Multiplier-less and table-less linear approximation for square and square-root

Square and square-root 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 and square-root 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.