Chaos-based coded modulations in ISI channels

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Chaos-based coded modulations in ISI channels

  1. 1. Effects of Intersymbol Interference on Chaos-Based ModulationsF.J. Escribano1, L. López2 and M.A.F. Sanjuán2 1 Universidad de Alcalá de Henares 2 Universidad Rey Juan Carlos Spain e-mail: francisco.escribano@ieee.org SCS 2008 Hammamet, Tunisia, 7th November 2008
  2. 2. Background & Motivation  Very often chaos based encoders/modulators had so far proved poor performing in terms of bit error rate (BER)  A big boost in BER has been obtained when using bad performing chaos-based encoders/modulators in multi- dimensional encoding systems [Kozic06,Escribano09]  Generalization of discrete chaotic map systems  Concatenation (serial, parallel)  The building blocks of such systems have been only evaluated in AWGN and fading channels [Escribano08]  It would thus be interesting to evaluate the effects of ISI  ISI can easily arise due to strict band constraints  The scope is not to equalize, but just characterize the2 behavior in this kind of distorting environment
  3. 3. General Setup: System Model  Similar setup to a Trellis Coded Modulation (TCM):  Chaos-Based Coded Modulation (CCM) block at the transmitter  ML or MAP sequence decoding at the receiver. We will perform MAP decoding, since a MAP SISO is the building block for the iterative decoders of the concatenated good performing systems [Escribano06]  Differences with TCM:  The CCM kind considered works at a rate of one symbol per bit  Parameters:  Type of CCM (underlying map)  Quantization level (Q)3
  4. 4. 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 z n = f ( z n −1 , bn ) + bn ⋅ 2 − Q xn = 2 z n − 1 xn ∈ [ − 1,+1]  1 2Q − 1 z n ∈ SQ = 0, Q ,  , Q 4  2 2 
  5. 5. General Setup: Parameters  Maps considered (previous slide)  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 negligible in practice  The channel consists in additive white Gaussian noise (AWGN) plus Intersymbol Interference (ISI).  The ISI model is a standard model in digital communications, where this impairment is simulated by means of a FIR filter.5
  6. 6. General Setup: ISI Channel  According to the model, the signal at the receiver input is N rn = yn + nn = ∑h n=− N m ⋅ xm + n +nn  Where hm are the 2N+1 FIR filter coefficients Coefficients for low and moderate ISI N=3  Since the scope is not to compensate, we only consider a low-to-6 moderate degree of ISI, where equalization is not always mandatory
  7. 7. Error Analysis: PEP  Pairwise error probability (PEP) under ISI with ML decoding (equivalent to MAP decoding in our case) [Schlegel91] 1  d2 E  Pe (x → x | x) = erfc ISI b  2  4P N0     P: power of the chaotic sequence (≅1/3)  d2ISI: equivalent pairwise distance under ISI 2  m + L −1 m + L −1   ∑ ( y n − x n ) − ∑ ( y n − xn ) 2  2 2 d ISI =  n =m n =m   dE      m + L −1 d = 2 E ∑ (x n=m n − x n ) 27  Sequences xn and x’n differ in an error loop of length L, related to a corresponding binary error loop
  8. 8. Error Analysis: Error Events  Under low-to-moderate ISI, the most probable error events of each CCM system can be easily evaluated for high Eb/N0  E.g.: BSM CCM will exhibit error events generated by a binary error with Hamming weight 2, L=Q+1, and structure 1,1,0,…,0  CCM’s do not meet the uniform error property [Biglieri91]  A binary error is related to 24N+Q+L pairs of output sequences leading to potentially different pairwise distances  The ISI BER degradation can be calculated using the spectrum of the related pairwise distances for a given CCM •d2ISI histogram of the BSM CCM for its most probable binary error event under low ISI for Q=5 •Minimum pairwise Euclidean distance8 in the non-ISI case for any pair of sequences ≅4/3
  9. 9. Error Analysis: Bounds  De = set of possible distances d2ISI associated to the corresponding most probable binary error event  w: Hamming weight of the most w  d2 E  probable binary error event Pb ≈ ∑ erfc 4ISI Nb  2 D d ISI ∈De  P 0 2    D: number of elements in De.  If following condition is met for some xn, x’n related through the mentioned binary error event, an error floor appears m + L −1 m + L −1  Now the decoder will chose x’n ∑(y n=m n − x n ) < 2 ∑(y n=m n − xn ) 2 regardless of the noise, and system will always need equalization  Be = number of xn, x’n sequences related through the corresponding binary error event that meet last inequality wBe Eb9 Pb floor ≈ →∞ 2 4 N +Q + L N0
  10. 10. Simulation Results  Comparison with a related TCM system with QPSK modulation, polynomials = 06 and 23 (octal) and constraint length = 5 -> rate 1 bit/symbol  Frames with M=10000 bits, results recorded for 100 frames on error TM low ISI Q=5 * Dotted/dash-dotted: simulation results * Continuous: bounds (except TCM) BSM TCM mTM10 mBSM
  11. 11. Simulation Results  Q=5, moderate ISI TM mTM * Dotted/dash-dotted: mBSM simulation results * Continuous: bounds (except TCM) BSM TCM11
  12. 12. Simulation Results  BSM CCM, low ISI, Q=5 * Dotted/dash-dotted: simulation results * Continuous: bounds12
  13. 13. Concluding Remarks  The bounds calculated using the spectrum distance induced by the ISI channel have shown to be accurate enough for low-to-moderate ISI  The effect of the ad-hoc quantization level is small and the system behavior shows to be rather linked to the underlying dynamics of the map involved  CCM systems keep the good properties of coded modulations in dispersive environments  In some situations, CCM systems can exhibit lower losses than with a TCM alternative  We have provided a condition to detect potential error floors  The principles shown can be readily applied to other examples of chaos-based coded modulations, whenever they can be represented in terms of a trellis  The CCM systems are nonlinear and send chaotic-like samples to the channel, which exhibit interesting properties  This kind of chaotic-like signal is easy to generate and can be decoded efficiently with known frameworks  The results obtained can help in the design and evaluation tasks of multi-dimensional good performing chaotic communications systems13 based on CCM systems
  14. 14. References  [Kozic06] S. Kozic, T. Schimming and M. Hassler, ‘Controlled One- and Multidimensional Modulations Using Chaotic Maps’, IEEE Transactions on Circuits & Systems - I, vol. 53, Sep 2006.  [Escribano09] F. J. Escribano, S. Kozic, L. López, M. A. F. Sanjuán and M. Hassler, ‘Turbo-Like Structures for Chaos Coding and Decoding’, IEEE Transactions on Communications, in Press, 2009.  [Escribano08] F. J. Escribano, L. López and M. A. F. Sanjuán, ‘Chaos- Coded Modulations over Rayleigh and Rician Flat Fading Channels’, IEEE Transactions on Circuits & Systems - II, vol. 55, Jun 2008.  [Escribano06] F. J. Escribano, L. López and M. A. F. Sanjuán, ‘Exploiting Symbolic Dynamics in Chaos Coded Comunications with Maximum a Posteriori Algorithm’, Electronics Letters, vol. 42, Aug 2006.  [Schlegel91] C. B. Schlegel, ‘Evaluating Distance Spectra and Performance Bounds of Trellis Codes on Channels with Intersymbol Interference’, IEEE Transactions on Communications, vol. 37, May 1991.  [Biglieri91] E. Biglieri and P. J. McLane, ‘Uniform Distance and Error Properties of TCM Schemes’, IEEE Transactions on Comunications, vol. 39, Jan 1991.14 Thanks for your attention

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