Efficient Factorization of DSP Transforms using Taylor Expansion Diagrams

Abstract

This paper describes an efficient method to perform factorization of DSP transforms based on Taylor ExpansionDiagram (TED). It is shown that TED can efficiently represent and manipulate mathematical expressions. We demonstrate that it enables efficient factorization of arithmetic expressions of DSP transforms, resulting in a simplification of the computation.

Optimizing data flow graphs to minimize hardware implementation

“…It has been shown that the TED structure allows for efficient factorization and decomposition of expressions modeled as linear multi-variate polynomials [3,4]. For example, a TED for expression F = ab + ac, for variable ordering (a, b, c) naturally represents the polynomial in its factored form, a(b + c).…”

“…The goal of TED decomposition described in this work is to find a factored form that will produce DFG with minimum hardware cost of the final, scheduled implementation. This is different than a straightforward minimization of the number of operations in the unsched-uled DFG, which has been the subject of the known previous work [3,4,12]. The TED decomposition method described here extends the work of the original cut-based decomposition of Askar [4], which was based on the identification and selection of admissible cut sequences.…”

“…TEDs have already been applied to functional optimization, such as factorization and common subexpression elimination (CSE). However, so far their scope was limited to linear expressions, such as linear DSP transforms and to the simplification of arithmetic expressions, without considering final scheduled implementation [3,4]. This paper describes how TEDs can be extended to handle the optimization of nonlinear polynomial expressions, using novel factorization and decomposition algorithms, to generate optimized data flow graphs (DFG), better suited for high-level synthesis.…”

Data-Flow Transformations using Taylor Expansion Diagrams

“…This factorization is manifested in the graph by the presence of a subexpression (B + C), rooted at node B, which can be extracted from the graph. This is an important feature of the TED representation, explored in factorization and common subexpression extraction [13].…”

“…This phase is intended to minimize the number of ADD/MULT operations in the resulting DFG. 2) TED Factorization and Extraction: Generate hierarchical TED representation by performing TED factorization and common subexpression extraction using techniques described in [13]. Hierarchical TED is a graph, whose nodes represent TEDs of local functions.…”