Fast Blind Recognition of Channel Codes

Fast Bind Recognition of Channe Codes Reza Moosavi and Erik G. Larsson Linköping University Post Print N.B.: When citing this work, cite the origina artice. 213 IEEE. Persona use of this materia is permitted. However, permission to reprint/repubish this materia for advertising or promotiona purposes or for creating new coective works for resae or redistribution to servers or ists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. Reza Moosavi and Erik G. Larsson, Fast Bind Recognition of Channe Codes, 213, IEEE Transactions on Communications, 62, 5, 1393-145. http://dx.doi.org/1.119/tcomm.214.5614.13297 Postprint avaiabe at: Linköping University Eectronic Press http://urn.kb.se/resove?urn=urn:nbn:se:iu:diva-12536

IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION 1 Fast Bind Recognition of Channe Codes Reza Moosavi and Erik G. Larsson Abstract We present a fast agorithm that, for a given input sequence and a inear channe code, computes the syndrome posterior probabiity SPP of the code, i.e., the probabiity that a parity check reations of the code are satisfied. According to this agorithm, the SPP can be computed bindy, i.e., given the soft information on a received sequence we can compute the SPP for the code without first decoding the bits. We show that the proposed scheme is efficient by investigating its computationa compexity. We then consider two scenarios where our proposed SPP agorithm can be used. The first scenario is when we are interested in finding out whether a certain code was used to encode a data stream. We formuate a statistica hypothesis test and we investigate its performance. We aso compare the performance of our scheme with that of an existing scheme. The second scenario deas with how we can use the agorithm for reducing the computationa compexity of a bind decoding process. We propose a heuristic sequentia statistica hypotheses test to use the fact that in rea appications, the data arrives sequentiay, and we investigate its performance using system simuations. Index Terms Bind code detection; Bind decoding; Contro signaing; Sequentia probabiity ratio test. I. INTRODUCTION TODAY S wireess access systems need to offer high throughput. At the same time, as more and more users join the system, the channe resources such as bandwidth have become scarce. Many sophisticated agorithms have been devised to cope with the situation. A common approach is to use adaptive moduation and coding AMC [2], i.e., instead of using fixed transmission parameters, the transmitter changes the moduation format and coding rate on the fy in order to adapt to a changing channe quaity. To support this, there is typicay a need for a contro channe on which the AMC parameters are signaed. However, it has recenty been shown that AMC can be achieved without expicity signaing the parameters, using so-caed bind decoding. The idea is that the receiver tries to bindy identify the AMC parameters from the data coected from the channe. For instance in [3] [5] nove schemes for bind cassification of moduation formats have been proposed. In [6] [9], bind identification of encoder The associate editor coordinating the review of this paper and approving it for pubication was Prof. T.-K. Truong. Manuscript received Apri 22, 213; revised November 15, 213, anuary 23 and March 2, 214. R. Moosavi was with the Dept. of Eectrica Engineering ISY, Linköping University, Linköping, Sweden. He is now with Ericsson Research, Linköping, Sweden e-mai: reza.moosavi@ericsson.com. E. G. Larsson is with the Dept. of Eectrica Engineering ISY, Linköping University, Linköping, Sweden e-mai: eg@isy.iu.se. This work was supported in part by the Swedish research counci VR and the Exceence Center at Linköping-Lund in Information Technoogy ELLIIT. Parts of the materia in this paper were presented at the IEEE GLOBECOM 211 conference [1]. Digita Object Identifier 1.119/TCOMM.214.9.13297 9-6778/1$25. c 214 IEEE parameters have been studied. In these studies, the receiver uses some properties of the channe code such as agebraic properties of the parity check matrix or a recursive structure of the encoder that happens for exampe for convoutiona codes to bindy identify the encoder parameters. For an iustration of how bind decoding is impemented in practice, see [1, Section 16.4], where the procedure for the physica downink contro channe PDCCH decoding in LTE is described. Adaptive moduation and coding using bind decoding comes at the price of a decoding deay and more importanty energy consumption in the decoder on the receiver side. Given that the receiver is a mobie device with imited battery capacity, the atter is of some concern and any reduction in the decoding compexity incurred by the bind decoding strategy woud be vauabe. In this paper, we are concerned with finding which channe code out of a possibe set of genera inear channe codes was used to encode the data. This probem is therefore different from [3] [9] in the sense that i we assume that the moduation format is known a priori, and ii the objective is to recognize or verify which one of the channe codes out of the possibe codes which is denoted by the candidate set from hereon was used to encode the data stream. To the best of our knowedge, there is very itte work in the iterature addressing this probem in its generaity. For instance in [11], [12], two bind schemes for recognition of space-time bock codes and LDPC codes were proposed, respectivey. The proposed agorithm therein, after intercepting a number of code bocks, computes the ikeihood of each code candidate and picks the most ikey one. However, these schemes can ony be appied for recognition of specific codes. In [13], an agorithm for bind recognition of a inear code in a binary symmetric channe BSC was proposed. In order to determine if a certain code with a given parity check matrix was used to encode the data, the author therein proposed to first take hard decisions on the received data to get a rough estimate of the transmitted codeword, and then to find the inner products between the estimated codeword and the rows of the parity check matrix and use this quantity to determine whether the data was encoded with the given channe code or not. We use this atter scheme as a benchmark in our comparison. A. Contribution We present a fast agorithm that, for a given input sequence and a given inear channe code, computes the probabiity that a the parity check reations of the code are satisfied. We ca this probabiity the syndrome posterior probabiity SPP of the code. Using this agorithm, the SPP can be computed bindy, i.e., given the soft information on a received sequence we can compute the SPP for the code without first decoding

2 IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION the bits. We show that the proposed scheme for obtaining the SPP is efficient by investigating the computationa compexity of the scheme. We aso show that under some conditions, we can approximate the og-ikeihood ratio LLR of SPP with a norma distribution and we then compute the mean and the variance of it. This approximation gives us quantitative insight on how the SPP behaves. We then consider two scenarios where our proposed SPP agorithm can be used. The first scenario is when we are interested in finding out whether a certain code was used to encode a data stream. We formuate a statistica hypothesis test and we investigate the performance of the proposed test. The second scenario deas with how we can use the proposed agorithm for reducing the computationa compexity of the bind decoding process. We propose a heuristic sequentia statistica hypotheses test SSHT to use the fact that in rea appications, the data arrives in a sequentia manner, and we investigate its performance using system simuations. The paper extends our conference paper [1], among others by proposing the optimum statistica hypothesis test for the detection probem and aso the proposed SSHT test. II. COMPUTING THE SYNDROME POSTERIOR PROBABILITY In this section, we provide a fast agorithm for computing the syndrome posterior probabiity SPP for a given code. More precisey, given the soft information for an arbitrary sequence of encoded bits c, we compute the probabiity that a the parity check reations for the code are satisfied. A preiminary, sighty different version of the proposed agorithm was introduced initiay in our conference paper [1] for detecting the presence of a channe code. A modified version of this agorithm has subsequenty been used to detect an additiona oney bit piggybacked on a ineary encoded data stream [14]. Aso, a simiar agorithm has been used in [15], [16] in the context of bind frame synchronization. Since this scheme provides a basis for our further discussions, we wi present it again in this section. We wi aso use the same technique as used in [16] to find anaytica cosed-form expressions for the probabiities of fase aarm and missed detection in Section III. However, in addition to the work in [16], we aso compute the cross-correations that we need in using the centra imit theorem see Appendices A and B. Consider a genera communication ink depicted in Figure 1. The information bits b =[b 1,...,b K ] are first encoded to obtain a sequence of coded bits c =[c 1,...,c N ], generay N>K. The coded bits are then transmitted using a certain moduation scheme. Upon reception of the received vector r, the receiver computes the soft information =[ 1,..., N ] for the transmitted bits c. The soft information i for the ith encoded bit c i is usuay presented as the posterior ogikeihood ratio LLR, that is, Prci = r i =og. Prc i =1 r The proposed scheme for computing the SPP for a given code uses the fact that any codeword c obtained from the channe code with parity check matrix H, satisfies Hc =. 1 That 1 The computations are carried out in binary fied F 2. is, a the codewords of this code satisfy N K parity check reations 2 of the form c hi =, for i =1,...N K, where h i is the index of the th nonzero eement of the ith row of the parity check matrix H. Since the encoded bits may be corrupted during the transmission, if we woud take hard decisions on to obtain estimates ĉ of the coded bits, Hĉ may not be a zero vector even if the channe code with parity check matrix H was used to encode the data. However, given the soft information on c, we can compute the SPP as foows. We are interested in finding Γ Pr a syndrome checks are satisfied r N K N K =Pr c hi = r Pr c hi = r, 1 where we assumed in the ast step that the syndrome checks are independent in the sense that the two events c hi = and c hi = are independent for i i, given r. This independence assumption shoud be justifiabe for ong observation sequences and for channe codes with sparse parity check matrices, but in any case it is not crucia for the upcoming discussion. The LLR associated with the ith syndrome check of the code is given by Pr c hi = r γ i og = h i 2 1 Pr c hi = r where denotes the box-pus operation [17]. 3 From 2, we have that e γ i Pr c hi = r =og. 4 1+e γi Using 4 in 1 and taking the ogarithm yieds ogγ N K e γ i og 1+e γi N K = og1 + e γi. 5 We ca Γ the syndrome posterior probabiity SPP of the code. A. Computationa Compexity of Computing SPP In order to compute the SPP for a given code, we need to compute 2 and 5, respectivey. We examine each computation separatey. First note that the box-pus operation can be equivaenty expressed as [18, Eq. 14] 1 2 = 6 sign 1 sign 2 min 1, 2 +f 1 + 2 f 1 2, 2 since H is an N K N matrix. 3 More precisey, the definition of is: 1+tanh1 /2 tanh 2 /2 1 2 og 3 1 tanh 1 /2 tanh 2 /2 with =, = and =, see [17] for more detais.

MOOSAVI and LARSSON: FAST BLIND RECOGNITION OF CHANNEL CODES 3 Transmitter information bits b Encoder c Moduator s Receiver Channe Γ SPP Demoduator r Fig. 1. Schematic for a genera communication ink. The proposed scheme for computing syndrome posterior probabiity works on the soft information and computes the probabiity that a the parity check reations for a specific code is satisfied. where fx og1 + e x. The function fx is very webehaved and can be computed efficienty. 4 Moreover, the boxpus operator has the associative property, that is 1 2 3 = 1 2 3, n k n matrices H: H H = H.... 7 H which can be used to compute 2 recursivey. This means that the computationa compexity of computing γ i is equa to the number of nonzero eements of the ith row of the parity check matrix, say i, and is generay sma. 5 Now ooking at the second computation 5, we observe that it can aso be computed efficienty using the same function fx described above. That is, the second operation aso has inear compexity with respect to the number of terms in the corresponding expression. Therefore, the tota number of computations required to obtain the SPP is N1 R i +1, where R = K/N is the code rate. The overa computationa compexity of finding the SPP is thus ON. B. Parity Check Matrices and the SPP We need to know the parity check matrix of the code in order to find the SPP for that code. The parity check matrix for bock codes can be obtained from the corresponding generator matrix. For convoutiona codes, the parity check matrix can be obtained from the syndrome former of the code [19]. For LDPC codes, the parity reations are obtained from the code graph. Note that is some situations the transmitted bits c consist of severa smaer codewords. This is especiay true for inear bock codes with ow dimensions such as Hamming codes. More specificay, consider an n, k inear bock code with a given parity check matrix H. For a received vector of ength N consisting of N/n codewords, we can define the overa parity check matrix H consisting of N/n smaer and identica 4 For x>, fx can be tabuated with arbitrary precision. For x<, we can write fx = x +og1+e x and then use the same tabe ookup. 5 In practice, the number of nonzero eements in different rows need not be the same, that is, i i, for i i. However in many situations, a i are indeed equa such as for reguar LDPC codes, where i is given by the degree distribution of the code, or for convoutiona codes. This is very usefu for the approximations in the next section. C. Approximation of the SPP Using 5, we have N K e γ i N K Γ=exp og = 1+e γi and thus the LLR associated with the SPP Γ is given by, Γ ΛΓ og 1 Γ = N K γ i og N K 1 + e γi exp e γi, 8 γi 1+e N K 9 γ i. If e γi which happens, for instance, when the operating SNR is high and the associated binary vaue is, the above expression can be approximated as ΛΓ N K γ i. 1 This approximation has been used in many works, mainy for reducing the computationa compexity, since it does not significanty affect the performance [15], [16]. The box-pus operation can aso be approximated using the we-known approximation [17] n i n sign i min i. 11,...,n We wi use the two approximations above ater on to simpify some of our expressions, see Section III. It is worth noting that in [18], other better methods to approximate box-pus were proposed. These approximation methods can be used to compute the equations invoving box-pus operations as in 2 more accuratey.

4 IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION III. USING SPP FOR BLINDLY IDENTIFYING A CHANNEL CODE In this section, we wi consider the probem of detecting whether a certain channe code was used to encode a sequence of received baseband data or not. That is, given a soft information associated with coded bits c, we woud ike to determine if a specific code, say the channe code C t with parity check matrix H, was used to encode the data or not. We study this probem by considering the foowing hypothesis test. We consider two a priori equay ikey hypotheses H and H 1. Under hypothesis H 1, we assume that the channe code C t was used to encode the data stream, whereas under H, we assume that the data stream is not encoded with any channe code but instead the transmitted bits are i.i.d. and may take or 1 with equa probabiity. The specific choices of moduation on the communication ink does not affect our discussion and thus we assume BPSK moduation over an AWGN channe throughout the rest of the paper 6, and we define the signa-to-noise ratio SNR to be the average transmitted power divided by the noise variance. The supporting rationae behind the assumption of i.i.d. bits under hypothesis H ies in the fact that if c is obtained from some channe code C not equa to C t with parity check matrix H H H, and if we construct the vector Hc, it wi have amost no structure such that we can assume that it contains i.i.d. bits that take or 1 with equa probabiity. The impicit assumption here is that the two channe codes are distinct in the sense that their corresponding parity check matrices do not have any identica rows. However, the proposed agorithm aso works when we are interested in distinguishing between two channe codes with parity check matrices that share a few identica rows. The way to tacke these scenarios is to excude the common rows since they impose the same parity check reations on the coded bits, and ony consider the distinct parity check reations. It is worth noting that many codes in practice have indeed different non-overapping parity check matrices. Another way to resove the situations with simiar channe codes is to use different intereavers for each code which essentiay resuts amost aways in distinct parity check matrices. As an exampe, consider the case with two channe codes: i the channe code C with a given parity check matrix H, and ii a randomy permuted version of C which is obtained by first encoding the information bits with C foowed by a random intereaving of the coded bits. Let H denote the parity check matrix of the intereaved code. Consider now the kth row of H and assume that the number of nonzero eements of this row is k, with the parity check reation c hk =. Assuming that a the N! possibe intereaver sequences are equay ikey, then by the union bound the probabiity that has an identica row to the kth row of H is at most I k!n k! k N!, where N denotes the number of coded bits H c, and I K denotes the number or rows that have k nonzero eements. Since I k is at most N K, this probabiity is upper 6 This aso simpifies the procedure that we wi use ater to derive approximate cosed-form expressions for the probabiities of fase aarm and missed detection. bounded by N K k!n k! N! and is vanishingy sma 7, and hence we can assume that the two codes have no overapping syndrome checks. Having computed the SPP for the code Γ, the optima test for detecting the presence of C t is ΛΓ H1 η. 12 H We can use the approximation 1 to simpify the test. Using this approximation, the resuting test becomes N K γ i H 1 H η. 13 This suboptima test coincides with the test that was proposed in our conference paper [1]. A. Anaysis of the Code Detection Performance In order to anayze the performance of our detection agorithm, we need to know the probabiity distribution of ΛΓ under each hypothesis. The distributions are not known in cosed-form. However, using the suboptima test 13 in combination with the approximations provided in Section II for the box-pus operation, we can use the centra imit theorem CLT to approximate the probabiity distribution of ΛΓ under the two hypotheses in some cases. More precisey, according to the CLT for a sequence of identicay distributed weaky stationary random variabes {Z k } with mean m Z and variance σz 2,if V σz 2 +2 covz 1,Z i <, 14 i=2 then 1 K K k=1 Z k approaches a Gaussian random variabe with mean m Z and variance V as K increases [2]. In the case of i.i.d. random variabes {Z k } this reduces to the aw of 1 K arge number, i.e., K k=1 Z k can be approximated with a Gaussian distribution with mean m Z and variance σz 2,asK increases. Now consider the ith syndrome check constraint. Approximating γ i using 11 and as our anaysis in Appendix A shows, the probabiity distribution of γ i depends ony on the operating SNR and on the number of eements in the corresponding box-pus operator 2, i.e., the number of ones in the ith row of the parity check matrix i. That is, a the rows of the parity check matrix that have the same number of nonzero eements produce identicay distributed syndrome check constraints. Therefore, if there are sufficienty many rows with i nonzero eements, then we can approximate their corresponding summation with a Gaussian distribution given that the condition for the CLT are satisfied. The idea is thus to spit the γ i into different sets where the γ i s in each set have the same number of terms in their corresponding box-pus operator in other words, they corresponds to the rows with the same number of nonzero eements and then appy the CLT to each set to approximate the summation with a Gaussian distribution. Note that the γ i s in each set 7 For instance for rate-1/2 code with N = 1 and k =7, the probabiity is roughy 1 1.

MOOSAVI and LARSSON: FAST BLIND RECOGNITION OF CHANNEL CODES 5 are identicay distributed. However in order to use the CLT, two additiona conditions are required: i the γ i s must be stationary in the weak sense, and ii the inequaity 14 must hod. As the anaysis in Appendix B shows, covγ i,γ i depends on the SNR, i, i and on the number of common terms in the corresponding box-pus operations for γ i and γ i, say λ i,i.asλ i,i decreases, covγ i,γ i aso decreases and hence, if λ i,i is sma, we can assume that γ i and γ i are uncorreated. As an exampe consider LDPC codes. For these codes, due to the sparsity of the parity check matrix, we can assume that the γ i s are independent 8 and hence based on the degree distribution, we can spit them into different sets as mentioned above with each set containing the portion of γ i s that correspond to the rows with equa i. Let there be T such sets and et N t, t =1,...,T denote the number of eements in each set obviousy N 1 +...+ N T = N. Then we can write ΛΓ T N t γ t j 15 t=1 where γ t j, j = 1,...,N t denotes γ i s in set t. If N t is sufficienty arge, 9 then we can approximate N t γt j with a Gaussian distribution with mean zero and variance N t σ t 2 under H and with mean N t m r t and variance N t σ r t 2 under H 1 using the aw of arge numbers. Consequenty, ΛΓ can aso be approximated as a Gaussian random variabe with mean zero and variance t=1 σ 2 = T t=1 N t σ t 2 under H, and with mean and variance, T T m r = N t m r t, σr 2 = N t σ r t 2 t=1 under H 1, respectivey. Note that σ t 2 t, m r and σ r t 2 can be obtained using the anaysis in Appendix A. Aso note that for reguar LDPC codes, T =1which simpifies the expressions above. As another exampe, et us consider convoutiona codes. Since convoutiona codes have memory, γ i are not independent and hence the approximation is sighty more compicated. However, we sti can approximate ΛΓ with a Gaussian random variabe in some situations. We expain this via an exampe. Consider the standard rate-1/2 convoutiona code with constraint ength C =4, depicted in Figure 2. Using the syndrome former of the code, we get the foowing parity check matrix 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 H =............, 1 1 1 1 1 1 1 16 8 In other words λ i,i is sma for any i and i, with i i. 9 in the order of 1 as we have seen from our numerica experiments. 1 The rest of the entries in the matrix are zero. b i D + + + D Fig. 2. The standard rate-1/2 convoutiona code with constraint ength 4. The generators for this code are g 1 =15and g 2 =17in octa. and hence by arranging the coded bits and defining a window S i as foows, S i c 1 i 1 c 1 i c 1 i+1 c 1 i+2 c 1 i+3 c 1 c 2 i 1 c 2 i c 2 i+1 c 2 i+2 c 2 i+3 c 2 we get N b syndrome check constraints + D + i+4... i+4... c j i =, for i =1,...,N b, i,j S i where N b denotes the number of information bits. Thus, by defining c 1 i c 2 i γ i = j i, i,j S i and using 5, the SPP for this code can be found directy. We use the foowing convention to specify the window S i, which wi be used ater in Section V, {c 1 i, 1, 2, 3, c 2 i, 2, 3} 17 meaning that the syndrome check constraint at time instant i consists of coded bits c 1 i+j and c2 i+j where j {, 1, 2, 3} and j {, 2, 3}. For this code, a the rows of the parity check matrix have = 7 nonzero eements, and thus a γ i are identicay distributed. Moreover, due to the recursive nature of the code, covγ i,γ i depends ony on the difference i i. In fact for this exampe, since S i and S i share no common coded bits when i i 4, and since the transmitted bits are statisticay independent of each other, γ i and γ i are independent for i i 4. That is, covγ 1,γ i =, for i 4. This enabes us to approximate ΛΓ with a Gaussian random variabe in this case too. 11 As the exampes above showed, we can in many situations approximate the distribution of ΛΓ with a Gaussian distribution under the two hypotheses. According to Appendix A, under H 1, E {γ i } depends on the operating SNR and is inversey proportiona to i see 38 42. This is intuitive, since both decreasing the SNR and having many terms in the ith parity check reation c hi =wi increase the risk of error in the received sequence and hence Pr c hi = r decreases. This is aso seen from Figure 3, where we have potted E {γ i } under H 1 as a function of the SNR for different vaues of i. Another important observation from equation 11 Note that in this case, γ i s are stationary as discussed above.

6 IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION E {γi H1} 35 3 25 2 15 1 5 i = 2 i = 7 i = 12 1 2 3 4 5 6 7 8 9 1 SNR [db] Fig. 3. E {γ i } under H 1 as a function of SNR for different vaues of i 38 is that at high SNR regime, E {γ i } scaes ineary with SNR. This can aso be seen from Figure 3. Having computed the mean and variance under each hypothesis, we can approximate the fase aarm and the detection probabiities as, P F Pr { K } η γ i >η H = Q σ, 18 { K } η mr P D Pr γ i >η H 1 = Q, 19 where σ 2 denotes the variance under H and m r and σr 2 denote the mean and the variance under H 1, and Qx is the Gaussian error integra Q- function, defined as 1 Qx = e t2 /2 dt. 2π x The receiver-operating-characteristics ROC is, therefore, given by σ Q 1 P F m r P D = Q. 2 Note that in the above expression, σ and σ r scae as N whereas m r scaes as N. We stress that whie the anaysis provided in this section is restricted to some specia cases, the proposed detection agorithm as such is appicabe to any inear channe code without any modification incuding the tai-biting convoutiona codes; the corresponding syndrome checks have to be written down in each specific case. The anaysis provided in this section can provide, for instance, a rue-of-thumb for the required number of observations in order to achieve certain fase aarm and detection probabiities. As for the performance with different rates, typicay ower-rate codes are easier to detect than higher-rate codes, because there are more syndrome checks for ower-rate codes, and hence more reiabe decisions can be made. σ r σ r B. Appication of the Code Detection Scheme to Convoutiona Codes In this section, we investigate the performance of our detection scheme for standard rate-1/2 convoutiona codes. In particuar, we consider three choices for the true channe code: a C 2 with constraint ength 4 depicted in Figure 2, b C 5 with constraint ength C 2 = 7, and c C 7 with constraint ength 9. The generators for these codes are given in Tabe I. For each scenario, we et hypothesis H denote the hypothesis under which the transmitted bits are i.i.d. and hypothesis H 1 denote the hypothesis under which the corresponding convoutiona code is used to encode the data. We assume that both hypotheses are equay ikey a priori and that the coded bits are transmitted over an AWGN channe using BPSK moduation. Let N b denote the number of information bits. The syndrome check constraints for each code is given in Tabe I. As a benchmark for comparison, we aso impemented the test proposed in [13]. However, since the agorithm therein can ony be used in binary symmetric channes BSC, we consider an equivaent BSC channe with cross-over probabiity equa to the error probabiity of the AWGN channe in our mode with BPSK moduation. Figures 4a 4c iustrate the ROC curves of the different schemes for the case where the SNR is db and for two different vaues of N b for the three convoutiona codes, respectivey. More precisey, we have provided ROC curves for i the statistica test given by 12, ii the suboptima test given by 13, iii the suboptima test where in addition to 13, we aso used the approximation given by 11 in computing γ i, iv our anaysis provided by 2, and v the test using the agorithm in [13]. 12 As the resuts show, the proposed statistica test i has better performance compared to the other tests, speciay compared to the test in [13]. This is reminiscent of hard-decision decoding in an AWGN channe which is known to be roughy 2 db worse than soft-decision decoding [21, pp. 612]. Additionay, we see that the anaysis provided by 2 is very cose, speciay when N b is arge, to the empirica performance of the corresponding suboptima test case iii. We aso see, as expected that by increasing the number of information bits, a better detection performance is achieved. Aso we see by comparing the ROC curves for the three scenarios that it is easier to recognize the convoutiona code C 2. This is expected since for this code the number of non-zero eements in the parity check matrix is smaer than for the other two convoutiona codes with constraint engths 7 and 9. The mean vaues according to our anaysis in Section III-A are.154,.411, and.17 for the three codes C 2, C 5 and C 7, respectivey. IV. USING SPP FOR REDUCING THE COMPUTATIONAL COMPLEXITY OF BLIND DECODING In a system where bind decoding is used, the receiver is interested in bindy detecting the channe code used by the transmitter as eary as possibe with a given probabiity 12 Simiar simuation resuts, but without the curves representing the optima test, the anaysis and the comparison with [13], were provided in our conference paper [1]. However, the definition of SNR given there contained an error.

MOOSAVI and LARSSON: FAST BLIND RECOGNITION OF CHANNEL CODES 7 Channe Code Const. Length Generators Syndrome Check Constraint C 1 3 5,7 {c 1 i, 1, 2, c 2 i 1, 2} C 2 4 15,17 {c 1 i, 1, 2, 3, c 2 i, 2, 3} C 3 5 23,35 {c 1 i, 2, 3, 4, c 2 i, 1, 4} C 4 6 53,75 {c 1 i, 2, 3, 4, 5, c 2 i, 1, 3, 5} C 5 7 133,171 {c 1 i, 3, 4, 5, 6, c 2 i, 1, 3, 4, 6} C 6 8 247,371 {c 1 i, 3, 4, 5, 6, 7, c 2 i, 1, 2, 5, 7} C 7 9 561,753 {c 1 i, 1, 3, 5, 6, 7, 8, c 2 i, 4, 5, 6, 8} TABLE I STANDARD RATE-1/2 CONVOLUTIONAL CODES WITH THEIR GENERATORS IN OCTAL REPRESENTATION AND ALSO WITH THEIR CORRESPONDING SYNDROME CHECK CONSTRAINTS. of error. Assume that there are in tota M candidate codes C 1,...,C M, with the corresponding parity check matrices H 1,...,H M. If a soft channe symbos are presented to the receiver at once, then the optima strategy woud be to compute the syndrome probabiities γ m 1,γ m 2,... for a candidate codes m = 1,...,M, and then compute the SPP for each candidate code and pick the candidate code that yieds the maximum SPP. However, in practice the data arrives sequentiay and the objective is to decide on the code candidate as soon as possibe, subject to some constraints on the detection performance. The performance criteria may be expressed in terms of probabiities of detection and fase aarm as foows: i. The probabiity of detection shoud be above a certain threshod, say P min D. ii. The probabiity of fase aarm shoud be smaer than a certain threshod, say P max F. This probem can be formuated as a sequentia procedure for mutipe hypothesis testing [22], [23]. When there are two aternative hypotheses H and H 1, that is, when there are two possibe candidate codes M =2, then the optima test is known and is found by the so-caed sequentia probabiity ratio test SPRT [24]. Here, the optimaity is in the sense that among a hypothesis tests satisfying the above constraints on detection and fase aarm probabiities, the SPRT requires the smaest number of observations N. However, when there are more than two aternative hypotheses M >2, then the optima test is often not known, or the optima test has a very compicated structure that imits its use in practice [22]. In these cases, one may opt for heuristic soutions. In what foows, we wi propose a sequentia test for our probem and evauate its performance via simuations. A. Proposed Sequentia Statistica Hypothesis Test Let H m, m =1,...,M denote the hypothesis under which channe code C m was used to encode the data, and et πm denote the prior probabiity of this hypothesis, i.e. the probabiity of picking channe code C m by the transmitter. Having observed the received vector r, the minimum probabiity-oferror detector chooses the hypothesis with the argest posterior probabiity Pr H m r [25]. That is, the decision is k =argmax m Pr H m r. 21 In our probem, we can find, for each candidate code C m, the syndrome posterior probabiity Γ m using the corresponding parity check matrix H m ; Γ m P a syndrome checks of H m are satisfied r. 22 If the transmission was error-free, then Γ m woud be 1 for the true code and it woud be zero for the rest of the code candidates. Due to errors introduced in the transmission, this wi not be the case in practice. However, we know that the SPP for the true code is ikey to be higher than that of the others, and that the difference wi be arger as the ength of the observed sequence increases. Therefore, we propose the foowing sequentia probabiity ratio test to detect the channe code. Proposed SSTH. Let Γ n m denote the syndrome posterior probabiity for candidate code C m obtained at stage n, i.e., after observing γ m 1,γ m 2,...,γ n m, and define the vector P n =[p n 1,...,pn M ], where p n m = Γ n m M. Γn i Given a threshod ζ to be expained beow, we use the foowing rue as the stopping criterion N A = first n 1, such that p n k ζ, for some k =1,...,M. 23 The decision rue at the stopping time is k =argmax m p NA m. 24 Note that according to the proposed SSHT, the required number of observations is finite. That is so because, we have exp n og 1+e γm j p n m = Γn m M Γ n i = n M = n M exp n og 1+e γi j 1+e γm j 1+e γi j 1. 25 Assume that the mth code was used by the transmitter to encode the data. For the true code m, the syndrome probabiity constraints γ m i are ikey to be greater than zero how much greater they are than zero generay depends on the operating SNR, whereas for a random code, the syndrome probabiity

8 IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION Probabiity of Detection Probabiity of Detection Probabiity of Detection 1.9.8.7.6.5.4.3.2.1 N b = 75 i Proposed Test given by 12 ii Suboptima Test given by 13 iii Suboptima Test with Approx. 11 iv Anaysis given by 2 v Proposed Test in [13].1.2.3.4.5.6.7.8.9 1 1.9.8.7.6.5.4.3.2.1 N b = 75 Probabiity of Fase Aarm a Constraint Length 4 N b = 1 N b = 1 i Proposed Test given by 12 ii Suboptima Test given by 13 iii Suboptima Test with Approx. 11 iv Anaysis given by 2 v Proposed Test in [13].1.2.3.4.5.6.7.8.9 1 1.9.8.7.6.5.4.3.2.1 N b = 75 Probabiity of Fase Aarm b Constraint Length 7 i Proposed Test given by 12 ii Suboptima Test given by 13 iii Suboptima Test with Approx. 11 iv Anaysis given by 2 v Proposed Test in [13].2.4.6.8 1 Probabiity of Fase Aarm c Constraint Length 9 N b = 1 Fig. 4. Receiver Operating Characteristic curves for the different schemes at an SNR of db. constraints woud take both positive and negative vaues with equa probabiity. This means that as n increases n n 1+e γm j, if j m, 1+e γi j and hence p n m 1 as n increases. Remark 1. The overa error probabiity P e of the proposed scheme depends on the vaue chosen for the threshod ζ. Whie it woud be desirabe to know the exact reation between P e and ζ, due to the unknown probabiity distribution of the γ m i s, this reation is not known. The error probabiity is, however, in the order of 1 ζ. In Section V, we use a greedy search agorithm for choosing ζ such that the desired probabiity of error is achieved. Remark 2. The impicit assumption is that the tota number of avaiabe observations is infinite. In other words, we may continue samping unti one of p n k is greater than ζ. In many situations, however, there is a finite number of observations. Those situations can be considered as sequentia hypothesis tests with finite horizon and the optima soution may be found using the method of backward induction [26]. More precisey, if we reach the fina stage N where no more observations are avaiabe, then the optima decision is known, resuting in a certain probabiity of error, say Pe N. The decision in the previous stage N 1 is thus to stop samping if the expected error probabiity in that stage is ess than Pe N and to continue and take the ast sampe, otherwise. The decision for the rest of the stages can be found simiary. As the simuation resuts in Section V show, the proposed SSHT scheme requires sma number of observations for most operating SNR and thus the assumption of infinite avaiabe observations is not crucia in this study. Remark 3. If the threshod ζ is greater than 1/2, then at the stopping time, ony one of the codes can satisfy 23, since m pn m =1. Aso, the threshod can be chosen differenty for different code candidates, i.e. if the cost of making a specific error is arger than the others, then the corresponding threshod may be chosen arger. B. A Rue-of-Thumb for the Required Number of Observation In this section, we provide an approximation of the error rate of the suboptima test with a fixed number of observations. This approximation can be used to obtain a rue-of-thumb for the required number of observations according to the different schemes as expained beow. Let γ m j, j =1,...,N denote the LLR associated with the jth parity check reation corresponding to the mth code candidate C m. Using the suboptima test 13, the error probabiity under hypothesis H m is,

MOOSAVI and LARSSON: FAST BLIND RECOGNITION OF CHANNEL CODES 9 P e Hm =Pr N =1 Pr 1 γ i j > N γ i j M N Pr i m N γ i j γ m j, for some i I m H m N γ m j, for a i I m H m N γ m j, 26 where I m {1, 2,m 1,m+1,...,M}. The approximation in the ast step is due to the independence assumption that we N have made among the events γi j > N γm j and N γi j > N γm j, for i i. 13 Using the anaysis in Appendix A and B and appying the CLT, we can write M P e Hm 1 1 Q m m r 27 2 i 2 + σ i m σ m r where m m r and σ r m 2 denote the mean and the variance of N γm j and σ i 2 N denotes the variance of γi j under H m. Here aso we assume that N γm j and N γi j are independent, which as expained before happens when the code candidates have distinct parity check matrices, see Section III. The overa error probabiity is therefore, P e 1 M P e Hm. 28 M m=1 The ratio m m r / σ r m 2 i 2 + σ scaes as N and hence we can use the above approximation to find a rough estimate of the required number of observations when appying the test with a fixed number of observations. More particuary, for scenarios where we use a fixed given base code with different intereavers to obtain different channe codes, then σ m r m m r = N a, for a i, m =1,...,M 2 i 2 + σ for some constant a that depends on and on the SNR and hence we have, NM 1 [ a 2 Q 1 ] 1 1 P e 1 2 M 1. 29 Equation 29 in combination with our findings via simuations suggest a rue-of-thumb approximation for the number of observations required by the different schemes, for scenarios where we have a base code with different intereavers as the code candidates. More precisey, if N M P e is the number of observations for a given candidate set size M and a given error probabiity P e, then for an arbitrary candidate set size M and arbitrary error probabiity P e,wehave N M P e N M P = d, e where d Q 1 1 1 P e 1 M 1 Q 1 1 1 P e 1 M 1 2. 3 13 Note that for the case with M =2, this approximation is exact. Using the above equation, if the required number of observations for specific choices of M and P e is known, then we can use that as a basis to compute the required number of observations for arbitrary candidate set sizes and arbitrary error probabiities. The important observation is that the increase in the number of required observations when we increase the candidate set size from M to M M >M, is independent of the choice of the base code, and in given by 3. Another important observation that we get from 3 is that the increase in the number of observations required when the candidate set size is increased is ogarithmic. For instance, the increase in the required number of observations when we increase the candidate set size from say 2 to 4 is arger than that when the candidate set size is increased from 8 to 16. This might not be immediatey cear from 3, however we can see this by first using the fact that for arge x, Qx 1 2 e x2 /2. Therefore, since QQ 1 x = x, we can write Q αq 1 x 1 2 2xα2. 31 Now, using 31 in 3 in combination with the Tayor series approximation 1 1 P e 1 M 1 og1 P e, M 1 we have d og M 1 og2og1 P e og M 1 og2og1 P e, 32 which confirms the above observation. V. SIMULATION RESULTS In this section, we provide some simuation resuts for the performance of our proposed SSHT scheme. We assume as before BPSK transmission over an AWGN channe. We consider two scenarios for the code candidates: i where we use the standard rate-1/2 convoutiona with constraint ength 4 depicted in Figure 2 in combination with different randomy chosen intereavers to obtain different code candidates, and ii where we consider a cass of standard rate-1/2 convoutiona codes with different constraint engths as our candidate codes [21]. In this case, we consider 7 different rate 1/2 convoutiona codes C 1,...,C 7 with constraint engths 3 to 9 respectivey. The octa representations of the generator poynomias for different codes are given in Tabe I. Using syndrome formers of the codes, we get N b syndrome check constraints for each of the codes N b is the number of information bits. The syndrome check constraints are aso presented in Tabe I using the convention specified in Section III. A the codes are equay ikey to be chosen by the transmitter. For the second scenario, the candidate set is constructed using the convoutiona codes C 1,...,C M. Tabe II and III show the average number of observations N SSHT required by the proposed SSHT scheme for two vaues of the error probabiity at an SNR of 3 db, for the first and the second scenario respectivey. For comparison, the average number of observations N FIX required to achieve the same error rate as with the optima test with a fixed observation ength and the corresponding reduction in the required number of observations are presented as we. As the resuts show, a

1 IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION 1 2 Error Rate 1 3 M =2 M =4 M =8 M =16 SSHT Test Fixed Test 4 6 8 1 12 14 16 18 2 22 Number of Observations Fig. 5. Error rate as a function of the number of observations for different vaues of the candidate set size M, at an SNR of 3 db. For comparison, the corresponding resuts predicted by 3 are aso presented by dashed ines. reduction of roughy 6% can be achieved with our proposed SSHT scheme. This reduction is in accordance with the expectations, as the number of required observations with SPRT is typicay about one-haf to one-third of that with the optima test with a fixed number of observations [22]. For the first scenario, we can appy the proposed rue-of-thumb to obtain a rough estimates for N SSHT and N FIX. The corresponding normaized vaues are aso presented in Tabe II. We can see that the resuts are very cose to those predicted by equation 3, where we use the case with M =2and P e =.1 as the basis for the computations. For a more detaied comparison of the two schemes, we have potted the empirica error rate curves as a function of the number of observations for the first scenario at SNR of 3 db in Figure 5. Here aso, the curves highighted by dashed ines represent the corresponding predictions via 3, where for each M we take the case with P e =.5 as the basis. We see that the proposed rue-of-thumb offers quite accurate predictions in a the cases. Aso, we see that the proposed SSHT scheme requires significanty fewer observations on the average compared to the optima test with a fixed number of observations to achieve a certain probabiity of error. Another observation is that as M increases, we need more observations on the average to achieve a given error probabiity for both tests, and that this increase is greater for smaer vaues of M, as predicted by 32. To see the effect of the SNR on the performance, we have potted the average required number of observations to achieve an error probabiity of 1% as a function of SNR for different vaues of M for the first and the second scenario in Figures 6 and 7, respectivey. For the first scenario, we aso potted the corresponding predicted resuts using 3, where we have used the resuts for M =2as the basis. Again, we see a cose match between the predicted resuts and the empirica resuts. Aso we see that for higher SNR, the proposed SSHT requires very few observations in order to work. We stress again that codes with parity check matrices that Average Number of Observations 1 2 M =2 M =4 M =8 M =16 Empirica Anaysis.5 1 1.5 2 2.5 3 3.5 4 SNR [db] Fig. 6. Required number of observations as a function of SNR with our proposed SSHT scheme and for the first scenario with error probabiity of 1%. The corresponding anaytica resuts obtained from 3 are aso presented by dashed ines. have a sma number of nonzero eements in each row provide better operating conditions for our proposed SPP scheme. This is so because for such codes, the sum in 1 has fewer terms. Since in this sum, each additiona term contributes an additiona risk of making an error, having fewer terms means ess overa probabiity of error. This expains the differences in the required number of observations for the two different scenarios in Figures 6 and 7. For instance, consider the case M = 2. In the first scenario, we have two randomy intereaved convoutiona rate-1/2 codes with constraint ength 4, whereas in the second scenario, we have two rate-1/2 convoutiona codes with constraint engths 3 and 4 respectivey. The syndrome check constraints consist of 5 and 7 terms for the constraint engths 3 and 4 respectivey see Tabe I. Therefore, we expect that the average required number of observations in the second scenario is smaer than in the first scenario, for a given error probabiity. This is aso the case in the presented resuts see Tabes II and III. VI. CONCLUSIONS AND FUTURE WORK In this paper, we presented a fast agorithm for bindy recognizing which channe code from a candidate set that was used to encode a data stream. The proposed agorithm uses the fact than any inear code satisfies a certain set of parity check reations, incuding convoutiona codes with and without tai-biting. Our agorithm obtains the probabiities that a parity check constraints are satisfied, caed the syndrome posterior probabiity SPP of the code here, for a code candidates and then compares these probabiities. We aso proposed a sequentia hypothesis test that makes decisions before coecting a avaiabe data, hence saving computationa compexity. Quantitativey, under typica operating conditions, the agorithm identifies the correct code out of 16 candidates in 99% of the cases by observing ess than 5 sampes, at an SNR of 4 db. The proposed scheme is potentiay usefu for compexity reduction of the PDCCH decoding in LTE. A detaied study

MOOSAVI and LARSSON: FAST BLIND RECOGNITION OF CHANNEL CODES 11 P e =.1 P e =.1 Set Size M =2 M =4 M =8 M =16 M =2 M =4 M =8 M =16 N SSHT 36 48 59 68 58 71 82 92 N FIX 82 18 129 148 146 173 21 217 Reduction 56% 55% 54% 54% 6% 59% 59% 58% Normaized N SSHT 1 1.33 1.64 1.89 1.62 1.98 2.28 2.56 Normaized N FIX 1 1.32 1.58 1.8 1.78 2.11 2.45 2.65 d 1 1.36 1.64 1.9 1.76 2.14 2.43 2.69 TABLE II REQUIRED NUMBER OF OBSERVATIONS BY THE PROPOSED SSHT SCHEME N SSHT AND BY THE OPTIMAL TEST WITH A FIXED OBSERVATION LENGTH N FIX FOR DIFFERENT VALUES OF M AND AN SNR OF 3 DB, FOR THE FIRST SCENARIO.IN THIS TABLE, NORMALIZED VALUES OF N SSHT AND N FIX ARE ALSO GIVEN TO FACILITATE EASY COMPARISON WITH d IN 3. THE NORMALIZATION FACTOR IS THE CORRESPONDING VALUES FOR THE CASE WITH M =2AND P e =.1. P e =.1 P e =.1 Set Size N SSHT N FIX Reduction N SSHT N FIX Reduction M =2 23 56 59% 36 1 64% M =3 33 81 59% 49 14 65% M =4 48 114 58% 69 196 65% M =5 62 157 6% 87 274 68% M =6 8 25 61% 19 348 69% M =7 99 27 63% 14 45 69% TABLE III REQUIRED NUMBER OF OBSERVATIONS WITH OUR PROPOSED SSHT SCHEME AND ACCORDING TO THE OPTIMAL TEST WITH A FIXED OBSERVATION LENGTH FOR DIFFERENT VALUES OF M AND AN SNR OF 3 DB, FOR THE SECOND SCENARIO. Average Number of Observations 1 3 1 2 M =2 M =3 M =4 M =6 M =5 M =7.5 1 1.5 2 2.5 3 3.5 4 SNR [db] Fig. 7. Average number of observations required as a function of SNR with our proposed SSHT scheme for different vaues of the candidate set size M, at error probabiity of 1%, for the second scenario. of this topic is a possibe direction for the extension of this work. Another potentia appication of our agorithm, that may aso be studied in future work, is to faciitate entirey bind mutipe access based on the terminas bindy recognizing their payoad data. In this case, the base station woud not signa any expicit contro information or AMC parameters at a. This may be faciitated either by assigning different terminas different codes, or different intereaving sequences. Since our agorithm tends to perform better for codes with ow variabe node degrees, in this foreseen appication appropriate consideration has to be made when choosing the channe codes. APPENDIX A COMPUTING THE MEAN AND THE VARIANCE OF γ k Assume, without oss of generaity, that the moduation scheme is BPSK, and consider the transmission of bits c 1, c 2,...,c over an AWGN channe with noise variance N /2 per rea dimension. The received symbo at time instance i is r i = s i + n i, where s i denotes the BPSK symbo binary is mapped to +1, and binary 1 is mapped to -1 and n i is the additive white Gaussian noise with mean zero and variance N /2. We consider two hypotheses: H 1 under which we know that c i =, and H under which the transmitted bits are i.i.d. and take or 1 with equa probabiity, which consequenty means that c i may take or 1 with equa probabiity no structure. We are interested in computing the mean and the variance of γ = sign i min i, under the two hypotheses, where i denotes the posterior conditiona LLR of c i. Let X sign i, Y min i. Since we assume BPSK transmission over an AWGN channe, we have [21] i =Λc i r i = 4r i. 33 N According to our system mode, r i has a mixture Gaussian distribution. This aows us to use the foowing emma to simpify the computations for finding E {Y }. Lemma. Consider two random variabes: 1 W with a mixture Gaussian probabiity distribution of the form pn m, σ 2 +1 pn m, σ 2, for some given p 1, m and σ 2.

12 IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION 2 A zero-mean Gaussian random variabe Z with variance σ 2. Then, W and Z +m have the same probabiity distribution. Proof. Let f Z z denote the probabiity distribution function pdf of Z, i.e., f Z z = 1 2πσ exp z2, 34 2σ 2 and et F Z z denote its cumuative distribution function cdf. We start by finding the cdf for W. Wehave, F W w =Pr{ W <w} 35 =Pr{ w <W<w} = F W w F W w for w, and for w<, F W w =. Thus, for w the pdf of W is given by f W w = d dw F W w =f W w+f W w = pf Z w + m+1 pf Z w m + pf z w + m+1 pf Z w m = f Z w + m+f Z w m, 36 since f Z z =f Z z. Therefore, { fz w + m+f f W w = Z w m, w, otherwise. 37 It is straight forward to check that Z + m has the same pdf too, which competes the proof. A direct concusion of this emma is that n i +1 and n i 1 have the same pdf, so E {Y } = 4 { } E min N r i = 4 { } E min N n i +1, 38 under both hypotheses. That is, we may work with n i rather than r i. The important observation is that r i, i =1,..., are i.i.d. under H whie under H 1, they are not independent. However as we wi see ater, the same technique can be used to simpify the computations for finding the statistica properties under H 1. Before we continue further, it is worth noting that since E { X 2} =1, E { γ 2} is aso the same under both hypotheses and is given by E { γ 2} = E { { Y 2} = 16 } 2 N 2 E min r i = 16 { } N 2 E min r i 2. 39 Now, we are ready to compute the means and the variances of γ under the two hypotheses. Under hypothesis H, since c i is equay ikey to be or 1 no structure, X and Y are i independent. More specificay, X wi be a binary random variabe that takes one of the vaues { 1, +1} with equa probabiity, and thus E {γ H } = E {XY H } = E {X H }E {Y H } =, 4 and therefore σ 2 E { γ 2 } { H = E Y 2 } = 16 { } N 2 E min r i 2. 41 Under hypothesis H 1, X and Y are not independent. Indeed, if there are no errors, then X =1. To compute the mean of γ under hypothesis H 1, we can write m t E {γ H 1 } = E {Y H 1,X =1} Pr{X =1 H 1 } E {Y H 1,X = 1} Pr{X = 1 H 1 }. 42 The event {X =1} impies that either there have been no errors or there have been an even number of errors in the received sequence. Simiary, the event {X = 1} impies that there have been an odd number of errors in the received sequence. Therefore, 2 Pr{X =1} = Pe 2i 1 P e 2i, 43 2i i= where P e is the bit error probabiity of the channe. Since, we assume BPSK moduation, an error in the received sequence r i occurs, when i n i < 1, and c i =, or ii n i > 1, and c i =1. Using this, and the resuts from the Lemma, we can write E {Y H 1,X =1} = 4 { } E min N n i +1 B 1 44 E {Y H 1,X = 1} = 4 { } E min N n i +1 B 2 45 where the event B 1 B 2 is defined as the event that among noise sampes, none or an even an odd, respectivey number of the sampes are smaer than -1. We can finay write σt 2 E { γ 2 } H 1 m 2 t = 16 N 2 E { min r i 2 } m 2 t. 46 Note that the above quantities a depend ony and on the noise variance N /2 and can be found numericay. Once they are computed, they can be saved in a ook-up tabe for future use. Aso note that by increasing, Pr{X =1} decreases and hence m t decreases too. APPENDIX B COMPUTING THE CORRELATION BETWEEN γ k AND γ k Consider again the same system mode as presented in Appendix A and consider the transmission of + α α 1 bits c 1, c 2,...,c +α. Let z c i and z = +α c i.we i=α+1 consider again two hypotheses: H 1 under which both z and z are zero, and H under which z and z may take or 1 with equa probabiity no structure. We are interested in computing the correation between γ and γ where as before, γ = sign i min i, and +α +α γ = sign i min i. i=α+1 i=α+1 Under hypothesis H, since the transmitted bits are independent of each other, γ and γ are independent and hence E {γ γ H } = E {γ H }E { γ H } =. 47

MOOSAVI and LARSSON: FAST BLIND RECOGNITION OF CHANNEL CODES 13 Under hypothesis H 1, if α, then γ and γ are independent and hence for α, E {γ γ H 1 } = E {γ H 1 }E { γ H 1 } = m 2 t. 48 For α<, by defining X sign i, +α X i=α+1 sign i, Y min i, Ỹ min +α i i=α+1 we have { } E {γ γ H 1 } = E X XY Ỹ H 1 { = E Y Ỹ H 1,X X } =1 Pr {X X } =1 H 1 { E Y Ỹ H 1,X X } = 1 Pr {X X } = 1 H 1 Under hypothesis H 1, since signx 2 =1, the event B {X X =1} impies that among +α received symbos, either there has been no error or there have been an even number of errors in the first and the ast α tota 2α sampes symbos. Therefore, α 2α Pr{B} = Pe 2i 1 P e 2α 2i, 49 2i i= where as before P e is the bit error probabiity of the channe. Using the resut of the Lemma in Appendix A, we can write { } E Y Ỹ H 1, B = 5 { } 16 +α N 2 E min n i +1 min n i +1 B, i=α+1 { E Y Ỹ H 1, B c} = 51 { } 16 +α N 2 E min n i +1 min n i +1 B c, i=α+1 where B c denotes the compement of the event B. [9]. Dinge and. 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Houcke, Bind frame synchronization for error correcting codes having a sparse parity check matrix, IEEE Trans. Commun., vo. 57, no. 6, pp. 1574 1577, une 29. [16] R. Imad and S. Houcke, Theoretica anaysis of a MAP based bind frame synchronizer, IEEE Trans. Wireess Commun., vo. 8, pp. 5472 5476, Nov. 29. [17]. Hagenauer, E. Offer and L. Papke, Iterative decoding of binary bock and convoutiona codes, IEEE Trans. Inf. Theory, vo. 42, pp. 429 445, Mar. 1996. [18] T. Cevorn and P. Vary, Low-compexity beief propagation by approximations with ookup-tabes, 5th Internantiona ITG Conf. Source Channe Coding SCC, Erangen, Germany, an. 24. [19] R. ohannesson and K. Sh. Zigangirov, Fundamentas of Convoutiona Coding, IEEE Press, 1999. [2] R. Durrett, Probabiity: Theory and Exampes, Cambridge University Press, 21. [21]. G. Proakis and M. Saehi, Digita Communication, 5th Edition, McGraw-Hi, 28. [22] C. W. Baum and V. V. Veeravai, A sequentia procedure for mutihypothesis testing, IEEE Trans. Inf. Theory, vo. 4, pp. 1994 27, Nov. 1994. [23] H. Chernoff, Sequentia Anaysis and Optima Design, Phiadephia: SIAM, 1972. [24] A. Wad, Sequentia Anaysis, NY: Wiey 1947. [25] H. L. Van Trees, Detection, Estimation, and Moduation Theory, Wiey, 21. [26] T. S. Ferguson, Optima Stopping and Appications, [Onine]. Avaiabe: http://www.math.uca.edu/ tom/stopping/contents.htm, Visited Dec. 212. REFERENCES [1] R. Moosavi and E. G. Larsson, A fast scheme for bind identification of channe codes, in Proc. IEEE Gobecom, Dec. 211. [2] A.. Godsmith and S. G. Chua, Adaptive coded moduation for fading channes, IEEE Trans. Commun., vo. 46, no. 5, pp. 595 62, May 1998. [3] S. Xi and H. C. Wu, Robust automatic moduation cassification using cumuant features in the presence of fading channes, in Proc. IEEE WCNC, pp. 294 299, Apr. 26. [4] H. C. Wu, M. Saquib and Z. Yun, Nove automatic moduation cassification using cumuant features for communications via mutipath channes, IEEE Trans. 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Reza Moosavi received his B.Sc. degree in Eectrica Engineering from Isfahan University of Technoogy, Isfahan, Iran in 25, his M.Sc. degree from Chamers University of Technoogy, Göteborg, Sweden, in 28 and his Ph.D. degree from Linköping University, Linköping, Sweden in 214. Since December 213, he is an experienced researcher with Ericsson Research at Linköping, Sweden. His research interests incude resource aocation, signaing protocos in ceuar systems, and future radio access technoogies.

14 IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION Erik G. Larsson received his Ph.D. degree from Uppsaa University, Sweden, in 22. Since 27, he is Professor and Head of the Division for Communication Systems in the Department of Eectrica Engineering ISY at Linköping University LiU in Linköping, Sweden. He has previousy been Associate Professor Docent at the Roya Institute of Technoogy KTH in Stockhom, Sweden, and Assistant Professor at the University of Forida and the George Washington University, USA. His main professiona interests are within the areas of wireess communications and signa processing. He has pubished some 8 journa papers on these topics, he is co-author of the textbook Space-Time Bock Coding for Wireess Communications Cambridge Univ. Press, 23 and he hods 1 patents on wireess technoogy. He is Associate Editor for the IEEE Transactions on Communications and he has previousy been Associate Editor for severa other IEEE journas. He is a member of the IEEE Signa Processing Society SAM and SPCOM technica committees. He is active in conference organization, most recenty as the Technica Chair of the Asiomar Conference on Signas, Systems and Computers 212 and Technica Program co-chair of the Internationa Symposium on Turbo Codes and Iterative Information Processing 212.