Truncation Schemes for Recursive Multipliers

This paper explores the effects of truncation schemes in recursive multiplier architectures in terms of the trade-off between circuit complexity versus introduced truncation errors. The recursive architecture is examined due to its inherent hierarchical structure whereby a larger multiplication is subdivided into a collection of smaller multiplications. Three data-dependent truncation schemes are proposed that exploit this multiplication architecture. Error analysis and complexity savings for each scheme are also discussed. Signal processing applications, in general, require a constant word size throughout the processing system. This poses a problem in basic integer arithmetic operations, where the result of each operation has a tendency of differing from the original operand size. Of these operations, multiplication is of the biggest concern since each operation results in a product that is twice as large as the initial operand widths. To alleviate the problem of expanding word widths, truncation and rounding methods are used, often in conjunction with the use of floating point arithmetic. In general, the operation of floating point multipliers may be summarized in two steps: the generation of a product from the integer multiplier in carry save format, and the rounding of this product according to a specified mode. A great deal of research has been done on the performance of the integer multiplier over the past four decades, and more recently there have been some advancements made in the area of the truncation schemes that help reduce the complexity of the arithmetic circuitry (1,2,3,4,5,6). Our focus in this paper is to analyze the post-rounding error associated with eliminating a portion of a recursive integer multiplier. We will present an in depth analysis of the error associated with truncation schemes directly targeting recursive multipliers that minimize the truncation error by carefully choosing data-dependent correction terms. We begin with a brief overview of the structure of the recursive multiplication algorithm, followed by a discussion on truncation schemes for multipliers. Finally we will present the results of our simulations and further, provide error analysis and complexity savings for these schemes.

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