Overview

The filtering operation performed by a linear, time invariant system (LTI) is mathematically described by a set of LTI equations as it has been shown several times in previous chapters. These equations relate two magnitudes: the input and the output of the system. However, they do not impose any mathematical constraint in internal variables, which implies that internal variables can be related in any manner and not necessarily in a linear way. This opens up a wider set of options in the internal processing of the signals. Among the nonlinear functions that can be chosen to process the internal variables, logarithmic and exponential functions have proven to be very interesting, mostly because they link directly to physical models of transistor devices.

The concept of log-domain signal processing was originally proposed by Adams ^[179] and rigorously formalised by Frey ^[182], ^[183], ^[184]. Frey proposed a nonlinear (exponential) mapping of the state variables in the state-space description of any linear transfer function that results in a set of nonlinear equations which can be interpreted according to the Kirchoff's current law. This class of circuits is known as Exponential State Space (ESS) circuits. Later on, Tsividis came up with a more generalised approach which included all the topologies that are Externally Linear Internally Nonlinear (ELIN) ^[21].

Log-domain filters can be found in many different applications due to the fact that an exponential function models the current of both a bipolar and a MOS transistor in the weak...