Jean-Marc Delosme

On Sigma-lossless transfer functions and related questions

Abstract This paper is concerned with a systematic approach to the properties of ∑-lossless rational transfer functions in the discrete as well as in the continuous time case. As a result, a unifying framework is revealed where several known results fit naturally. Special attention is given to the embedding problem of the Lyapunov equation in view of its direct application to generalized Levinson algorithms.

VLSI implementation of rotations in pseudo-Euclidean spaces

Digital feedback is at the heart of implementations of many functions and can be used for the evaluation and application of rotations in definite and indefinite metrics. The CORDIC algorithms exploit this concept to perform rotations in two dimensions. These algorithms are analyzed as a first step before extending them to higher dimensions. A side result of this analysis is the simultaneous reduction of the number of steps and the enlargement of the domain of convergence of the algorithms. A few guidelines for their extension and a worked out three-dimensional example are then presented.

Application of reconfigurable CORDIC architectures

Reconfiguration enables the adaption of Coordinate Rotation DIgital Computer (CORDIC) units to the specific needs of sets of applications, hence creating application specific CORDIC-style implementations. Reconfiguration can be implemented at a high level, taking the entire CORDIC unit as a basic cell (CORDIC-cells) implemented in VLSI, or at a low level such as Field-Programmable Gate Arrays (FPGAs). We suggest a design methodology and analyze area/time results for coarse (VLSI) and fine-grain (FPGA) reconfigurable CORDIC units. For FPGAs we implement CORDIC units in Verilog HDL and our object-oriented design environment, PAM-Blox. For CORDIC-cells, multiple reconfigurable CORDIC modules are synthesized with state-of-the-art CAD tools. At the algorithm level we present a case study combining multiple CORDICs based on a geometrical interpretation of a normalized ladder algorithm for adaptive filtering to reduce latency and area of a fully pipelined CORDIC implementation. Ultimately, the goal is to create automatic tools to map applications directly to reconfigurable high-level arithmetic units such as CORDICs.

Application of Reconfigurable CORDIC Architectures

Reconfiguration enables the adaption of Coordinate Rotation DIgital Computer (CORDIC) units to the specific needs of sets of applications, hence creating application specific CORDIC-style implementations. Reconfiguration can be implemented at a high level, taking the entire CORDIC unit as a basic cell (CORDIC-cells) implemented in VLSI, or at a low level such as Field-Programmable Gate Arrays (FPGAs). We suggest a design methodology and analyze area/time results for coarse (VLSI) and fine-grain (FPGA) reconfigurable CORDIC units. For FPGAs we implement CORDIC units in Verilog HDL and our object-oriented design environment, PAM-Blox. For CORDIC-cells, multiple reconfigurable CORDIC modules are synthesized with state-of-the-art CAD tools. At the algorithm level we present a case study combining multiple CORDICs based on a geometrical interpretation of a normalized ladder algorithm for adaptive filtering to reduce latency and area of a fully pipelined CORDIC implementation. Ultimately, the goal is to create automatic tools to map applications directly to reconfigurable high-level arithmetic units such as CORDICs.

Scattering Arrays For Matrix Computations

Several new mesh connected multiprocessor architectures are presented that are adapted to execute highly parallel algorithms for matrix alge-bra and signal processing, such as triangular- and eigen-decomposition, inversion and low-rank updat-ing of general matrices, as well as Toeplitz and Hankel related matrices. These algorithms are based on scattering theory concepts and informa-tion preserving transformations, hence they exhibit local communication, and simple control and memory management, all properties that are ideal for VLSI implementation. The architectures are based on two- dimensional scattering arrays, that can be folded into linear arrays, either through time-sharing, or due to simple computation wave-fronts, or due to special structures of the matrices involved, such as Toeplitz.

Source location from time differences of arrival: Identifiability and estimation

A new framework is presented for the problem of estimating source location from a set of time difference of arrival (TDOA) measurements. First the conditions under which the source coordinates are identifiable are derived and geometrically interpreted. Then it is shown that the source location can be found, when it is identifiable, by estimating the coefficients of a set of linear relationships, with the number of equations being equal to the number of stations.

Redundant Constant-Factor Implementation of Multi-Dimensional CORDIC andIts Application to Complex SVD

Redundant Implementations of Multi-dimensional CORDIC algorithms are presented where the carry-ripple additions are replaced by carry-free signed-digit additions. Both folded (iterative) and unfolded (pipelined) architectures are considered in the redundant implementation. Furthermore, the scaling iterations are merged with the unscaled CORDIC iterations in the folded CORDIC architecture in order to reduce the overall computation time of one CORDIC operation. The redundant multidimensional CORDIC is then applied to the singular value decomposition of complex matrices, with either a folded or an on-line architecture. The resulting processing speed is higher than with alternative approaches based on 2-D CORDIC.

Fast Cholesky algorithms and adaptive feedback filters

In this paper, the Fast Cholesky algorithms, both by columns and by rows, are reviewed. It is shown that the algorithms lead naturally to a prediction error feedback filter. In addition, if this filter is used as the whitening filter for a moving average process, it is of fixed order but has time-varying coefficients. Simulation results for the case when the data came from the output of a moving average process driven by white Gaussian noise confirms theoretical results on convergence and stability of the triangular factors. In addition, the bandedness of the process being identified is revealed. Finally, from a VLSI implementation standpoint, it is shown that an array of CORDIC processors may be configured and controlled to factor a covariance matrix. In particular, there exists a method of factorization where the partial correlations associated with the given matrix are stored within the processors.

Concurrent Implementations Of Matrix Eigenvalue Decomposition Based On Isospectral Flows

In this paper we evaluate several techniques for solving the symmetric tridiagonal problem based on the method of isospectral flow. Architectures which result from these considerations are discussed. Their advantages and disadvantages from the viewpoints of numerical accuracy and ease of implementation in VLSI are also investigated.

Normalized doubling algorithms for finite shift-rank processes

Recently, various fast doubling procedures have been sketched or developed for the inversion of matrices with low shift-rank, e.g., Toeplitz matrices. The subclass of symmetric positive definite matrices is of particular interest in linear estimation, these matrices having the interpretation of covariances of finite shift-rank processes. This paper describes a doubling procedure for such covariance matrices. The procedure evaluates in O(n log2n) operations, both the inverse of an order n covariance and an associated set of parameters of great importance in linear filtering (the reflection coefficients if the covariance is Toeplitz).