Finite-state Sequences Research Articles

Randomness of finite‐state sequence machine over GF(4) and quality of hopping turbo codes

In this study, the authors study a turbo-coding (TC) scheme, whose constituent codes are designed using convolutional encoders. These encoders are finite-state sequence machines (FSSMs) operating over the Galois Field, GF(4). The scheme includes encryption polynomials whose coefficients are selected every L steps, from the set of optimal polynomials of GF(4). Two cases are considered for the polynomial selection: periodical and random. This kind of encoder was studied in a previous study and a correspondence between the randomness of the encoded sequence and performance of the TC was conjectured. The main contribution of this study is to systematically confirm this correspondence, by analysing the randomness of the output and performance of the TC using several randomness quantifiers. Three of the quantifiers are defined on the basis of recurrence plots. Other two quantifiers are defined on the basis of the information theory. All the quantifiers allow one to justify why the proposed TC works better with random selection of the optimal polynomials and with small values of L. In summary, it is shown that a random selection of polynomials and a small L produce FSSMs with enhanced randomness properties and it is also shown that they produce the best quality of the TC, measured by means of the corresponding bit error rate.

Trellis-hopping turbo coding

We present a turbo coding scheme whose constituent codes are designed using convolutional encoders with time-varying coefficients. These encoders are finite state sequence machines that operate over the Galois field GF(q). The scheme includes an encryption polynomial whose coefficients are changed periodically by means of a user key. The trellis coding procedure thus hops from one trellis to another, following a random sequence taken over a set of subtrellises which correspond to different convolutional encoders. The proposed scheme is introduced as a turbo code with encryption properties, and is presented in two forms, the systematic and the nonsystematic schemes.

Not all (possibly) "random" sequences are created equal.

The need to assess the randomness of a single sequence, especially a finite sequence, is ubiquitous, yet is unaddressed by axiomatic probability theory. Here, we assess randomness via approximate entropy (ApEn), a computable measure of sequential irregularity, applicable to single sequences of both (even very short) finite and infinite length. We indicate the novelty and facility of the multidimensional viewpoint taken by ApEn, in contrast to classical measures. Furthermore and notably, for finite length, finite state sequences, one can identify maximally irregular sequences, and then apply ApEn to quantify the extent to which given sequences differ from maximal irregularity, via a set of deficit (def(m)) functions. The utility of these def(m) functions which we show allows one to considerably refine the notions of probabilistic independence and normality, is featured in several studies, including (i) digits of e, pi, radical2, and radical3, both in base 2 and in base 10, and (ii) sequences given by fractional parts of multiples of irrationals. We prove companion analytic results, which also feature in a discussion of the role and validity of the almost sure properties from axiomatic probability theory insofar as they apply to specified sequences and sets of sequences (in the physical world). We conclude by relating the present results and perspective to both previous and subsequent studies.

Discrimination of restrictions upon auditory sequentially encoded information: Block parity

Finite-state sequences were generated bya digital computer, converted to interval-coded pulse sequences, and transduced to sound by means of earphones. Within each block of B successive intervals, (B − 1) intervals were randomly selected and the Bth interval was selected to achieve a fixed parity sum over the block. Discriminability among the sequences—as measured by interstate interval thresholds-reflects both the specific form and the extent of the sequential restrictions introduced by parity. Parity restrictions upon interval-coded sequences may bediscriminable through modification of distributions of runs.

Some Structure Theorems for Stationary Probability Measures on Finite State Sequences

Some Structure Theorems for Stationary Probability Measures on Finite State Sequences