On list decoding of wavelet codes over finite fields of characteristic two

Доказывается, что вейвлет-код над полем GF(2m) c длиной кодовых и информационных слов n = 2m — 1 и (n — 1)/2 соответственно, у которого среди коэффициентов спектрального представления порождающего многочлена имеется d + 1 последовательных нулей, 0 < d < (n — 3)/2, допускает списочное декодирование за полиномиальное время. Шаги алгоритма, осуществляющего списочное декодирование с исправлением до e < n — д/n(n — d — 2) ошибок, реализованы в виде программы. Приведены примеры её применения для списочного декодирования зашумленных кодовых слов. Отмечено, что неравенство Варшамова — Гилберта при достаточно больших n не позволяет судить о существовании вейвлет-кодов c максимальным кодовым расстоянием (n — 1) /2

Cyclic LTI systems in digital signal processing

Cyclic signal processing refers to situations where all the time indices are interpreted modulo some integer L. In such cases, the frequency domain is defined as a uniform discrete grid (as in L-point DFT). This offers more freedom in theoretical as well as design aspects. While circular convolution has been the centerpiece of many algorithms in signal processing for decades, such freedom, especially from the viewpoint of linear system theory, has not been studied in the past. In this paper, we introduce the fundamentals of cyclic multirate systems and filter banks, presenting several important differences between the cyclic and noncyclic cases. Cyclic systems with allpass and paraunitary properties are studied. The paraunitary interpolation problem is introduced, and it is shown that the interpolation does not always succeed. State-space descriptions of cyclic LTI systems are introduced, and the notions of reachability and observability of state equations are revisited. It is shown that unlike in traditional linear systems, these two notions are not related to the system minimality in a simple way. Throughout the paper, a number of open problems are pointed out from the perspective of the signal processor as well as the system theorist

Списочное декодирование вейвлет-кодов

В работе обсуждается возможность списочного декодирования вейвлет-кодов и приводится утверждение, согласно которому вейвлет-коды над полем $GF(q)$ нечетной характеристики с длиной кодовых и информационных слов $n=q-1$ и $n/2$ соответственно, а также над полем четной характеристики с длиной кодовых и информационных слов $n=q-1$ и $(n-1)/2$ соответственно допускают списочное декодирование, если среди коэффициентов спектрального представления их порождающих многочленов имеется $d+1$ последовательных нулей, $0$ &lt; $d$ &lt; $n/2$ для полей нечетной характеристики и $0$ &lt; $d$ &lt; $(n-3)/2$ для полей четной характеристики. Также описывается алгоритм, позволяющий выполнять списочное декодирование вейвлет-кодов при соблюдении перечисленных условий. В качестве демонстрации его работы приводятся пошаговые решения модельных задач списочного декодирования зашумленных кодовых слов вейвлет-кодов над полями четной и нечетной характеристики. Помимо этого, в работе построена вейвлет-версия квазисовершенного троичного кода Голея, длины его кодовых и информационных слов равны 8 и 4 соответственно, кодовое расстояние равно 4, минимальный радиус шаров с центрами в кодовых словах, покрывающих пространство слов длины 8, равен 3

Pyramid scheme for constructing biorthogonal wavelet codes over finite fields

Конструктивным образом доказывается существование биортогонального разбиения векторного пространства V размерности n над полем GF(q), а именно двух его представлений в виде прямых сумм подпространств V = W0 ®W1 ф.. .®Wj®Vj и V = Wо ф W1 ф ... ф Wj ф Vj, таких, что на j-м уровне разложения (0 < j J) Vj-1 = Vj ф Wj, Vj-1 = Vj ф Wj, подпространство Vj ортогонально Wj, а подпространство Wj ортогонально Vj. Для этого используются пары биортогональных фильтров (h,g) и (h,g). Разбиение пространства на j-м уровне разложения осуществляется при помощи пар уровневых фильтров (hj ,gj) и (hj ,gj), для построения которых разработаны и теоретически обоснованы соответствующие алгоритмы. На основе многоуровневой схемы вейвлет-разложения строится новое семейство биортогональных вейвлет-кодов со скоростью кодирования 2-L, где L — количество использованных уровней разложения, и приводятся примеры таких кодов

Morphological subband decomposition structure using GF(N) arithmetic

Linear filter banks with critical subsampling and perfect reconstruction (PR) property have received much interest and found numerous applications in signal and image processing. Recently, nonlinear filter bank structures with PR and critical subsampling have been proposed and used in image coding. In this paper, it is shown that PR nonlinear subband decomposition can be performed using the Gallois Field (GF) arithmetic. The result of the decomposition of an n-ary (e.g. 256-ary) input signal is still n-ary at different resolutions. This decomposition structure can be utilized for binary and 2k (k is an integer) level signal decompositions. Simulation studies are presented

Mapping Equivalence for Symbolic Sequences: Theory and Applications

Processing of symbolic sequences represented by mapping of symbolic data into numerical signals is commonly used in various applications. It is a particularly popular approach in genomic and proteomic sequence analysis. Numerous mappings of symbolic sequences have been proposed for various applications. It is unclear however whether the processing of symbolic data provides an artifact of the numerical mapping or is an inherent property of the symbolic data. This issue has been long ignored in the engineering and scientific literature. It is possible that many of the results obtained in symbolic signal processing could be a byproduct of the mapping and might not shed any light on the underlying properties embedded in the data. Moreover, in many applications, conflicting conclusions may arise due to the choice of the mapping used for numerical representation of symbolic data. In this paper, we present a novel framework for the analysis of the equivalence of the mappings used for numerical representation of symbolic data. We present strong and weak equivalence properties and rely on signal correlation to characterize equivalent mappings. We derive theoretical results which establish conditions for consistency among numerical mappings of symbolic data. Furthermore, we introduce an abstract mapping model for symbolic sequences and extend the notion of equivalence to an algebraic framework. Finally, we illustrate our theoretical results by application to DNA sequence analysis

Paraunitary Filter Banks over Finite Fields

real and complex fields, unitary and paraunitary (PU) matrices have found many applications in signal processing. There has recently been interest in extending these ideas to the case of finite fields. In this paper, we will study the theory of PU filter banks (FB’s) in GF(y) with y prime. Various properties of unitary and PU matrices in finite fields will be studied. In particular, a number of factorization theorems will be given. We will show that i) all unitary matrices in GF(y) are factorizable in terms of Householder-like matrices and permutation matrices, and ii) the class of first-order PU matrices (the lapped orthogonal transform in finite fields) can always be expressed as a product of degree-one or degree-two building blocks. If Q&gt; 2, we do not need degree-two building blocks. While many properties of PU matrices in finite fields are similar to those of PU matrices in complex field, there are a number of differences. For example, unlike the conventional PU systems, in finite fields, there are PU systems that are unfuctorizable in terms of smaller building blocks. In fact, in the special case of 2 x 2 systems, all PU matrices that are factorizable in terms of degree-one building blocks are diagonal matrices. We will derive results for both the cases of GF(2) and GF(y) with q&gt; 2. Even though they share some similarities, there are many differences between these two cases. I

Wavelet transforms associated with finite cyclic groups

Abstmct- Multiresolution analysis via decomposition on wavelet bases has emerged as an important tool in the analysis of signals and images when these objects are viewed as sequences of complex or real numbers. An important class of multiresolution decompositions are the so-called Laplacian pyramid schemes, in which the resolution is successively halved by recursively low-pass filtering the signal under analysis and decimating it by a factor of two. Generally speakhg, the principal framework within which multiresolution techniques have been studied and applied is the same as that used in the discrete-time Fourier analysis of sequences of complex numbers. An analogous framework is developed for the multiresolution analysis of finite-length sequences of elements €mm arbitrary fields. Attention is restricted to sequences of length 2 &amp;quot; for n a positive iuteger, so that the resolution may be recursively halved to completion. As in finite-length Fourier analysis, a cyclic group structure of the index set of such sequences is exploited to characterize the transforms of interest for the particular cases of complex and finite fields. This development is motivated by potential applications in areas such as digital signal processing and algebraic coding, in which cyclic Fourier analysis has found widespread applications. Index Terms- Multiresolution analysis, wavelet transforms, Laplacian pyramid, finite fields, cyclic group, quadrature mimr filters. I