Partial Inner Product Research Articles

Partial-LUT Designs for Low-Complexity Realization of DA-Based BLMS Adaptive Filter

This brief presents two-optimized partial look-up table (LUT) designs for low-complexity realization of distributed arithmetic (DA) based block least-mean-square (BLMS) adaptive filter (ADF). These are based on the partial add-store (PAS) and partial store-add (PSA) methods. A novel optimization scheme is presented which exploits the redundancies between the partial inner-products with sliding-window of input-block. It is found that the PAS method provides shorter critical-path-delay than the PSA method. A case study on echo-cancellation is demonstrated for architectural trade-off between the proposed designs. Synthesis results show that the proposed PSA based BLMS ADF for 64 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> order offers 41.04% lesser area and 39.38% lesser power, while the PAS based BLMS ADF provides 52.08% lesser area and 50.05% lesser power as compared to the best existing work.

An Analytical Framework and Approximation Strategy for Efficient Implementation of Distributed Arithmetic-Based Inner-Product Architectures

Distributed arithmetic (DA)-based approximate structures are used for efficient implementation of inner-products in various error-resilient applications. In the existing literature, most of these approximate architectures are developed by truncating the least significant bits (LSBs) of the inputs and/or the multiplying coefficients. The existing works do not provide any analytical study to evaluate and design an approximate structure. To address this issue, an analytical framework is proposed in this paper. It is shown that the analytical results match very closely with the Monte Carlo simulation results. The proposed framework reveals that the truncation of the LSBs of partial inner-products is a promising alternative to design more efficient DA architectures with less error. Following these observations, a novel approach to truncate the LSBs of partial inner-products, namely, a weight-dependent truncation strategy and its two variants with a suitable error compensation function are presented in this paper. Synthesis results, accuracy analysis, and evaluation in two commonly used error-tolerant applications demonstrate the superiority of the proposed architectures over the state-of-the-art DA-based approximate structures.

Some unbounded commutants of a set of operators on a partial inner product space

This paper continues our systematic study of unbounded commutants within the framework of operators on a partial inner product (PIP) space. The most general commutant of a set of operators on a PIP space is considered and its behavior with respect to a topology finer than the weak and quasiweak *-topologies used in previous investigations is studied. The relationship between a bicommutant introduced by Shabani and a bicommutant introduced by Araki and Jurzak for closed Op*-algebras satisfying some countability conditions is given.

Partial inner product spaces. IV. Topological considerations

Whereas the third paper in this series dealt with the algebraic structure of partial inner product (PIP) spaces, the present one explores systematically their topological properties. A slightly more restricted object is introduced, that we call an indexed PIP-space: it consists of a PIP-space together with a distinguished family of assaying subspaces. The upshot of the analysis is the characterization of two types of indexed PIP-spaces, called type (B) and type (H), respectively, as the most likely candidates for practical applications; they are simply lattices of Banach, resp. Hilbert spaces. Operators on indexed PIP-spaces are discussed and conditions are given that guarantee that the domain of any such operator is a vector subspace. Finally, we examine the question of the existence of a central Hilbert space, in the case of a positive definite partial inner product.