Sorter-Based Arithmetic Circuits for Sigma-Delta Domain Signal Processing—Part II: Multiplication and Algebraic Functions

We construct arithmetic modules for signal processing with sigma-delta modulated signal form which has advantage in signal quality over other pulsed signal forms. In the second part of this paper, multi-input multipliers are presented first. Secondly, dividers and square root function modules with the multiplier on their internal feedback path are constructed. Combined use of the multipliers, dividers, and the square root functions creates various algebraic functions including polynomial and rational functions. Only two bit-manipulations, bit-permutation with sorting networks and bit-reversal with NOT gates, have built up all the algebraic operations on any form of SD modulated signals. These modules, together with transcendental functions presented in the first part of this paper, organize an extensive module library for the sigma-delta domain signal processing. The multiplier output contains noise components which originate from quantization. The noise power can decrease in exchange for circuit complexity. A time-division multiplexing technique based on N-tone sigma-delta modulation is applied to the multipliers for reducing the complexity. Signal processing circuits built of nanometer-scale quantum effect devices must be equipped with fault tolerance of transient device error. By computer simulation of a multiplier built of single-electron tunneling devices, we found that the multiplier decreased its output SNDR from 43 to 27 dB at an OSR of 2⁸ as the device error rate increased from 0 to 10^-3. However, the multiplier was never functionally failed during the simulation.