Analytical Evaluation of Adaptive Codingfor Markov Models of Nakagami FadingJason D. Ellis, Michael A. Juang, and Michael B. Pursley

Clemson University

Abstract—Packet radio systems that communicate overchannels with slow fading experience changes in prop-agation loss from one packet to the next. In orderto maintain high throughput, it is necessary for thepacket radios to adapt their transmissions in responseto changes in the channel. Practical adaptive codingprotocols can respond to such changes by adjustingthe code rate to maximize the expected throughputfor each packet transmission. Performance evaluationsfor practical protocols typically require simulation ofboth the time-varying fading process and the adap-tive protocol. We employ recently developed finite-state Markov models of Nakagami-m fading to obtainanalytical evaluations of the throughput of an adaptivecoding protocol. In our approach, neither the fadingprocesses nor their Markov models are simulated.

I. INTRODUCTION

Methods are provided in [1] for the evaluation ofthe state probabilities and transition probabilities fora finite-state Markov chain model of a Nakagami-mfading process [2]. We employ a Markov-chain modelto develop an accurate analytical approximation tothe throughput of an adaptive coding protocol forpacket radio systems. Our method for performanceanalysis avoids the need for simulations of the pro-tocol and its derivation of the statistics it uses to adaptthe code; furthermore, neither the fading channel northe Markov chain is simulated in our approach.We evaluate the performance of a practical pro-

tocol [3] that relies on a count of symbol errorsto control the adaptation of the rate of the error-control code. The error count that is obtained fromthe previous packet is used to choose the code ratefor the next packet. In order to provide a benchmarkfor the performance of the practical protocol, weconsider a hypothetical ideal protocol that has perfect

This research was supported by the Office of Naval Researchunder Grant N00014-12-1-0060, the Army Research Office underGrant W911NF-11-1-0427, and the MIT Lincoln Laboratory.

knowledge of the state of the channel when the pre-vious packet was received and perfect knowledge ofall the parameters of the Markov chain model. Withthis knowledge, the protocol selects the code thatmaximizes the conditional expected throughput forthe next packet. The throughput of the hypotheticalprotocol gives an upper bound on the throughputof any practical protocol for which the code rateadaptation is based on channel state information thatwas obtained when the previous packet was sent.

II. MARKOV CHAIN MODELS

The envelope of the fading process is a nonnega-tive stationary random process denoted by V (t). Itsinstantaneous power is W (t)=V 2(t) and its averagepower is 2=E{V 2(t)}, which is normalized to unityfor our numerical results. Thus, we let =1 in theequations that we use from [1]. The intensity of thefading process is Y (t) = 10log10[V 2(t)]. The rangefor the Nakagami fading parameter m is 1

2 !m< .The relative intensity is Z(t) =Y (t)"y0, where y0is a reference level for the intensity whose value ischosen to give convenient relative intensity levels forthe states of the Markov chain.For the equal step-size J-state Markov chain mod-

els of [1], the relative intensity intervals are

Z = {(µj,µj+1] : 0! j!J"1},

where µ0 =" , µj= j for 1! j!J"1, and µJ = .The parameter is the step size in decibels (dB).Each state is represented by the relative intensitythat is /2 from the corresponding interval’s finiteendpoints. If j is the relative intensity that representsstate j, then j =µj+ 1

2 for 1! j! J"1 and 0=µ1" 1

2 . In effect, the relative intensity is quantized,and our model uses a sequence of quantized valuesto represent the fading process. The corresponding

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intervals for the intensity are

Y = {(µj+y0,µj+1+y0] : 0! j!J"1},

and the intensity for state j is j+y0. Thus, forboth the relative intensity and the intensity, the rep-resentative levels for the J states are dB apart. If=10( /10), 0=0, j= j for 1! j!J"1, and J= ,then the partition R = {( j, j+1] : 0! j!J"1} forthe relative power is equivalent to the partition Zfor the relative intensity and the partition Y for theintensity. The corresponding partition for the actualpower is W = {( j, j+1] : 0 ! j ! J"1}, wherej= jw0 for 0! j!J and w0=10(y0/10).The density function for the envelope of the

Nakagami-m fading process is [2]

fV (v) =2mmv2m"12m (m)

exp{"mv2/ 2}, v#0. (1)

The corresponding distribution function FV (v) isobtained by integrating the density function of (1),and the state probabilities for the Markov chain are

j = FV ($

j+1 )"FY ($

j ) (2)

for 0! j! J"1. The transition probabilities for theMarkov chain model can be determined from thejoint density function for samples V1 =V (t1) andV2=V (t2) of the fading envelope, which is [2]

fV,2(v1,v2) =4mm+1(v1v2)m

4(1" )( 2$ )m"1 (m)

exp!"m(v21+ v22)

2(1" )

"Im"1

#2mv1v2

$

2(1" )

$(3)

for v1 # 0 and v2 # 0. For numerical results, thecorrelation coefficient in (3) is = J20 (2 fdTs), whereJ0 is the Bessel function of the first kind of orderzero, fd is the maximum Doppler frequency, andTs= |t2"t1|, the time between samples of the inten-sity or envelope. The bivariate distribution functionFV,2(v1,v2) is obtained by numerical integration ofthe bivariate density.A time at which the Markov chain has an oppor-

tunity to change its state is referred to as a transitiontime. We consider transition times that are integermultiples of Ts, which is appropriate for any fadingprocess that is approximately constant over intervalsof length Ts. The transition probability q(k| j) is theprobability that the channel is in state k at time

(i+1)Ts given that it was in state j at time iTs. For anyvalues of m, k, and j, the state transition probabilitiescan be computed from

q(k| j) = P( k<Wi+1! k+1, j<Wi! j+1)/ j. (4)

where W! =W (t!) for each integer ! and

P( k<Wi+1! k+1, j<Wi! j+1)

= FV,2($

k+1,$

j+1)"FV,2($

k+1,$

j)

"FV,2($

k,$

j+1)+FV,2($

k,$

j). (5)

Because there is no restriction on the relationshipbetween k and j, the transition probability betweenany two states, adjacent or not, can be evaluated from(4) and (5).

III. AN ADAPTIVE CODING PROTOCOLSuppose a source must send a sequence of pack-

ets indexed by i= 1,2, . . . over the fading channelto a destination. The codes the source has avail-able are denoted by C(1), C(2), . . . , C(N), whichare in order of increasing rates r1 < r2 < · · ·< rN .For notational convenience, we define r0 = 0. Aspacket i is being demodulated and decoded, thedestination’s receiver makes measurements, estimatesparameters, or extracts statistics from the demodula-tor and decoder. From the measurements, parameterestimates, or statistics, the adaptive coding protocolderives information about the state of the channelwhen packet i was sent, which the protocol uses toselect the code for packet i+1.One of the practical adaptive coding protocols

introduced in [3] for frequency-hop transmission usesthe error count from packet i to select the code forpacket i+1. The error count for a packet that isdecoded correctly is the number of binary symbolerrors that occur from hard-decision demodulation ofthe individual symbols, even if the decoder uses softdecisions. Methods for obtaining the error count aredescribed in [3]. The code selection in the error-countprotocol is accomplished by applying an intervaltest to the error count. The adaptation parametersfor the interval test are the integers 0, 1, . . . , N , indecreasing order, where N is the number of codes.The extreme parameters are 0=Np, the number ofbinary code symbols per packet, and N =0, whichis the only lower bound on the number of binary

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symbol errors. The remaining adaptation parameters1, 2, . . . , N"1 are referred to as the intermediateparameters. The intervals for the error-count intervaltest are In = ( n, n"1] for 1! n!N"1 and IN =[ N , N"1]. According to the error-count protocol, ifthe binary symbol error count for the previous packetis and if % In, then code C(n) is chosen for thenext packet transmission. If the previous packet usedcode C(m) and the packet was not decoded correctly,then the error count cannot be determined and theprotocol chooses for the next packet the code thathas rate max{rm"1,r1}Let s(n|k) denote the average throughput that is

achieved when a packet using code C(n) is sentwhile the channel is in state k. Let Pc(n|k) denotethe probability the packet is decoded correctly whencode C(n) is used and the channel is in state k. IfKn is the number of information bits carried by apacket that uses code C(n), then s(n|k)=KnPc(n|k).For our numerical results, we use five turbo productcodes [6] that provide K1 = 968, K2 = 1331, K3 =2028, K4 = 2704, and K5 = 3249 information bitsper block of 4096 binary code symbols; therefore,r1 = 0.236, r2 = 0.325, r3 = 0.495, r4 = 0.660 andr5 = 0.793 are the approximate rates for the fivecodes. Bit-interleaved coded modulation is employed,S-random interleaving [4] is applied to the codesymbols prior to modulation, and the log-likelihoodbit metric [5] is used for soft-decision decoding. Theextreme parameters are fixed at 0=4096 and 5=0.

IV. PERFORMANCE ANALYSIS

In typical performance evaluations of adaptivetransmission protocols, simulations are required forboth the fading channel and the operation of the pro-tocol, including the procedure by which the protocolextracts whatever information it requires about thestate of the channel. The performance results of [3]are based on simulations of the adaptive coding pro-tocol; however, the Markov chain model is simulatedrather than the fading channel. Although simulationsof the Markov chain model are considerably simplerthan simulations of the fading process, it is desirableto replace simulations by analyses whenever possible.In this section, we provide analytical results thatavoid the need for simulations of the error-count pro-

tocol, the fading process, or the Markov chain modelfor the fading channel. Our performance results aregiven in terms of the average throughput. For theerror-count protocol, the average throughput for theMarkov chain model of the fading channel is

S̄ec =N

n=1

J"1

k=0s(n|k)

J"1

j=0Q(n| j)q(k| j) j, (6)

where Q(n| j) is the probability that code C(n) ischosen for the next packet given that the channelstate was j for the previous packet.We illustrate our analytical method for QPSK

modulation, but the extension to QAM is straightfor-ward. A packet consists of Np binary code symbolsthat are represented by Np/2 QPSK symbols. LetE j denote the received energy per QPSK symbolwhen the channel state is j and let N0 be the one-sided thermal noise density. For channel state j,the probability of error for the hard decision thatis made for a given binary symbol in the packetis p j =Q(

%E j/N0 ), where Q is the complementary

standard Gaussian distribution function. The QPSKsymbol-energy to noise-density ratio in dB for statej is QSENR j=10log10(E j/N0).Our analytical approximation of S̄ec is obtained by

approximating the probabilities Q(n| j) according to

Q(n| j) &!%In

#Np

!

$p!j (1"p j)Np"!. (7)

Although the actual number of binary symbol errorsfor a packet that is sent when the channel is in state jhas the binomial distribution with parameters Np andp j, the receiver can determine the number of binarysymbol errors only for packets that are decodedcorrectly; in addition, the choice of the code for thenext packet is not determined by an error count ifthe previous packet did not decode correctly. Thus,there are two reasons that the binomial distribution isonly an approximation. First, the conditional distri-bution of a packet’s error count given that the packetdecoded correctly is not binomial [7]. Second, if thepacket is not decoded correctly, then the interval testis not employed in the determination of the codefor the next packet; instead, the protocol changesto the code of the next lower rate, if there is one.Nevertheless, we demonstrate that for purposes ofcalculating the average throughput of the protocol,

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(7) is a good approximation, especially if the packeterror probability is low.For our illustration of the application of (6) and

(7), we make the common assumption for slow fadingthat the fade level is constant over the duration of apacket. The QPSK symbol-energy to noise-densityratio in dB is QSENR= 10log10(E/N0), where Eis the received energy per QPSK symbol and N0 isthe one-sided thermal noise density. The value of Ein the absence of fading is denoted by E', and thecorresponding QPSK symbol-energy to noise-densityratio in dB is QSENR' =10log10(E'/N0). Thus, inthe presence of fading, QSENR=QSENR'+Y , whereY is a sample of the intensity Y (t) of the fadingprocess. As described in Section II, the jth stateof the Markov chain model for the fading processis represented by relative intensity j, which meansthat the signal-to-noise ratio for state j is QSENR j=QSENR'+ j.As a performance benchmark for our practical

adaptive coding protocol, we consider the ideal pro-tocol from [3] that has perfect previous-state infor-mation (PPSI) and knows all the parameters of theMarkov chain model. Conditioned on its knowledgeof the state that the channel was in when the previouspacket was sent, the PPSI protocol chooses the codethat maximizes the conditional expected throughput;therefore, the average throughput for the hypotheticalPPSI protocol is

S̄ps =J"1

j=0jmax{

K"1

k=0s(n|k)q(k| j) : 1!n!N}. (8)

Because the hypothetical PPSI protocol knowsexactly which state the channel was in when the pre-vious packet was sent and maximizes the conditionalexpected throughput, then S̄ps is an upper bound onthe average throughput achieved by any protocol,including our error-count protocol, that measures thechannel, estimates channel parameters, or extractsstatistics during the reception of the previous packetin order to decide what code to use for the nextpacket.

V. PERFORMANCE RESULTSWe demonstrate the accuracy of the analysis pre-

sented in Section IV by comparing the throughputobtained from the analytical results in (6) and (7)

with the throughput obtained from a simulation ofthe adaptive coding protocol and the Markov chainmodel of the fading channel that is described inSection II. For this demonstration, we chose m= 1for the Nakagami-m channel, but the conclusionsthat we draw from the comparisons of the analyticaland simulation results are the same for other valuesof m. Nakagami-m fading for m=1 corresponds toRayleigh fading, which is the most severe fading thatis encountered in typical outdoor wireless commu-nication systems. Channels with m< 1 (worse thanRayleigh fading) have been observed, but only rarely.For m= 1, the parameters of the Markov chain

model are as follows: The reference level for theintensity is y0 ="17 dB, the step size is = 2 dB,and the number of states is J= 12. The parameterTs defined in Section II is the amount of timefrom the start of one packet transmission to thestart of the next packet transmission. Results arepresented for fdTs=0.05, which represents relativelyfast fading, and fdTs = 0.005, which gives slowerfading. The intermediate parameters that are used forcode adaptation are 1 = 763, 2 = 508, 3 = 299,and 4 = 172. For these values of the channel andprotocol parameters, the throughput curves obtainedfrom analysis are compared with simulation resultsin Figs. 1 and 2. Also shown in each figure is thethroughput curve for the hypothetical PPSI protocol.Both figures demonstrate good agreement betweenanalysis and simulation.Even for the relatively fast fading used for Fig. 1,

the throughput levels obtained from simulation of thepractical adaptive coding protocol are only approxi-mately 2.9% less than the throughput levels achievedby the hypothetical PPSI protocol, which establishesthat the error count alone is sufficient for good perfor-mance of an adaptive coding protocol. Measurementsor estimates of fade levels or other channel parame-ters are not required. The throughput levels of Fig. 1that are obtained by analysis are only approximately1.8% greater than those obtained by simulation ofthe channel and the adaptive coding protocol, andthey are approximately 1.1% less than those for thehypothetical PPSI protocol. The source of error in theapproximation is the use of the binomial distributioneven when the previous packet fails to decode and the

4

0

500

1000

1500

2000

2500

3000

3500

-10 -5 0 5 10 15 20 25

PPSIAnalyticalSimulation

Thro

ughp

ut

QSENR* (dB)

PPSIAnalytical

Simulation

Fig. 1. Throughput for fdTs = 0.05 (relatively fast fading).

protocol is not able to determine the error count. Forslower fading, the packet error probability is smallerso the error in the analytical approximation shouldbe even less than in Fig. 1. Indeed this is what wesee from Fig. 2.

VI. CONCLUSIONWe have developed an analytical approximation for

the throughput that is achieved by the error-countprotocol for adaptive coding of transmissions overa fading channel that is modeled by a finite stateMarkov chain. Comparisons of the analytical resultswith the results obtained from simulations of the pro-tocol and the channel show good agreement betweenanalysis and simulation. We also demonstrated thatthe performance of the low-complexity error-countprotocol is nearly as good as the performance of ahypothetical protocol that is given perfect channelstate information for the previous packet.

0

500

1000

1500

2000

2500

3000

3500

-10 -5 0 5 10 15 20 25

PPSIAnalyticalSimulation

Thro

ughp

ut

QSENR* (dB)

Simulation

AnalyticalPPSI

Fig. 2. Throughput for fdTs = 0.005 (slower fading).

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[4] S. Dolinar and D. Divsalar, “Weight distributions for turbocodes using random and nonrandom permutations,” JPLTDA Progress Report 42–122, pp. 56–65, August 1995.

[5] S. Le Goff, A. Glaviuex, and C. Berrou, “Turbo-codes andhigh spectral efficiency modulation,” Proceedings of the1994 IEEE International Conference on Communications,pp. 645–649, 1994.

[6] Advanced Hardware Architectures, Inc., Product Specifica-tion for AHA4501Astro 36 Mbits/sec Turbo Product CodeEncoder/Decoder. Available: http://www.aha.com

[7] J. M. Frye and M. B. Pursley, “Simplified methods forperformance evaluations of adaptive transmission proto-cols,” Proceedings of the 2009 International Symposiumon Communication Theory and Applications, July 2009.

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