Nonlinear Digital Encoders for Data Communications

Introduction

During the last few decades, several chaotic sequence generators have been investigated for secure and efficient digital communication systems. Because of their generators’ sensitive dependence on the initial state, these sequences present pseudo-random properties and offer an enhanced security.

As is well known, chaotic sequence generators are nonlinear dynamical systems and their finite precision or quantized approximations affect the chaotic regimes. Hence, this problem can be overcome by developing digital generators. In [FRE 93], Frey proposed a chaotic digital infinite impulse response (IIR) filter for a secure communications system. The Frey filter contains a nonlinear function called the left-circulate (LCIRC) function, which provides the chaotic properties of the filter. The LCIRC function performs a bit left circulation over the N bits representation word. In the same paper, Frey showed that this nonlinear recursive filter possesses quasi-chaotic properties, both for autonomous and non-autonomous modes. In [WER 98], Werter improved this encoder in order to increase the randomness between the output sequence samples. The performances of a pulse amplitude modulation (PAM) communication system using the Frey encoder, with additive white Gaussian noise (AWGN), were analyzed in [AIS 96]. A modified state feedback decoder was proposed in [AIS 96] and performs better than the Frey inverse filter decoder, in terms of bit error rate (BER) at a high signal-to-noise ratio (SNR).

In all these previous works, the Frey encoder has been considered as a digital filter, operating over Galois field GF(2N) and was used to increase the security of the transmissions. During the last few decades, the nonlinear functions have been used extensively in chaotic sequence generators to increase the security of communications systems.

Nowadays, channel-encoded transmissions are used in all systems. Several types of channel encoding methods have been proposed during the last few decades. Almost all coding methods known in the literature use linear functions. Barbulescu et al. made one of the first approaches regarding the possible use of the Frey encoder in a turbo-coded communication system [BAR 00]. Nevertheless, in [BAR 00], the authors only mentioned as an advantage the intrinsic randomness of the encoder, which eliminates the use of an interleaver from the typical turbo encoding/decoding schemes. Despite the promising idea, this book did not consider any information theory approach for performance evaluation and, above all, it did not prove the advantages of these turbo encoders. In [ZHO 01], Zhou et al. made a similar analysis and again, the paper lacks proof for the stated performance enhancement. Another more recent work [NG 08] addressed different methods for using chaotic sequence generators to enhance the coding gain or the security of several coded schemes. More recently, in [ESC 09], a turbo trellis-coded modulation (TTCM) scheme using digital chaotic encoders with binary inputs and chaotic outputs is proposed. This work and the references therein introduce a different family of nonlinear encoders than the encoders analyzed in this book. However, we consider that the work of Escribano et al. [ESC 09] presents many similarities to our work, especially in the encoders’ design and performance analysis solutions.

In [VLA 09a], a different perspective was offered for the chaotic digital encoder proposed by Frey in [FRE 93]. Mainly, it was observed that this encoder with finite precision possesses a trellis that describes its deterministic functioning. This led to the possibility of improving the performances of the chaotic sequence transmission over a noisy channel by using sequence detection. In fact, this is the reason why this encoder should perform well in turbo coding schemes. To demonstrate this performance improvement as compared to the non-encoded system, a new TCM scheme was developed for obtaining better performance in the presence of noise. This method partially follows the rules proposed by Ungerboeck in [UNG 82] for defining optimum TCMs by proper set partitioning (SP). The main idea is to use a different word length in the output as compared to the input. In [CLE 06], Clevom et al. developed a method for separating a recursive systematic convolutional (RSC) encoder into subencoders with only a single delay element. This equivalency makes a GF(2) RSC equivalent to a simpler RSC that works inside over a higher order field, while its input and output still work over GF(2). Even if a different problem was addressed in [CLE 06], this idea led to the fact that different representation word lengths can be used in the input and output, and a higher order field nonlinear encoder is equivalent to a linear GF(2) RSC encoder. Therefore, in [VLA 09a], it was demonstrated that the Frey encoder with finite precision (word length of N bits) presented in [FRE 93] is a recursive convolutional (RC), but non-systematic, encoder operating over GF(2N) . In the same work [VLA 09a], a new method is proposed for enhancing the performances of the chaotic PAM-TCM transmission over a noisy channel.

A generalization of the optimum one-delay GF(4) encoder in [VLA 09a] was made, for any output word length N and for any possible encoding rate, in the cases of PAM-TCM [VLA 09b] and phase-shift keying TCM (PSK-TCM) [VLA 10b] transmission schemes. The development of optimum GF(2N) encoders for the quadrature amplitude modulation (QAM) TCM scheme is more difficult than in the case of PAM and PSK modulations, due to the larger constellations and non-uniform power per symbol. In [VLA 11b], a generalization of the optimum one-delay GF(4) encoder in [VLA 09a] is performed, for any output word length N and for any possible encoding rate in QAM-TCM schemes. These encoders follow the rules proposed by Ungerboeck [UNG 82] for defining optimum TCM by proper SP. However, all the previously mentioned encoders are non-systematic, making them unfeasible for TTCM schemes.

Two-dimensional (2D) TCM schemes using a different trellis optimization method for Frey encoder were proposed in [VLA 09c]. Hence, the filter scheme is modified to have an additional output, which transforms it into a rate 1/2 encoder. The second output is derived in such a manner as to transmit a 2D TCM signal, following the Ungerboeck SP rules [UNG 82]. Despite these changes, the filter representation word length is not changed.

The coding gain was estimated theoretically for all the considered encoders, for different values of N. In fact, exact expressions of the minimum Euclidean distance were determined for all these TCM schemes. Frey encoders with small representation word length (N = 1 and N = 2) were considered for simulations. There are two reasons for simulating encoders with small word lengths. First, the trellis size increases exponentially with N; second, the coding gain reduces when the word length increases. This last property is explained by the fact that the increase in the constellation size decreases the signal minimum Euclidean distance, more than the encoding can cope with. It is also noted that the linear encoders corresponding to the considered nonlinear encoders (obtained by eliminating the nonlinear block) do not have good trellises. For all these schemes, the simulated performances confirm the theoretically determined Euclidean distances.

The turbo coding scheme introduced by Berrou and Glavieux in their seminal paper [BER 96] allows communications systems’ performances close to the Shannon limit, by concatenating in parallel RC encoders in the transmitter and using iterative decoding algorithms in the receiver. Turbo schemes were developed for the TCM schemes as well [OGI 01, ROB 98].

In [PAU 10a], the RC-LCIRC encoder from [VLA 10b] is adapted for, and introduced into, a parallel turbo PSK-TCM transmission scheme, and the performances of this scheme are analyzed in case of transmitting over a channel with AWGN. Similarly, in [PAU 10b], the same RC-LCIRC is introduced into a parallel turbo QAM. The QAM-TCM transmission scheme and the performances of this scheme are analyzed in case of transmitting over a channel with AWGN. The performances of the RC-LCIRC-TTCM schemes introduced in [PAU 10a] and [PAU 10b] were analyzed in [VLA 10a] assuming a transmission over a non-selective Rayleigh fading channel.

In [VLA 11a], an improved version of the RC-LCIRC encoder from [VLA 10b] is proposed. The main encoder improvement consists of making it systematic. In fact, this was the only disadvantage of the previous LCIRC encoders which were not fully compatible with the corresponding binary encoders. Further to this new advantage, the encoder designing process aimed to keep all previous advantages of the LCIRC encoders, such as optimum performances in terms of Euclidean distance, the reduced complexity in the memory usage (for any encoding rate, only one delay element is used) and a compact expression of the Euclidean distance for a specific modulation. The optimum SP method is used both for PSK and QAM-TCM schemes. The symbol error rate (SER) performances of these new schemes are analyzed in case of transmitting over a channel with AWGN. Corresponding binary encoders, i.e. with the same values for the encoding rate, the number of trellis states and the minimum Euclidean distance, are considered as a reference for SER comparison.

A family of nonlinear encoders for the TTCM scheme was analyzed in [VLA 11c]. The systematic encoders introduced in [VLA 11a] were adapted for a parallel TTCM transmission scheme. As compared to the TTCM scheme analyzed in [PAU 10b] operating at low coding rates due to the lack of puncturing, the...