The robust Chinese remainder theorem (CRT) has been recently proposed for robustly reconstructing a large nonnegative integer from erroneous remainders. It has found many applications in signal processing, including phase unwrapping, and frequency estimation under sub-Nyquist sampling. Motivated by the applications in multidimensional (MD) signal processing, in this article we propose the MD-CRT and robust MD-CRT for integer vectors. Specifically, by rephrasing the abstract CRT for rings in number-theoretic terms, we first derive the MD-CRT for integer vectors with respect to a general set of integer matrix moduli, which provides an algorithm to uniquely reconstruct an integer vector from its remainders, if it is in the fundamental parallelepiped of the lattice generated by a least common right multiple of all the moduli. For some special forms of moduli, we present explicit reconstruction formulae. Moreover, we derive the robust MD-CRT for integer vectors when the remaining integer matrices of all the moduli left divided by their greatest common left divisor (gcld) are pairwise commutative, and coprime. Two different reconstruction algorithms are proposed, and accordingly, two different conditions on the remainder error bound for the reconstruction robustness are obtained, which are related to a quarter of the minimum distance of the lattice generated by the gcld of all the moduli or the Smith normal form of the gcld.

Exact and Robust Reconstructions of Integer Vectors Based on Multidimensional Chinese Remainder Theorem (MD-CRT)

The Chinese remainder theorem is a fundamental technique widely employed in digital signal processing for designing fast algorithms for computing convolutions. Classically, it has two versions. One is over a ring of integers and the second is over a ring of polynomials with coefficients defined over a field. In our previous papers, we developed an extension to this well-known theorem for the case of a ring of polynomials with coefficients defined over a finite ring of integers. The objective was to generalize number-theoretictransforms, which turn out to be a special case of this extension. This paper focuses on the extension of the Chinese remainder theorem for processing complex-valued integer sequences. Once again, the present work generalizes the complex-number-theoretic transforms. The impetus for this work is provided by the occurrence of complex integer sequences in digital signal processing and the desire to process them using exact arithmetic.

The AICE-CRT and digital signal processing algorithms: the complex case

In our recent work, we solved the word sequence length constraint problem associated with number theoretic transforms defined in finite integer rings. This is based on the American-Indian-Chinese extension of the Chinese remainder theorem. This work builds further on the results by extending them to the domain of integer polynomial rings. The theory of polynomial factorization and the resulting direct sum property are studied in depth. The emphasis is on the theory of computational algorithms for processing sequences defined in finite integer and complex integer rings.

On the factorization of polynomials and direct sum properties in integer polyomial rings

A polynomial remainder code, derived from the Chinese remainder theorem, is a class of linear block codes, where the Reed–Solomon (RS) code is a special case. In this letter, an extended version of polynomial remainder codes is introduced, where the class of doubly extended RS codes is a special case. Furthermore, the extended version of Chinese remainder codes is also presented. The erasure decoding methods for both the codes are proposed. Finally, an application of the extended polynomial remainder codes is discussed.

Extended Versions of Polynomial Remainder Codes and Chinese Remainder Codes

An irregular MDS array code is a class of array codes allowing unequal sizes of each columns. In this paper, two classes of irregular MDS array codes are proposed to reduce the number of parity symbols at certain cases. The first code is based on polynomial remainder codes, and the second code uses the code bundling method. In addition, a systematic encoding approach of the polynomial remainder code is also presented.

Irregular MDS Array Codes With Fewer Parity Symbols

For pt. I, see ibid., vol. 41, no. 10, p. 641-55 (1994). In Part I of the research work, we introduced an extension to the well known Chinese remainder theorem for processing polynomials with coefficients defined over a finite integer ring. We term this extension as the American-Indian-Chinese extension of the Chinese remainder theorem. A systematic procedure for factorizing a monic polynomial into pairwise relatively prime monic factor polynomials over integer rings was presented. This factorization is based on the corresponding factor polynomials, monic and relatively prime, over the associated finite field containing prime number of elements. In this paper, we study the application of the theory developed in Part I to deriving computationally efficient algorithms for performing tasks having multilinear form. Especially, we focus on the cyclic and acyclic convolution as they are two of the most frequently occurring computationally intensive tasks in digital signal processing. >

B. Krishna

Papers

Rings, fields, the Chinese remainder theorem and an extension-Part II: applications to digital signal processing