performance evaluation of a class of chaos-based coded modulations
TRANSCRIPT
F.J. Escribano, L. López and M.A.F. Sanjuán
Universidad Rey Juan Carlos
Spain
e-mail: [email protected]
Parallel Concatenated Chaos Coded Modulations
Split, Croatia, 27Split, Croatia, 27thth September 2007 September 2007
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Background
In most cases, chaos based encoders/modulators had so far proved poor performing in terms of bit error rate (BER) and usually not very robust
Previous work hinted towards a potential boost in BER when parallel concatenating bad performing chaos based modulators
Turbo-like Structures for Chaos Coding and Decoding,F. J. Escribano, S. Kozic, L. López, M. A. F. Sanjuán and M. Hasler (submitted)
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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 Coded Modulations (PCCCM)
Differences to Turbo-TCM: Individual CCMs work at a rate of one symbol per bit
Bit interleaver instead of symbol interleaver
Parameters: Kind of CCM (underlying map) + quantization level (Q)
Size of interleaver π (N) + structure of permutation
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General Setup: CCM block
Map view: one chaotic map (f0(z)=f1(z)), or switched maps
Trellis encoder view: quantized version of the switched map setup driven by small perturbations, with a feedback connection.
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General Setup: parameters
Maps considered Bernoulli shift map (BSM)
Switched version of the BSM, multi-BSM (mBSM)
Tent map (TM)
Switched version of the TM, multi-TM (mTM)
Quantization level Q>4 is enough to make quantization effects neglibigle in
practice
Interleaver S-random interleaver
The channel consists in additive white Gaussian noise (AWGN channel).
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General Setup: iterative decoder
The trellis coded characteristics of the chaotic signal allows the use of known decoding frameworks for concatenated coding
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).
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Error floor analysis: binary error events
Each CCM has minimal binary error event loops with structure 10…01, Hamming weight 2 and length L*=Q+n
n=1,2 depending on the kind of chaos coded modulation
Euclidean distance between CCM sequences xn and xn’
If S>3L*, the dominant error events when Eb/N0->∞ for the PCCCM consist in the concatenation of two of said error events
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Error floor analysis: Euclidean distance
PCCCM 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 dE
2≈4/3, regardless of xk and xk’
For the rest of CCM’s, each individual Euclidean distance depends on the values of xk and xk’ (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
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Error floor analysis: Euclidean distance spectra
Histograms for the individual error events (Q=5)
mBSM TM mTM
Histogram for the compund error events (mTM)
Distance spectra does not change basically with growing Q
1010
Error floor analysis: error floor 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 total dE2
N: size of the interleaver
w4=4: Hamming weight of the related binary error
N4: combinations of two 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 coded sequence
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Simulation results and bounds
N=10000, Q=5, S=23 (left plot), 20 decoding iterations.
mBSM, different N, QSame parameters, different maps
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Concluding remarks
BER performance of the PCCCM system can be comparable to the attainable with the turbo-TCM or binary turbocode related systems: Steep waterfal at low Eb/N0
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 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 The CCM’s not complying with the uniform error property and with
complex distance spectra lead to lower error floors The effect of the quantization level is small and the system
behaviour shows to be rather linked to the underlying dynamics of the map involved
The PCCM system is nonlinear and sends chaotic-like samples to the channel, which exhibit desirable properties. This chaotic-like signal is easy to generate and can be decoded efficiently with known frameworks
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Future work
We have shown that chaos based digital communications can attain a similar performance to other successful coding schemes, and that standard analysis techniques can be applied to predict the BER
But there is still a number of questions to be addressed:
Study other possible encoding structures, based upon other chaotic maps, and try to find general properties and design criteria
Verify the exact influence of the design parameters (S, N, Q…)
Consider other kind of channels (e.g. Rayleigh fading channels) and verify the robustness or suitability of the system
Try to find the link between map dynamics and final performance
Try to exploit in the best possible way the chaotic nature of the system in analysis and performance
Thanks for your attention