Published on: **Mar 4, 2016**

- 1. Performance Evaluation of ParallelConcatenated Chaos-Based Coded Modulations Francisco J. Escribano Universidad de Alcalá de Henares Spain e-mail: francisco.escribano@ieee.org Politecnico di Torino, Italia, 18th September 2009
- 2. Background I In most cases, chaos based encoders/modulators for digital communications had so far proved poor performing in terms of bit error rate (BER). Recent work shows that poor performing chaos-based systems can benefit from the performance boost of parallel concatenation. F. J. Escribano, S. Kozic, L. López, M. A. F. Sanjuán and M. Hasler, Turbo-like Structures for Chaos Coding and Decoding, IEEE Transactions on Communications, March 2009. We will analyze here how such concatenated chaos-based systems work, with the aim to gain insight in their2 possibilities.
- 3. Background II Chaos-based systems for digital communications used to work: At the waveform level. E.g.: Chaos Shift Keyng. At the coding level. E.g.: sequences for spread spectrum communications.3
- 4. CCM basics I Chaos-Based Coded Modulated (CCM) systems. They work at a joint waveform and coding level. They are based on expanding piecewise linear chaotic maps. S. Kozic, T. Schimming and M. Hasler, Controlled One- and Multidimensional Modulations Using Chaotic Maps, IEEE Transactions on Circuits & Systems I, September 2006. Map example: Bernoulli shift map (BSM). f ( z ) : [ 0,1] → [ 0,1] z n = f ( z n −1 ) = 2 z n −1 mod 14 Uncontrolled dynamics
- 5. CCM basics II Control by small perturbations through binary sequence bn: bn z n = f ( z n −1 ) + Q 2 Q (integer>=1) gives the amplitude of the perturbation. Symbolic dynamics of the uncontrolled BSM: 1 Simbolic state sn = z n + sn ∈ { 0,1} 2 ∞ Binary expansion z n = ∑ 2 −( i +1) sn +i of the chaotic sequence i =05 The perturbation manifests itself after Q-1 iterations.
- 6. CCM basics III Let’s define the set SQ as m SQ = Q m = 0,1,...,2 − 1 Q 2 bn When z0 ∈ SQ , then z n ∈ SQ if z n = f ( z n −1 ) + 2Q Therefore, with the small perturbations setup and choosing, e.g., z0=0, the iteration leaves the set SQ invariant -> we get a quantized chaos-based sequence.6 The process can be described by a finite state machine.
- 7. CCM basics IV Trellis encoder view of the BSM driven by small perturbations, restricted to SQ: In the concatenated setup, we will need feedback to get interleaver gain: z n = f ( z n −1 ) + g (bn , z n −1 ) ⋅ 2 −Q bn z n −1 < 1 / 2 g (bn , z n −1 ) = b n z n −1 ≥ 1 / 27
- 8. CCM basics V We can extend the CCM concept to build a system based on switched maps driven by small perturbations: f ( z ) bn = 0 z n = f ( z n −1 , bn ) = 0 n −1 f 0,1 ( z ) : [ 0,1] → [ 0,1] f1 ( z n −1 ) bn = 1 f0 and f1 can be the same (case seen with BSM), or different maps (switched maps). The whole framework can be generalized to cover multidimensional maps (Kozic et al, TCAS-I, 2006). The encoding process finishes with a scaling process: z n = f ( z n −1 , bn ) + g (bn , z n −1 ) ⋅ 2 − Q One-dimensional switched maps driven z0 ∈ S Q → z n ∈ S Q by small perturbations xn = 2 z n − 1 xn ∈ [ − 1,+1] with feedback8
- 9. General Setup: CCM constituent blocks Maps and corresponding CCM’s considered in the examples (expanding maps with slope ±2): Bernoulli shift map (BSM) Switched version of the BSM, multi-BSM (mBSM) Tent map (TM) Switched version of the TM, multi-TM (mTM)9
- 10. General Setup: concatenated modulator Similar setup to the parallel concatenation of Trellis Coded Modulations (TCM): Two Chaos Coded Modulation (CCM) blocks + Interleaver Parallel Concatenated Chaos-Based Coded Modulations (PCCCM) Differences with respect to traditional parallel cocatenated systems: Individual CCMs (one-dimensional) work at a rate of one symbol per bit Bit interleaver instead of symbol interleaver Strongly nonlinear in general10
- 11. General Setup: parameters Parameters: Kind of CCM blocks (underlying map) Quantization level (Q) Size of interleaver π (N) Structure of permutation Quantization level: Q>=4 is normally enough to make quantization effects neglibigle in practice Interleaver: S-random interleaver without any optimization The channel considered for the test simulations is the standard AWGN channel.11
- 12. General Setup: iterative decoder The trellis coded characteristics of the chaos-based signal allows the use of known decoding frameworks for concatenated coding. The decoder consists on two SISO (soft-input soft-output) decoding blocks working iteratively. The decoders interchange soft information in the form of log likelihood ratios (llr’s).12
- 13. Convergence analysis The similitude with a Turbo-TCM system allows the use of known principles and tools for analysis of convergence (pinchoff point) and performance (error floor). Convergence of the iterative decoder: EXIT charts. mBSM CCM’s Q=5 They are able to predict the pinchoff point with a mismatch of some tenths of dB13
- 14. Error floor analysis: binary error events I Each CCM kind considered has weight 2 binary error event loops with structure 10…01, and length L*=Q+n. n=1,2 depending on the kind of CCM.14 Error loop in the BSM CCM trellis, Q=3
- 15. Error floor analysis: binary error events II The important parameter is the Euclidean distance between CCM sequences xn and xn’ corresponding to said error loops: L* + m −1 d = 2 E ∑ (x n=m n −x n) 2 In our setup, if S>3L*, the dominant error events for high Eb/N0 mainly consist in the concatenation of two of said error events:15
- 16. Error floor analysis: Euclidean distance The sequences xk and xk’ related through such compound binary error event exhibit four chaos coded subsequences of length L* with non-zero difference d = ∑ ( xk − xk ) = d E1 + d E2 + d E3 + d E4 2N 2 2 2 2 2 2 E k =1 For the BSM CCM, each individual Euclidean distance has the same value, regardless of xk and xk’: dE2≈4(4/3). For the other CCM’s, the Euclidean distance depends on the values of xk and xk’ (exact path through the trellis). They do not comply with the uniform error event property. The evaluation of the corresponding distance spectrum requires evaluating the distance spectrum of the16 individual error events and of their combinations.
- 17. Error floor analysis: Euclidean distance spectra I Histograms for the individual error events (Q=5) mTM mBSM TM17
- 18. Error floor analysis: Euclidean distance spectra II Histogram for the compund error events (mTM, Q=5): Main contribution given by the few sequences leading to this few values Distance spectra basically does not change with Q (>=4).18
- 19. Error floor analysis: bound The bound for the bit error probability in the error floor region can be given in the general case by numerical integration over the probability density function (pdf) of the Euclidean distance spectrum dE2 as estimated through the histogram: 2 d E max w4 N 4 v Eb Pb floor ≈ 2N ∫ p (v) ⋅ erfc R dv 2 d E min 4P N0 p(v): pdf of the overall dE2 N: size of the interleaver w4=4: Hamming weight of the related binary error event N4: number of combinations of pairs of individual binary errors of Hamming weight 2 and length L* allowed by the interleaver19 R=1/2: overall rate of the PCCCM P≈1/3: power of the chaos-based coded sequence
- 20. Simulation results and bounds N=10000, Q=5, S=23 (left plot), 20 decoding iterations. Same parameters, different maps mBSM, different N, Q20
- 21. Concluding remarks I BER performance of the PCCCM system is comparable to the attainable with other turbo related systems (Turbo-TCM): Steep waterfal at low Eb/N0 (exception: BSM) Relatively high error floor for high Eb/N0 The error floor decreases as 1/N By examining the permutation structure of the interleaver, it is possible to approximately bound the BER at the error floor region. The PCCCM system based on a quasi-linear CCM (BSM) complies with the uniform error property, but the final behaviour is poor (weak coding structure and poor distance properties). The CCM’s not complying with the uniform error property and with complex distance spectra lead to lower error floors.21
- 22. Concluding remarks II The effect of the quantization level is small. The system behaviour seems to be rather linked to the dynamics of the underlying map. The PCCM system is nonlinear and sends chaotic-like samples to the channel. This chaotic-like signal is easy to generate and can be decoded efficiently with known frameworks. Iterative decoding helps to avoid the possibility of catastrophic decoding. H. Andersson, Error-Correcting Codes Based on Chaotic Dynamical Systems, PhD Thesis, Linköping University, Sweden.22
- 23. Open issues We have shown that: Chaos-based digital communications systems can attain similar performance to standard communications schemes Well known analysis techniques could be applied to predict the final behavior But there is still a number of important questions to be addressed: Study other encoding structures, based on a more general framework, and give general properties and design criteria Try to find an optimized interleaver structure Consider other kind of channels Try to find the link between chaotic dynamics and performance Try to get higher spectral efficiency Unsolved question: chaos in the channel seems to be not so bad performing after all…, but what is it really good for?23 Thanks for your attention