Addition Aware Quantization for Low Complexity and High Precision Constant Multiplication

Abstract

Abstract-Multiplication by constants can be efficiently realized using shifts, additions, and subtractions. In this work we consider how to select a fixed-point value for a real valued, rational, or floating-point coefficient to obtain a low-complexity realization. It is shown that the process, denoted addition aware quantization, often can determine coefficients that has as low complexity as the rounded value, but with a smaller approximation error by searching among coefficients with a longer wordlength.

Alternatives for low-complexity complex rotators

“…On the other hand, for the CORDIC algorithm the error is based on the fact that the angle cannot be exactly approximated with a finite sequence of micro-rotation. The results for constant multiplication and scaled constant multiplication have been obtained by applying the addition aware quantization methodology [12]. For the case of the CORDIC, all the possible combinations of micro-rotations have been calculated considering δ i ∈ {−1, 0, 1}, and the sequence of micro-rotations that best approximate the angle has been chosen.…”

“…For the constant multiplication cases it is often possible to optimize the error with the same complexity compared with rounding by addition aware quantization [12]. In [12], E additional fractional bits are used to realize that there are exactly 2 E different representable coefficients for which ǫ ≤ 2 −(N +1) , including the one obtained by rounding to N fractional bits.…”

“…In [12], E additional fractional bits are used to realize that there are exactly 2 E different representable coefficients for which ǫ ≤ 2 −(N +1) , including the one obtained by rounding to N fractional bits.…”

4k-point FFT algorithms based on optimized twiddle factor multiplication for FPGAs

“…This twiddle factor multiplication can be implemented with the dedicated constant multiplier of sin . In [15] authors proposed the low complexity in terms of adder with minimum error based on aware quantization method. In the proposed architectures we implement dedicated constant multiplier for W 16 twiddle factor multiplication.…”

Low-Complexity Multiplierless Constant Rotators Based on Combined Coefficient Selection and Shift-and-Add Implementation (CCSSI)

“…The success of previous approaches is mainly due to an efficient shift-and-add implementation: On the one hand, the techniques used in multiplier-based approaches to implement constant multiplications as shifts and additions are widely developed [6], [28]- [37]. On the other hand, CORDICbased approaches rely on using elementary angles that allow for an efficient shift-and-add implementation.…”