Series, ad Products. Sa Diego, CA: Academic Press, fifth ed., [7] N. M. Temme, Special Fuctios A Itroductio to the Classical Fuctios of Mathematical Physics. New York, NY: Joh Wiley & Sos, [8] G. L. Stüber, Priciples of Mobile Commuicatio. Norwell, MA: Kluwer Academic Publishers, [9] T. Ue, S. Sampei, N. Moriaga, ad K. Hamaguchi, Symbol rate ad modulatio levelcotrolled adaptive modulatio/tdma/tdd system for high-bit-rate wireless data trasmissio, IEEE Tras. Veh. Techol., vol. 47, pp , Nov

Diversity Combining Techniques

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