Abstract
The field of neuromorphic silicon synapse circuits is revisited and a parsimonious mathematical framework able to describe the dynamics of this class of log-domain circuits in the aggregate and in a systematic manner is proposed. Starting from the Bernoulli Cell Formalism (BCF), originally formulated for the modular synthesis and analysis of externally linear, time-invariant logarithmic filters, and by means of the identification of new types of Bernoulli Cell (BC) operators presented here, a generalized formalism (GBCF) is established. The expanded formalism covers two new possible and practical combinations of a MOS transistor (MOST) and a linear capacitor. The corresponding mathematical relations codifying each case are presented and discussed through the tutorial treatment of three well-known transistor-level examples of log-domain neuromorphic silicon synapses. The proposed mathematical tool unifies past analysis approaches of the same circuits under a common theoretical framework. The speed advantage of the proposed mathematical framework as an analysis tool is also demonstrated by a compelling comparative circuit analysis example of high order, where the GBCF and another well-known log-domain circuit analysis method are used for the determination of the input-output transfer function of the high (4th) order topology.
Highlights
Almost 20 years ago, a novel, systematic, transistor-level formalism for the analysis and synthesis of externally-linear, internally-nonlinear (ELIN) (Tsividis, 1997) log-domain filters was introduced
For intrinsically non-linear log-domain circuits. When it comes to the synthesis of purely non-linear log-domain circuits, a variant of the BCF, termed Non-linear Bernoulli Cell Formalism (NBCF), is able to implement challenging non-linear dynamics, based on the “Coupled BC Formation,” where the input and output currents of the BCs are interconnected in a nonsequential way, in contrast to the cascaded Log-Domain State Space” (LDSS) topology
An extended version of the BCF proved the existence of BC-operators when a linear capacitor is connected to the emitter/source of a transistor and when a linear capacitor is connected to the base/gate of a diode-connected transistor
Summary
Almost 20 years ago, a novel, systematic, transistor-level formalism for the analysis and synthesis of externally-linear, internally-nonlinear (ELIN) (Tsividis, 1997) log-domain filters was introduced. A review of past literature reveals that the BCF constitutes a complete, systematic mathematical framework for ELIN and for intrinsically non-linear log-domain circuits When it comes to the synthesis of purely non-linear log-domain circuits, a variant of the BCF, termed Non-linear Bernoulli Cell Formalism (NBCF), is able to implement challenging non-linear dynamics, based on the “Coupled BC Formation,” where the input and output currents of the BCs are interconnected in a nonsequential way, in contrast to the cascaded LDSS topology. With the help of this work it is genuinely hoped that the interested reader will develop a deep understanding for the functionality of this class of low-power circuits and will appreciate the systematic nature of the formalism by consolidating the advantages of using one single framework to describe multiple, different, but in principle similar, log-domain synaptic topologies This alternative treatment of aVLSI synaptic circuit succeeds in unifying the past analysis approaches of the same circuits under a common aegis and underlines the tutorial value of this paper. The compelling comparison results stress the advantages of using a single mathematical formalism for the description of any log-domain circuit, regardless of its linearity or order of complexity
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