Dual code method for blind identification of convolutional encoder for cognitive radio receiver design
M. Marazin1-2 , R. Gautier1-2, Member, IEEE and G. Burel1-2, Senior Member, IEEE
1Université Européenne de Bretagne, France
2Université de Brest; CNRS, UMR 3192 Lab-STICC, ISSTB, 6 avenue Victor Le Gorgeu, CS 93837, 29238 Brest cedex 3, France
roland.gautier@univ-brest.fr http://www.labsticc.fr/cacs/equipe/comint/
Abstract—Digital communication systems are in perpetual evolution in order to respond to the new user expectations and to new applications transmissions constraints, in term of data rate or reliability. With this fast development of new communication standards, it becomes very difficult for users and also for communications devices producers, to stay compatible with all standards used and with the oncoming ones. For that reason, cognitive radio systems seem to provide an interesting solution to this problem. The conception of intelligent receiver able to adapt itself to a specific transmission context and to blindly estimate the transmitter parameters is a promising solution for the future. In such context, new coding schemes like turbocodes, Low-Density Parity-Check (LDPC) or concatenated codes are developed to increase transmission robustness without significant degradation of the data rate. It is why we have developed a method, described in this paper, dedicated to the blind identification of convolutional encoders usually used in many standards. Moreover, an analysis of the method performances is detailed.
Index Terms—Cognitive radio, intelligent receiver, blind iden- tification, convolutional encoder, dual code.
I. INTRODUCTION
Modern digital communication systems and new Broadband Wireless Access (BWA) technologies are dedicated to allow high data rate transmissions from a transmitter to a receiver with the best reliability and Quality of Service (QoS). On the transmitter side, a communication system is composed of sev- eral different blocks, each one having a specific function, like source encoder, channel encoder or modulator... The channel coding block is essential to achieve a good immunity against channel impairments, but due to the redundancy introduced by a correcting code, it produces a decrease of the transmission rate. For that reason, new correcting codes like turbocodes, LDPC or concatenated codes are always developed to reduce the impact of the amount of redundancy introduced and to improve the correction capabilities. The arrival of new high- performance coding schemes induces that communication sys- tems are in perpetual evolution and it becomes more and more difficult for users to follow all the changes to stay up-to-date and also to have an electronics communications device always compatible with every standard in use all around the world. In such context, cognitive radio systems provide an obvious solution to these problems. In fact, a cognitive radio receiver is an intelligent receiver able to adapt itself to a specific transmission context and to blindly estimate the
transmitter parameters for self-reconfiguration purpose with the only knowledge of the received data stream.
In this paper, we consider the problem of blind identification of error-correcting codes for cognitive radio receiver design and more precisely for convolutional codes usually used in many actual communication standards. There is a few methods which have been published in this domain, for example in [1], a method based on Euclidean algorithm is proposed to identify a rate 1/2 convolutional encoder in the noiseless case. In [2], a different approach is presented to identify a rate 1/n encoder in the noisy case which is based on the Expectation Maximization algorithm. In [3], we have developed a method for blind recovery of convolutional encoder in turbocode configuration and a perfect transmission.
In this paper, we present an iterative method for blind recognition of rate(n−1)/nconvolutional encoder from noisy data which is based on algebraic properties of convolutional encoder [4], [5] and dual code [6]. The paper is organized as follows. In section II, some properties of convolutional codes are presented. Then, the principle of blind recognition via a dual code is described in section III in the noiseless case and the method of blind identification in the noisy case is explained in section IV. Finally, a discussion on the performances of the method is proposed in section V, and some conclusion and prospects are given in section VI.
II. CONVOLUTIONAL ENCODER AND DUAL CODE
A. Principle and mathematical model
Before trying to explain the method for blind reconstruction of convolutional encoder, it is important to give the main mathematical notions to describe a convolutional code and the interesting properties to achieve our goal of blind encoder recovery.
A convolutional code, denoted C(n, k, K), is first defined by three parameters,kthe number of inputs,nthe number of outputs, K the constraint length, and also by hisk×n code generator matrix given by:
G(D) =
⎡
⎢⎣
g1,1(D) · · · g1,n(D) ... . . . ... gk,1(D) · · · gk,n(D)
⎤
⎥⎦ (1)
where gi,j(D), ∀i = 1,· · · , k, ∀j = 1,· · ·, n, are generator polynomials andDrepresents the delay operator. The memory of thei-th input, denotedμi, is defined by:
μi= max
j=1,···,ndeggi,j(D) ∀i= 1,· · · , k (2) which permits to define the overall memory of convolutional code as:
μ= max
i=1,···,kμi=K−1 (3)
For such code, the encoding process can be described by:
c(D) =m(D).G(D) (4)
where m(D) is an input sequence and c(D) the codewords sequence.
One of the most important properties in error correction theory is the equivalent encoder notion. A simple way to explain this property is to say that the codewords, or by extension the encoded sequences, generated by two equivalent encoders belong to the same codeword set, but that the codewords are not generated in the same order.
Theorem 1: Two rate r = k/n code generator matrices G(D)andG(D)are equivalent if and only if there is ak×k nonsingular matrixL(D)such that
G(D) =L(D).G(D) (5) There exist many convolutional codes, but most of them do not have the good properties in term of correction performances on the receiver side. Especially, some generator matrices of a code are called catastrophic generator matrices and must be avoided.
With such catastrophic matrix, a small number of channel errors may generate an unlimited number of errors after the decoding process. In practice, only optimal convolutional encoders are usually used because they have the highest error correction capabilities, and good algebraic properties ( [4], [5]) which can be judiciously exploited for blind identification.
To take into account perturbation generated by the global transmission system (modulation, propagation channel, de- modulation...) we consider the binary symmetric channel (BSC) to modelize the errors. If the codewords sequencec(D) is sent over a BSC, then the received sequence, denotedy(D), is:
y(D) =c(D) +e(D) (6) where e(D) is an error pattern. We denote P e the error probability of the channel ande(i)the i-th bits ofe(D)such as: P r(e(i) = 1) = P e and P r(e(i) = 0) = 1−P e. The errors are assumed independent.
In turbocodes or parallel concatenated convolutional codes, rates 1/n or (n−1)/n optimal convolutional encoders are used. For this reason, we have decided to study the problem of the blind recovery of rate (n−1)/n convolutional codes.
B. The dual code of the rate(n−1)/nconvolutional encoder A convolutional encoder can also be described by a dual code generator matrix, called parity check matrix, which has also excellent properties in blind recovery of convolutional encoder context. This parity check matrix is an (n−k)×n polynomial matrix which verifies the following property:
Property 1: LetG(D)a generator matrix ofC. If an(n−
k)×npolynomial matrix,H(D), is a parity check matrix of C, then:
G(D).HT(D) = 0 (7)
Corollary 1: Let H(D) a parity check matrix of C. The output sequence c(D) is a codeword sequence of C if and only if:
c(D).HT(D) = 0 (8) For a rater= (n−1)/n code, the parity check matrix is a simple row vector defined by:
H(D) =
h1(D) · · · hn−1(D) h0(D) (9) where hj(D) are the generator polynomials of H(D), ∀j = 0,· · ·, n−1.
According to properties of optimal convolutional encoder and dual code ( [6], [7]), the memory of H(D), denotedμ⊥, is such as:
μ⊥=
n−1
i=1
μi (10)
and the generator polynomials are:
hj(D) =
µ⊥
l=0
hj(l).Dl
=hj(0) +hj(1).D+· · ·+hj(μ⊥).Dµ⊥
(11)
According to [7], a convolutional encoder is realizable if the polynomial h0(D) is a delayfree polynomial. A polynomial f(D) = ∞
i=0f(i).Di is called a delayfree polynomial if f(0) = 1. Indeed, the generator polynomialh0(D)is such as:
h0(D) = 1 +h0(1).D+· · ·+h0(μ⊥).Dµ⊥ (12) In practice, for implementation consideration, the inter- cepted sequence is not usually in the polynomial form: it is considered as a binary vector denoted c. So, the parity check matrix can be rewritten under binary form:
H =
⎛
⎜⎝
hµ⊥ hµ⊥−1 · · · h1 h0 hµ⊥ hµ⊥−1 · · · h1 h0
. .. . .. ... ... ...
⎞
⎟⎠ (13)
where hi, ∀i = 0,· · · , μ⊥, are row vectors of size n and defined by:
hi=
h1(i) · · · hn−1(i) h0(i)
(14) We denoteha vector defined by:
h=
hµ⊥ hµ⊥−1 · · · h0
(15)
From (13), the parity check matrix is composed of moved versions of this same vector, h. This vector (15) of size n.(μ⊥+ 1) is called a parity check of the code. From (14) and (12), the vectorh0 is:
h0=
h1(0) · · · hn−1(0) 1
(16) Thus, we can deduce that the degree ofh, denotedna, is equal to:
na =n.(μ⊥+ 1) (17) and is of great interest in the blind convolutional code identi- fication process.
III. PRINCIPLE OF BLIND IDENTIFICATION OF THE RATE
(n−1)/nCONVOLUTIONAL CODE IN THE NOISELESS CASE
In this part, we describe the principle of our method of blind identification when the intercepted sequence is not corrupted by errors defined here as the noiseless case. In the case of block code, an approach to recovery of the code rate has been presented in [8]. In [3], we have adapted this method to be applied to the convolutional code in the case of a perfect transmission. Its principle is to reshape columnwise the intercepted data bit stream under matrix form. This matrix, denoted Rl, is computed for different values of l, wherel is the number of columns. If the codewords sequence length is L, then, the rows number of each matrix Rl, noted M, is M =Ll, where.stands for the integer part. An example of this construction is presented in Fig. 1. For each matrixRl,
0 1 2 3 4 5 6 7 8 1 2 0 3 4 5 6 7 8 M
l c
l
Figure 1. Example of matrixRl
the rank in Galois Field, GF(2), is computed and has two possible values:
• Ifl=α.nandl≥na,∀α∈N rank(Rl) =l.n−1
n +μ⊥< l (18)
• Ifl=α.norl < na,∀α∈N
rank(Rl) =l (19) We can note that if a block code C(n, k) is used, the relation (18) becomes:
rank(Rl) =l.n−1
n < l ∀l=α.n (20) In this case, the matrix Rl would have presented a rank deficiency for alll =α.n. But, for a convolutional encoder, the memory introduced by the code implies that the rank of the matrixRl (18), forl=α.n < na, is always equal toα.n.
So, the first matrix Rl which exhibits a rank deficiency is obtained for l=na and its rank is given by:
rank(Rna) = (n−1).(μ⊥+ 1) +μ⊥=na−1< na (21) Equation (21) shows that there exists only one linear relation between the columns ofRna.
It is obvious that it is possible to identify all parameters of the code with only two consecutive rank deficiency matrix.
The different steps of this identification process are described below.
• Identification of n:
The codewords size,n, is equal to the difference between two values of l corresponding to two consecutive falls of rank of Rl.
• Identification of μ⊥ andμ:
Using one matrix with rank deficiency, from (18), it is possible to recovery the memory of the dual code and with (10) we obtain the memory of the code.
• Identification of na:
From (17), if we knownandμ⊥, it is easy to obtain the value of na.
Now, we must have to identify the parity check h for recovery the code generator matrix. We have shown that the degree of this parity check (17) is equal to na and that the rank of the matrix Rna (21) is equal tona−1. Indeed, if we build this system:
Rna.hT = 0 (22) it is possible to solve it to identify the parity check h.
Finally, the generator matrix is obtained by solving the system described in prop. 1. Nevertheless, the resolution of such system may led to multiple solutions which can be reduced by taking into account the particular properties of optimal encoders. Even so, it is always possible to fail finding the true generator matrix of the code, and, instead, to find only one equivalent matrix. In this case it is theoretically impossible to distinguish equivalent matrices.
This method exhibits excellent performances if the inter- cepted sequence is not corrupted but, it becomes inefficient if data is severely corrupted. In the next section a method for blind recovery of convolutional encoder is presented when the intercepted sequence is severely corrupted.
IV. BLIND IDENTIFICATION OF THE RATE(n−1)/n
CONVOLUTIONAL CODE IN THE NOISY CASE
Let us note Rl the M ×l matrix constructed in the same way as Rl in section III, but with data taken from corrupted codewords sequence. In this case, y = c, and in spite of the redundancy introduced by the code, all these matrices are full rank matrices even in the configuration with l = α.n and l ≥ na. The dependency between columns is lost due to the erroneous bits introduced by the transmission channel.
So, in such erroneous context, the previous method cannot be applied to identify the parameters of the code. In [9], a method for blind detection of the interleaver size from a block coded
and interleaved sequence, corrupted by the channel, has been presented. We have adapted this method for blind identification of the size of codewordsnand of the value ofna in the case of convolutional codes. In this section, we propose to replace the rank criterion which indicates the number of independent columns of the matrixRl, by a method permitting to evaluate the columns which are probably dependent. With this method, it becomes possible to detect values of l where the matrices Rl seem to be degenerated rank.
The principle of this method is to convert each matrix Rl into a lower triangular matrix notedGlwith the Gauss Jordan Elimination Through Pivoting (adapted inGF(2)):
Al.Rl.Bl=Gl (23) whereAlis a rows permutation matrix of sizeM×M andBl a matrix describing the columns combination of sizel×l. In the noiseless case and forl=na, the last column of the lower triangular matrixGlis a null column. In this way, we will try to detect a linear form according to the number of ones in the lower part of each column in the triangular matrix. LetNl(i) be the number of ones in the lower parts of columni in the matrix Gl. We will show that this variable has two different behaviors depending on l.
• Ifl=α.norl < na
In this case, Nl(i) follows a Binomial distribution with pa- rameters B(M−l,0.5).
• Ifl=α.nandl≥na
In this configuration, because of redundancy introduced by the code, it exists at least one column of Gl such that Nl(i) follows a Binomial distribution with parametersB(M−l, P).
Here,P is a probability that one element of thei-th column of Gl is equal to “1”:
P = 1−1 + (1−2.P e)wi
2 (24)
where wi is the Hamming weight of the i-th column of Bl. This value, wi, represents the necessary number of columns ofRlto obtain thei-th column ofGl. For the other columns, Nl(i)follows a Binomial distribution with parametersB(M− l,0.5)
According to the laws of probability ofNl(i), it is possible to define a threshold, denoted γ, which will allow to decide if thei-th column of the matrixRl is dependent on the other columns. We decided that thei-th column ofRlis dependent if:
Nl(i)≤(M−l).γ
2 (25)
and this column is independent if:
Nl(i)>(M−l).γ
2 (26)
For obtaining at least one dependent column, we realize the Gaussian Elimination process (23) for different value ofl and with γ = 0.6. After this step, it is possible to estimate an optimal threshold [9]. This optimal threshold, denoted γopt, is chosen to minimize the sum of the missing probabilities.
The numbers of dependent columns detected, denoted Q(l), are such as:
Q(l) =Card
i∈ {1,· · · , l} |Nl(i)≤(M −l).γopt 2
(27) So, the gap between two non-zero cardinals, Q(l), is equal to the size of codeword, denoted n. Letˆ I be a set of values of l where the cardinal is non-zero.
From matrix Bi,∀i∈ I, it is possible to build a basis of dual code, that is to say a set of linear form. Let us denote Zi, a sub-matrix of Ri, where Zi is a (M −i)×i matrix composed of the M−i last rows ofRi. If bj represents the j-th column of Bi, we decide that bj is a linear form close to the dual code if:
d(Zi.bj)≤(M−i).γopt (28) We noted Da set of all detected linear forms.
For an optimal convolution code, we have seen that the parity checkhis the linear form of dual code of the smallest degree. Among the set of detected linear form, D, we shall try to find that of smaller degree. We notedhˆ this linear form andnˆa this degree.
To identify the other parameters, we follow the same steps described in section III. So, according to (17) and (10), with ˆ
n, hˆ and nˆa, it is possible to estimate the memory of the code and according to prop. 1, we will be able to estimate a generator matrix of codeC.
At the output of this algorithm, two situations may occur.
The first one arise when the true encoder or an optimal encoder is identified and the second one, when no optimal encoder is identified. In the second situation, it is possible to increase the probability of detect an encoder by realizing a new iteration of this algorithm. For each iteration, the same data set is used but with a new permutation of the rows of the matrix R, these permutations yielding a virtual new realization of the intercepted sequence. Indeed, during the Gauss Elimination process (23), if the pivots are erroneous these errors will be propagated in the matrixGl. Doing permutation increases the probability to obtain a non-erroneous pivots.
V. ANALYSIS AND PERFORMANCES
In this section, for clarity and concision of the presentation, we will restrict the analysis to optimal convolutional encoder:
C(2,1,7)andC(3,2,3). Moreover, we build all the matrices Rl with l = 2,· · · ,100 and M = 200, which implies a received bit stream of only 20000 bits. For example, for the UMTS standard [10] which allows reach a bit rate up to 2Mbps, 20000 bits represents only a duration of 10ms and the future standards will reach higher data rates. Taking into account the amount of data required to perform the blind identification is very important to prove that our algorithm is well adapted for implementation in a realistic context. For each simulation, 1000 Monte Carlo Trials are run. Before analyzing in detail the global performances in term of probability of detection of our algorithm (in this context the detection includes the complete identification of the encoders) we will
investigate the influence of the number of iterations to obtain an optimal probability of detection.
A. The detection gain produced by the iterative process The number of iterations will be a compromise between detection performances and processing delay introduced in the reception chain. Moreover, due to the relative small amount of data used by the iterative process to generate virtual new realizations, the number of iterations is also theoretically limited to preserve statistical independence during the pivots selection process. To evaluate this number of iterations, let λx→y be the gain between the x-th iteration and the y-th iteration defined by:
λx→y= Pdet(y)−Pdet(x)
Pdet(x) (29)
where Pdet(i) is the probability to detect the true encoder at the i-th iteration. This gain, λx→y, will be expressed in percentage.
• C(2,1,7)convolutional encoder:
Fig. 2 shows the probability to detect the true encoder, denoted Pdet and called probability of detection, versus Pe, for 1, 5 and 50 iterations. For this code, we can see that the optimal
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0
0.2 0.4 0.6 0.8 1
Channel error probability: Pe
Probability of detection
Iteration 1 Iteration 5 Iteration 50
Figure 2. C(2,1,7): Probability of detection versusPe for 1, 5 and 50 iterations
performances are obtained with 5 iterations. The gain between the 5th iteration and the 50th iteration is close to zero. The gain between the 1st iteration and the 5th iteration, represented in the table Tab. I, is very large. Indeed, atPeequal to3.10−2, λ1→5 is close to112%. After 1 iterationPdet is close to0.4, and0.8 after 5 iterations.
Table I
C(2,1,7)ANDC(3,2,3)DETECTION GAIN
Pe 0.01 0.02 0.03
C(2,1,7):λ1→5(%) 0.1% 13.5% 112%
C(3,2,3):λ1→10(%) 1.6% 150% 315%
• C(3,2,3)convolutional encoder:
For this code, the probability of detection versusPefor 1, 10 and 50 iterations are represented in Fig. 3. With this encoder, 10 iterations are necessary to obtain optimal performances
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0
0.2 0.4 0.6 0.8 1
Channel error probability: Pe
Probability of detection
Iteration 1 Iteration 10 Iteration 50
Figure 3. C(3,2,3): Probability of detection versusPe for 1, 10 and 50 iterations
(the gain between the 10th iteration and the 50th iteration is close to zero). As for C(2,1,7) code, with Pe above 10−2, the performances increase significantly with the number of iterations. In table Tab. I, the gain between the 1st iteration and 10th iteration is represented.
The blind identification of this encoder is more difficult than with the former (C(2,1,7)) at the same channel error probability, but it is essentially due to the code properties. The C(3,2,3) code introduces less redundancy and consequently a lower correction capability.
It is clear that the algorithm performances depend on the number of iterations. As explained in V-A, in practice the maximum number of iterations has to be bounded to limit the processing delay. A method allowing to determine the optimal number of iterations as a function of parameters of the code andPeis under consideration
B. Probability of detection performances
To analyze performances of our method, we define three probabilities:
1) Probability of detection Pdet: Probability to identify the true encoder;
2) Probability of false-alarmPf a: Probability to identify an optimal encoder but not the true one;
3) Probability of missPm: Probability to identify no opti- mal encoder
In order to evaluate the relevance of our results, it is important to compare them with the correction capability of the code. For that, we denote BERr, the theoretical residual bit error rate obtained after decoding of the corrupted data stream with a hard decision [7]. Regarding the current standards, for example UMTS, for QoS reason of real time service the residual bit error rates is considered acceptable if it is between10−3and 10−7in urban zone and between10−3and10−4in rural zone.
Thus, we consider here that the acceptable BERr must be close to10−5.
For the C(2,1,7) code, the different probabilities are rep- resented in Fig. 4 after 5 iterations. To show the good perfor- mances of our algorithm, the limit of the 10−5 acceptable BERr is also plotted. We can see that the probability to
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0
0.2 0.4 0.6 0.8 1
BERr<10−5 BERr>10−5
Channel error probability: Pe
Probabilities
Pdet Pfa Pm
Acceptable BERr
Figure 4. C(2,1,7): Detection probabilities for 5 iterations
identify the true encoder is close to 1 for allPewhich involve aBERr after decoding lower than 10−5 (which correspond to a Pe = 0.0189). For Pe greater than 0.02, Pf a increases faster thanPm, which implies that our algorithm may find an optimal encoder different from true one, butPdetis still greater than 0.8 for Pe included in [0.02,0.03] which corresponds to a BERr of 10−4. For Pe > 0.038, the Pf a becomes lower thanPdet, but in this caseBERrbecomes greater than 3.10−4 which is not acceptable for QoS consideration, which means that this encoder will not be used in practice in such configuration.
The same probabilities for theC(3,2,3)code are shown in Fig. 5 for 10 iterations. With this encoder, at first sight our
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0
0.2 0.4 0.6 0.8 1
Channel error probability: Pe
Probabilities BERr<10−5 BERr>10−5
Pdet Pfa Pm
Acceptable BERr
Figure 5. C(3,2,3): Detection probabilities for 10 iterations
algorithm seems to be less effective, but we must not forget that the correction capability of this code is also lower. If we compare the BERr of 10−5 bounds and the associated Pdet, this algorithm also gives an excellent performances.
Indeed, with a Pe which corresponds to BERr < 2.10−4, the probability to identify the true encoder is close to 1.
VI. CONCLUSION
In this paper, we have presented an iterative method for blind identification of rate (n−1)/n convolutional encoder.
This method is based on algebraic properties of the optimal convolutional encoders and of their dual codes. In section V,
an analysis of the proposed algorithm was provided and we have shown that this algorithm exhibits good performances.
If the channel error probability is lower than the value cor- responding to BERr <10−5, the probability to identify the true encoder is close to 1. Our method requires a very small amount of received bit stream, but if we increase the amount of data available, the method is improved. In this case, the probability to obtain non-erroneous pivots increases and the probability to identify the true encoder too.
Moreover, errors introduced by mobile communication channels are usually burst errors an in this configuration, it is possible to obtain a noiseless portion of data and to easily recover the encoder. Our future work will be to extend this approach to the case of rate k/n convolutional encoder and also to adapt this method for blind recognition of punctured convolutional encoder.
ACKNOWLEDGEMENTS
This work was supported by the Brittany Region (France).
REFERENCES
[1] F. Wang, Z. Huang, and Y. Zhou, “A method for blind recognition of convolution code base on euclidean algorithm,” inproc. IEEE WiCom, China, 2007, pp. 1414–1417.
[2] J. Dingel and J. Hagenauer, “Parameter estimation of a convolutional encoder from noisy observations,” in proc. IEEE ISIT, France, June 2007, pp. 1776–1780.
[3] M. Marazin, R. Gautier, and G. Burel, “Blind recovery of the second convolutional encoder of a turbo-code when its systematic outputs are punctured,”Military Technical Academy (MTA) Review, vol. XIX, no. 2, pp. 213–232, June 2009.
[4] G. D. Forney, “Convolutional codes I: Algebraic structure,”IEEE Trans.
on Inf. Theory, vol. 16, no. 6, pp. 720–738, November 1970.
[5] R. McEliece, “The algebraic theory of convolutional codes,” inHand- book of coding theory. Elsevier Science, 1998, vol. 2, ch. 12, pp.
1065–1138.
[6] G. D. Forney, “Structural Analysis of Convolutional codes via Dual Codes,”IEEE Trans. on Inf. Theory, vol. 19, no. 4, pp. 512–5188, July 1973.
[7] R. Johannesson and K. S. Zigangirov,Fundamentals of Convolutional Coding. Wiley-IEEE Press, 1999.
[8] G. Burel and R. Gautier, “Blind estimation of encoder and interleaver characteristics in a non cooperative context,” inProc. IASTED, Scotts- dale, AZ, USA, Novembre 2003.
[9] G. Sicot and S. Houcke, “Etude statistique du seuil dans la détection d’entrelaceur,” inProc. GRETSI 2005, Belgium, 2005.
[10] 3GPP TS 25.212 v6.5.0, Multiplexing and channel coding (FDD) (release 6), The3rd Generation Partnership Project, Technical Specifi- cation Group Radio Access Network, June 2005, http://www.3gpp.org.
