Improved Frey Chaotic Digital Encoder for Trellis-Coded Modulation

In this brief, nonlinear digital filters with finite precision are analyzed as recursive systematic convolutional (RSC) encoders. An infinite-impulse-response (IIR) digital filter with finite precision (wordlength of $N$ bits) is a rate-1 RSC encoder over a Galois field $GF(2^{N})$. The Frey chaotic filter is analyzed for different wordlengths $N$, and it is demonstrated that the trellis performances can be enhanced by proper filter design. Therefore, a modified definition for the encoding rate is provided, and a trellis design method is proposed for the Frey filter, which consists of reducing the encoding rate from 1 to 1/2. This trellis optimization partially follows Ungerboeck's rules, i.e., increasing the performances of the encoded chaotic transmission in the presence of noise. In fact, it is demonstrated that for the same spectral efficiency, the modified Frey encoder outperforms the original Frey encoder only for $N = 2$. To show the potential of these nonlinear encoders, it is demonstrated that a particular nonlinear digital filter over $GF(4)$ is equivalent to a $GF(2)$ conventional optimum RSC encoder. The symbol error rate (SER) is estimated for all the proposed schemes, and the results show the expected coding gains as compared to their equivalent nonencoded and linear versions.