Spl Zopf Research Articles

A 2 x 2 antennas ultra-wideband system with biorthogonal pulse position modulation

The Golden code is a 2/spl times/2 space-time code that achieves the best known performance with all constellations carved from /spl Zopf/[i]. In this letter, we present the construction of a new coding scheme for 2M-ary biorthogonal pulse position modulations (BPPM) with M/spl ges/4. The proposed code satisfies all of the construction constraints of the Golden code and it has the additional advantage of being totally real making it suitable for low cost carrier-less ultra-wideband terminals. Namely, this totally real construction achieves full rate and full diversity with the best known coding gain and without any shaping losses for 2M-BPPM with M/spl ges/4.

Generalization of the Σ/sub 0/-rank criteria for space-time codes

Recently, Liu et al., developed the /spl Sigma//sub 0/-rank criteria for space-time codes. It provides a sufficient condition on codeword and generator matrices defined over finite rings /spl Zopf//sub 2k/(j) to ensure full spatial diversity with 2/sup 2k/-QAM modulation. Here, we generalize the /spl Sigma//sub 0/-rank criteria and derive a sufficient condition on the generator matrices defined over finite rings /spl Zopf//sub 2l/(j) to ensure full spatial diversity with 2/sup 2k/-QAM modulation for any positive integer l/spl les/k. We also show that generator matrices defined over GF(2) satisfying the BPSK stacking construction constraint of Mammons and El Gamal achieve full spatial diversity when used with 2/sup 2k/-QAM modulation.

Coded modulation schemes based on partial Gray mapping and unpunctured high rate turbo codes

The paper deals with coding for the band-limited AWGN channel. Coded modulation schemes based on high rate unpunctured turbo codes are proposed. It is shown that, owing to efficient decoding based on the trellis of the dual code and to the optimised code design, coded modulations based on this class of turbo codes allow excellent performance with moderate complexity for practical values of block size and redundancy. A mapping rule already envisaged in previously published papers, based on the partition /spl Zopf//2/sup l//spl Zopf/ with Gray mapping of l bits is exploited. Schemes based on the partition chain /spl Zopf//2/spl Zopf//4/spl Zopf/ are also presented in a comparative way. From this comparison it is concluded that, when constraints are imposed on the number of dimensions and on the redundancy of the coded modulation, partial Gray mapping based on /spl Zopf//4/spl Zopf/ should be preferred to pure mapping by set partitioning and, with QAM constellations with more than 16 points, it should also be preferred to pure Gray mapping.

On the additive (/spl Zopf//sub 4/-linear and non-/spl Zopf//sub 4/-linear) Hadamard codes: rank and kernel

All the possible nonisomorphic additive (/spl Zopf//sub 4/-linear and non-/spl Zopf//sub 4/-linear) Hadamard codes are characterized and the rank and the dimension of the kernel are computed for each one.

LDPC Codes Over Rings for PSK Modulation

This paper describes the design and analysis of low-density parity-check (LDPC) codes over rings and shows how these codes, when mapped onto appropriate signal constellations, can be used to effect bandwidth-efficient modulation. Specifically, LDPC codes are constructed over the integer rings /spl Zopf//sub m/ and G/sub m//sup 2/ and mapped onto phase-shift keying (PSK)-type signal sets to yield geometrically uniform signal space codes. This paper identifies and addresses the design issues that affect code performance. Examples of codes over /spl Zopf//sub 8/ and G/sub 64/ mapped onto 8-ary and 64-ary signal sets at a spectral efficiency of 1.5 and 2.0 bits per second per hertz (b/s/Hz) illustrate the approach; simulation of these codes over the additive white Gaussian noise (AWGN) channel demonstrates that this approach is a good alternative to bandwidth-efficient techniques based on binary LDPC codes-e.g., bit-interleaved coded modulation.

Worst-case error analysis of lifting-based fast DCT-algorithms

Integer DCTs have a wide range of applications in lossless coding, especially in image compression. An integer-to-integer DCT of radix-2-length n is a nonlinear, left-invertible mapping, which acts on /spl Zopf//sup n/ and approximates the classical discrete cosine transform (DCT) of length n. All known integer-to-integer DCT-algorithms of length 8 are based on factorizations of the cosine matrix C/sub 8//sup II/ into a product of sparse matrices and work with lifting steps and rounding off. For fast implementation one replaces floating point numbers by appropriate dyadic rationals. Both rounding and approximation leads to truncation errors. In this paper, we consider an integer-to-integer transform for (2/spl times/2) rotation matrices and give estimates of the truncation errors for arbitrary approximating dyadic rationals. Further, using two known integer-to-integer DCT-algorithms, we show examplarily how to estimate the worst-case truncation error of lifting based integer-to-integer algorithms in fixed-point arithmetic, whose factorizations are based on (2/spl times/2) rotation matrices.

Quaternary Reed–Muller Codes

The class of quaternary Reed-Muller codes is introduced as a generalization of Reed-Muller codes to /spl Zopf//sub 4/-linear codes. Properties of this class of codes are established including the rank and kernel of the Gray map image of codes in this class. The class includes all /spl Zopf//sub 4/-linear Kerdock-like and Preparata-like codes.

Unimodular Perfect Sequences of Length<tex>$p^s$</tex>

A new class of unimodular perfect sequences of length p/sup s/, where p is a prime, is proposed. An explicit secondary construction is provided. This construction includes most of the previously known unimodular perfect sequences of length p/sup s/ as special cases. Also the proposed construction is extended to sequences with autocorrelation over /spl Zopf//sup s//sub p//sup k/, V/sub m2//spl times//spl Zopf//sub p//sup m1/, V/sub k/.

Interleavers for turbo codes using permutation polynomials over integer rings

A class of deterministic interleavers for turbo codes (TCs) based on permutation polynomials over /spl Zopf//sub N/ is introduced. The main characteristic of this class of interleavers is that they can be algebraically designed to fit a given component code. Moreover, since the interleaver can be generated by a few simple computations, storage of the interleaver tables can be avoided. By using the permutation polynomial-based interleavers, the design of the interleavers reduces to the selection of the coefficients of the polynomials. It is observed that the performance of the TCs using these permutation polynomial-based interleavers is usually dominated by a subset of input weight 2m error events. The minimum distance and its multiplicity (or the first few spectrum lines) of this subset are used as design criterion to select good permutation polynomials. A simple method to enumerate these error events for small m is presented. Searches for good interleavers are performed. The decoding performance of these interleavers is close to S-random interleavers for long frame sizes. For short frame sizes, the new interleavers outperform S-random interleavers.

A Mass Formula and Rank of<tex>$BBZ_4$</tex>Cyclic Codes of Length<tex>$2^e$</tex>

In this correspondence, we study cyclic codes of length n=2/sup e/ over the ring R/sub 4/=/spl Zopf//sub 4/[x]/(x/sup n/-1). In particular, we derive a closed-form expression for the number of these codes for a given length n. We also study the rank of these codes and derive an expression for that. Furthermore, we give an example in which we study all cyclic codes of length 8. We also study all self-dual codes of length 8 and 16 and classify them according to their type.

Generalized MacWilliams Identities for<tex>$BBZ _4$</tex>-Linear Codes

In this correspondence, the generalized MacWilliams identities between the support weight distributions of a /spl Zopf//sub 4/-linear code of type 4/sup k/ and that of its dual are established.

Input–Output Stabilization of Linear Systems on<tex>$BBZ$</tex>

A formal framework is set up for the discussion of generalized autoregressive with external input models of the form Ay=Bu, where A and B are linear operators, with the main emphasis being on signal spaces consisting of bounded sequences parametrized by the integers. Different notions of stability are explored, and topological notions such as the idea of a closed system are linked with questions of stabilizability in this very general context. Various problems inherent in using /spl Zopf/ as the time axis are analyzed in this operatorial framework.

Cyclic and Negacyclic Codes Over Finite Chain Rings

The structures of cyclic and negacyclic codes of length n and their duals over a finite chain ring R are established when n is not divisible by the characteristic of the residue field R~. Some cases where n is divisible by the characteristic of the residue field R~ are also considered. Namely, the structure of negacyclic codes of length 2/sup t/ over /spl Zopf//sub 2//sup m/ and that of their duals are derived.

Feedback With Carry Shift Registers Synthesis With the Euclidean Algorithm

Feedback with carry shift registers (FCSR) were introduced by Klapper and Goresky (1994). They are very similar to classical linear feedback shift registers (LFSR) used in many pseudorandom generators. The main difference is the fact that the elementary additions are not additions modulo 2 but with propagation of carries. The mathematical models for LFSR are equivalently linear recurring sequences over GF(2) or rational series in the set GF(2)[[x]]. For FCSR, the good model is the one of rational 2-adic numbers. It is well known, that the series generated by a LFSR can be synthesized by either the Berlekamp-Massey algorithm for binary linear recurring sequences or the extended Euclidean algorithm in the set GF(2)[x] of binary polynomials. Klapper and Goresky (1997) give an algorithm for the FCSR synthesis. This algorithm is similar to those of Berlekamp-Massey and is based on De Weger and Mahler's rational approximation theory. In this correspondence, we prove that it is possible to synthesize the FCSR with the extended Euclidean algorithm in the ring /spl Zopf/ of integers. This algorithm is clearly equivalent to the previous one, however, it is simpler to understand, to implement, and to prove. Our algorithm is still valid in the case of g-adic integers where g is a positive integer. We also give a near-adaptative version of this algorithm.

M-channel lifting factorization of perfect reconstruction filter banks and reversible M-band wavelet transforms

An intrinsic M-channel lifting factorization of perfect reconstruction filter banks (PRFBs) is presented as an extension of Sweldens' conventional two-channel lifting scheme. Given a polyphase matrix E(z) of a finite-impulse response (FIR) M- channel PRFB with det(E(z))=z/sup -K/, K/spl isin//spl Zopf/, a systematic M-channel lifting factorization is derived based on the Monic Euclidean algorithm. The M-channel lifting structure provides an efficient factorization and implementation; examples include optimizing the factorization for the number of lifting steps, delay elements, and dyadic coefficients. Specialization to paraunitary building blocks enables the design of paraunitary filter banks based on lifting. We show how to achieve reversible, possibly multiplierless, implementations under finite precision, through the unit diagonal scaling property of the Monic Euclidean algorithm. Furthermore, filter-bank regularity of a desired order can be imposed on the lifting structure, and PRFBs with a prescribed admissible scaling filter are conveniently parameterized.

On Z/sub 4/-linear preparata-like and kerdock-like codes

We say that a binary code of length n is additive if it is isomorphic to a subgroup of /spl Zopf//sub 2//sup /spl alpha// /spl times/ /spl Zopf//sub 4//sup /spl beta//, where the quaternary coordinates are transformed to binary by means of the usual Gray map and hence /spl alpha/ + 2/spl beta/ = n. In this paper, we prove that any additive extended Preparata (1968) -like code always verifies /spl alpha/ = 0, i.e., it is always a /spl Zopf//sub 4/-linear code. Moreover, we compute the rank and the dimension of the kernel of such Preparata-like codes and also the rank and the kernel of the /spl Zopf//sub 4/-dual of these codes, i.e., the /spl Zopf//sub 4/-linear Kerdock-like codes.

On the generators of Z/sub 4/ cyclic codes of length 2/sup e/

Results are presented on the generators of ideals in the ring /spl Zopf//sub 4/[x]/(x/sup n/-1). In particular, each ideal (cyclic code) has a unique distinguished set of generators that characterizes any cyclic code. Some results about dual codes are also included.

The rank and kernel of extended 1-perfect Z/sub 4/-linear and additive non- Z/sub 4/-linear codes

A binary extended 1-perfect code of length n + 1 = 2/sup t/ is additive if it is a subgroup of /spl Zopf//sub 2//sup /spl alpha// /spl times/ /spl Zopf//sub 4//sup /spl beta//. The punctured code by deleting a /spl Zopf//sub 2/ coordinate (if there is one) gives a perfect additive code. 1-perfect additive codes were completely characterized and by using that characterization we compute the possible parameters /spl alpha/, /spl beta/, rank, and dimension of the kernel for extended 1-perfect additive codes. A very special case is that of extended 1-perfect /spl Zopf//sub 4/-linear codes.

Curves on a sphere, shift-map dynamics, and error control for continuous alphabet sources

We consider two codes based on dynamical systems, for transmitting information from a continuous alphabet, discrete-time source over a Gaussian channel. The first code, a homogeneous spherical code, is generated by the linear dynamical system s/spl dot/=As, with A a square skew-symmetric matrix. The second code is generated by the shift map s/sub n/=b/sub n/s/sub n-1/(mod 1). The performance of each of these codes is determined by the geometry of its locus or signal set, specifically, its arc length and minimum distance, suitably defined. We show that the performance analyses for these systems are closely related, and derive exact expressions and bounds for relevant geometric parameters. We also observe that the lattice /spl Zopf//sup N/ underlies both modulation systems and we develop a fast decoding algorithm that relies on this observation. Analytic results show that for fixed bandwidth expansion, good scaling behavior of the mean squared error is obtained relative to the channel signal-to-noise ratio (SNR). Particularly interesting is the resulting observation that sampled, exponentially chirped modulation codes are good bandwidth expansion codes.

Complete joint weight enumerators and self-dual codes

We define the complete joint weight enumerator in genus g for codes over /spl Zopf//sub 2k/ and use it to study self-dual codes and their shadows. These weight enumerators are related to the theta series of the associated lattices and Siegel and Jacobi forms are formed from these series.