ter's chaos generator. The presented simulation re- sults show that the classical Binary Phase Shift Key- ing (BPSK) modulation scheme can be slightly out-.
Maximum Likelihood Detection of Symbolic Dynamics in Communication Systems with Chaos Shift Keying Andreas Abel and Wolfgang Schwarz Technical University Dresden Institute for Fundamentals of Electrical Engineering and Electronics Mommsenstraße 13, D-01062 Dresden, Germany e-mail: abel,schwarz @iee1.et.tu-dresden.de Abstract— The paper describes the application of the Viterbi algorithm in the implmentation of a maximum likelihood receiver in a chaos communication scheme. The Viterbi decoder is constructed based on a symbolic state representation of the transmitter’s chaos generator. The presented simulation results show that the classical Binary Phase Shift Keying (BPSK) modulation scheme can be slightly outperformed this way. I. I NTRODUCTION In classical communications channel coding is divided in two large classes – block encoding methods and convolutional encoding methods. In a similar way chaos communication approaches may be classified into blockwise (or static) encoding methods exploiting free-running chaos generators and dynamic encoding methods, where the message to be transmitted affects the dynamics of the generator. In this paper we will present a digital chaos communication scheme with dynamic encoding. In the receiver we exploit a method developed for the reception of convolutionally encoded messages – the Viterbi decoder [11], [3] – and analyze its performance on an Additive White Gaussian Noise (AWGN) channel by simulation. The scheme is studied in different transmitterreceiver configurations, some of which reach and even slightly outperform comparable (non-encoded) classical approaches. However, this goal is achieved by a considerable implementational effort only. The paper is structured as follows: Section II discusses the system under consideration, modeling procedures, and the system components. In section III the results of the system analysis are presented and discussed. Conclusions are drawn in section IV. II. S YSTEM D ESCRIPTION We examine a communication system, which by its nature is a relative of the convolutional (i.e. dynamic) encoding methods found in classical communications [3]. The binary message to be transmitted switches a parameter of a chaos generator, selecting one of two
possible configurations for the generation of the samples to be sent to the channel. By this the message is fed into the dynamics of the chaos (resp. code) generator affecting the evolution of the chaotic trajectory in the future. This makes the approach different from e.g. Chaos Shift Keying (CSK – as initially proposed in [4]), Differential Chaos Shift Keying (DCSK) [8], or Chaotic On Off Keying (COOK) [5], which use a static (block) encoding, mapping a bit to a chaotic sample function without affecting the generator dynamics. The presented communication scheme will be designed and analyzed in a discrete-time baseband representation. The motivation and properties of baseband models are shortly discussed in the sequel, followed by a description of the communication scheme components. A. Discrete-Time Baseband Models of Communication Schemes The analysis or simulation of a communication scheme is confronted with two core problems – 1. Different time constants due to the coexistence of the time scales of the message to be transmitted and the radio-frequency (RF) channel and 2. Continuous time due to the inherently continuous-time physical channel. A fully discrete-time system, composed of lowfrequency components only, would ease the analysis considerably. So in communications one tries to derive an equivalent discrete-time model whenever possible. We do not describe this modeling procedure in detail (it is found in the literature e.g. [3], [10]), but a few properties are worth to be noted: 1. Baseband models are complex-valued. 2. An RF channel with band-limited (bandwidth ) Additive White Gaussian Noise (two-sided power spectral density ) is represented in the model by a channel with a discrete-time complex Additive White Gaussian Noise (AWGN) with independent real and imaginary parts, each of variance . 3. Signals transmitted via the channel transform to discrete-time signals . Their energies are related as
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B. Chaotic Encoding The proposed system exploits a one-dimensional discrete-time chaos generator
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fig. 1. The scheme forms a close relative of CSK, being different solely by the fact, that there is only one state space on which both of the generators act intermittendly. C. The Modulation Scheme The chaotically encoded message shall phase modulate a carrier (at a certain frequency ). The phase modulation (PM) is represented in the discrete-time baseband model by a mapping of the sequence of states to the unit circle in the complex
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thus forming a polyphase chaos generator. and is additionThe usage of the two maps ally motivated in a PM scheme, since for a given output value the possible successors and are antipodal, i.e. have maximum distance on the complex unit circle. D. Symbolic System Representation Instead of continuous-state trajectories we may use a symbolic representation to describe a generator’s behaviour. For this we partition the generator state space, i.e. the interval into equally sized subintervals with . The trajectory of the system is then converted to a sequence of states taking values if . For (binary case) , which possesses a uniform invariant probability density function (pdf) [2], is known to produce balanced sequences of independent, identically distributed (i.i.d.) binary values (see e.g. [6]). The map also possesses a uniform invariant pdf [2] and produces, for an identical initial condition, binary sequences which are inverse to the sequences from , hence i.i.d. also. So we may represent the state sequence generators by Markov chain models with equally likely states and two equally probable transitions diverging from each state. One observes, that one bit of information (the selection of a transition branch, performed the bits of the i.i.d. sequences) is generated in each time step. Fig. 2 shows state transition diagrams for both maps if 8 symbolic states are introduced. If we let the binary message (which we assume to be balanced and i.i.d. also) switch the map, we obtain a merged version of the two state transition diagrams with 4 branches starting at each of the states according to the possible combinations of the message bit and the mapgenerated bit (fig. 2). All transitions are equally probable.
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For the transmission over the channel we may consider either the continuous-valued chaotic sequence or its symbolic state representation in form of a representative value. E. Receiver Structure and Message Decoding For the receiver side we assume a coherent demodulation of the PM signal. The state representation derived in subsection IID resembles the representation we find in convolutional encoders, which are in widespread use in modern communication systems [3]. So we may consider to exploit the decoding methods established in this field. Assuming that only the state sequence is transmitted, and received as a noise-corrupted sequence r, the receiver has to perform a maximum likelihood decision based on all possible state sequences, which are for states and the observation of subsequently transmitted samples
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F. Application of a Viterbi Decoder The full search of all possible symbolic sequences requires a computational effort which grows exponentially with . However, this is more than necessary. An elegant solution for which the effort only grows linearily with the observation time, is the Viterbi decoder [11], [3]. It exploits the fact, that (5) may be evaluated successively (incrementally) in time.
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G. System Implementation G.1 Transmitter 1. The transmitter emits a continuous-valued signal. The receiver then may be implemented with an arbitrary state resolution, as long as it is able to represent all possible state transitions (minimum of 4 states in the example). 2. The transmitter emits quantized values, i.e. a representation of the symbolic state sequence. The number of states used in the transmission should match the number of states in the receiver. G.2 Receiver 1. The receiver implements a pre-quantization (so called hard decision [1]) of the received signal in order to produce a symbolic state sequence as the decoder input, allowing a fully digital implementation of the decoder. 2. The decoder uses the values as they are received from the demodulator, i.e. it implements a so called soft decision [1]. III. P ERFORMANCE E VALUATION A. Measuring the AWGN Performance The established measure for the evaluation and comparison of the performance of digital communication schemes on AWGN channels is the bit error ) as a function of the ratio of the energy rate ( spent for the transmission of one bit ( ) and the noise power spectral density ( ) [3]
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simulation performed a transmission of 10000 random bits, using Viterbi-decoders which traced paths of length 13 (with 10 segments decoded in order to avoid the fan out of the paths in the last steps). Compared to the results below, hard decision is clearly outperformed by soft decision. Nevertheless, it may be beneficial since the pre-discretization allows a fully digital decoder implementation, which may lead to a low-cost and efficient solution. B.2 Soft Decision Reception The application of soft decision leads to better results as seen in fig. 4. If only symbolic states are transmitted, the performance is not affected by the chosen state resolution, i.e. one may stick to the lowest possible resolution. If the chaotic signal itself is transmitted, with an increasing number of states in the receiver a convergence towards the performance of the discretized system is observed. In general, the reached performance compares to orthogonal modulation with coherent detection, even leading to marginally better results. The simulations were performed transmitting ), decoded 100000 bits (500000 for with a 10-stage Viterbi decoder. Still, compared to BPSK the performance curve of the described method is by a multiplier (i.e. ) shifted along the axis. The reason for this is
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pretty obvious: The Viterbi-decoder is constructed to retrieve the message bit stream as well as the bit generated by the chaotic map from the transmitted signal. Since in the example we use maps generating 1 bit per iteration, half of the transmitted amount of information (the map bits) gets thrown away at the receiver side. Nevertheless it has to be transmitted via the channel, i.e. it eats up channel capacity. So it is straightforward reasoning to exploit these bits as message carriers also, either by suitably setting initial conditions or controlling the motion of the trajectory by small disturbances and thus forcing it to a prescribed symbolic sequence. In order to analyze the performance of such a system we assumed the map-generated sequence of bits to be prescribed and measured the quality of its reconstruction at the receiver side, together with the reconstruction quality of the map-switching message stream. The results are shown in fig. 5. Compared to the result obtained by switching the maps only, the impact of a continuousvalue transmission is significantly larger. This can be attributed to the fact, that the map-generated bits result in possible transitions to two neighbouring states (see fig. 2), so the chaotic deviation from the state sequence itself leads to significant errors in the reconstruction of the map-generated bits. In contrary the map switching leads to two possible antipodal points on the unit circle. Transmitting symbolic states only, the obtained performance is slightly better than in BPSK. This result coincides with [9] where BPSK performance was achieved when coding information into symbolic sequences by prescribing initial conditions of a generator (so the map bits were the only information to be transmitted). The simulation results were obtained by transmitting 100000 bits (for the 8state symbolic transmission: 500000 bits at and , and 2000000 bits at ) and using a 10stage decoder.
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The presented simulation results show once more, that it is possible to achieve the performance of classical communication approaches using chaos. However, there is virtually no coding gain obtained so far with the described approach, which would be present in the classical encoding systems (convolutional coding) with comparable implementational complexity. For the given approach the limitation to the transmission of the symbolic state sequence results in a better system performance. The symbolic states carry the complete amount of information necessary at the receiver side, whereas the chaotic reminder acts as an additional noise. An analytical confirmation of the simulation results still has to be provided. Also, a proper design of the encoding method may lead to further improvements and more significant coding gains. Both points are subject to further research. R EFERENCES [1]
B. Friedrichs. Channel Coding. Springer Verlag, Berlin, 1995. (in German). [2] M. G¨otz. Analysis of the Frobenius-Perron-Operator and Correlation Theory of Piecewise Linear Discrete-Time Systems. Shaker Verlag, Aachen, 1998. (PhD Thesis, in German). [3] S. Haykin. Communication Systems. John Wiley & Sons, New York, 1994. [4] M. P. Kennedy and H. Dedieu. Experimental demonstration of binary chaos-shift-keying using self-synchronising chua’s circuits. In A. C. Davies and W. Schwarz, editors, Nonlinear Dynamics of Electronic Systems – Proceedings of the Workshop NDES’93, pages 262–275. World Scientific, Singapore, 1994. [5] M. P. Kennedy and G. Kolumb´an. Chaos communications: From theory to implementation. In Proc. ECCTD, volume 1, pages 272–277, Budapest, 1997. [6] T. Kohda and A. Tsuneda. Statistics of chaotic binary sequences. IEEE Trans. IT, 43(1):104–112, January 1997. [7] G. Kolumb´an, M. P. Kennedy, and G. Kis. Performance improvement of chaotic communications systems. In Proc. ECCTD, volume 1, pages 284–289, Budapest, 1997. [8] G. Kolumb´an, B. Vizvari, W. Schwarz, and A. Abel. Differential chaos shift keying: A robust coding for chaos communication. In Proc. NDES, pages 87–92, Seville, 1996. [9] T. Schimming and J. Schweizer. Chaos communication from a maximum likelihood perspective. In Proc. NOLTA, volume 1, pages 77–80, Crans Montana, 1998. [10] W. Schwarz, M. G¨otz, K. Kelber, A. Abel, T. Falk, and F. Dachselt. Statistical analysis and design of chaotic systems. In P. Kennedy, R. Rovatti, and G. Setti, editors, Applications of Chaotic Electronics to Telecommunications. CRC Press, Boca Raton, FL, 2000. (to appear). [11] A.J. Viterbi. Error bounds for convolutional codes and an asymptotically optimum decoding algorithm. IEEE Trans. IT, 13(2):260–269, April 1967.
