CRC-Free Hybrid ARQ System Using Turbo Product Codes

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI10.1109/TCOMM.2014.2366753, IEEE Transactions on Communications

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CRC-Free Hybrid ARQ System using TurboProduct Codes

H. Mukhtar, Student Member, IEEE, A. Al-Dweik, Senior Member, IEEE,and M. Al-Mualla, Senior Member, IEEE

Abstract—This paper presents a hybrid automatic repeatrequest (HARQ) system using turbo product codes (TPC). Theinherent word-error detection capability of TPC is exploited toreplace the conventional cyclic redundancy check (CRC) used forpacket error detection in conventional HARQ systems. Therefore,the TPC is used for joint bit error correction and packet errordetection. Consequently, the HARQ system efficiency is improvedby increasing the system throughput when short packets aretransmitted, or by reducing the computational complexity/delaywhen the packets transmitted are long. Monte Carlo simulationresults reveal that the CRC-free TPC-HARQ system consis-tently provides equivalent or higher throughput than CRC-basedHARQ systems. Moreover, numerical results show that the TPCself-detection has lower computational complexity than CRCdetection, especially for TPC with high code rates. In particularscenarios, the relative complexity of the self-detection approachwith respect to popular CRC techniques is about 0.3%.

Index Terms—Hybrid automatic repeat request, turbo productcodes, error detection, iterative decoding, complexity.

I. INTRODUCTION

Most modern communication systems such as LTE [1] andWiMAX [2] employ a combination of forward error correction(FEC) and automatic repeat request (ARQ) to provide therequired quality of service (QoS) for various applications.When ARQ is combined with FEC codes, the compositesystem is commonly referred to as hybrid ARQ (HARQ). Inconventional HARQ systems, the error correction is performedin two phases. In the first phase, the FEC codes are appliedto correct the bit errors in the received packets. In the secondphase, the receiver verifies if the FEC process was successful,and then sends a message to the transmitter to retransmit thepacket if the FEC process was unsuccessful; otherwise, themessage informs the transmitter to send a new packet. Ingeneral, most of the work reported in the literature performsthe bit error correction and packet error detection (PED) astwo independent processes where the FEC is implemented atthe Physical Layer (PHY) while the packet error detection isperformed at the Data Link Control (DLC) layer [3], [4].

In the literature, the performance of HARQ systems hasbeen evaluated using various FEC codes such as low densityparity check codes (LDPC) [5]-[6], turbo product codes (TPC)[4], convolutional codes [7] and turbo convolutional codes [8].The PED is usually performed using cyclic redundancy check

H. Mukhtar, A. Al-Dweik and M. Al-Mualla are with the Department ofElectrical and Computer Engineering, Khalifa University, Abu Dhabi, UAE,e-mail: {husameldin.mukhtar, dweik, almualla}@kustar.ac.ae).

A. Al-Dweik is also with the School of Engineering, University of Guelph,Guelph, ON, Canada, e-mail: [email protected].

(CRC) codes at the DLC layer [3], [4]. Because the FEC andPED are implemented at two different layers, the interactionbetween the two FEC and PED operations is considered as across-layer cooperation [3].

The CRC operation can be realized using different softwareand hardware implementations. However, speed requirementsusually make software schemes impractical and hence, ded-icated hardware is required [9], [10]. The generic hardwareimplementation of the CRC process is based on low com-plexity linear feedback shift register (LFSR) that performs thepolynomial division process of the serial data input. In thepresence of wide data buses, the serial computation can beextended to parallel versions that process large number of bitssimultaneously at the expense of additional hardware com-plexity [11], [12]. Moreover, the energy consumed by CRCcircuits becomes non-negligible for systems with continuousand high data rate transmission [13].

In the literature, the problem of designing CRC-free errordetection schemes has received noticeable attention with theaim of reducing the adverse effects of the CRC process onthe computational complexity, delay, throughput and energyconsumption. For example, Coulton et al. [14] proposed asimple PED by comparing the ratio of the standard deviation(STD) to the mean of the soft information energy at the outputof the decoder. However, the results given in [14] indicatethat a reliable performance requires a large sample space tocompute the mean and STD accurately. Moreover, the systemrequires a substantial number of operations to compute theenergy, and then the mean and STD of the soft information.Buckley and Wicker [15] proposed to use neural networksfor CRC-free error detection. However, training the neuralnetwork requires very large number of samples, which mightcause severe performance degradation when the channel istime variant.

In [16], Zhai and Fair proposed an error detection criterionby continuously monitoring log-likelihood ratio (LLR) of thesoft information at the decoder output. However, the systemperformance depends on signal-to-noise ratio (SNR), framesize and code rate. Moreover, computing the LLR requiresaccurate knowledge of the channel statistics, which are hardto compute and track. Fricke et al. [17] proposed an errordetection technique using the word and bit error probabilities.However, this technique is unreliable and usually causesthroughput inflation.

As it can be noted from the above discussion, the perfor-mance of CRC-free techniques is usually limited by the properselection of certain thresholds, which are dependent on channel



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statistics, codeword size, code rate and sample space. Theperformance of such techniques has been evaluated for HARQsystems with various FEC codes such as LDPC, convolutional,and turbo codes. However, to the best of our knowledge,there is no work reported in the literature that considersCRC-free HARQ systems with TPC. TPC are powerful FECcodes with high coding gain, reasonable complexity, andsupport wide range of codeword sizes and code rates. TPCare currently included in the IEEE-802.16 for fixed and mobilebroadband wireless access systems [18] and the IEEE-1901 forbroadband powerline networks [19]. TPC has low complexityconvergence detection characteristics that are commonly usedfor aborting the iterative decoding process [20]. However,such characteristics were never used in the context of HARQsystems.

In this work, we consider a TPC-based HARQ scheme(TPC-HARQ) where the error self-detection capability of TPCis exploited. The TPC inherent error detection capability isused to provide CRC-free PED, which can have a significantimpact on the overall system performance. The performanceof the CRC-free system is evaluated in terms of complexityand throughput.

The complexity analysis demonstrated that a significantcomplexity reduction can be achieved using the proposedCRC-free system. However, the achieved complexity reductionis proportional to the TPC code rate and number of CRC bits.In addition to the complexity reduction, the CRC-free systemoffers throughput enhancement because the CRC bits can bereplaced by other redundant bits to support the error correctionprocess, which can be observed at low SNR. It is worthnoting that the throughput enhancement is more apparent forshort TPC where the number of CRC bits is non-negligiblecompared to the codeword size.

The rest of the paper is organized as follows. SectionII describes TPC-HARQ system model. The complexity ofthe CRC-based and CRC-free error detection techniques areanalyzed and compared in Section III. The system throughputis defined in Section IV. In Section V, the performance of theTPC-HARQ with self-detection is evaluated through extensiveMonte Carlo simulation and compared to CRC-based TPC-HARQ. Finally, conclusions are provided in Section VI.

II. TPC HARQ SYSTEM MODEL

TPC are two-dimensional linear block codes constructed byserially concatenating two linear block codes Ci (i = 1, 2).The elementary code Ci has the parameters (ni, ki, d

(i)min)

which describe the codeword length, number of informationbits, and minimum Hamming distance, respectively [20]. Tobuild a product code, k1 × k2 information bits are placedin a matrix of k1 rows and k2 columns. The k1 rows areencoded by code C1 and a matrix of size k1 × n1 bitsis generated. Then, the n1 columns are encoded by theC2 code and a two-dimensional codeword of size n2 × n1bits is obtained. The parameters of the product code C are(n1× n2, k1× k2, d(1)min× d

(2)min). Without loss of generality, we

consider in this paper squared TPC where n1 = n2 , n,k1 = k2 , k and d(1)min = d

(2)min , dmin. Therefore, the product

code is denoted as (n, k, dmin)2.

TPC are powerful FEC codes that can provide high codinggain. Nevertheless, the complexity of TPC decoders can bevery high when maximum likelihood decoding (MLD) isused. Therefore, sub-optimum iterative decoding methods arealternatively used to reduce the complexity while providingsatisfactory performance [21].

Consider an information data block d = [d1, d2,· · · , dκ]which is applied to a CRC encoder where lcrc bits areappended to d for error detection purposes at the re-ceiver side. The CRC encoder output can be written asd= [d1, d2, · · · , dκ, c1, c2, · · · , clcrc ], which is then applied toa TPC encoder that appends lp = n2−k2 bits to d as describedin [21] to form a TPC codeword matrix C. Therefore, the k2

bits at the input of the TPC encoder comprise of κ informationbits and lcrc CRC bits, and hence k2 = κ + lcrc. On theother hand, when TPC self-detection is used, the CRC bitsare replaced by information bits such that κ = k2.

In this work, we consider an HARQ system where eachtransmitted packet is a TPC codeword with size n2 , N =k2 + lp. The TPC encoder output is interleaved to decorrelatethe channel fading effects. The interleaved bits are thenmodulated using binary phase shift keying (BPSK) modulationand transmitted through a Rayleigh fading channel. Afterdeinterleaving, the received packet during the ith transmissionround can be expressed as

R(i) = F(i) ◦U + W(i) (1)

where F is the channel matrix, U is the transmitted packet,the symbol ‘◦’ denotes the Hadamard product and W is theadditive white Gaussian noise (AWGN). Each of the matricesF, U, and W in (1) consists of n × n elements denoted asfx,y , ux,y and wx,y where x and y denote the row and columnindices, respectively. The channel and AWGN componentsare independent and identically distributed (iid) zero-meancomplex Gaussian random variables with variance σ2

f and σ2w,

respectively.At the receiver side, the received packet is decoded and

checked for errors. If the packet is error-free, an acknowl-edgment (ACK) is sent to the transmitter to proceed with thetransmission of the next packet. Otherwise, the soft version ofthe erroneous packet is stored and a negative acknowledgment(NACK) is sent to instruct the transmitter to retransmit theerroneous packet. Once a packet is retransmitted, the newreceived version and the stored versions of that packet canbe combined using maximal ratio combining (MRC).

It is worth noting that conventional MRC is not optimalin HARQ systems because it ignores the fact that one ofthe combined packets is severely affected by the channelconditions [22]. However, the results reported in [22] showthat the optimal combiner outperforms the MRC only for veryshort packet lengths, less than 10 symbols per packet, or ifthe SNR changes substantially between different transmissionrounds. Otherwise, the MRC and optimal combiner provideroughly the same throughput. Practically speaking, a typicalpacket length is much larger than 10 symbols, and extremelyrapid SNR changes do not usually occur even at high Dopplervalues. Consequently, we adopt MRC to implement the packetcombining.



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The output of the combiner after the `th ARQ round is equalto

RC =∑i=1

A(i) ◦R(i) (2)

where A is the MRC weights matrix, a(i)x,y =(

f(i)x,y

)∗ [∑`i=1

∣∣∣f (i)x,y∣∣∣2]−1 and (.)∗ denotes the complex

conjugation process [23]. The combiner output RC is feddirectly to the TPC decoder for soft input soft output (SISO)decoding [21]. If hard input hard output (HIHO) decoding isdesired, the matrix RC is converted to a binary matrix B thatis fed to the HIHO decoder, where B = 0.5 (sign [RC] + 1)and sign(.) is the signum function.

The TPC decoder takes as input the matrix RC or B, andperforms a series of soft/hard decoding iterations to ultimatelyproduce the decoded binary n × n matrix D. In the iterativedecoding of TPC, all rows of the code matrix are decodedsequentially followed by column decoding. A full iterationcorresponds to the decoding of all set of rows and columnswhile a half iteration corresponds to the decoding of eitherall the rows or all columns. All row/column component code-words are decoded independently. The row/column decodingin every iteration is performed using MLD when HIHO isconsidered, and it is performed as described in [21] whenSISO is considered. The decoding process is terminated if themaximum number of iterations is reached, or if all rows andcolumns are valid codewords of their respective elementarycodes. In the latter case, the decoding process is declaredsuccessful. Consequently, error detection after the last columndecoding process can be performed by checking the syndromesof the first k rows in D. If all k syndromes are equal tozero, then the TPC packet is declared error-free; otherwise,the packet is declared erroneous. There is no need to checkthe syndromes of the last n−k rows because they only containparity bits.

In general, the last column decoding process guarantees thatall columns in the matrix are valid codewords. However, if therow/column decoding process does not necessarily produce avalid codeword [20], then the last two half iterations shouldbe checked.

The transmission process is repeated until the packet isdeclared error-free, or the maximum number of ARQ roundsL is reached after which the erroneous packet is dropped.

III. COMPLEXITY ANALYSIS

A. Complexity Analysis of TPC Self Detection

In this paper, extended Bose-Chandhuri-Hocquenghen(eBCH) is used as the elementary code to produce the TPCcode matrix. The eBCH codes are cyclic codes which can bedecoded using efficient algebraic decoding algorithms such asthe Berlekamp-Massey algorithm [23], [24], [25].

Let dx = [dx,1, dx,2, · · · , dx,n] refer to the xth row in Dwhere x = 1, 2, · · · , n. Therefore, assuming the last decodingprocess is column-wise, then dx = cx + ex, where cx and exare the corresponding rows in the transmitted binary matrixC and post-decoding error pattern matrix E, respectively.

For error detection, the syndromes of the rows in D can beobtained using:

• matrix multiplication S = DH> where H is (n−k)×nparity check matrix for the used eBCH component codeand {.}> is the matrix transpose. Each row sx in Scorresponds to the syndrome of dx; or

• polynomial division of dx(X) by g(X) where dx(X)is the polynomial representation of dx and g(X) isan (n − k)-order generator polynomial. The syndromepolynomial sx(X) is the remainder of the polynomial

division sx(X) = Rem

[dx(X)

g(X)

]. The coefficients of

sx(X) correspond to the binary vector sx.

To evaluate the computational complexity of TPC self-detection and CRC detection, the number of operations carriedout by each technique is counted. For TPC error self-detection,the number of operations NS depends on the used eBCH codeparameters n and k. Moreover, NS varies randomly based onthe index of the first row that has an error. Because the rowsyndromes sx are computed sequentially, once a non-zero rowsyndrome is found the self-detection process is aborted and thereceived matrix is declared erroneous.

Binary matrix multiplication can be implemented using onlyexclusive-OR (XOR) operations [26, p. 340]. Therefore, thenumber of XOR operations required to perform TPC self-detection using matrix multiplication is given by

NmtxS = x

[n−k∑x=1

(‖hx‖1 − 1)

](3)

where ‖hx‖1 is the Manhattan norm of the xth row in H andx is the index of the row in which the first error is discovered,x ∈ {1, 2, · · · , k}. If no error is detected until the kth row,the self-detection process is aborted and the TPC packet isdeclared error-free.

Alternatively, TPC self-detection can be performed usingpolynomial division which is implemented using a low com-plexity (n−k)-stage LFSR. As reported in [10], polynomial di-vision requires k iterations/shifts per row. Moreover, each shifthas an average of (2m−1)

2 XOR operations where m = ||g||1and g is the vector representation of g(X). Therefore, thenumber of XOR operations required to perform one TPC self-detection using LFSR is

N lfsrS = 1

2 (2m− 1)kx. (4)

As it can be noted from (3) and (4), the complexity of theTPC self-detection is a random variable. Therefore, expressing(3) and (4) numerically can be achieved using the averagevalue of x, which requires the knowledge of the distribution ofx. In TPC, the errors that cannot be corrected after performinga sufficient number of iterations are the closed chains of errors[20]. The location, number and size of such chains are allrandom variables that depend on the TPC code used and theSNR. Consequently, a general expression for the distributionof x is not easily obtained, and hence, we use upper and lower



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bounds to indicate the system complexity, wheren−k∑x=1

(‖hx‖1 − 1) ≤ NmtxS ≤ k

[n−k∑x=1

(‖hx‖1 − 1)

](5)

and

12 (2m− 1)k ≤ N lfsr

S ≤ 12 (2m− 1)k2. (6)

Fig. 1 compares NmtxS with N lfsr

S for all possible values of kand g(X) as tabulated in [23, p. 466] for n = 128 and 64.The figure shows that the LFSR implementation has a slightadvantage over matrix multiplication. Therefore, we adopt theLFSR implementation for TPC self-detection. In the figure,UB and LB denote the upper and lower bound, respectively.

101

102

102

103

104

105

k parameter of eBCH code

NumberofXOR

Operations,

Nmtx

SandN

lfsr

S

n=128 − mtxn=128 − lfsrn=64 − mtxn=64 − lfsr

UB

LB

Fig. 1: Upper and lower bounds on the complexity of theTPC self-detection process for the eBCH(128, 120, 4)2 andeBCH(64, 57, 4)2.

It is worth noting that matrix multiplication method hashigher computational complexity as compared to the serialimplementation of the LFSR, but it may have lower compu-tational delay due to the fact that matrix multiplication cannaturally achieve high level of parallelism. In the literature, thecomputational delay inherent in serial LFSR implementationcan be reduced by using parallel architectures at the cost ofadditional complexity [11].

B. Complexity Analysis of CRC-Based Systems

Using CRC codes that comprise of 16 to 32 bits is gener-ally sufficient to provide adequate PED accuracy for variousapplications. As reported in [27], the misdetection probabilityfor a CRC code that consists of lcrc bits approaches 2−lcrc overbinary symmetric channels (BSC) with large error probabil-ities. Therefore, the impact of the CRC bits on the system

throughput might be negligible for packets with large numberof bits. However, the packet size is an important optimizationparameter that depends on the channel conditions [7]. In burstytraffic conditions, transmitting small packet sizes is necessaryto maximize the system throughput and energy efficiency.Consequently, the value of lcrc becomes the dominant factorthat determines the system efficiency [7].

On the other hand, the CRC-based TPC-HARQ systemrequires a different number of operations for error detection,denoted as NC. CRC encoding and decoding are also im-plemented using LFSR. The degree of the CRC generatorpolynomial g(X) is lcrc; therefore, an lcrc-stage LFSR is usedto implement CRC encoding or decoding. Polynomial divisionfor CRC detection requires κ = k2 − lcrc shifts. Moreover,each shift has an average of (2m − 1)/2 XOR operationswhere m = ||g||1 and g is the vector representation of g(X).Therefore, the total NC including CRC encoding and decodingoperations is given by

NC = (2m− 1)κ. (7)

Unlike the TPC self-detection, it can be noted from (7) thatNC is deterministic because it depends only on g(X) and κ.

Comparing the computational complexity of the TPC self-detection to other CRC-based detection standards can beachieved using the relative complexity CS, where

CS =N lfsr

S

NC. (8)

Table I presents upper and lower bounds of CS forthe codewords (128, 120, 4)2, (64, 57, 4)2, (32, 26, 4)2 and(16, 11, 4)2. The corresponding g(X) for each codeword isobtained from [23, p. 466]. Moreover, three commonly usedCRC codes, namely, 16-bit CRC-8005, 16-bit CRC-8BB7, and32-bit CRC-1EDC6F41 are used in the comparison. It can beobserved that TPC self-detection is remarkably less complexthan CRC detection. The complexity reduction is proportionalto the codeword size and number of CRC bits. For exam-ple, the eBCH(128, 120, 4)2 TPC self-detection complexity isbounded by 3 × 10−3 ≤ CS ≤ 0.3575 of the 16-bit CRC8005 case, and it is 6 × 10−4 ≤ CS ≤ 0.0715 of 32-bit CRC1EDC6F41 case. It is also worth noting that the CRC processrequires additional hardware for encoding and decoding unlikethe TPC self-detection which uses the existing error correctionhardware for packet error detection. Therefore, the eliminationof CRC overhead can reduce the system complexity andimprove the system throughput, particularly for small packetsizes.

Although the obtained results demonstrate that TPC er-ror detection introduces substantial complexity reduction ascompared to conventional CRC techniques, more comparisonswith the overall system complexity are required to evaluatethe significance of the gained complexity reduction. Thecomplexity of SISO and HIHO TPC decoders is mainlydetermined by the number of hard decision decoding (HDD)operations performed [20]. For SISO TPC, the number ofHDD operations per half iteration used to decode the receivedmatrix R is n × 2p where p is the number of least reliableelements used to generate the test patterns in the Chase-



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TABLE I: The relative complexity CS of the TPC error detection versus CRC error detection using three common CRCstandards.

CRC (128, 120, 4)2 (64, 57, 4)2 (32, 26, 4)2 (16, 11, 4)2

LB UB LB UB LB UB LB UBCRC 8005 0.0030 0.3575 0.0063 0.3589 0.0141 0.3658 0.0374 0.4116CRC 8BB7 0.0010 0.1192 0.0021 0.1196 0.0047 0.1219 0.0125 0.1372

CRC 1EDC6F41 0.0006 0.0715 0.0013 0.0718 0.0028 0.0732 0.0075 0.0823

II decoder [28]. For HIHO decoding, the number of HDDoperations per half iteration is n. The complexity of eachHDD operation depends on the adopted HDD algorithm. Forexample, complexity analysis of BCH-HDD using Burton’salgorithm is given in [29] where the computational complexityfor each HDD operation NHDD is bounded by

45λ2n2(log10 n)2 < NHDD < (45λ+ 4)λn2(log10 n)2 (9)

where λ = t/n, and t is the number of errors that can becorrected by the code. By noting that t = 1 for all the codesused in this work, the total HDD complexity of SISO andHIHO TPC per half iteration is bounded by

(n× 2p)NminHDD < NS

TPC < (n× 2p)NmaxHDD (10)

andn Nmin

HDD < NHTPC < n Nmax

HDD (11)

for SISO and HIHO TPC, respectively.

The error detection process is usually required to stopthe iterative decoding, and hence each half TPC decodingiteration is followed by an error detection process. Therefore,the number of iterations and transmission rounds can beignored and the relative computational complexity of the CRCdetection to TPC decoding per half iteration is given by

0.5NC

nNmaxHDD

< CHC <

0.5NC

nNminHDD

. (12)

Similarly, the relative complexity when SISO is used can beexpressed as

0.5NC

2pnNmaxHDD

< CSC <

0.5NC

2pnNminHDD

. (13)

The CHC given in (12) is depicted in Fig. 2 for the CRC 8005,

8BB7 and 1EDC6F41. The codes considered are the eBCH(128, 120, 4)2, (64, 57, 4)2, (32, 26, 4)2 and (16, 11, 4)2. Thex-axis in the figure represents the value of k for these codes.As it can be noted from the figure, the CRC complexity canhave a significant impact on the overall system complexity.For example when considering eBCH(128, 120, 4)2, the com-plexity of the CRC-1EDC6F41 can be as high as 0.89NH

TPC,and for the CRC-8BB7 it can be 0.54NH

TPC.

For SISO decoding, CSC = CH

C/2p, which implies that the

significance of the CRC complexity is scaled by a factor of 2p.Since typically p ≤ 4, then CS

C sustains non-negligible values.For example, CS

C for the CRC-1EDC6F41 is upper boundedby 0.11 and 0.055 for p = 3 and 4, respectively.

11 26 57 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

k parameter of eBCH code

RelativeComplexity,CH C

8005, Max8005, Min8BB7, Max8BB7, Min1EDC6F41, Max1EDC6F41, Min

Fig. 2: The relative complexity of the CRC error detection toTPC HIHO decoding.

IV. SYSTEM THROUGHPUT

The transmission efficiency or throughput η, is defined asthe ratio of the number of information bits received success-fully to the total number of transmitted bits [30, p. 461]. Giventhat Q packets are transmitted, then

η =κz1 + κz2 + · · ·+ κzQNρ1 +Nρ2 + · · ·+NρQ

=κ∑Qi=1 zi

N∑Qi=1 ρi

(14)

where 1 ≤ ρi < L is a random number that represents the totalnumber of transmissions per packet and zi is a random numberthat indicates if the packet is dropped, zi = 0 if the ith packetis dropped, and 1 otherwise. However, because zi ∈ {0, 1},then 1

Q

∑Qi=1 zi is just the ratio of the non-zero elements to the

total number of transmitted packets, i.e., the complement ofthe packet drop rate PD. Given that Q→∞, using the law oflarge numbers 1

Q

∑i ρi → E {ρ} and 1

Q

∑Qi=1 zi → (1−PD).

Therefore (14) can be written as

η =1

E {ρ}κ

N(1− PD) (15)



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TABLE II: False alarm rate of TPC self-detection versus 16-bit CRC detection in Rayleigh fading channels when SISO decodingis used and L = 1. The hyphen ‘–’ means that the number of error-free packets after transmitting 106 packets is not enoughto compute the FAR reliably. The symbol

⟨10−5

⟩means FAR < 10−5.

Eb/N0 (dB) (16, 11, 4)2 (32, 26, 4)2 (64, 57, 4)2 (128, 120, 4)2

TPC CRC TPC CRC TPC CRC TPC CRC0 3.26× 10−1 – – – – – – –1 2.85× 10−1 1.33× 10−1 – – – – – –2 1.37× 10−1 8.22× 10−2 – – – – – –3 7.52× 10−2 4.17× 10−2 – – – – – –4 2.81× 10−2 1.82× 10−2 – – – – – –5 7.40× 10−3 6.54× 10−3 6.05× 10−2 2.26× 10−3 – – – –6 9.86× 10−3 1.41× 10−3 1.80× 10−2 8.35× 10−4 – – – –7 9.01× 10−5 1.66× 10−4 2.93× 10−3 1.29× 10−4 – – – –8

⟨10−5

⟩ ⟨10−5

⟩1.40× 10−4

⟨10−5

⟩2.07× 10−2 9.38× 10−5 – –

9⟨10−5

⟩ ⟨10−5

⟩ ⟨10−5

⟩ ⟨10−5

⟩3.95× 10−3 1.05× 10−5 – –

10⟨10−5

⟩ ⟨10−5

⟩ ⟨10−5

⟩ ⟨10−5

⟩5.21× 10−5

⟨10−5

⟩– –

11⟨10−5

⟩ ⟨10−5

⟩ ⟨10−5

⟩ ⟨10−5

⟩ ⟨10−5

⟩ ⟨10−5

⟩1.43× 10−2 –

12⟨10−5

⟩ ⟨10−5

⟩ ⟨10−5

⟩ ⟨10−5

⟩ ⟨10−5

⟩ ⟨10−5

⟩4.34× 10−4

⟨10−5

⟩13

⟨10−5

⟩ ⟨10−5

⟩ ⟨10−5

⟩ ⟨10−5

⟩ ⟨10−5

⟩ ⟨10−5

⟩ ⟨10−5

⟩ ⟨10−5

⟩where PD can be computed by noting that a packet is droppedif the first L transmissions fail. Thus, the packet drop proba-bility can be expressed as [7]

PD =

L∏i=1

P(i)E (16)

where P(i)E is the probability of packet error during the ith

transmission round. In other words, P (i)E can be expressed as

P(i)E =

{P (NACK1), for i = 1

P (NACKi |NACK1, · · · ,NACKi−1 ), for i > 1(17)

It is worth noting that the throughput analysis can be obtainedusing the renewal reward theory as well [31], [32].

Since the number of retransmissions takes only non-negativeinteger values, E {ρ} can be written as [32], [33]

E {ρ} =

L∑i=1

P (ρ ≥ i)

= 1 +

L−1∑i=1

P (ρ > i) = 1 +

L−1∑i=1

i∏j=1

P(j)E . (18)

Therefore, η is given by

η =κ

N1−

∏Li=1 P

(i)E

1 +∑L−1i=1

∏ij=1 P

(j)E

. (19)

Unfortunately, computing P (.)E analytically is not easy because

TPC error correction capability depends on the error patternrather than the number of errors [20]. Nevertheless, P (.)

E canbe estimated using a semi-analytical solution as described in[34].

A. Throughput using TPC Detection

In practical HARQ systems, the error detection mechanismis not perfect where some correct packets are falsely rejectedand some erroneous packets are misdetected. The ratio offalsely rejected packets to the total number of received packetsis defined as the false alarm probability PF, whereas, the ratioof misdetected errors to the total number of received packetsis defined as the misdetection probability PM. The probabilitythat a packet is declared erroneous during the ith transmissionround

(P

(i)E

)by the receiver is therefore defined as

P(i)E = P

(i)E + P

(i)F − P (i)

M . (20)

As it can be noted from (20), false rejections increase theerror probability PE and the number of retransmissions whichdegrades the system throughput. On the other hand, misdetec-tions reduce PE and allow erroneous packets to be forwardedto upper layers causing a throughput inflation. Therefore, theHARQ throughput using TPC error detection can be obtainedby substituting (20) in (19).

V. NUMERICAL RESULTS

The performance of the CRC-based and CRC-free errordetection schemes is first evaluated in terms of false alarmrate (FAR) and misdetection rate (MDR). The FAR is thenumber of false alarms divided by the number of correctpackets, whereas, the MDR is the number of misdetectionsdivided by the number of erroneous packets. The FAR andMDR are obtained using Monte-Carlo simulation of one-shottransmission to gain an insight into the basic properties of botherror detection schemes. Different TPC codes are considered inthe evaluation of the CRC-free and CRC-based schemes whereTPC self-detection and 16-bit CRC code are used, respectively.

The FAR and MDR are given in Table II and III, respec-tively. The results are generated for different Eb/N0 values inRayleigh fading channels. Eb is the average energy per bit and



7

TABLE III: Misdetection rate of TPC self-detection versus 16-bit CRC detection in Rayleigh fading channels when SISOdecoding is used and L = 1. The hyphen ‘–’ means that the number of erroneous packets after transmitting 106 packets is notenough to compute the MDR reliably. The symbol

⟨10−5

⟩means MDR < 10−5.

Eb/N0 (dB) (16, 11, 4)2 (32, 26, 4)2 (64, 57, 4)2 (128, 120, 4)2

TPC CRC TPC CRC TPC CRC TPC CRC0 1.19× 10−4 2.30× 10−5

⟨10−5

⟩1.30× 10−5

⟨10−5

⟩1.50× 10−5

⟨10−5

⟩2.00× 10−5

1 5.62× 10−4 1.40× 10−5⟨10−5

⟩1.30× 10−5

⟨10−5

⟩2.20× 10−5

⟨10−5

⟩1.10× 10−5

2 4.34× 10−3 1.11× 10−5⟨10−5

⟩1.70× 10−5

⟨10−5

⟩1.00× 10−5

⟨10−5

⟩1.10× 10−5

3 1.71× 10−2 2.75× 10−5⟨10−5

⟩1.90× 10−5

⟨10−5

⟩1.00× 10−5

⟨10−5

⟩1.80× 10−5

4 4.73× 10−2 2.35× 10−5⟨10−5

⟩1.90× 10−5

⟨10−5

⟩2.10× 10−5

⟨10−5

⟩2.10× 10−5

5 1.00× 10−1 2.65× 10−5 2.04× 10−4 2.23× 10−5⟨10−5

⟩1.40× 10−5

⟨10−5

⟩1.20× 10−5

6 1.80× 10−1 1.78× 10−5 3.26× 10−3 1.29× 10−5⟨10−5

⟩1.70× 10−5

⟨10−5

⟩1.60× 10−5

7 3.74× 10−1 – 1.90× 10−2 1.69× 10−5⟨10−5

⟩1.80× 10−5

⟨10−5

⟩2.00× 10−5

8 4.63× 10−1 – 8.41× 10−2 –⟨10−5

⟩1.42× 10−5

⟨10−5

⟩1.60× 10−5

9 – – 3.64× 10−1 – 4.73× 10−4 4.17× 10−5⟨10−5

⟩1.70× 10−5

10 – – – – 1.78× 10−2 –⟨10−5

⟩1.30× 10−5

11 – – – – – –⟨10−5

⟩1.62× 10−5

12 – – – – – – – –13 – – – – – – – –

N0 is the noise power spectral density. For each SNR value,the simulation is performed using 106 transmitted packets. Atlow SNR values, the number of correctly received packets isnot enough to reliably obtain the FAR especially for largepacket sizes. On the other hand, for high SNR values, thenumber of erroneous packets is not enough to reliably obtainedthe MDR. Therefore, a hyphen ‘–’ is used in the tables toindicate that the FAR or MDR cannot be obtained reliablyusing 106 transmitted packets.

Table II shows that the FAR rate decreases as the channelSNR increases. A false alarm happens when the informationbits are correct but the parity bits related to error detectionare in error. Therefore, when the channel SNR increases theprobability of having an error in the parity bits but not in theinformation bits reduces and the FAR subsequently decreases.The table also shows that the CRC-based detection has lowerFAR as compared to the TPC self-detection, which is due tothe fact that the number of parity bits of the TPC is large andhence the probability of error within these bits is large as well.

Table III shows that the MDR of TPC self-detection issmaller than 16-bit CRC code for all SNR values whenlarge TPC are used such as (128, 120, 4)2. For smaller TPCcode sizes, the MDR of TPC self-detection is smaller thanCRC detection only at low SNR values. The MDR of TPCself-detection deteriorates as the TPC code size is reduced.Small TPC codewords have less number of component codesinvolved in TPC self-detection as compared to larger TPCcodewords. Therefore, the joint error detection capability ofthese component codes decreases when smaller TPC code-words are used.

Moreover, for small TPC codeword sizes, we observe thatthe MDR of TPC self-detection increases as the SNR in-creases. At low SNR values, the number of errors in thedecoded TPC is high with various patterns; therefore, theprobability of having undetected errors in all component code-words is small. As the channel SNR increases, the number of

errors decreases; however, the errors tend to have closed chainpatterns. These types of errors are likely to be misdetected inall affected component codewords if the error pattern is largerthan their individual error detection capability. Hence, theTPC misdetection probability increases. However, it should benoted that at high SNR the probability of having an erroneouspacket is low which alleviates the effect of high MDR onthe system throughput. On the other hand, the MDR ofCRC detection is almost constant and matches the theoreticalapproximation 2−16 ≈ 1.53× 10−5.

The HARQ system with MRC described in Section II issimulated for different TPC codeword sizes and code rates.The packet is TPC encoded with code eBCH(n, k, dmin)2.For the CRC-based HARQ, 16-bit CRC 8005 code is usedbefore encoding with TPC for error correction. Moreover,Rayleigh fading is assumed as described in Section II. Themaximum number of ARQ rounds per packet is L = 4.For each simulation run (i.e. for a given SNR value) 1000packets are transmitted. The TPC decoder is configured toperform a maximum of four iterations. Moreover, the numberof reliability bits for the Chase-II decoder [28] is set to 4 inthe SISO decoder [21].

Fig. 3 shows the simulated relative complexity of TPCdetection to CRC 8005 for a range of SNR values in Rayleighfading channels. The relative complexity decreases as the SNRdecreases. That is because the number of bit errors becomeshigher at low SNR and an error is more likely to be detectedat the beginning of the TPC codeword. The figure also showsthat the relative complexity decreases as the codeword sizeincreases. Fig. 3 and Table I show the complexity advantageof TPC self-detection over CRC-based detection.

Fig. 4 compares the performance of TPC-HARQ whencombining and SISO decoding are used to the theoreticallimit of reliable communication in AWGN channels for BPSK.In this work, the TPC encoder output is interleaved makingthe individual symbols experience independent channel fading.



8

Therefore, the effect of the channel would be equivalent to fastfading where the standard capacity can be used to describe thelimits of reliable communications [23, p. 905]. The capacity ofAWGN channel is shown in the figure as an upper bound forthe throughput of TPC-HARQ in both AWGN and Rayleighfading channels.

0 5 10 150.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Eb/N0 (dB)

RelativeComplexity,CS

(16,11,4)2

(32,26,4)2

(64,57,4)2

(128,120,4)2

Fig. 3: Simulated relative complexity of TPC detection to 8005CRC detection for HARQ with SISO decoding and L = 4 inRayleigh fading channels.

−5 0 5 10 15

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Eb/N0 (dB)

Throughput,

η

AWGN Capacity

(32,26,4)2

(64,57,4)2

(128,120,4)2

AWGNFading

Fig. 4: TPC-HARQ throughput versus channel capacity whenBPSK and SISO decoding are used in AWGN and Rayleighfading channels.

Fig. 5 compares the system throughput when perfect de-tection and TPC self-detection are used. The TPC-HARQsystem is simulated for various TPC codes with SISO de-coding in Rayleigh fading channels. The figure shows thatthe throughput with TPC self-detection is almost the sameas the throughput with perfect detection which means thatthe misdetections and false alarms in TPC self-detection hasnegligible effect on the system throughput.

0 2 4 6 8 10 12 140.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Eb/N0 (dB)

Throughput,

η

(16,11,4)2

(32,26,4)2

(64,57,4)2

(128,120,4)2

Perfect DetectionTPC Detection

Fig. 5: TPC-HARQ throughput using perfect detection andTPC detection with SISO decoding and L = 4 in Rayleighfading channels.

In addition to the complexity advantage of TPC self-detection over CRC-based detection, TPC self-detection hasan advantage in terms of system throughput. Fig. 6 comparesthe throughput of the CRC-based and CRC-free HARQ us-ing equal code rates in both systems. In CRC systems, theredundant bits are composed of the CRC bits and the paritybits added by the TPC encoder. Therefore, to obtain equalcode rates in both systems, the number of TPC parity bitsshould be reduced (punctured) by the number of CRC bitmultiplied by the inverse of the code rate. The simulatedthroughput using equal code rates is depicted in Fig. 6 forrelatively long TPC. The number of bits punctured is 24,20 and 18 for the eBCH(32, 26, 4)2, eBCH(64, 57, 4)2 andeBCH(128, 120, 4)2, respectively. As it can be noted fromthe figure, the throughput with TPC detection is higher thanthe CRC-based system. However, the difference becomes verysmall as the TPC code size increases and almost vanishes athigh SNRs. Such behavior is expected because the puncturingprocess reduces the error correction capability of the TPC.



9

0 2 4 6 8 10 12 140.2

0.3

0.4

0.5

0.6

0.7

0.8

Eb/N0 (dB)

Throughput,

η

TPC DetectionCRC 16−bit

(128,120,4)2

(64,57,4)2

(32,26,4)2

Fig. 6: TPC and CRC throughput using equal numbers ofredundant bits with SISO decoding and L = 4 in Rayleighfading channels.

0 2 4 6 8 10

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Eb/N0 (dB)

Throughput,

η

TPC DetectionCRC 8−bitCRC 16−bit

Fig. 7: Throughput of eBCH(16, 11, 4)2 with SISO decodingand L = 4 in Rayleigh fading channels using TPC detectionand CRC with 8 and 16 bits. Both systems have equal numberof redundant bits.

For short TPC codes, the impact of the CRC bits lengthis more significant, and hence, smaller numbers of CRCbits can be employed. However, smaller number of CRCbits might result in unreliable performance because of the

limited capability of such codes to detect the errors. Moreover,the puncturing process has a more significant effect on theerror correction capability of the TPC, which is exhibited asthroughput reduction at low SNRs. Fig. 7 shows the throughputof the eBCH(16,11,4)2 TPC using 16 and 8-bit CRC usingequal code rates, where the number of punctured bits are 34and 17 for the CRC with 16 and 8 bits, respectively. As itcan be noted from the figure, the throughput results of theCRC-free and CRC-based systems converge to the same valueat high SNRs. However, the puncturing process reduces thethroughput at low SNR because the decoding process is lesseffective in terms of error correction. Overall, using CRC-8 isa reasonable compromise.

VI. CONCLUSIONS

A CRC-free HARQ scheme based on TPC error correctionand self-detection is proposed and evaluated in this paper. TheTPC inherent error detection provides additional degrees offreedom by eliminating the CRC overhead. The false alarmrate, misdetection rate and transmission efficiency are evalu-ated using extensive Monte Carlo simulations. The obtainedresults show that the CRC-free system consistently providesequivalent or higher throughput than the CRC-based HARQwith a noticeable advantage when small packet sizes areused. However, it was observed that the misdetection ratefor short codeword lengths increases at high SNR. Althoughsuch performance should not be of significant impact formany applications because the probability of packet error issmall at high SNR, for applications which do not tolerateerror misdetection, CRC-based transmission might be pre-ferred. In addition, the numerical results show that the TPCself-detection has lower computational complexity than CRCdetection especially for TPC with high code rates. Unlike CRCdetection, the complexity of TPC self-detection scales with thelocation of error where significant speedup is achieved whenerrors are located at the beginning of the TPC packet whichis usually the case at low SNR.

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[34] H. Mukhtar, A. Al-Dweik, M. Al-Mualla, and A. Shami, “Low com-plexity power optimization algorithm for multimedia transmission overwireless networks,” In press, DOI:10.1109/JSTSP.2014.2331915, June2014.

H. Mukhtar (S’08), received the B.Sc. and theM.Sc. degrees in electrical engineering from theAmerican University of Sharjah (AUS), Sharjah,United Arab Emirates (UAE), in 2004 and 2010, re-spectively. From 2005 to 2008, he worked as a tech-nical support and project engineer in an engineeringcompany in Dubai, UAE. He is currently workingtowards the Ph.D. degree in electrical and computerengineering at Khalifa University (KU), Abu Dhabi,UAE. Since October 2011, he is associated withthe Visual Signal Analysis and Processing Research

Center (VSAP) as a Graduate Teaching Assistant at KU. His current researchinterests include wireless communications, video streaming, and modeling andsimulation of communication systems.

A. Al-Dweik (S’9–M’01–SM’04), received the M.S.and Ph.D. degrees in electrical engineering fromCleveland State University, Cleveland, OH, USAin 1998 and 2001, respectively. Dr. Al-Dweik hasseveral years of industrial experience in the USA,recipient of the Fulbright Scholarship, and has beenawarded several awards and research grants. He isalso a Senior Member of the IEEE and AssociateEditor of the IEEE Transactions on Vehicular Tech-nology. The main research interests of Dr. Al-Dweikinclude Wireless Communications, synchronization

techniques, OFDM technology, modeling and simulation of communicationsystems, error control coding, and spread spectrum systems.

M. Al-Mualla (S’97–M’00–SM’06) holds a PhD de-gree in Electrical and Electronics Engineering and anMSc degree in Communication Systems and SignalProcessing both from the University of Bristol, UK.He also holds a BEng degree in CommunicationsEngineering from Etisalat College of Engineering,UAE. Since 2000 he has been with Khalifa Univer-sity where he is currently the Senior Vice Presidentfor Research and Development and the Director ofthe Visual Signal Analysis and Processing (VSAP)Research Center. He is a Senior Member of the IEEE

and author of the book “Video Coding for Mobile Communication: Efficiency,Complexity and Resilience,”Academic Press, 2002. His main research interestsare focused on multimedia communication and processing with particularemphasis on the problems of higher coding efficiency, reduced computationalcomplexity, improved error resilience, and security.

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