Multiplier-less and Table-less Linear Approximation for Square-Related Functions

Abstract

SUMMARYSquare-related functions such as square, inverse square, square-root and inverse square-root operations are widely used in digital signal processing and digital communication algorithms, and their efficient realizations are commonly required to reduce the hardware complexity. In the implementation point of view, approximate realizations are often desired if they do not degrade performance significantly. In this paper, we propose new linear approximations for the square-related functions. The traditional linear approximations need multipliers to calculate slope offsets and tables to store initial offset values and slope values, whereas the proposed approximations exploit the inherent properties of square-related functions to linearly interpolate with only simple operations, such as shift, concatenation and addition, which are usually supported in modern VLSI systems. Regardless of the bit-width of the number system, more importantly, the maximum relative errors of the proposed approximations are bounded to 6.25% and 3.13% for square and square-root functions, respectively. For inverse square and inverse square-root functions, the maximum relative errors are bounded to 12.5% and 6.25% if the input operands are represented in 20 bits, respectively.

FPGA Design for Statistics-Inspired Approximate Sum-of-Squared-Error Computation in Multimedia Applications

“…Approximate squarers from Risojevic [5] and Cho et al [6] achieve 76% and 20% savings, respectively. Park and Kim [7] and Zhang et al [8] do not report hardware implementation details. The hardware savings for our proposed approximate squarers show clear cost advantage over the existing ones.…”

“…MRE data for the proposed designs have been compared with existing 8-bit approximate squarer designs [3]- [8] in Table IV. The existing designs [3]- [5], [7], [8] only report theoretical MRE data for uniformly distributed random data, whereas our work specifically targets SSE computation in practical ME and DE applications and reports MRE results for these well. These results show that the proposed approximate squarers, AS1 (M = 6) and AS3 (M = 6), perform better than the existing designs in terms of MRE when practical data are considered.…”