Designing a Winner–Loser Gap for WTA in Subthreshold. Resolution Performance Revisited

Abstract

AbstractThe paper pioneers a thorough mathematical approach for the Lazzaro variant of the W(inner) T(ake) A(ll) maximum rank and amplitude analog selector. Two exact levels of output which split the maximum and determine the resolution, are found for the first time. At the input, a list of currents $$\left( I_{1},I_{2},\ldots ,I_{N}\right) $$ I 1 , I 2 , … , I N from a large family $${{\mathcal {L}}}$$ L with smallest relative distance $$\Delta $$ Δ on a $$\left[ 0,I_{M}\right] $$ 0 , I M scale is applied. To distinguish the largest current $$I_{w}$$ I w (the winner) from the second largest $$I_{l}$$ I l (the loser), the paper proposes two decision levels, $${\overline{D}}$$ D ¯ and $${\underline{D}}$$ D ̲ , for the output voltage list $$\left( U_{1},U_{2},\ldots ,U_{N}\right) $$ U 1 , U 2 , … , U N . The upper level $${\overline{D}}$$ D ¯ is surpassed only by the $$U_{w}$$ U w winner and encodes the winning rank w. All other ranks are placed under the lower level $${\underline{D}}$$ D ̲ . Two rigorously treated optimization problems with inequality constraints lead to the identification of two input lists that yield the levels $${\overline{D}}$$ D ¯ and $${\underline{D}}$$ D ̲ as outputs. They are valid for processing any list in the $${\mathcal {L}}$$ L family. The index $$\left( {\overline{D}}-{\underline{D}}\right) /U_{M}$$ D ¯ - D ̲ / U M —“the output resolution”—expresses how large the gap between the first and the second component on the $$\left[ 0,U_{M}\right] $$ 0 , U M scale is. It exceeds “the input resolution,” i.e., the similar index $$\Delta /I_{M}$$ Δ / I M at the input and the two depend monotonically on each other. Widely commented numerical examples are presented.