performance of reed–solomon codes in concatenated schemes with nonideal interleaving

EUROPEAN TRANSACTIONS ON TELECOMMUNICATIONSEur. Trans. Telecomms. 2007; 18:693–703Published online 2 May 2007 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/ett.1217

Information Theory

Performance of Reed–Solomon codes in concatenated schemeswith nonideal interleaving

Marco Ferrari1,∗, Sandro Bellini2, Fabio Osnato3, Massimiliano Siti3 and Stefano Valle3

1CNR - IEIIT, DEI, Politecnico di Milano, P.za L. da Vinci, 32, 20133 Milano, Italy2DEI, Politecnico di Milano, P.zza L. da Vinci, 32, 20133 Milano, Italy

3AST - STMicroelectronics Srl, Via C. Olivetti 2, 20041 Agrate Brianza, Italy

SUMMARY

The prediction of the performance of a Reed–Solomon (RS) code has an analytical solution in case ofstatistical independence of the errors at the input of the RS decoder (RSD). In concatenated schemes,this condition is often obtained through an interleaving device disrupting the correlation between erroneoussymbols. Sometimes the ideal depth of such interleaver is too large to implement and the RSD must operate insub-optimal conditions, for which no analytical formulas are available. In this paper, we propose a statisticalmodel that can manage under-dimensioned interleavers. With a mild set of hypotheses on the behaviourof the inner decoder (ID), we derive analytical expressions for the performance of the concatenated code.Input data for the model can be analytical or can be obtained by simulation. We apply the method to twodifferent types of inner codes, namely turbo codes and RS codes, and we compare the results predictedby the model with those obtained through simulation, when available, showing a very good agreement.Copyright © 2007 John Wiley & Sons, Ltd.

1. INTRODUCTION

Concatenated codes have been widely used in manyapplications since Forney’s analysis [1], usually combiningan inner convolutional code with an outer Reed–Solomon(RS) code. Early, the need of an interleaver between thetwo decoders has been recognised, with the purpose ofscrambling the error bursts at the output of the convolutionaldecoder. This interleaver is designed to obtain independenterrors, and some authors worked on this task [2, 3] todetermine the minimum required depth. More recently,other authors tried to predict bounds [4] or performance[5] in the case of under-dimensioned interleaver depths (i.e.correlated errors at the RS decoder (RSD) input), in the caseof inner convolutional codes.

Since their appearance in the literature [5], turbo codeshave astonished for their excellent performance for mediumbit error rate (BER) values (above 10−5), outperformingthe best convolutional codes by more than 2 dB. Early soon

∗ Correspondence to: Marco Ferrari, CNR - IEIIT, DEI, Politecnico di Milano, P.za L. da Vinci, 32, 20133 Milano, Italy. E-mail: [email protected]

it has been thought of concatenating them to RS codes, inapplications requiring very low BERs. To perform this task,we need appropriate interleavers scrambling the errors at theoutput of the turbo decoder. Unfortunately, errors withinone turbo frame are correlated [6], hence the only way toachieve independent symbols at the input of the RSD is aninterleaver depth as large as the turbo frame, which may beunfeasible.

Some authors have investigated the concatenation ofturbo codes and block codes such as RS [7] or BCH [8,9] using shorter, sometimes ‘ad hoc random’ interleavers[8]. In any case, the outer algebraic decoder is fed withcorrelated symbols, hence the error floor is impaired andthere is no simple means to predict it.

In this paper, we propose a statistical model that predictsthe performance of the concatenated scheme in the case ofunder-dimensioned interleavers. The method requires theanalysis of the statistical behaviour of the inner decoder(ID). If certain (mild) properties are met, the analytical

Received 1 March 2004Revised 22 June 2005

Copyright © 2007 John Wiley & Sons, Ltd. Accepted 20 November 2006

694 M. FERRARI ET AL.

evaluation of the output symbol error rate is possible. Weshow two examples in which these conditions are verified,that is the cases of an inner turbo code and of an innerRS code. Some further results of the application of themodel have already been shown in References [11, 12].In most cases, a computer simulation is sufficient to collectthe required statistics, enabling the designer to foresee theRS behaviour and to define a trade-off between interleaverdepth and other constraints, without a costly hardwareimplementation of the system.

2. OVERVIEW

The performance of an algebraic RSD can be calculatedfrom the probability PI (i) of the random variable I =number of erroneous symbols in a codeword at the decoderinput.

The symbol error probability at the output of the RSD isalways bounded by

Po �M∑

i = t + 1

min (i + t, M)

MPI (i) (1)

where M is the codeword length [symbols] and t isthe correction power. If proper checks are performed,codewords with more than t erroneous symbols arerecognised with very high probability and left uncorrected.Hence, the output symbol error probability is

Po =M∑

i = t + 1

i

MPI (i) (2)

Let ek = 1 denote an error in position k, while ek =0 corresponds to no error. If errors at the input ofthe decoder are independent identically distributed (i.i.d.)random variables, that is

P (ek = 1 | ei = 1) = P (ek = 1) = P ∀ i �= k (3)

the probability PI (i) has a binomial distribution withparameter P , that is:

PI (i) =(

M

i

)Pi (1 − P)M−i (4)

If errors are not independent, interleaving is used tomake the input sequence of the outer decoder fit the

above hypothesis. This is common in concatenated codeconfigurations, for example, convolutional inner codes withRS outer codes. The interleaver depth is chosen on the basisof the maximum length of error events of the ID, in orderto spread correlated errors in different RS codewords.

However, there are also cases where the interleaverdepth is a trade-off between several design constraints thatoblige to give up guaranteeing input error independence.The statistical model we propose in this paper allows tocalculate PI (i) for varying interleaver depths, also whenerrors at the input of the RSD are not independent. Themodel requires some basic assumptions on the statistics ofinput errors.

In the following sections we will introduce the model andits application to two cases of interest. The first one is thedesign of a concatenated scheme with an inner turbo code[1]. As shown in Section 4, the hypothesis of independenterrors does not hold in this case. Errors are correlatedall over the turbo code frame. Thus the best performanceis obtained only with an interleaver depth equal to theturbo frame, which might be unacceptable. This does notmean that a concatenated scheme with an interleaver depthsmaller than the turbo frame will not work. It means thatwe must expect a performance degradation and we cannotpredict the output byte error probability using Equations(2) and (4), hence we need a different model. Applying themodel proposed in this paper, the designer can achieve thetarget BER with the best trade-off between interleaver sizeand SNR extra cost.

In Section 5 we will describe the results obtainedapplying the model to the case of two concatenated RScodes, an error correction technique that has applications,for instance, in optical fibre transmission systems.

3. THE MODEL

For the sake of simplicity, we consider a block interleaverbetween the ID and the RSD. Some comments onconvolutional interleaving will be given later on. The IDwrites the output stream row-wise and the RSD reads thecodewords column-wise. Besides, we assume that:

1. the inner frame length is N bytes2. the RS codeword length is M bytes3. the interleaver depth is d bytes, hence the interleaver size

is d × M bytes4. each ID frame fills exactly r rows, where r = N/d is an

integer

Copyright © 2007 John Wiley & Sons, Ltd. Eur. Trans. Telecomms. 2007; 18:693–703DOI: 10.1002/ett

CONCATENATED INTERLEAVED CODES 695

Full interleaving is achieved if d = N. Furtherhypotheses about statistical properties of the ID outputstream are:

5. symbol errors are uniformly distributed, that is P(ek =1) does not depend on the symbol position k in the IDframe

6. symbol errors in an ID frame are uniformly correlated,that is

P (ek = 1 | eh = 1) ={

1 h = k

c h �= k

where c is some constant, independent of k and h, to bedetermined

7. the number n of erroneous symbols in an erroneousframe, given that the frame has errors, is distributedaccording to the (known) probability PN (n), n =1, 2, . . . , N

8. frames with errors occur independently, with frequencyf , hence the measured frame error rate (FER) is anestimate of f .

Note that Hypotheses 5 and 6 may not hold if the innercode is a turbo code and the decoder operates in the errorfloor region, where the performance is dominated by fewcodewords with low Hamming weight. Our model doesnot handle this situation. However, in a well designedconcatenated code the inner turbo code must not operatein the error floor, not to incur in a large SNR penalty (toreduce the FER to the required value).

For the time being, we further assume that the interleaversize corresponds to an integer number m of inner codeframes, that is dM = mN. Now we pick a column (i.e. an RScodeword) from the interleaver. The number of erroneoussymbols is

i =e∑

j=1

kj (5)

where e is the number of erroneous ID frames in theinterleaver frame under observation, and kj is the numberof erroneous symbols among the r belonging to the jtherroneous frame occurring in the column. Both e and kj arerandom variables. As to e, Hypothesis 8 implies a binomialprobability distribution:

PE (e) =(

m

e

)f e (1 − f )m−e (6)

It also follows from Hypothesis 8 that the kj are independentrandom variables, and from Hypotheses 4–6 that they areidentically distributed. Their probability distribution, givenn, follows from Hypotheses 5–7: k is a hyper-geometricdistributed random variable of parameters {N, n}, with r

draws:

PK|N (k | n) =(nk

)(N−nr−k

)(Nr

) (7)

Hence PK(k) can be calculated as:

PK(k) =∑n

PK|N (k | n) PN (n) (8)

where PN (n) specifies the ID. Given the number e oferroneous frames, the conditional density is the convolution(e − 1 times) of the probabilities PK(k):

PI|E (i | e) = PK (i) ∗ PK (i) ∗ . . . ∗ PK (i) (9)

Finally we insert Equation (6) to evaluate

PI (i) =m∑

e = 0

PI|E (i | e) PE (e) (10)

A refinement is necessary when M is not a multiple of N/d,since the interleaver does not contain an integer number ofID frames. In this case the first ID frame, or the last one, orboth fill less than r rows. As an (oversimplified) example, letM = 7 and r = 3. In three consecutive interleaver frameswe have respectively, 3+3+1, 2+3+2 and 1+3+3 rows. Ifinstead M = 8 and r = 3, we fill 3+3+2, 1+3+3+1 and2+3+3 rows, respectively. The number of ID frames isalways equal to m = �Md/N� or m + 1. In any case it isa simple matter to find all possible subdivisions and theirfrequencies of occurrence. Note that, for instance, 3+3+2and 2+3+3 are equivalent subdivisions, that can be analysedjust once. Equations (6)–(9) can be easily modified to takecare of m or m + 1 ID frames and of one or two edge frameswith different r, hence different distributionPK(k). The finalstep is to weigh all cases according to their frequencies ofoccurrence.

In the examples that we have tried so far, also avery simple approximation worked fine: analyse just onesubdivision, namely m − 1 ID frames filling r rows and oneID frame filling the remaining M − r(m − 1) rows.



3.1. Refinement of the model for application toconvolutional interleavers

Similar reasonings allow to manage interleavers with N/d

not an integer, or even more general interleavers, like, forexample convolutional interleavers (which require abouthalf the memory, and latency, of block interleaving). In fact,what is changed compared to the previous subsection, is thatthe lth RS codeword is composed of rh,l symbols drawnfrom the hth ID frame and rh,l may vary with both h and l.Actually what is of interest for our model is not the exactindexes h, but the distribution of the values rh,l for a given l.For convolutional interleavers, there exists a period S suchthat the lth and the (l + S)th RS codewords share the samenumber ml of ID frames, and the same distribution PRl

(r)of the values r [13]. S different cases must be considered.For each case we evaluate the probability PIl

(i) of havingi errors and then we average the S results. S may be quitelarge, yet the evaluation is straightforward.

To get PIl(i), we start from the conditional PIl|E(i | e)

and then we apply the total probability theorem as inEquation (10). Now the e independent contributions are nolonger identically distributed, because in Equation (7) theparameter r may vary according to the distribution PRl

(r).Equation (9) now reads

PIl|E (i | e) =∑r1,r2...re

P (r1, r2 . . . re) PK1 (i)∗PK2 (i)∗. . .∗PKe (i) (11)

where Equation (8) still holds, with PK|N (k | n) replaced by:

PKj |N (k | n) =(nk

)(N−nrj−k

)(Nr

) (12)

and the distribution P(r1, r2 . . . re) is easily obtained asthe probability of picking the e values r1, r2 . . . re froma population with mlPRl

(r) samples for each r, withoutre-insertion.

For a short comparison between block and convolutionalinterleavers see Reference [13]. A convolutional interleavermay be (a little bit) less efficient than a row–columninterleaver in spreading errors as evenly as possible. Yet,the degradation (if any) is very small, whereas the memoryand latency advantages of convolutional interleaving aremaintained.

Figure 1. Inner turbo coder and decoder.

4. COMPARISON BETWEEN ANALYTICALRESULTS AND NUMERICAL SIMULATIONS

In this section we present an application of the model,including its numerical validation. Here we focus on theconcatenation of code RS(204, 188) with an inner ParallelConcatenated Convolutional Code (PCCC), consisting oftwo 16 states, R = 1/2, terminated convolutional codes,2048 information bit block, 8PSK modulation, puncturedto spectral efficiency 2 bit/s/Hz. Refer to Figure 1 for ablock diagram of the coding/decoding scheme; as to themissing details, their description can be found in Reference[10]. The turbo code performances are shown in Figure 2.The Eb/N0 values include the 0.35 dB loss due to the outerRS(204, 188) code.

4.1. Statistical properties of the output of theturbo decoder

We have collected a great deal of turbo frames (143 010erroneous frames at Eb/N0 = 4.45 dB, 61 159 erroneousframes at Eb/N0 = 4.6 dB, 17 679 erroneous frames atEb/N0 = 4.75 dB, 2063 erroneous frames at Eb/N0 =5.35 dB). Since the outer codewords are composed of eight-bit-symbols (bytes), the most meaningful statistics are:

1. the byte error occurrence for each specific byte positionk in the frame. From these data we estimated theprobability of error for each byte, given that the framecontains errors, which is shown in Figure 3(a), atEb/N0 = 4.45 dB.

2. the correlation coefficient ρ(k, n) of the probabilities oferrors in positions k and n, estimated as follows:

ρ(k, n) = P (ek = 1, en = 1) − P (ek = 1) P (en = 1)√(P (ek = 1) − P2 (ek = 1)

) (P (en = 1) − P2 (en = 1)

) (13)



3 3.5 4 4.5 5 5.510

−4

10−3

10−2

10−1

100

Eb/N0 [dB]

FE

R

Figure 2. Frame Error Rate (FER) of the turbo decoder; block size 2048 bits, four iterations, 8-PSK modulation, 2 bit/s/Hz.

0 50 100 150 200 25010

−2

10−1

100

Byte position k

ρ(e n,e

k)

(b)

0 50 100 150 200 25010

−4

10−3

10−2

Byte position k

Pro

babi

lity

of e

rror

(a)

measuredmodel

Figure 3. Histogram (a) of byte error positions at the turbo decoder output and correlation coefficient (b) for n = 128 (Eb/N0 =4.45 dB).



0 50 100 150 200 25010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

n = number of erroneous bytes in a frame

Pro

babi

lity

pN(n

)measuredfitted

Figure 4. Probability of the number of erroneous bytes in an erroneous frame at the turbo decoder output (Eb/N0 = 4.45 dB).

where P(ek = 1, en = 1) is the estimate of the joint byteerror probability in the nth and kth positions in the frame.An example of ρ(k, n), for Eb/N0 = 4.45 dB and n =128, is shown in Figure 3(b).

3. the probability PN (n) of the number of erroneous bytesin an erroneous frame, which is shown in Figure 4, forEb/N0 = 4.45 dB.

From our analysis we conclude that:� The probability density of errors is roughly uniform over

the frame. The slight differences at the edges are due tocode termination.

� Errors are correlated all over the frame. The correlationis higher for bytes far apart less than 20 positions,then settles at a fixed (nonvanishing) value. The highercorrelation for almost adjacent bytes is not reproducedin our model.

� PN (n) is a decreasing function of n. Our simulationsdid not gather any event with probability PN (n)below 10−5–10−6. A conservative approach is to fitthe measures, assigning a value 10−6 to all missingPN (n). An optimistic approach is to let those missingprobabilities be zero (see Figure 4). We made ourevaluation in both cases, to see the influence on the finalresults.

4.2. Evaluation and validation of the modelwith simulations

We evaluated analytically PI (i) and Po for differentinterleaver depths, according to the model of the previoussection. We also simulated the concatenated scheme withthe same interleaver depths, using the stored erroneousframes.

In Figures 5 and 6 we sketch the results at the input andoutput of the RSD (Eb/N0 = 4.45 dB). The results for depth256 fully agree with the theoretical values, as the bytesare actually independent. As to the difference between thetwo versions of PN (n), we found it negligible: in Figure 6the curves of Po obtained from the fitted PN (n) are alsosketched.

The same statistics and curves were computed for otherEb/N0 values. The results are resumed in Figure 7, wherethe output BER is plotted versus Eb/N0. The superimposedcircles were obtained through the emulation of the RSD: theagreement is very good.

We conclude that

� The model gives almost the same results as the simu-lations do, down to BER ≈ 10−9. The analytical modelis slightly conservative. This is due to the fact that thehigher correlation between almost adjacent bytes, which



Figure 5. Byte error probability PI (i) at the input of the RS decoder (RSD), for various interleaver depths: analytical results (solid)and simulation (symbols).

100

101

102

10−10

10−8

10−6

10−4

10−2

Interleaver depth [bytes]

Out

put B

yte

Err

or P

roba

bilit

y P

o

simulationmodel with original p

N(n)

model with fitted pN(n)

Figure 6. Byte error probability at the output of the RSD versus the interleaver depth.



4.2 4.4 4.6 4.8 5 5.2 5.4

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

Eb/N0 [dB]

BE

Rdepth = 2 " = 4 " = 8 " = 16 " = 32 " = 64 " = 128 " = 256 Turbo Codesimulation

Figure 7. Bit Error Rate out of the RSD versus Eb/N0 for different interleaver depths; circles are obtained emulating the RSD. Thedashed line refers to the turbo code alone.

are spread in different RS codewords, is not reproducedin our model. This extra correlation is beneficial, but ouranalysis misses the (little) improvement. It is interestingto note that another half iteration of the turbo decoderwould spread the errors more evenly, and make ourmodel more precise. This would not be an advantage,yet. Hence, it is (just a little) better to stop the turbodecoder after an integer number of iterations.

� The curve with d = 256 perfectly agrees with Equation(2), because in this case the model is exact.

� The interleaver depth can be reduced, if we accept toincrease Eb/N0; the analytical model is suitable todefine a trade-off between interleaver depth and Eb/N0.

� There seems to be a good agreement between analysisand simulations also at SNR values well in the error floor(say, Eb/N0 > 5 dB), even if we are not aware of anysimple reason why our model should fit the ID behaviourin this region. However, as stated above, a good codedesign should avoid to operate in the error floor region.

� This method provides good estimates of the RSperformance at very low BERs with a set of statisticsobtainable via reasonable computer simulations,avoiding complex and expensive real-time simulationwith FPGA board or dedicated processors.

An interesting question, suggested by a reviewer, is thefollowing. Suppose we want to halve the interleaver size(and latency) with respect to full interleaving (d = 256, inour example). Then, d = 128. We can keep the inner blocksize N = 256 bytes (i.e. 2048 bits) unchanged, or we canreduce the block length to N = 128 (1024 bits). In the lattercase we have full interleaving, but the performance of theinner turbo code is worse since N is reduced. The questionis, what is the best strategy.

In general, we need to simulate turbo codes withseveral block lengths and repeat the analysis for each case.However, sometimes we can obtain a quick answer by basicinformation theory principles, like the Shannon sphere-packing bound.

In our example, Figure 7 shows that nonideal interleaving(N = 256) costs about 0.07 dB (at BER = 10−12). Figure 7also shows that with full interleaving we need Eb/N0 =4.45 dB, which corresponds to FER = 3 × 10−3 accordingto Figure 2. At block error rates around this value, thesphere-packing bound predicts an SNR loss around 0.18 dB,if N is halved. Then, for our example we see with little effortthat the best solution is nonideal interleaving (N = 256).

If we want to further halve the memory, we can keep N =256 and read the degradation from Figure 7 (about 0.17 dB),



or repeat the analysis with N = 128 and d = 64 and withN = 64 (and full interleaving). According to the sphere-packing bound, N = 128 costs about 0.18 dB. Besides,there is some extra loss due to nonideal interleaving. Thesphere-packing bound also predicts that N = 64 would costabout 0.4 dB. Then, N = 256 is again the best solution.

5. APPLICATION TO CONCATENATEDREED–SOLOMON CODES

In this section, we describe an application that does noteven need preliminary simulations to calculate the inputstatistics.

We consider a concatenated code with nonidealinterleaving, where both the inner and the outer codes areRS, of parameters (N1, K1) and (N2, K2), respectively.

5.1. Calculation of the statistical propertiesof the ID frames

To apply the model described in Section 3, we assume thateach RSD detects with certainty input words containingerrors exceeding the correction capability, and leaves themuncorrected. Hypothesis 6 is certainly met, in fact thesymbol error occurrences out of the channel are i.i.d.random variables; moreover, under our assumptions on the

ID behaviour, no dependence of the symbol errors on theirposition in the ID frames can be expected.

Therefore, all we need is to calculate the distribution ofthe erroneous symbols in the erroneous inner code framesPN (n) and the frame error rate f .

The probability of i symbol errors at the input of the ID is:

PI (i) =(

N1

i

)Pi (1 − P)N1−i (14)

where P is the channel symbol error rate. If i � t1 allthe errors are corrected by the ID. Otherwise, since theerrors are uniformly distributed in the ID frame, theprobability of n errors in the K1 information symbols,given i (uncorrectable) errors, is:

PN|I (n | i) =(in

)(N1−iK1−n

)(N1K1

) , max {0, i − 2t1} � n � i (15)

Then the probability of one or more errors in the informationbits is given by

f = 1 −t1∑

i = 0

PI (i) −2t1∑

i = t1 + 1

PI (i)PN|I (0 | i) (16)

where we have subtracted also the (small) probability that allthe errors are in the parity symbols. Finally, the distribution

Figure 8. Output BER versus Eb/N0 of two concatenated RS codes, with several interleaver depths.



Figure 9. Net coding gain with respect to uncoded binary transmission versus interleaver depth.

PN (n) of errors, conditional to at least one error, is

PN (n) = 1

f

n + 2t1∑i = t1 + 1

PI (i)PN|I (n | i) , 1 � n � K1 (17)

Note that if the inner RSD does not recognise uncorrectablewords (not a decoder design to be recommended), it addsits own errors. Then, PN|I (n | i) must be evaluated bysimulation, instead of Equation (15), and the performancegets worse, of course.

5.2. Performance results

We show the results obtained for the concatenation ofan inner RS(255, 224) and an outer RS(255, 239) withinterleaving. Figure 8 shows the output BER versus Eb/N0for various interleaver depths. A significant performancedegradation comes up only for interleaver depths smallerthan 56. These results are very helpful to define a goodtrade-off between a target performance and the interleaversize in a practical implementation of the concatenated code.

Figure 9 shows the Net Coding Gain (NCG) of the codingsystem, that is the Eb/N0 advantage with respect to uncodedantipodal binary transmission, for a given BER level. Itcan be seen, for instance, that the NCG at BER = 10−15

decreases by 0.25 dB when passing from d = 224 to 56,while a further decrease by 1.5 dB occurs from d = 56 to 2.

6. CONCLUSIONS

We have introduced and validated a statistical modelpredicting the RS code performance in the case of correlatedinput errors scrambled by a row–column interleaver. Themodel enables the designer to define the trade-off betweeninterleaver depth and other design constraints. The modelassumes that the errors at the interleaver input are uniformlycorrelated and uniformly distributed along the input frame,and requires the knowledge of the probability of the numberof erroneous symbols in the inner frame.

The analysis can be generalised, to manage, for examplealso convolutional interleavers.

The two applications presented in this paper show that themodel is applicable when such statistics of the ID come frommeasurements or simulations, or are known analytically.In these cases we can predict the performance of theconcatenated code at very low BER (<10−10), when verylong computing time would be required for simulations. Forthis reason the model is an attractive tool for the design ofsystems working at very low BER like digital TV, opticallinks and Hard Disk drives.

ACKNOWLEDGEMENTS

The authors are grateful to the anonymous reviewers for their criti-cism, that helped to improve the paper content and the presentation.



REFERENCES

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2. Berman T, Freedman F, Kaplan T. An analytical analysis of aconcatenated Reed-Solomon (RS)/Viterbi coding system both withand without RS interleaving. Proceedings of 11th IEEE InternationalPhoenix Conference on Computers and Communications (IPCCC),Phoenix, Arizona, 1992; pp. 260–266.

3. Yuan D, Zhang L, Gao C. Performance Analysis of RS-BCHConcatenated Codes in Rayleigh Fading Channel. ProceedingsAPCC/OECC’99, 5th Asia-Pacific Conference on Communications,Beijing, China, 1999; pp. 677–679.

4. Wadayama T, Wakasugi K, Kasahara M. An upper bound on biterror rate for concatenated convolutional codes. IEICE TransactionsFundamentals 1997; E80-A(11): 2123–2129.

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AUTHORS’ BIOGRAPHIES

Marco Ferrari was born in 1971, and he graduated at the Politecnico di Milano in Telecommunications Engineering (cum laude) in1996. In 1997 he started his Ph.D. program at the Dipartimento di Elettronica e Informazione, where he received his title in 2000. In2001 he joined the National Research Council (CNR) as a Research Associate, in the group on Space Communications (CSTS). He iscurrently with the Electronics, Information and Telecommunications Engineering Institute (IEIIT) of the CNR. His research interestsare information theory and digital transmission systems, mainly channel coding, iterative decoding and modulation.

Sandro Bellini got the Laurea (cum laude) in Electronic Engineering in 1971. He is Full Professor of Telecommunications since1990. His main research interests have been many aspects of digital transmission and signal processing, like continuous phasemodulation (CPM), equalisation, synchronisation, multicarrier demodulation, mobile satellite systems, wireless LANs, optical storageof information, and error correcting codes for various applications (with an emphasis on turbo codes).

Fabio Osnato graduated in Electronics Engineering in 1993 at Politecnico di Milano. Since 1995 he is part of System ArchitectureR&D of STMicroelectronis with specific focus on digital transmission techniques: OFDM transmission and Error Correction codes. Heactively participated to the definition of the DVB-S2 and DVB-RCT, European standard for Digital Satellite and Interactive TerrestrialTelevision. He is currently part of Advanced System Technology, the System R&D of STM, as the team manager investigating advancedtechnologies for the physical layer of next generation wireless transmission systems like IEEE 802.11n (channel coding, multi-carriertransmission, Multiple Antenna—MIMO).

Massimiliano Siti was born in 1972. He received his Electrical Engineering degree from Politecnico di Milano, Milan, Italy in 1996.In 1998–2000 he joined Alcatel, Italy, working on Gbit/s optical communications. Since 2000 he is with STMicroelectronics where heworked on modulation, error correction codes and algorithms for multiple antenna wireless systems. In 2004–2005 he was a visitingresearcher in the wireless laboratory at the University of California Los Angeles with the support of a EUC (Marie Curie) fellowship.His research interests include space-time codes, MIMO detection, error correction codes.

Stefano Valle was born in 1969. He received his Electrical Engineering degree (1995) and the Ph.D. in Electrical and CommunicationsEngineering (1999) from Politecnico di Milano, Milan, Italy. Until 2000 he was with Dipartimento di Elettronica e Informazione ofPolitecnico di Milano (Digital Signal Processing group), where he worked on radar imaging. In 2000, he joined STMicroelectronics(Italy)—Advanced System Technology, where he worked on modulation, channel modelling and error correction codes for wirelesssystems. In 2006 he joined the Data Storage Division working on signal processing and error correction codes for HDD Read WriteChannels. His research interests are digital transmission systems, mainly channel coding, modulation and estimation.

7. Liu Y, Tang H, Lin S, Fossorier M. An interactive Concatenated TurboCoding System. Proceedings ISIT 2000, Sorrento, Italy, 2000; p. 367.

8. Narayanan KR, Stuber GL. Selective serial concatenation of turbocodes. IEEE Communications Letters 1997; 1(5):136–140.

9. Kim HC, Lee PJ. Performance of Turbo Codes with a Single-ErrorCorrecting BCH Outer Code. Proceedings ISIT 2000, Sorrento, Italy,2000; p. 369.

10. Benedetto S, Montorsi G. Versatile Bandwidth-efficient Paralleland Serial Turbo-trellis-coded Modulation. Proceedings of 2ndInternational Symposium on Turbo Codes & Related Topics, Brest,France, 2000; pp. 201–208.

11. Ferrari M, Osnato F, Valle S, Siti M, Bellini S. Performance ofconcatenated Reed-Solomon and Turbo codes with non idealinterleaving. Proceedings Globecom 2001, San Antonio, Texas, Vol.2, 2001; pp. 911–915.

12. Osnato F, Scalise F, Siti M, Ferrari M. FEC codes for optical transmis-sion systems at 40 Gbit/s. Proceedings of Wireless and Optical Com-munications conference (WOC) 2002, Banff, Alberta, Canada, 2002.

13. Siti M, Gatti D, Osnato F. A statistical model of convolutionalinterleavers for concatenated codes. Proceedings of 11th InternationalConference on Software, Telecommunications & Computer Networks(SoftCOM’03), Split-Venice-Ancona-Dubrovnik, Italy-Croatia, 2003;pp. 208–212.


performance of reed–solomon codes in concatenated schemes with nonideal interleaving

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