Performance evaluation of a class of chaos-based coded modulations
Parallel Concatenated Chaos Coded Modulations F.J. Escribano, L. López and M.A.F. Sanjuán Universidad Rey Juan Carlos Spain e-mail: firstname.lastname@example.org Split, Croatia, 27th September 2007
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)2
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 permutation3
General Setup: CCM block Map view: one chaotic map (f0(z)=f1(z)), or switched maps f ( z ) bn = 0 z n = f ( zn −1 , bn ) = 0 n −1 f1 ( z n −1 ) bn = 1 f 0,1 ( z ) : [ 0,1] → [ 0,1] Trellis encoder view: quantized version of the switched map setup driven by small perturbations, with a feedback connection. z n = f ( z n −1 , bn ) + g (bn , z n −1 ) ⋅ 2 − Q xn = 2 z n − 1 xn ∈ [ − 1,+1] 1 2Q +1 − 1 z n ∈ Q +1 , , Q +1 4 2 2
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).5
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). p (cn = 1 y1 y3 y2 N −1 ) 1, O llr = log p (c = 0 y y y n 1 3 2 N −1 ) 6
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’ L* + m −1 ∑ (x − xn ) 2 dE = 2 n n=m If S>3L*, the dominant error events when Eb/N0->∞ for the PCCCM consist in the concatenation of two of said error events7
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 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 dE2≈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 the8 individual error events and of their combinations
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 change9 basically with growing Q
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 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 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 interleaver10 R=1/2: overall rate of the PCCCM P≈1/3: power of the chaos coded sequence
Simulation results and bounds N=10000, Q=5, S=23 (left plot), 20 decoding iterations. Same parameters, different maps mBSM, different N, Q11
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 known12 frameworks
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 performance13 Thanks for your attention
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