OPTIMIZED LIK ADAPTATIO FO WIELE PACKET COMMUICATIO BAED O DICETE-ATE MODULATIO AD CODIG CHEME Jalil eifali Harsii, ad Farshad Lahouti Wireless Multimedia Commuicatios Laboratory chool of Electrical ad Computer Egieerig, Uiversity of Tehra, Ira ABTACT Adaptive modulatio ad codig (AMC is a powerful techique for improvig the spectral efficiecy or the error performace i wireless packet etworks over fadig chaels. I this article, usig a fixed umber of modulatio ad codig modes we propose a variable power trasmissio scheme for trasmissio of packets over block-fadig wireless chaels. We obtai the geeral form of optimal power adaptatio ad optimum AMC mode switchig levels that maximize spectral efficiecy uder prescribed packet error rate (PE costrait while satisfyig a average trasmit power costrait. umerical results reveal that i wireless packet systems with limited packet legth, the proposed scheme provides substatial average spectral efficiecy gai whe compared to the o-adaptive case. The optimizatio framework preseted i this article would be useful for cross-layer optimizatio of physical layer i packet etworks. Idex Terms Adaptive modulatio ad codig, block fadig chaels, optimizatio, packet commuicatios. ITODUCTIO The demad for high data rate ad quality of service (Qo based services is icreasig i moder wireless commuicatio systems. However, wireless liks are subject to various physical impairmets such as chael fadig, which limits the performace of such systems. Lik adaptatio at the trasmitter, i particular adaptive modulatio ad codig (AMC, is a promisig approach towards high throughput ad power efficiet wireless commuicatios. Today, AMC schemes are already proposed for implemetatio i wireless systems such as HIPELA/, IEEE 80.a ad IEEE 80.6e stadards [], []. I order to optimize the wireless system performace, a lik adaptatio algorithm selects a suitable chael code ad modulatio costellatio, ad sets the trasmit power based o the time-varyig chael coditios []. Goldsmith ad Varaiya [] showed that the hao capacity of a flat-fadig chael ca be achieved by employig both power ad rate adaptatio. Moreover, i [] it is show that adaptatio of both these factors leads to a egligibly higher gai i capacity over a scheme with rate adaptatio aloe. It is oteworthy that to achieve the hao capacity, codig schemes have ubouded legth ad complexity, ad o delay costrait is assumed. I cotrast, practical systems are delay-limited ad must use fiite-legth codewords. I particular, for a practical system with bouded delay, better throughput is achieved cosiderig both power ad rate adaptatio i [5]. I the traditioal AMC-based lik adaptatio, system parameters are adapted o a symbol-by-symbol basis, to icrease spectral efficiecy while satisfyig a target bit error rate (BE performace metric [6][7]. However, i moder wireless packet etworks, trasmissio is performed o a frame by frame basis at the physical layer, where each frame cotais a fixed umber of symbols ad a variable umber of packets from the data lik layer []. O the other had, packet error rate (PE is a more relevat physical layer performace measure tha BE due to CC-based AQ mechaism i data packet trasmissio []. As show i [8], relatig BE ad PE is ot straight forward especially for coded trasmissios. Therefore, the authors i [8] used a approximate expressio for PE i order to desig the AMC scheme to meet directly the required PE. However, i [8] a simple costat power AMC scheme is cosidered ad the problem of optimal rate adaptatio is ot addressed. This paper follows the same approach. We derive the optimal lik adaptatio strategy, o a frame by frame basis, to meet directly the required target PE. We cosider joit power ad data rate adaptatio i the physical layer, aimig at improvemet of system spectral efficiecy i wireless packet etworks subject to a prescribed packet error rate costrait. Towards this ed, the optimizatio algorithm specifies the appropriate AMC mode switchig levels ad the correspodig power. umerical results reveal that i wireless packet systems with limited packet legth, the proposed scheme provides substatial average spectral efficiecy gai, whe compared to its o-adaptive couterpart. The remaider of this paper is orgaized as follows. ectio itroduces the system model, icludig a approximate expressio for the PE of AMC scheme. We drive the optimal solutios for power adaptatio ad optimum AMC mode switchig levels that maximizes system spectral efficiecy uder prescribed target PE ad average trasmit power costraits i ectio. umerical --0955-/07/$5.00 007 IEEE.
Iput Packets from higher layers Block Fadig AWG Chael h(k Output Packets Buffer AMC Power Cotrol (k + Coheret receiver Buffer h ˆ( k Feedback Chael Chael estimator h ˆ( k Fig.. ystem ad chael model results are provided i ectio... ystem Descriptio. YTEM MODEL As show i Fig., we cosider a wireless packet commuicatio lik betwee a sigle-atea trasmitter ad a sigle-atea receiver. It cosists of a AMC ad power cotrol module. Iput packets, arrived from higher layers of stack, are queued at a ifiite buffer, divided ito frames ad trasmitted over the wireless chael. At the trasmitter, AMC provides multiple trasmissio modes, where each mode is specified by a modulatio ad forward error correctio (FEC code pair. The trasmitter selects a AMC mode for trasmissio ad adapts trasmit power o a frame-by-frame basis based o the feedback chael state iformatio (CI from the receiver. We assume perfect chael estimates at the receiver ad perfect CI at the trasmitter. We also assume that coheret demodulatio ad maximum-likelihood decodig are used at the receiver. The decoded bit streams are coverted to packet structure ad the are pushed towards the upper layers of stack. Followig the approach i [8], at the physical layer frame by frame trasmissio is cosidered. Each frame cotais a fixed umber of symbols ( s ad a variable umber of packets ( p from the data lik layer. Each packet cotais a fixed umber of bits ( b, which iclude packet header, payload, ad cyclic redudacy check (CC bits. Applyig modulatio ad codig with rate (bits/symbol i mode, b bits of a packet are mapped ito a block of b / symbols. A frame comprises of multiple such symbol-blocks as well as c pilot symbols ad cotrol parts, as i HIPELA/ ad IEEE 80.a stadards []. I mode, the umber of symbols per frame is s = c + p b /, which idicates that p depeds o the chose AMC mode. I calculatio of spectral efficiecy of our system model, we igore the effect of the header ad CC bits of each packet. We also assume strog CC code so that packet error detectio usig CC bits is perfect... Chael Model ad AMC Modes We assume a wireless chael with statioary ad ergodic time-varyig real gai h with average chael power gai h =, ad additive white Gaussia oise with zero mea ad variace σ. The chael is assumed to follow a block fadig model, i.e., the gai remais ivariat durig a frame, but varies from frame to frame. This model is suitable for slowly-varyig fadig chaels [9]. We deote the average trasmit sigal power by. Trasmittig with costat power, the istataeous pre-adaptatio received sigal to oise ratio ( i trasmittig kth frame is ( k = ( h( k / σ. We deote the trasmit power durig trasmittig kth frame, which is a fuctio of (k, by ( (k. Thus the received post-adaptatio whe trasmittig the kth frame is ( k ( ( k /. By virtue of statioary assumptio of h(k, the distributio of (k is idepedet of k, ad we deote this distributio by p (. To simplify the otatio we will omit the frame idex k relative to ad (. I Fig., AMC is performed o a frame-by-frame basis by dividig the rage of the chael ito + o-overlappig cosecutive itervals, deoted by [, +, =0,,...,, where 0 = 0, + =,, ad is the umber of AMC modes. Wheever the CI fed back to the trasmitter falls withi the iterval [, +, the mode is chose, data is trasmitted with rate (bits/symbol ad power ( (watt. o data is set whe [ 0, correspodig to deep chael fades or the outage mode with rate 0 =0 (bits/symbol ad 0 ( = 0. I this paper, the trasmissio modes of AMC scheme are adopted from the HIPELA/ stadard []. These modes are costructed by covolutioally coded M -ary rectagular or square schemes. The ecoder cosists of a / rate mother code with geerator polyomial g = [ 7] ad subsequet pucturig. Table I presets the trasmissio modes of AMC scheme... PE Approximatio for AWG Chael I order to maximize the spectral efficiecy of system model i Fig., we eed a expressio for PE that is
ivertible ad differetiable i terms of the received. To this ed, for each mode, we first obtai the exact PE through Mote Carlo simulatios; the we use a fittig expressio to approximate PE. I order to approximate the PE of coded discrete-rate M- scheme over AWG chael, two fittig expressios, mi(, a exp( g ad g / ( + (a are suggested by [8] ad [0], respectively; where, a, g are costats that deped o the chael codig ad modulatio. I this paper, we use the followig expressio as a fuctio of post-adaptatio received ( / to approximate PE i mode (, 0 < Γ PE ( = ( ( a exp(-g, Γ where is the pre-adaptatio received, ( is the allocated power i mode, ad parameters, {a, g, Γ } are mode ad packet-size depedet costats. These parameters ca be obtaied by least square fittig the expressio of ( to the exact PE. Uder a simple costat power allocatio strategy with ( =, =,,...,, the PE expressio i ( reduces to that used i [8]; therefore we select the trasmissio modes ad the set of correspodig fittig parameters {a, g, Γ } similar to [8], as show i Table I. Accordig to such AMC modes, we ca compare our results with that reported i [8]. It is ecessary to ote that the fittig parameters i Table I, are also valid uder ay power allocatio strategy that is related to PE through (. TABLE I AMC Trasmissio Modes ad their Correspodig Fittig Parameters [8] Mode ( 5 6 Modulatio BPK QPK QPK 6-6- 6- Codig 9 rate 6 0.5.5.5.5 a 7.7 90.5 67.6 50. 5.99 5.5 g 7.99.998.688 0.66 0.756 0.0900 Γ (db -.5.09.97 7.70 0.9 5.978 I this table, the AMC trasmissio rates,, are specified uder assumptio that the ideal yquist pulses are used for data trasmissio.. POWE ADAPTATIO AD AMC AALYI I this sectio, we cosider our system model i two cases: ( cotiuous power ad discrete-rate adaptatio with a istataeous PE (IPE costrait ad ( discrete-rate adaptatio usig costat power with a IPE costrait. We illustrate the derivatio of optimal power adaptatio ad optimum AMC mode switchig levels that maximize spectral efficiecy uder a average trasmit power costrait. Assumig that there are always sufficiet packets available at the trasmitter buffer to be trasmitted, the spectral efficiecy of our system model is the average data rate per uit badwidth /W, where W [Hz] deotes the received sigal badwidth. The average spectral efficiecy of coded discrete-rate M- is the sum of the data rates associated with the idividual + regios, weighted by the probability that falls i the th regio [] + ηw = =. p ( d bits/sec/hz ( W = We also assume a average trasmit power costrait over AMC modes give by = + ( p ( d (.. Optimal Power ad ate Adaptatio We ow maximize spectral efficiecy as preseted i equatio ( subject to a target PE ad a average power costrait as i equatio (. Cosider the case of IPE costrait, so that PE ( = Pt, + ; =,...,, where P t deote the target PE. Usig (, we fid the followig expressio for power adaptatio i mode ( a = l(, + g Pt, Γ Uder the above power adaptatio strategy, the desired optimizatio problem ca be formulated as follows Maximize C C { } = : = :( + p ( d a + l(. p d g P ( t : Γ, =,,..., = + subject to where C coditio shows the average trasmit power costrait ad the last coditios esures that for each mode, the power is allocated based o (. The problem i (5 is a stadard costraied optimizatio problem, which we use the Karush-Kuh-Tucker (KKT coditios [] to determie its optimal solutio. I order to do so, we first costruct the Lagragia of (5 as L(,..., λ, β,..., β λ = g a l(. P t + = = + p ( d + p ( d + = β ( Γ where ( β,..., β, λ are the Lagragia multipliers. Usig KKT coditios, the optimal solutio (,,..., ad the correspodig Lagragia multipliers ( β,..., β, λ, must satisfy the followig coditios ( (5
L = λ β (,...,,λ,β,...,β a l(. g P Γ 0 β (, =,,..., 0, =,,..., Γ t +! With (6 we have L p( d = 0, =,,..., (,,...,,λ,β,β,...,β = 0, =,,..., l( a ( / P p t p ( λ + β = = 0, g l( a ( / p Pt p ( p ( + λ g l( a / P p ( t λ + β = 0, g = Usig (7, if > Γ, the β = 0. Followig (8, if > Γ the it satisfies the followig coditio l(a/pt = λ g g l( a g l( a = λ, =,..., g g ( As a result, the geeral form of optimal mode switchig levels ca be writte as l(a/pt = Max λ, Γ g g l( a g l( a = Max g g ( λ, Γ (6 (7 (8, =,..., (9 where the costat λ ca be foud umerically such that the AMC mode switchig levels i (9 satisfy the costrait C i (5, while the maximum average trasmit power is used. We refer to this sceario as adaptive power-ipe... Costat Power Adaptive ate AMC cheme A simple costat power AMC scheme is used i [8], assumig ( =, {,,..., }, ad satisfyig a target IPE costrait, P t. This meas that the istataeous PE is guarateed to be o greater tha P t for each chose AMC mode. Uder such assumptio, the AMC mode switchig levels { } are set to the miimum required to achieve P t, i.e. = / g.l( a. However, all available average power i ( is ot used by this scheme. A improved solutio is to set ( / = α, {,,..., }, ad fid the costat α such that equality coditio i ( is satisfied. Accordigly, we fid α = (0 p ( d To satisfy the costraits PE( Pt, ;, =,,...,, it is sufficiet to satisfy the PE costrait at each boudary poit. Assumig Γ ad substitutig ( / = α, i (, results i a = l, =,,..., g. P ( α t Therefore, ca be obtaied by the followig equatio l( a = ( g p ( d I this sceario, which we refer to as the costat power- IPE, if (,..., are obtaied by the above procedure, the fial mode switchig levels are selected as = Max, Γ, =,,...,. (. UMEICAL EULT I this sectio, we preset umerical results for spectral efficiecy, power adaptatio ad PE usig the solutios derived i the previous sectios. We use the PE approximatio parameters listed i Tables I, obtaied for packet legth b =080 bits. Varyig b would yield differet umerical results, however similar observatios are expected []. Accordig to IEEE 80.a stadard, a PE of 0 percet idicates a reasoable poit of operatio for packet services without delay costrait, whe packet legth is 500 byte []. However, based o approximate equatio PE ( BE p (if each bit iside the packet has the same BE ad bit-errors are ucorrelated, this equatio is exact, for packet legth of about 000 bits, this traslates to a PE requiremet of about 0.00 to 0.0. Therefore, i our experimets, we select the target PE, P t =0.00. Although our derivatios are for geeral fadig distributios, for the followig umerical results, a ayleigh fadig chael model has bee assumed, i.e. p ( = (/ exp( /, 0, where deote the average received. Uder such assumptio, the objective ad costraits fuctios i (5 are covex fuctios of { } variables; however, the problem i (5 is ot a covex optimizatio problem as defied i []. evertheless, umerical experimets show that the proposed solutio always coverges to a uique solutio resultig i a desired level of performace gai. The average spectral efficiecy for AMC scheme is depicted i Fig.. This plot shows that the optimal rate ad power adaptatio has a sigificat effect o system spectral efficiecy. I particular, if we compare power adaptive
scheme with the simple costat power scheme used i [8], we observe that employig the proposed power ad rate adaptatio algorithm leads to at least 0.5 b/s/hz spectral efficiecy gai. This shows a sigificat rate improvemet. For istace, i the HIPELA/ stadard, where the symbol rate is Msymbols/s [], this gai leads to a approximate 5. Mb/s icrease i trasmissio rate. Fig. also reveals that, the proposed costat power AMC scheme provides a higher spectral efficiecy i compariso with that suggested i [8], especially whe the average is low. Fig. demostrates the actual average PE correspodig to differet cases depicted i Fig., respectively. The pealty of spectral efficiecy i costat power-ipe sceario is predictable, because the PE adaptatio ability of the adaptive power IPE scheme is o loger available. I fact, for costat power-ipe sceario, the average achievable PE is much lower tha our target IPE as show i Fig.. From a practical poit of view, costat power AMC scheme simplifies the hardware complexity of the system, at the cost of a smaller spectral efficiecy. Fig. shows that the allocated trasmit power by adaptive power- IPE policy follows the iverse water-fillig patter with respect to istataeous withi each rate regio iterval. ACKOWLEDGEMET This work is supported by Ira Telecommuicatios esearch Ceter (ITC. 5. EFEECE [] Mobile WiMAX Part II: A Comparative Aalysis, WiMAX Forum, May, 006. [] A. Doufexi,. Armour, M. Butler, A. ix, D. Bull, J. McGeeha, ad P. Karlsso, A compariso of the HIPELA/ ad IEEE 80.a wireless LA stadards, IEEE Commu. Mag., vol. 0, pp. 7 80, May 00. [] M.. Alouii ad A. J. Goldsmith, Adaptive modulatio over akagami fadig chaels, Kluwer Joural o Wireless Commuicatios, Vol., o. -, pp. 9-, May 000. [] A. Goldsmith ad P. Varaiya, Capacity of Fadig Chaels with Chael ide Iformatio, IEEE Tras. Ifor. Theory, vol., o. 6, ov 997. [5]. Ahmed, M.A. Khojestapour ad.g. Baraiuk, Deleylimited Throughput Maximizatio for Fadig Chaels usig ate ad Power Cotrol, IEEE Globecom 00. [6]. T. Chug ad A. J. Goldsmith, Degrees of freedom i adaptive modulatio: A uified view, IEEE Tras. o Commuicatios, pp. 56 57, ept 00. [7] B.J. Choi ad L. Hazo, Optimum mode-switchig-assisted costat-power sigle- ad multicarrier adaptive modulatio, IEEE Tras. Vehicular Tech., vol. 5, o., pp. 56-560, 00. [8] Q. Liu,. Zhou, ad G. B. Giaakis, Cross-layer combiig of adaptive modulatio ad codig with trucated AQ over wireless liks, IEEE Tras. o Wireless Commuicatios, vol., o. 5, pp. 76-755, ept. 00. [9] E. Biglieri, G. Caire, ad G. Taricco, Limitig performace of block fadig chaels with multiple ateas, IEEE Tras. Iform. Theory, vol. 7, pp. 7 89, May 00. [0] K. B. og, ad. A. Mujtaba, O the code-diversity performace of bit-iterleaved coded OFDM i frequecyselective fadig chaels, i Proceedig of IEEE VTC, 00. [] J. ocedal ad. J. Wright, umerical Optimizatio, priger-verlag, ew York, 999. []. Boyd ad L. Vadeberghe, Covex Optimizatio, Cambridge Uiversity Press, Cambridge, Eglad, 00. Average pectral Efficiecy (bits/sec/hz Average Achievable PE.5.5.5.5 0.5 Adaptive Power - IPE Costat Power - IPE imple Costat Power AMC i [ 8]-IPE 0 0 5 0 5 0 5 Average (db 0-0 - Fig. pectral efficiecy for AMC schemes. Adaptive Power - IPE Costat Power - IPE imple Costat Power AMC i [ 8]-IPE 0 5 0 5 0 5 Average (db Fig.. Actual average PE s for AMC schemes. ormalized Power.5 0.5 Fig. Optimal Avarage =5 db 0 0 5 0 5 0 5 Istataeous (db ( for adaptive power-ipe AMC cheme.