Parallel Concatenated Chaos Coded Modulations

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1 Parallel Concatenated Chaos Coded Modulations Francisco J. Escribano, M. A. F. Sanjuán Departamento de Física Universidad Rey Juan Carlos 8933 Móstoles, Madrid, Spain Luis Lópe Departamento de Ingeniería Telemática y Tecnología Electrónica Universidad Rey Juan Carlos 8933 Móstoles, Madrid, Spain Abstract In this article we propose a parallel concatenated encoder similar to a turbo trellis coded modulation, but where the constituent encoders are chaos coded modulators. We show that, even when the uniform error property does not hold for the kind of constituent chaos coded modulations employed, it is still possible to draw a reasonable bound for the bit error probability at the error floor region based on the interleaver structure. The simulations validate the bounds and show that the dynamics of the underlying chaotic maps, rather than the quantiation level of the constituent encoders, is the most important factor to account for the error rate behaviour at the error floor region.. INTRODUCTION In recent times, we have witnessed the arising of proposals in chaos coded communications where the drawbacks of the systems based on one-dimensional discrete chaotic maps [] have been overcome by means of an increase in the dimensionality of the associated encoding map and by a generaliation of the related scheme []. According to this trend, we will try to exploit the good properties of concatenated coded systems [3] in order to build equivalent higher dimensional encoders consisting on one-dimensional chaos-based encoding modules. In Section we will review the principles of such modules and propose a parallel concatenated encoding scheme with iterative decoding. In Section 3 we will analye the resulting coded modulation in additive white Gaussian noise (AWGN) and show the possibility to draw a bound for the bit error probability at the error floor region. In Section 4 we show some simulation results and validate the corresponding bounds. Section 5 is devoted to the conclusions.. SYSTEM MODEL.. Encoder The concatenated encoder consists in the parallel concatenation of two chaos coded modulations (CCM s) [], [3], and, accordingly, we will call these systems parallel concatenated chaos coded modulations (PCCCM s). The first CCM receives an iid binary input word b = (b,,b N ) of sie N. The second CCM receives as input a scrambled sequence c =(c,,c N ),whichis n n n n Fig.. Maps for the constituent CCM encoders. The continuous line corresponds to f ( ); the dotted line, to f ( ). mtm (continuous line: TM); mbsm (continuous line: BSM). the word b interleaved by an S-random interleaver [4] of sie N. Each CCM belongs to the class of discrete chaotic switched maps driven by small perturbations []. They follow the general expression n = f ( n,b n )+g(b n, n ) Q, () x n = n, () where f (, ) = f ( ) and f (, ) = f ( ) are chaotic maps that leave the interval [, ] invariant. They are piecewise linear maps with slope ± whereverit is defined. The natural number Q indicates the number of bits to represent n (and thus x n ), and g (b n, n ) {, } is the small perturbations term, given by { b < g (b, ) = b, (3) where b = b and is the binary XOR operation. If f ( ) =f ( ) is the Bernoulli shift map, g(b, ) is equivalent to a precoder defined by the polynomial +D Q.In any case, the function g (b, ) ensures that two binary input words (b, b ) differing in only bit do not lead to output sequences (x, x ) for any individual CCM with a low squareuclidean distance d E = N n= (x n x n ).This recursive term is included to ensure a good interleaver gain, as is mandatory in any concatenated encoder [5]. With these definitions, it is easy to see that the recursion of () leaves the finite set S Q = {i Q i =,, Q

2 } invariant and, therefore, we can restrict () to S Q by taking as initial condition =. We shall consider the following pairs of maps f ( ) and f ( ): ) Bernoulli shift map (BSM). f () =f () = mod. (4) ) Tent map (TM), corresponding to equations { < f () =f () =. (5) 3) The BSM and a shifted version of the same (multi- Bernoulli shift map, mbsm), following f () = mod, (6) + < 4 f () = 4 <. (7) ) The tent map and a shifted version of the same (multi-tent map, mtm), following { < f () =, (8) < 4 f () = 4 < 3. (9) < b r n r r 3 r Q / Q / Q / Q / R= Fig.. Encoding structure for the mtm CCM. In Fig. we have depicted the corresponding maps. These CCM systems, when restricted to S Q, allow an equivalent representation in terms of a trellis encoder, closely related to a trellis coded modulation (TCM) system []. In Fig., for example, we can see the equivalent trellis encoder structure for the mtm CCM. The constituent encoders seen work at a rate of one chaotic sample per bit, and therefore, the parallel concatenation of two CCM s will have a coding rate of R =/. The output words have thus sie N, x =(x,,x N ), with x k, k =,, N, taking values alternatively from each of the CCM s: when k is odd (k =n, n =,,N), the sample corresponds to the first CCM; when k is even (k = n, n =,,N), the sample corresponds to the second CCM. Note that the overall system is similar to a turbo TCM system, but with the difference that the sequence x n is a chaos-like broadband signal not intended for spectral efficiency. n.. Channel The channel corresponds to the well known AWGN model. The sequence arriving at the decoder side, r = (r,,r N ), will then be r k = x k + n k, () where n k are iid samples of a Gaussian RV with ero mean and power σ. Since the joint PCCCM system proposed has rate R =/, the relationship between the power of the noise and the signal to noise ratio in terms of bit energy to noise spectral density will be σ = R E b = E b, () P N P N where P is the power of the chaos coded modulated signal and R =/..3. Decoder The decoder consists on two soft-input soft-output (SISO) decoding blocks working iteratively over each set of received samples (the ones coming from the first encoder, r n, n =,,N, and the ones coming from the second one, r n, n =,,N). This decoder reproduces the well known structure of the SISO iterative decoders for parallel concatenated binary channel codes [6], but with channel metrics adapted to the CCM setup p(r n x n )= exp ( (r n x n ) ), () πσ σ where x n = n, n S Q, is the candidate quantied chaotic sample for the corresponding branch metric [7]. 3. MINIMUM DISTANCE ANALYSIS The CCM s of the kind described do not comply in general with the uniform error property [8], and so a maximum likelihood (ML) bound for the bit error probability based on the PCCCM transfer function is almost unfeasible. Nevertheless, we will see that we can give a reasonable bound in the error floor region. In binary turbocodes or turbo TCM systems, the minimum distance of the resulting parallel concatenated code or coded modulation is usually employed to provide this error floor bound [3]. Each individual CCM has a characteristic binary input error event e = b b associated to its output minimum squareuclidean distance. For example, in the case of the TM CCM, the minimum squareuclidean distance is given by an input error event with length L = Q + and Hamming weight w(e) =Q +. This event produces Q different output values for the resulting encoded sequences x n and x n, and a minimum squareuclidean distance d E = m+l n=m (x n x n ) tending to for Q []. For the kind of piecewise linear maps considered, the invariant density of x k is uniform in [, ] and, therefore, we can assume P /3 when Q>4.

3 3 3 8 x Fig. 3. Histogram of the d E i corresponding to the Hamming weight binary input error events in the case of a CCM system with Q = 5 for all the possible input sequences. mbsm CCM; TM CCM. Nevertheless, due to the presence of the parallel concatenated scheme with the S-random interleaver, the input binary error events leading to the output minimum squared Euclidean distance for the joint PCCCM system are normally to be found among the Hamming weight input binary error events (or concatenations of the same). Such error events e consist in two s separated by L s, and have length L = Q+ for the BSM and mbsm CCM s, and L = Q+ for the TM and the mtm CCM s. The Hamming weight binary error events with length p(l ) +, p N will lead to increasing values in the associated squareuclidean distances by a factor of p. In the case of the BSM, each pair of sequences produced by a pair of input binary words differing only on a Hamming weight error event of length L will determine a squared Euclidean distance given by [] L +m d min =4 ( n n) =4 n=m Q 4 i = 4 ( ) 3 4 Q. i= (3) For the rest of CCM s, the squareuclidean distances associated to such error events will depend generally on the input words b and b. In Figs. 3, 3 and 4 we can see the distance spectrum for the corresponding maps, produced for all possible b and b sequences differing on a Hamming weight error event with length L. Let us denote as D L the set of all the possible squared Euclidean distances given by such binary error events in a CCM, that is to say D L = { d (x, x ) x b, x b, b b = e } d (x, x )= N n= (x n x n ). (4) If the S-random interleaver is designed with S taking a value at least 3L, the input binary error events with Hamming weight will not be the dominant ones in the error floor region, since they would lead to squared Euclidean distances consisting on the sum of the distances Fig. 4. Histogram of d E i for the Hamming weight binary input error events in the case of an mtm CCM with Q = 5 for all the possible input sequences; Histogram of d E when the binary error events consist on the concatenation of binary input error events with Hamming weight and length L in the case of a PCCCM with two mtm CCM s with Q =5. associated with an error event of length L for one CCM, and of length p(l ) + > S for the other CCM (p >3). That is to say, the squareuclidean distances in the PCCCM system will be around d E + pd E, with d E i D L. Since the S-random interleaver does not put any restriction about the concatenation of error events, in the case of S > 3L, the dominant error events will have Hamming weight 4 and will consist in two Hamming weight binary error events differing in b at indexes i, i + L and j, j + L, and leading to an analogous combination in c, so that, for example, π(i) π(j) = L and π(i + L ) π(j + L ) = L. These Hamming weight 4 compound binary error events will lead to squareuclidean distances in the PCCCM system, d E = 4 i= i, d E i D L, lower than the squareuclidean distances associated to the Hamming weight binary error events allowed by the interleaver structure. In the case of two concatenated BSM CCM s, all Hamming weight binary error events of length L have the same associated squareuclidean distance (D L has only one element), so that, in analogy with turbocodes, the error floor determined by the concatenation of two of such events will be given by P bfloor w 4N 4 N erfc 4d min 4P R E b, (5) where d min is given in (3), w 4 =4isthe number of error bits, and N 4 is the number of compound binary error events with Hamming weight 4 consisting in the mentioned concatenation of Hamming weight binary error events with length L allowed by the interleaver N

4 structure. In the rest of cases, such compound binary error events will lead to squareuclidean distances d E = + d E + d E 3 + d E 4,whered E i D L is the individual squareuclidean distance induced by each individual Hamming weight binary error event in each of the CCM s, which depends generally on b and b.for the mbsm CCM and the TM CCM, these distances d E i have the same frequency (see Fig. 3), and so it is enough to consider the combinations with repetition of 4 elements in D L to account for all possible values of d E between two PCCCM words. Therefore, the bound is given as P bfloor w ( ) 4N 4 t +3 erfc d E N 4 4P R E b, N d E (6) where t is the number of elements in D L and d E = 4 i= i, d E i D L.Theterm ( ) t+3 4 is the number of said combinations of the elements of D L. When we have two mtm CCM s, the number of distances in D L is remarkably larger and their frequency is not uniform (see Fig. 4), so that the overall squared Euclidean distance spectrum for the PCCCM outputs, d E = 4 i= i, d E i D L, follows the almost Gaussian distribution of Fig. 4. To provide a bound based on the related binary error events, we can resort to computing the probability density function (pdf) of d E with the help of the related histogram and calculate numerically P bfloor w d ( ) 4N 4 Emax v p(v) erfc N d 4P R E b dv, N E min (7) where p(v) is the estimated pdf for d E. 4. SIMULATION RESULTS For all the simulations, decoding iterations were performed and the BER results were recorded after finding frames on error. In Fig. 5 we have depicted the BER results for the parallel concatenation of two equal CCM s with an S-random interleaver of length N = and S = 3. All the CCM s have a quantiation factor of Q = 5. We also show the bounds for the BER at the error floor region calculated according to the principles and expressions seen in the previous section. Though they only take into account the kind of error events mentioned there, they are reasonably tight. In the case of BSM and mbsm, the interleavers employed allowed a total of N 4 = 5 possible concatenations of pairs of Hamming weight binary error events with L = L = Q +, and the interleavers of the TM and mtm cases allowed a total of N 4 =4of such compound events (this time with L = L = Q +). It is remarkable the fact that the theoretical error floor, together with the error floor shown by the simulations, decreases as the regularity in the spectrum of the squareuclidean distances associated BER mbsm mtm TM BSM BSM CCM, Q=5 mbsm CCM, Q=5 TM CCM, Q=5 mtm CCM, Q= E b /N (db) Fig. 5. BER and error floor bounds for different parallel concatenation of two equal CCM s in AWGN, Q =5, N = and S =3. Bounds are depicted with dashdotted lines. to the individual Hamming weight binary error events decreases. In fact, the mtm case, which has the most complex squareuclidean distance spectrum as seen in Fig. 4, exhibits a very good behaviour respecting the error floor. The rest of cases, excepting the concatenation of BSM CCM s, agree well with the principles of parallel concatenation [3]: a general good behaviour for lowmid E b /N once reached the turbo cliff threshold, but a relatively high BER error floor for E b /N.Note that the results shown here are comparable to the ones attained by binary turbocodes or turbo TCM systems of similar complexity [3]. In Fig. 6 we show the BER and the error floor bounds for the mbsm PCCCM with different values of N, S and Q. The difference between Q =5and Q =6is small, and seems to affect mainly to the turbo cliff, determining a slightly steeper slope for the Q =6case. Nevertheless, the error floor is very close: though for Q =6there are some important differences (the minimal loop length is now L =7), the values of the squareuclidean distances are practically the same, since the changes in such values are small as Q when Q>4. Therefore, the main change with a change in Q is in the multiplicity N 4 of the compound error events allowed by the interleaver permutation. This makes it clear that the dynamics of the map, which determines the spectrum of the squared Euclidean distances between pairs of chaotic sequences, is a more influential factor at the error floor region than the ad-hoc quantiation level Q. On the other hand, when we change the value of N,

5 BER N=, Q=5 N=65536 N=, S=3, Q=5 N=, S=, Q=5 N=, S=67, Q=5 N=65536, S=37, Q=5 N=, S=3, Q=6 is similar to that of binary turbocodes or turbo trellis coded modulations. Moreover, the analysis of the error events leading to low output square Euclidean distances has allowed us to draw bounds for the bit error probability at the error floor region. The same analysis has provided a valuable insight into the properties of the concatenated system and has stressed the fact that the dynamics of the underlying map is a major factor in the system design, since this dynamics controls the squareuclidean distance spectrum. In fact, the best results were attained when the uniform error property was furthest from being met. 7 N= N=, Q= E b /N (db) Fig. 6. BER and error floor bounds for the parallel concatenation of two equal mbsm CCM s in AWGN. Bounds are depicted with dash-dotted lines. we can appreciate in Fig. 6 that the interleaver gain in the error floor region changes as N. This is the expected behaviour for any parallel concatenated system with interleavers [3]. There is not even a noticeable improvement in the E b /N threshold for the waterfall region with a growing value of N with respect to the case with N =. Only when N< (see the N = case), the situation is clearly worse and the behaviour starts resembling the pure BSM case. In fact, with N = the interleaver requires a lower value for S, and, since it takes now the value S =,the dominant error events in the error floor region are those with overall Hamming weight (instead of 4) and given by one individual binary error loop with L = L in one encoder and L =3(L ) + = 6 >Sin the other. These error events have associated squareuclidean distances lower than the squareuclidean distances for the compound Hamming weight 4 error events of the cases with S>3L. Note finally that the maximum possible value for S is limited by the sie of the interleaver N, since S< N/. This also puts a limit on the value of Q once we have a pair S and N, because L = Q + a, a N, and we require that L < S to avoid low squareuclidean distance weight error events in the concatenated code. 5. CONCLUSIONS In this article we have reviewed the class of chaos coded modulations with switched discrete chaotic maps and small perturbations control, and we have proposed a parallel concatenated system including such modulations. The results have shown that the performance attainable ACKNOWLEDGEMENTS We acknowledge financial support from the Spanish Ministry of Education and Science under Projects No. FIS6-855, and TSI6-7799; from the Government of Madrid under Project No. S-55/TIC/85, and from the Research Program of the Universidad Rey Juan Carlos under Project No. URJC- CM-6-CET-643. REFERENCES [] S. Koic, K. Oshima, and T. Schimming, How to Repair CSK Using Small Perturbation Control - Case Study and Performance Analysis, in Proceedings of ECCTD, Krakow, Poland, August 3. [] S. Koic, T. Schimming, and M. Hasler, Controlled One- and Multi-dimensional Modulations Using Chaotic Maps, IEEE Trans. Circuits Syst. I, vol. 53, pp , September 6. [3]C.B.SchlegelandL.C.Pére, Trellis and Turbo Coding. New York: John Wiley & Sons, Inc., 4. [4] S. Dolinar and D. Divsalar, Weight Distributions for Turbo Codes Using Random and Nonrandom Permutations, Jet Propulsion Laboratory, Tech. Rep. 4-, August 995. [5] V. Gulati and K. R. Narayanan, Concatenated Codes for Fading Channels Based on Recursive Space-Time Trellis Codes, IEEE Trans. Wireless Commun.,vol., no., pp. 8 8, January 3. [6] S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, Serial Concatenation of Interleaved Codes: Performance Analysis, Design and Iterative Decoding, IEEE Trans. Inform. Theory, vol. 44, no. 3, pp , May 998. [7] F. J. Escribano, L. Lópe, and M. A. F. Sanjuán, Iteratively Decoding Chaos Encoded Binary Signals, in Proceedings of ISSPA, vol., Sydney, Australia, August 5, pp [8] E. Biglieri and P. J. McLane, Uniform Distance anrror Properties of TCM Schemes, IEEE Trans. Commun., vol. 39, no., pp. 4 53, January 99.