Kamen R. Ralev

Realization of block floating-point digital filters and application to block implementations

Realization issues of block floating-point (BFP) filters such as complexity, roundoff noise, and absence of limit cycles are analyzed. Several new results are established. Under certain conditions, BFP filters perform better than fixed-point filters at the expense of a slight increase in complexity; convex programming can be used to minimize the roundoff noise; limit cycles will not be present if the underlying fixed-point system is free of quantization limit cycles. It is shown that BFP arithmetic can be efficiently combined with block implementations to further improve the roundoff noise and stability of the implementation and reduce the complexity of processing BFP data.

Comments on "Stability and absence of overflow oscillations for 2-D discrete-time systems

In their article Xiao and Hill (see ibid., vol.44, p.2108-10, 1996) show via an example that the conditions of the Theorem 3 in an earlier paper by Tzafestas et al. (1992) may not guarantee stability of 2-D linear systems. The authors incorrectly conclude that the BIBO unstable 2-D linear system may have a stable finite wordlength implementation using saturation arithmetic. The authors comment that this incorrect conclusion is due to an erroneous result in the earlier paper by Tzafestas et al.

Implementation options for block floating point digital filters

Different options for block floating point filter implementation are introduced and their efficiency determined. The efficiency is quantified by the additional number of operations over those required for fixed point operation. Some of the implementations are new. It is shown that they are more efficient than the existing ones. Examples are given in which the processing time per recursion of a block floating point implementation on a fixed point processor is approximately the same as the recursion time of the corresponding fixed point implementation. Application of block floating point arithmetic to block implementations is also considered.

Limit cycles elimination in delta-operator systems

The existence of nonzero equilibria in /spl delta/-operator fixed point and block floating point (BFP) systems is investigated and methods for avoiding such equilibria are proposed. In the fixed point case these methods work by mapping the region in which nonzero equilibria may appear to zero. This is possible if the region is small. It is also shown that nonzero equilibria and limit cycles of any period can always be avoided by using BFP arithmetic with a sufficiently large mantissa wordlength.

Asymptotic error bounds for block floating point digital filters

This paper analyzes the register length requirements for block-floating point digital filters. Both, shift and delta operator systems are considered. In particular the block mantissa length requirements are investigated, if the asymptotic response of the filter is to converge to underflow or to be bounded. Examples which illustrate the conservatism of the result are also provided.

A limit cycle suppressing arithmetic format for digital filters

In this paper it is be shown that any digital filter with a stable transfer function can be implemented free of limit cycles, if block floating point arithmetic is used in conjunction with the so-called exponent saturation and flush-to-zero option. A certain system dependent minimum block mantissa length is required. Limit cycle suppression is achievable regardless of the structure and the quantization format. This format is simple to implement even on fixed point processors and offers high dynamic range.

Asymptotic behavior of block floating-point digital filters

The asymptotic behavior of block floating-point and floating-point digital filters is analyzed. As a result, mantissa wordlength conditions are derived guaranteeing the absence of limit cycles in the regular dynamic range. Explicitly, the requirements are given for block floating-point state space filters with different quantization formats. Although these conditions are only sufficient, examples are given in which they are also necessary. In most cases the conditions are easily satisfied.

New tools for localization of limit cycles in recursive block floating point systems

Abstract Some important properties of recursive block floating point systems are derived. On their basis an algorithm for searching limit cycles in such systems is proposed. For any finite wordlength implementation of linear systems, an ellipsoidal set of initial conditions is found which generates all limit cycle in the system. It is applicable for any exhaustive search algorithm and also yields a sufficient condition for absence of limit cycles in (block) floating point systems.

Exhaustive searching for limit cycles in recursive block floating point systems

Some important properties of recursive block floating point (BFP) systems are derived. On their basis an algorithm for searching limit cycles in such systems is proposed. For any finite wordlength implementation of linear systems an ellipsoidal bound on the set of initial conditions that may result in a limit cycle is derived. The bound also yields a sufficient condition for absence of limit cycles in (block) floating point systems.