A Parallel Prefix Modulo-(<b>2<sup> <i>q</i> </sup> </b> + <b>2<sup> <i>q</i>−1</sup> </b> + 1) Adder via Diminished-1 Representation of Residues

Diminished-1 representation of modulo- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(2^{q} + 1)$ </tex-math></inline-formula> residues provides for delay-balanced adders for the popular moduli set <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tau = \{ 2^{q},2^{q} \pm 1\}$ </tex-math></inline-formula> . Besides, conjugacy of moduli <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{q} \pm 1$ </tex-math></inline-formula> leads to efficient multiple-residue to binary conversion. Nevertheless, cardinality of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tau $ </tex-math></inline-formula> does not cover the required dynamic range of some applications that also require small <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> -values for gaining the desired arithmetic speed (e.g., convolutional neural networks). The recently proposed parallel prefix modulo- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(2^{q} + 2^{q - 1} - 1)$ </tex-math></inline-formula> addition scheme provides for balanced performance with similar adder architectures for the moduli of doubly-wide <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tau $ </tex-math></inline-formula> (i.e., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tau _{2} = \{ 2^{2q},2^{2q} \pm 1 \}$ </tex-math></inline-formula> ). Should there be a compatible adder for the conjugate modulo <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{q} + 2^{q - 1} + 1$ </tex-math></inline-formula> , the new moduli set <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tau _{2}^{+} = \{ 2^{2q},2^{2q} \pm 1,2^{q} + 2^{q - 1} \pm 1\}$ </tex-math></inline-formula> can provide more than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(8q - 3)$ </tex-math></inline-formula> bits of dynamic range, with equally fast adders as those of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tau _{2}$ </tex-math></inline-formula> , whose dynamic range is only <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$6q$ </tex-math></inline-formula> bits. Therefore, we were motivated to propose the new modulo <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{q} + 2^{q - 1} + 1$ </tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$[1,2^{q} + 2^{q - 1}]$ </tex-math></inline-formula> are encoded as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$[{ 0,2^{q} + 2^{q - 1} - 1}]$ </tex-math></inline-formula> and a zero indicator bit is set for 0-valued residues. The VLSI simulation and synthesis results, especially for the common cases of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q = 2^{p}$ </tex-math></inline-formula> , confirm the analytical evaluations regarding the balanced performance of the conjugate moduli <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{q} + 2^{q - 1} \pm 1$ </tex-math></inline-formula> .