A method for the approximate synthesis of cellular non‐linear networks—Part 1: Circuit definition

Abstract

SUMMARYIn this paper, we face the problem of the synthesis of cellular non-linear networks (CNNs) able to ÿnd approximate solutions of either partial di erential equations or functional minimization problems. In particular, the paper deals with the approximate synthesis of the multi-terminal resistors that include all the non-linearities of such dynamic circuits. The proposed approximation method is based on the piecewise-linear (PWL) approach developed by Julià an et al. in the last few years, but it resorts to a di erent deÿnition of the inner product and hence to other PWL basis functions. The method is applied to a case-study concerning a CNN devoted to image processing.

A method for the approximate synthesis of cellular non‐linear networks—Part 2: Circuit reduction

“…If, in the near future, a direct and suitable circuit implementation of another basis for the linear space PWL[S H ] is realized, it will be possible to obtain the corresponding set and reduced sets of weighting coe cients by exploiting the same method as outlined in Reference [1] and in this paper.…”

“…* * The 'level' p is the number of arguments of the function p (see Reference [1]), which is used to construct the ÿ-basis. We recall that, when n = 2, for the not reduced PWL model we have a single function ( coe cient of level 0 is never annihilated.…”

Towards analog implementations of PWL two-dimensional non-linear functions

“…The latter problem has recently been faced [2][3][4] by resorting to the piecewise-linear (PWL) approximation technique proposed by Julià an et al [5,6]. By means of this technique, each constitutive equation y = f(x) for a non-linear resistive element of a circuit (where y is a generic dependent descriptive variable and x is the vector of independent descriptive variables) is replaced with a proper canonical PWL expression.…”

“…For instance, there are bases that are more convenient either from a computational or a circuit-synthesis point of view. In particular, for the circuit synthesis we shall refer to the so-called Q-basis and R-basis [2,3]. The Q-basis seems to be more suitable for mixed-signal realizations (an example is given in References [7,8], concerning an electronic implementation for an image processing CNN based on PWL coupling elements).…”

“…The N -dimensional space PWL[S H ] associated with the class of continuous PWL functions f PWL is deÿned by the domain S, its simplicial partition H , and a proper inner product (see [3,5,6] for details). Each function belonging to PWL[S H ] can be represented as the sum of N basis functions (arbitrarily organized into a vector), weighted by an N -length coe cient vector c, which determines the shape of f PWL uniquely.…”