1. Politecnico di Torino, Italia, 18th September 2009
Performance Evaluation of Parallel Concatenated Chaos-Based Coded Modulations
Francisco J. Escribano
Universidad de Alcalá de Henares
Spain
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 their 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.

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).
Uncontrolled dynamics

5. of the chaotic sequence
CCM basics II
Control by small perturbations through binary sequence bn:
Q (integer>=1) gives the amplitude of the perturbation.
Symbolic dynamics of the uncontrolled BSM:
The perturbation manifests itself after Q-1 iterations.
Simbolic state
Binary expansion
of the chaotic sequence

6. CCM basics III Let’s define the set SQ as When , then if
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.
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:

8. by small perturbations
CCM basics V
We can extend the CCM concept to build a system based on switched maps driven by small perturbations:
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:
One-dimensional
switched maps driven
by small perturbations
with feedback

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)

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 general

11. General Setup: 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.

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).

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 dB

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.
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:
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:

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
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 the 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 TM

18. Error floor analysis: Euclidean distance spectra II
Histogram for the compund error events (mTM, Q=5):
Distance spectra basically does not change with Q (>=4).
Main contribution
given by the few
sequences
leading to this
few values

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:
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 interleaver
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, Q

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.

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.

23. Thanks for your attention
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?
Thanks for your attention