adaptive turbo-coded hybrid-arq in ofdm systems over

Adaptive Turbo-Coded Hybrid-ARQ in OFDMSystems over Gaussian and Fading Channels

Kingsley Oteng-Amoako and Saeid Nooshabadi

Abstract— In this paper, an analytical approach for spectralefficiency maximization of coded wideband transmissions ispresented based on OFDM. The approach exploits Type-IIIHybrid-ARQ, enabling all sub-carriers to be employed incodeword transmission regardless of the sub-carrier conditions.The effects of imperfect sub-channel estimation are characterizedand compensated for during code rate and signal constellationoptimization. The results of the paper highlight that byindependently adapting the code rate and signal constellation toindividual OFDM sub-carriers based on an estimated sub-carrierCSI, the overall spectral efficiency of the system is maximized.

Index Terms - Hybrid-ARQ, AMC, adaptive, fading, diversitycombining and turbo codes.

I. I NTRODUCTION

In fading channels, an increasing method employed inwideband communication is Orthogonal-Frequency-Division-Multiplexing (OFDM) [1] [2]. OFDM in frequency-selectivechannels effectively presents a set of sub-carriers each with achannel impulse response corresponding to a flat fading chan-nel behavior in nature [2]. Hence, Forward-Error-Correction(FEC) codes employed in flat fading scenarios can be em-ployed in an OFDM system. Turbo codes as a FEC scheme,provide close to capacity performance through iterative de-coding of Recursive Systematic Convolutional (RSC) codes[3]. The Turbo encoder is a parallel concatenation of multiplestatistically independent RSC codes. The statistical indepen-dence between encoders, is provided through separation ofRSC encoders by random interleavers, an encoding structurethat enhances the overall performance of the iterative decodingscheme.

Hybrid Automatic-Repeat-reQuest (Hybrid-ARQ) combinesthe flexibility of an ARQ with the error correction capabilitiesof FEC to provide significant coding and energy gains bycombining multiple transmit attempts across a communicationchannel [4]. The combination of Turbo Hybrid-ARQ withOFDM, provides significant bandwidth at close to capac-ity rates of the channel. This paper proposes an AdaptiveModulation and Coding (AMC) approach with incrementalredundancy employed independently within each sub-carrierin order to maximize the spectral efficiency [5] [6] [7]. Thescheme proposes Type-III Hybrid-ARQ with OFDM in orderto utilize the complete set of sub-carriers.

In the proposed scheme, Type-III Hybrid-ARQ is alsoemployed within each sub-carrier in order to minimize the Bit

Manuscript received November 11, 2005; revised February 23, 2006.K. Oteng-Amoako and S. Nooshabadi are with the School of ElectricalEngineering and Telecommunications, University of New South Wales, NSW,2052, Australia (email: [email protected]; [email protected]).

Error Rate (BER) over multiple transmit attempts and thus en-able all sub-carriers to be used regardless of the Channel-State-Information (CSI) [8], [9]. Type-III Hybrid-ARQ combinesself-decodable transmissions into a single low rate codeword[10]. The aim of the proposed scheme of combining Type-III Hybrid-ARQ with OFDM across a given sub-carrierk, isto select on thet-th instantaneous transmit attemptRk,t andMk,t, for t = 0, 1, .., l such that the throughput of individualsub-carriersηk,t is maximized whilst not exceeding the target-BER. Given thet-th transmission in a sub-carrier of an OFDMscheme of a totalτ sub-carriers, the optimization by selecting(Rk,t, Mk,t) is a τ discrete maximization expressed as

[(R∗

0,0,M∗

0,0), ..., (R∗

τ−1,0, M∗

τ−1,0), ..., (R∗

0,t, M∗

0,t), ...,

(R∗

τ−1,t, M∗

τ−1,t), ..., (R∗

0,l, M∗

0,l), ..., (R∗

τ−1,l, M∗

τ−1,l)]

= arg max(Rk,1,Mk,1),...,(Rk,t,Mk,t)

ηk,t

(1)

whereR∗

k,tandM∗

k,tare the code rate and signal constellation

respectively of thet-th transmission of thek-th sub-carriergiven a system subject to the constraints of powerS and theBER Pφ.

The remainder of the paper is organized as follows. InSection II, the OFDM system description along with channelcapacity expressions are presented. In Section III, a Marko-vian based channel state quantization expression is given. InSection IV, utilizing the Markovian description the proposedadaptation algorithm is described. In Section V, simulationresults are presented. The conclusion is given in Section VI.

II. SYSTEM MODEL

The system model in Figure 1 consists of the Turbo codedOFDM transmitter and receiver. The data source feeds a binaryencoder. The blocks of the data are subsequently divided intoτ data blocks and encoded intoτ separate transmit blocks.

The t-th transmit block is initially encoded by a Cyclic-Redundancy-Check (CRC) code. The output of the CRC en-coder is fed into a Rate Compatible Punctured Turbo (RCPT)encoder consisting of parallel concatenated RSC codes sepa-rated by an interleaver. The resulting rate 1/3 code of lengthn, is buffered before being punctured [8], [11]–[14]. Thus theset of generated codewords for the set of sub-carriers,c canbe represented as a(τ × 1) matrix, c = [c0, c1, ..., cτ−1]

T .The resulting codewords are modulated based on either M-aryPSK or M-ary QAM, such that each transmitted symbol hasQ-bits. The OFDM scheme results in an orthogonal channelresponse unique to each sub-carrier that is applied to respective

JOURNAL OF COMMUNICATIONS SOFTWARE AND SYSTEMS, VOL. 2, NO. 1, MARCH 2006 3

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FESB

Typewritten Text

Original scientific paper

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C e

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c o d e

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G u

a r d

B a

n d

P r e

f i x

B u

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n c t

u r i n

g M

a t r

i x

S i g

n a l

C o

n s t

e l l a

t i o n

M a

p

OFDM Block

channel estimate

ACK/NAK feedback from RX

B i n

a r y

e n

c o d e r

P r e f i x r e m

o v a l

D F

T

S i g n a l d e m

o d u l a t i o n

T u r b o d e c o d e r

OFDM Block

C R

C d e c o d e r

B u f f e r

channel estimate

ACK/NAK feedback to TX

Fig. 1. Turbo coded OFDM system with parallel encoder and decoder structures

codeword blocks by passage of individual sub-carriers throughan Inverse Discrete Fourier Transform (IDFT). The IDFT ofthe encoded symbols on thek-th sub-carrier is given by

xk,q =1√Q

Q−1∑

n=0

cn,kej 2πqQ (2)

wherecn,k is the encoded bit,xk,q is theq-th bit on thek-thsub-carrier andQ is the bit size of the transmitted symbol.

A. Channel Estimation And Imperfect Channel State Informa-tion

In OFDM, pilot symbols are employed in transmitted framesin order to estimate the channel state in addition to determin-ing the carrier frequency offset. OFDM can employ a two-dimensional pilot-assisted modulation in order to determinethe channel state [15], [16]. Thus given the probability densityfunction of both an ideal CSI and the estimated CSI, the adap-tive scheme can be employed in an imperfect CSI scenario.

If the channel is assumed modeled by the classical Jakeschannel model, the normalized power spectrum is given as

h(f) =

{ 1πfd

1q

1−( ffd

)2

if |f | < fd

0 otherwise(3)

wherefd is the Doppler frequency. In estimating the channelat the receiver, Weiner filtering is employed

hk,(t=0) = Wp = RhpR−1pp p (4)

where Wp is the linear minimum mean square error(LMMSE), p is a vector of back-rotated observations atdifferent pilot positions,Rhp is the cross co-variance betweenthe estimated channel attenuationhk,t and the observationsp, andRpp is the auto-covariance matrix of observations. Ifthe channel remains constant, this estimate can be assumedaccurate and used for the detection of the symbols during thewhole information frame.

However, the channel is rapidly time varying in nature. Inorder to track the divergence of the channel away from theinitial estimate, the Per-Survivor-Processing (PSP) technique isexploited in this paper [17]. The divergence tracking algorithmis written as

hk,(t+1) = hk,(t) + βek,(t→(t+1))xk,t (5)

wherexk,t is the transmitted bit at a timet on the k-th sub-carrier,β is an adaptation parameter selected as a compromisebetween the speed of convergence and a stable estimate.The expressionek,(t→(t+1)) is estimated from the followingequation

ek,(t→(t+1)) = yk,t+1 − hTk,(t)

xk,(t+1) (6)

whereyk,t is the mean of the received bit, at a timet on thek-th sub-carrier.

Assuming that the channel estimate can be consideredcomplex Gaussian, the correlation coefficient is given by [18],

δ = I0

(

2πfdTd)

(7)

whereI0 is the zeroth order Bessel function andTd the abso-lute magnitude difference in time between channel estimationand the instance at which the codeword is transmitted.

Given the instantaneous and estimated sub-carriers SNRcharacteristic CSI,Jγ and J′

γ respectively, the averagethroughput obtained across all sub-carriersηt is defined as

ηt = EJγJ′

γ

{

1

τ

τ−1∑

i=0

ηi,t

}

(8)

As the estimated CSI,J′

γ , differs from that of the actual CSIdue to the error effects of an imprecise estimation, it leadsto adegradation in the obtainableηt. The empirical Mean SquareError (MSE) of an imperfect CSI on thek-th carrier is givenas

ek , E

{

1

τ

τ−1∑

i=0

∣

∣J ′

k− Jk

∣

∣

2

}

(9)

whereJk andJ ′

k are the instantaneous and estimated SNR ofthe k-th sub-carrier respectively, such thatJk = J ′

k−ek. Thus

provided that the effects of imperfect CSI can be estimatedto determineek, the adaptive scheme can accordingly selectparameters to counter the effects. From (9) and (7), thecorresponding error in a CSI estimate considering the effectsof delay and Doppler spread, is expressed asek = 2 − 2δ.Correspondingly, the target BER in the presence of imperfectCSI is given asP ′φ = Pφ

(

(1 − ek)Hk

)

[16].

4 JOURNAL OF COMMUNICATIONS SOFTWARE AND SYSTEMS, VOL. 2, NO. 1, MARCH 2006

B. Receiver

In the analysis, assume that the OFDM signal is designedsuch that the effects of inter-symbol interference and the Inter-Carrier Interference (ICI) can be ignored. In order to achievethis, the cyclic prefix (CP) appended to the beginning ofeach block of transmit symbols is assumed larger than thechannel memory. It is assumed that the channel state remainsconstant during the transmission of an OFDM symbol, withno contributing effects of Inter-Symbol Interference (ISI) orICI at the receiver. The received OFDM signal is demappedby passage through a Discrete Fourier transform (DFT) beforedemodulation. The output of the decoder is fed into a bufferfor codeword ordering. In the event of a decoding error, aNAK is generated and the codeword is buffered for later codecombining, otherwise a positive ACK is sent to the receiver.

Thus the received symbol after demodulation of the DFToutput and removal of the CP in thek-th sub-carrier can beexpressed as [1]

y(k,t) = H(k,t)x(k,t) + n(k,t) (10)

wherey(k,t) is the received signal ofQ bits on thek-th sub-carrier, H(k,t) is the time response,x(k,t) is the generatedOFDM signal andn(k,t) is the zero-mean Gaussian noise withE{|n|} = No. Assuming that the channel remains static duringthe transmission of a codeword, the ensemble received signalover τ sub-carriers can be expressed as

y = Hx + n (11)

such that thesub-carrier SNR characteristics, Jγ , of τ sub-carriers is given by

Jγ =Es

No

|H0|2 0 ... 00 |H1|2 ... 0...

.... . .

...0 0 ... |Hτ−1|2

.

whereEs is the single sided energy. The capacity overτ sub-carriers of the transmitter is then given by [18],

C = (Bw,k)

τ∑

k=0

log2

[

1 + Jk

]

(12)

whereBw,k is the available bandwidth per sub-carrier andJk

is the estimated CSI characteristic of a sub-carrier.

C. Channel State Quantization

In order to assign a signal constellation and code rate toeach sub-channel, the range of possible SNRs are quantizedinto multiple channel state intervals. A channel state interval,is characterized by having a mean SNR. In the analysis, therange of possible SNRs is divided intoκ groups, resulting inκchannel quantization intervals. The resultingi-th quantizationstate interval, denotedΓi refers to a channel with a mean SNR,J k

J k =

∫

∞

oJk|H |2ρ(Jk)dJk (13)

where ρ(J k) is the probability density function of the sub-carrier. ThusJ k is a random valued variable of probabil-ity density functionρ(J k) and cumulative density functionF (J k). The channel quantization intervalΓi is determinedprior to transmission and the range of quantization intervalsremains constant over the period of a transmitted codeword.Given that the channel consists ofτ sub-carriers, the completecharacterization of the channel is obtained by determiningΓi for each of theτ sub-carriers at timet. Thus at a giventime interval, the complete characterization of the channel isexpressed as

Γ = [Γ(0,0), ..., Γ(κ−1,0), ..., Γ(0,k),..., Γ(κ−1,k), ...,

Γ(0,τ−1), ..., Γ(κ−1,τ−1)](14)

whereΓ(i,k) is the sub-channel interval for thei-th quantiza-

tion state of thek-th sub-carrier. The system analysis assumesthat each sub-carrier is subject to the same range in SNRs,correspondingly the subscriptk can be dropped andΓi usedto refer to thei-th quantization interval of a given sub-channel.

III. O PTIMIZATION PROBLEM

The goal of the adaptation algorithm is to maximize thethroughput of the overall system by maximizing the numberof bits sent error-free on a sub-carrier during each transmitinstant. The use of OFDM allows the transmitted sequence tominimize the effects of fading across a wide-band channel andthus maximize the overall throughput of the system. Given thatthe throughput per sub-carrierη, the optimization problem canbe expressed as

∫

∞

Jk=0

η(Jk)ρ(Jk)dJk (15)

Given that the upper-bound in throughput can be expressed as,

max η(Jk) = R(Jk)m(Jk) (16)

hence, the transmit maximization problem is a joint-maximization ofR and 2m = M across the channel quan-tization state of a sub-carrierΓi. Thus the required meanthroughput is given by

max

[

∫

∞

Jk=0

R(Jk)m(Jk)ρ(Jk)dJk

]

(17)

Communication systems generally operate with an expectationthat received codeword will be lower bounded by a maximumamount of error, termed the target error rate of the system.Thus an additional criteria in the design of the system is thetarget BERPφ.

IV. CODE RATE AND SIGNAL CONSTELLATION

OPTIMIZATION

The goal of the following optimization approach is tooptimize the code rate and signalling constellation on eachsub-carrier separately, such that the spectral efficiency ismaximized on the first transmit attempt in the Hybrid-ARQscheme.

OTENG-AMOAKO AND NOOSHABADI: ADAPTIVE TURBO-CODED HYBRID-ARQ IN OFDM SYSTEMS 5

A. Constellation adaptation

In selecting the constellation parameter for the Turbo codedsystem, it is assumed that the code has close to ergodiccapacity performance. Thus,Pφ given an AWGN channel isgiven by Equation (18) [19], [20]

Pb ≤(

2 − 21−m/2)Neff,free

Kexp

(

− 1.5dmin

(2m − 1)γ

)

(18)

where Neff,free is the multiplicity of words generating aneffective free distance anddmin is the minimum distance.Hence the signal constellation required on thek sub-carrieris given as

m(γ) = log2

(

1 − 1.5dmin

log2

(

KPb2Neff

)γS(γ)

S

)

, (19)

The transmitter in the system, has the option of not sendingany codewords on a sub-carrier if the required code rateresults in a channel outage [21]. In our adaptation scheme,codewords are transmitted under all conditions and henceduring optimization, the signalling constellation is assumedlower bounded asM = 4 (m=2).

Type-III combining is employed on retransmit attempts ofcodewords in order to minimize the effects of any residualerrors. In the event that the transmitted codeword is receivedin error despite the parameter optimization approach, thereceiver code combines the multiple transmit attempts basedon a Type-III combining scheme. This approach of combiningAMC with code combining, allows all sub-carriers to be usedfor transmission thus enabling higher throughput levels thanpreviously suggested schemes [22].

B. Code Rate Adaptation

The problem of obtaining the optimal code rate to achieveergodic capacity is now addressed. Given the quantizationinterval Γi, a unique code rate and signal constellation pairis assigned in order to achieve the capacity.

Without loss of generality, the channel capacity can beshown to be a monotonically increasing function ofγ =Es/No subject to a constrainedRi and Mi as stated inEquation (12). The required SNR to achieve capacity for agiven signal constellationM is denoted asγcap. Hence, theinstantaneous SNRγs that guarantees capacity is given as,γs ≥ γcap.

Thus the requiredR bound to achieve capacity in a base-band and spread-spectrum scenarios can be generalized as [12]

1 ≥ R ≥ min

{

r

Υ log2 MB,

[

γcap

γs

r

log2 MB

]

}

(20)

V. SIMULATION RESULTS

In this section, numerical results are presented to verifythe scheme and analyze the performance of the optimizationalgorithm. The example presented considers an OFDM systembased on Turbo coding, the parameters of which are given inTable I. The co-channel interference between sub-carriersisapproximated as Gaussian noise, in addition to the AWGN

TABLE I

TURBO CODEDHYBRID-ARQ IN OFDM SYSTEM PARAMETERS

Encoder,(g1/g2) ([37/23]8

Code Rate,R 4/5,3/4,1/2,2/5,1/3,1

Codeword length 1020 bits

Modulation Scheme,M Gray coded QPSK,8-PSK,

16-QAM,64-QAM

No. of OFDM sub-carriers 80

Channel model 2-path Rayleigh, AWGN

Bandwidth, BW 1.25 MHz

Turbo decoder log-MAP

Max. iteration of Turbo decoder 8 iterations

Max no of retransmit attempts,Tr 4

TABLE II

CODE RATE AND SIGNAL CONSTELLATION CARDINALITIES FOR SYSTEM

ADAPTATION

Partitions Code Rate Modulation

2-state 2/3,1/3 QPSK, 16-QAM

4-state 2/3,1/2,2/5,1/3 QPSK, 8-PSK, 16-QAM, 64-QAM

6-state 4/5, 3/4, 2/3, 1/2 QPSK, 8-PSK, 16-QAM, 64-QAM

2/5, 1/3

present. The information length of received codewords is thusK = 4096 bits. The scheme employs Gary-coded M-ary PSKand M-ary QAM modulation with QPSK, 8-PSK, 16-QAMand 64-QAM constellations. The scheme is analyzed for theBER requirements ofPφ = 10−2 andPφ = 10−4 respectively.

The effect of increasing the number of available parameterson the performance of a discrete adaptive scheme is examinedin Figures 2(a) and 2(b). The cardinalities of the adaptive pa-rameters used in the adaptive algorithm ”2-state”, ”4-state” and”6-state” are detailed in Table II. The performance differencebetween the 2-state and 4-state cardinalities, are negligible. Toachieve any appreciable gain in the performane a mimimum of6-state cardinality is required. The 6-state cardinality achievesup to 1 bps/Hz gain in performance over the 2-state cardinalitysystem for the given system configuration over a range,Jk.It is observed that effectively increasing the cardinalityduringadaptation corresponds to the spectral efficiency approachingergodic capacity. The effect ofPφ on the performance of 2-state, 4-state and 6-state adaptive cardinalities is observed bycomparing Figures 2(a) and 2(b). The application of a lowerPφ results in a lower spectral efficiency in the adaptive scheme.The effect of constrainingPφ results in a lower performanceconstraint when the available parameters are limited, thus6-state cardinality responds more favorably toPφ constraint than2-state cardinality.

The performance of the adaptive scheme is consideredacross the set of 80 sub-carriers in Figures 3,4,5,6,7 and 8 forvarying code rate and signal constellation sets given the BERconstraints ofPφ = 10−2 andPφ = 10−4. It is observed thatthe overall spectral efficiency of the system increases acrossall sub-carriers as the set of available code rate and signalconstellations increase.

In addition, the spectral efficiency decreases with a lowering


0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

Cha

nnel

effi

cien

cy (

bps/

Hz)

SNR(dB)

AWGN ergodic capacity2 state4 state6 state

(a) Performance of the adaptive Turbo-coded Hybrid-ARQ in OFDM given discrete code ratesand signal constellations withPφ = 10−2

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

Cha

nnel

effi

cien

cy (

bps/

Hz)

SNR(dB)

AWGN ergodic capacity2 state4 state6 state

(b) Performance of the adaptive Turbo-coded Hybrid-ARQ in OFDM given discrete code ratesand signal constellations withPφ = 10−4

Fig. 2. Throughput performance of adaptive algorithm in a single carrier of an OFDM system in an AWGN channel


0

50

100

150

0

20

40

60

800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

transmission time frame (time)Multicarrier index

Spe

ctra

l Effi

cien

cy (

bps/

Hz)

Fig. 3. Throughput performance of 2-state parameter sets of(R, M) employed in adaptation across an 80 carrier OFDM symbol fortransmission across aRayleigh channel ofJγ = 0dB with Pφ = 10−2

0

50

100

150

0

20

40

60

800

0.1

0.2

0.3

0.4

0.5

0.6

0.7


Spe

ctra

l Effi

cien

cy (

bps/

Hz)



0

50

100

150

0

20

40

60

800

0.2

0.4

0.6

0.8

1


Spe

ctra

l Effi

cien

cy (

bps/

Hz)


0

50

100

150

0

20

40

60

800

0.2

0.4

0.6

0.8

1


Spe

ctra

l Effi

cien

cy (

bps/

Hz)



of the BER constraint applied to the system.

VI. CONCLUSION

In this paper, a throughput maximization algorithm given arequired BER constraint was presented for an OFDM systememploying Type-III Turbo Hybrid-ARQ at the receiver. Thethroughput of the system was evaluated for fading channelswith varying BER constraints. It is shown that an adaptiveapproach based on OFDM and Type-III Turbo Hybrid-ARQachieves near capacity for wideband channels.

Kingsley Oteng-Amoako received the B.Eng withhonors in electrical engineering from the Universityof Canterbury, Christchurch, New Zealand in 2000.In 2005, he completed a Ph.D. degree in electricalengineering at the University of New South Wales,Sydney, Australia. He is a recipient of the Faculty ofEngineering Scholarship and the School of ElectricalEngineering and Telecommunication Scholarship,amongst other awards.

His principal research interest include wirelessinformation and coding theory, with a focus on

forward error correction codes, hybrid automatic repeat request schemes,cross-layer wireless network design, modulation and the implementation ofcommunication systems.

Saeid Nooshabadi received the BSc. and MSc.degrees in physics and nuclear physics from AndhraUniversity, India, in 1982 and 1984, respectively, andthe MTech and PhD degrees in electrical engineeringfrom the India Institute of Technology, Delhi, India,in 1986 and 1992, respectively.

Currently, he is a Senior Lecturer in microelec-tronics and digital system design in the Schoolof Electrical Engineering and Telecommunications,University of New South Wales, Sydney, Australia.Prior to his current appointment, he held academic

positions at the Northern Territory University and the University of Tasmaniabetween 1993 to 2000. In 1992, he was a Research Scientist at the CADLaboratory, Indian Institute of Science, Bangalore, India, working on thedesign of VLSI chips for TV ghost cancellation in digital TV.In 1996and 1997, he was a Visiting Faculty and Researcher, at the Centre forVery High Speed Microelectronic Systems, Edith Cowan University, WesternAustralia, working on high performance GaAs circuits; and Curtin Universityof Technology, Western Australia, working on the design of high speed-highfrequency modems. His research interests include very high-speed integratedcircuit (VHSIC) and application-specified integrated circuit design for high-speed telecommunication and image processing systems, low-power design ofcircuits and systems, and low-power embedded systems.

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