F.J. Escribano1, L. López2 and M.A.F. Sanjuán2

1Universidad de Alcalá de Henares2Universidad Rey Juan Carlos

Spain

e-mail: francisco.escribano@ieee.org

Effects of Intersymbol Interference on Chaos-Based

Modulations

SCS 2008SCS 2008Hammamet, Tunisia, 7Hammamet, Tunisia, 7thth November 2008 November 2008

22

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 the

behavior in this kind of distorting environment

33

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)

44

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

1 )(

0 )(),(

11

101

nn

nnnnn bzf

bzfbzfz

Q

Q

QQn

nnn

Qnnnn

Sz

xzx

bbzfz

2

12,,

2

1,0

1,1 12

2),( 1

0,10,1:)(1,0 zf

55

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.

66

General Setup: ISI Channel

According to the model, the signal at the receiver input is

Where hm are the 2N+1 FIR filter coefficients

Since the scope is not to compensate, we only consider a low-to-moderate degree of ISI, where equalization is not always mandatory

n

N

Nnnmmnnn nxhnyr

Coefficients for low and moderate ISI

N=3

77

Error Analysis: PEP

Pairwise error probability (PEP) under ISI with ML decoding (equivalent to MAP decoding in our case) [Schlegel91]

P: power of the chaotic sequence (1/3) d2

ISI: equivalent pairwise distance under ISI

Sequences xn and x’n differ in an error loop of length L, related to a corresponding binary error loop

0

2

42

1)|'(

N

E

P

derfcP bISI

e xxx

122

21 122

2

)'(

)()'(

Lm

mnnnE

E

Lm

mn

Lm

mnnnnn

ISI

xxd

d

xyxyd

88

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 distance in the non-ISI case for any pair of sequences 4/3

99

Error Analysis: Bounds

De = set of possible distances d2ISI associated to the

corresponding most probable binary error event w: Hamming weight of the most

probable binary error event 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

Now the decoder will chose x’n 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

eISI Dd

bISIb N

E

P

derfc

D

wP

2 0

2

42

1

21

2 )()'(Lm

mnnn

Lm

mnnn xyxy

0

4

2 N

EwBP b

LQNe

b floor

1010

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

low ISIQ=5

TM

mTM

mBSM

BSMTCM

* Dotted/dash-dotted: simulation results* Continuous: bounds (except TCM)

1111

Simulation Results

Q=5, moderate ISI

TM

mTM

mBSM

BSM

TCM

* Dotted/dash-dotted: simulation results* Continuous: bounds (except TCM)

1212

Simulation Results

BSM CCM, low ISI, Q=5

* Dotted/dash-dotted: simulation results* Continuous: bounds

1313

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 systems based on CCM systems

1414

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.

Thanks for your attention