Francisco J. Escribano

Universidad de Alcalá de Henares

Spain

e-mail: francisco.escribano@ieee.org

Performance Evaluation of Parallel Concatenated Chaos-Based Coded Modulations

Politecnico di Torino, Italia, 18Politecnico di Torino, Italia, 18thth September 2009 September 2009

22

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.

33

Background II

At the waveform level. E.g.: Chaos Shift Keyng.

At the coding level. E.g.: sequences for spread

spectrum communications.

Chaos-based systems for digital communications used to work:

44

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).

1 mod 2)( 11 −− == nnn zzfz

[ ] [ ]0,10,1:)( →zf

Uncontrolled dynamics

55

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.

Qn

nn

bzfz

2)( 1 += −

{ }

∑∞

=+

+−=

∈

+=

0

)1(2

1,0 2

1

iin

in

nnn

sz

szsSimbolic state

Binary expansionof the chaotic sequence

66

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.

−== 12,...,1,0

2Q

QQ mm

S

QSz ∈0 Qn Sz ∈Qn

nn

bzfz

2)( 1 += −

77

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:

≥<

=

⋅+=

−

−−

−−−

2/1

2/1 ),(

2),()(

1

11

11

nn

nnnn

Qnnnn

zb

zbzbg

zbgzfz

88

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:

[ ]1,1 12

2),(),(

0

11

+−∈−=

∈→∈⋅+= −

−−

nnn

QnQ

Qnnnnn

xzx

SzSz

zbgbzfz

==

==−

−− 1 )(

0 )(),(

11

101

nn

nnnnn bzf

bzfbzfz [ ] [ ]0,10,1:)(1,0 →zf

One-dimensionalswitched maps drivenby small perturbations

with feedback

99

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)

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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

1111

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.

1212

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).

1313

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’sQ=5

They are ableto predict thepinchoff point

with a mismatchof some tenths of dB

1414

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

1515

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:

( )∑−+

=

−=1

2'2

* mL

mnnnE xxd

1616

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’: dE

2≈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.

( )∑=

+++=−=N

kEEEEkkE ddddxxd

2

1

22222'2

4321

1717

Error floor analysis: Euclidean distance spectra I

Histograms for the individual error events (Q=5)

mBSM TM

mTM

1818

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 contributiongiven by the few

sequencesleading to this

few values

1919

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 dE

2 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

∫

⋅≈

2max

2min

0

44

4erfc)(

2

E

E

floor

d

d

bb dv

N

ER

P

vvp

N

NwP

2020

Simulation results and bounds

N=10000, Q=5, S=23 (left plot), 20 decoding iterations.

mBSM, different N, QSame parameters, different maps

2121

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.

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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.

2323

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