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A Parallel Prefix Modulo-(2q + 2q−1 + 1) Adder via Diminished-1 Representation of Residues | IEEE Journals & Magazine | IEEE Xplore

A Parallel Prefix Modulo-(2q + 2q−1 + 1) Adder via Diminished-1 Representation of Residues


Abstract:

Diminished-1 representation of modulo- (2^{q} + 1) residues provides for delay-balanced adders for the popular moduli set \tau = \{ 2^{q},2^{q} \pm 1\} . Besides, c...Show More

Abstract:

Diminished-1 representation of modulo- (2^{q} + 1) residues provides for delay-balanced adders for the popular moduli set \tau = \{ 2^{q},2^{q} \pm 1\} . Besides, conjugacy of moduli 2^{q} \pm 1 leads to efficient multiple-residue to binary conversion. Nevertheless, cardinality of \tau does not cover the required dynamic range of some applications that also require small q -values for gaining the desired arithmetic speed (e.g., convolutional neural networks). The recently proposed parallel prefix modulo- (2^{q} + 2^{q - 1} - 1) addition scheme provides for balanced performance with similar adder architectures for the moduli of doubly-wide \tau (i.e., \tau _{2} = \{ 2^{2q},2^{2q} \pm 1 \} ). Should there be a compatible adder for the conjugate modulo 2^{q} + 2^{q - 1} + 1 , the new moduli set \tau _{2}^{+} = \{ 2^{2q},2^{2q} \pm 1,2^{q} + 2^{q - 1} \pm 1\} can provide more than (8q - 3) bits of dynamic range, with equally fast adders as those of \tau _{2} , whose dynamic range is only 6q bits. Therefore, we were motivated to propose the new modulo 2^{q} + 2^{q - 1} + 1 and its equally fast parallel prefix adder with compatible cost as of its conjugate modulo. This is actually achieved via diminished-1 representation of residues, where those in [1,2^{q} + 2^{q - 1}] are encoded as [{ 0,2^{q} + 2^{q - 1} - 1}] and a zero indicator bit is set for 0-valued residues. The VLSI simulation and synthesis results, especially for the common cases of q = 2^{p} , confirm the analytical evaluations regarding the balanced performance of the conjugate moduli 2^{q} + 2^{q - 1} \pm 1 .
Page(s): 3104 - 3108
Date of Publication: 14 March 2023

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