“Log-domain state-space”: a systematic transistor-level approach for log-domain filtering

In this paper the properties of a low-level nonlinear continuous-time circuit element-termed a Bernoulli Cell (or Operator)-are described in a systematic way. This cell is composed of an n-p-n BJT and an emitter-connected grounded capacitor, and is governed by a differential equation of the Bernoulli form. Although this cell has the potential for application in both linear and nonlinear analog signal processing, this paper focuses on the field of input-output linear log-domain filtering. The Bernoulli Cell can be utilized in both the analysis and synthesis of log-domain circuits. The Bernoulli Cell approach leads to the creation of a system of linear differential equations with time dependent coefficients and state variables nonlinearly related to currents internal to the circuit; this set of equations is termed “log-domain state-space”, and can be used for the synthesis of linear log-domain filters. Four design examples-including a bandpass biquad-are presented