Fourier-like transforms over the Galois field, GF(q2), and the direct sum of such Galois fields, analogous to the field of complex numbers, were developed and used to design round-off noise free nonrecursive filters (1), (2). It was shown (3), (4) that recursive realizations of finite impulse response (FIR) filters on the finite field or ring are also possible by techniques, analogous to those recursive realizations used on the usual complex number field. This paper developes an algorithm to reduce the number of recursive equations needed to realize a filter when recursive equations are restricted to be first order. The recursive filters developed here do not have the accumulation of round-off or truncation error one usually expects in recursive computations. The FIR filter without error is more naturally a digital" filter than the usual FIR digital filter. By the very finiteness of the finite field or ring, only finite bit words are needed to realize the FIR filter. Such digital filters when realized as discrete transducers are information lossless in the sense of Shannon (5, pp. 26).

This paper examines performance study items for ATM connections in B-ISDNs. We consider the characteristics of B-ISDN performance and describe the current status in ITU-T and the ATM Forum. On this basis, we propose a new performance framework and performance criteria. We also describe objectives for ATM cell transfer performance.

This paper introduces a new recursive factorization of the polynomial, 1-zN, over the real numbers when N is an even composite integer. The recursive factorization is applied for efficient computation of the discrete Fourier transform (DFT) and the cyclic convolution of real sequences with highly composite even length.

Filtering techniques on a finite ring are paralleled to the techniques on the usual complex number field. A z-transform is defines over a finite ring which assimilates the complex number field. This z-transform is related to the usual z-transform and shown to have many properties in common. This z-transform can uniquely represent a filter on the finite ring with a given impulse response or frequency sample response function. The z-transform representation enable one to design recursive FIR (Finite Impulse Response) filter on the finite ring by the analogous methods of the usual z-transform. An interesting property of the recursive filter design on the finite ring is that stability problems due to particular design configulations need not to be considered so long as the inputs are limited to a certain value so that the design problem becomes primarily a problem of complexity reduction.

This paper introduces a generalized cyclic convolution which can be implemented via the conventional cyclic convolution system by the discrete Fourier transform (DFT) with pre-multiplication for the input and post-multiplication for the output. The generalized cyclic convolution is applied for computing a negacyclic convolution. Comparison shows that the proposed implementation is more efficient and simpler in structure than other methods. The modified Fermat number transform (MFNT) is known to be useful for computing a linear convolution of integer-valued sequences. The generalized cyclic convolution is also applied for generalizing the linear convolution system by MFNT, and easing the signal length restriction imposed by the system.

THe decimation-in-time (DIT) and the decimation-in-frequency (DIF) algorithms are the most well-known fast algorithms for computing the discrete Fourier transform(DFT). These algorithms constitute the basis of the fast Fourier transform (FFT) implementations, including the pipeline implementation and other parallel configurations. This paper derives an alternative generalization of the algorithms which applies for sequences whose lengths are not a power of two. The treatment is consistent with the radix-two DIF and DIT algorithms, and the generalization is useful for utilizing the accumulated technologies of the FFT algorithm for such sequences.