IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 57, NO. 2, FEBRUARY 2010 131 Digital Circuit Realization of Piecewise-Affine Functions With Nonuniform Resolution: Theory and FPGA Implementation Tomaso Poggi, Francesco Comaschi, and Marco Storace, Member, IEEE Abstract—This brief proposes a digital circuit architecture im- plementing a class of continuous piecewise-affine (PWA) functions. The work rests on a previous architecture realizing PWA functions with uniform resolution. By using PWA mapping that can be implemented through a few simple functional blocks, it is possible to extend the representation capabilities of the architecture to PWA functions with nonuniform resolution. After defining the mapping and the corresponding functional blocks, the proposed architecture is implemented in a field-programmable gate array, and a simple example is shown. Index Terms—Digital architectures, nonlinear circuits, nonuni- form resolution, piecewise-affine (PWA) functions. I. I NTRODUCTION P IECEWISE-LINEAR or piecewise-affine (PWA) multival- ued functions are the core of many recent and less recent works in the circuit and system [1]–[7], control [8], [9], fuzzy system [10]–[13], and neural network [14]–[16] communities. Such an interest in PWA functions can surely be ascribed to their conceptual simplicity, which allows one to exploit the large body of knowledge provided by linear system theory, and to the very good accuracy in the representation of nonlinear functions, even multivalued, achievable through approxima- tion/interpolation with sufficiently dense sampling [17]. Basically, there are two main research lines (often strictly re- lated) concerning PWA functions: modeling (i.e., identification or approximation) and implementation of nonlinear systems. This brief is concerned with the digital circuit implementation of PWA functions. Then, we do not focus on modeling prob- lems, and we assume to have a PWA function to implement. In most cases, this function is the result of a preliminary modeling step providing the optimal PWA function for a given approximation problem. This step is usually taken offline by using a computer, whereas we deal with the online computation of the PWA function by using dedicated digital circuits. In many cases, PWA functions are implemented (often through interpolators) on computers or digital signal processor boards. When embedded real-time small-size low-power non- Manuscript received June 1, 2009; revised September 9, 2009. Current version published February 26, 2010. This work was supported in part by the European Community through the MOBY-DIC Project (FP7-IST-248858) and in part by the University of Genoa. This paper was recommended by Associate Editor N. Takahashi. The authors are with the Biophysical and Electronic Engineering Depart- ment, University of Genoa, 16145 Genova, Italy (e-mail: marco.storace@ unige.it). Digital Object Identifier 10.1109/TCSII.2010.2040316 linear (not necessarily PWA) function evaluation is required, the circuit implementation of PWA functions might be of interest by resorting to either programmable hardware such as field-programmable gate arrays (FPGAs) or even dedicated integrated circuits. In this context, many architectures have been proposed in recent years for the implementation of PWA functions [11], [12], [18], [19]. For a review, see [20]. Most of the architec- tures proposed so far are based on a uniform partition (called simplicial partition) of the domain of the PWA function to be implemented. The resulting PWA function is linear over each simplex and is completely defined by the values it takes at the vertices of the partition. The main limit of such an approach (called uniform-resolution approach) is that the implementable functions are defined with uniform resolution over the whole domain since all the simplices are identical. Quite often, one needs to implement a PWA function with different resolution requirements in different domain subre- gions. In [21], a method has been proposed to obtain a multires- olution PWA approximation of a given function. This method is efficient from a computational point of view but not suitable for an efficient circuit implementation. In this brief, we use a different approach, based on invert- ible mapping between uniformly and nonuniformly partitioned domains, that allows one to implement PWA functions with nonuniform resolution. We show that there is a simple and efficient way to implement nonuniform-resolution PWA func- tions by extending the results reported in [19] and [22]. The FPGA implementation of a 3-D PWA function is proposed and discussed. II. CIRCUIT I MPLEMENTATION OF UNIFORM-RESOLUTION PWA FUNCTIONS Here, we briefly summarize the elements of the uniform resolution approach that are essential to introduce the nonuni- form resolution one. For a deeper treatment of this topic, the reader is referred to [22]. We deal with a continu- ous PWA function f PWA : S z → R, defined over a properly scaled n-dimensional compact domain S z = {z ∈ R n :0 ≤ z i ≤ m i ,i =1,...,n,m i ∈ N}. f PWA can be easily circuit- implemented by introducing a regular partition of the domain S z [22]: Each dimensional component z i of the domain S z is di- vided into m i subintervals of unitary length. As a consequence, the domain S z is partitioned into  n i=1 m i hypersquares and contains N =  n i=1 (m i + 1) vertices v z k collected in a set V z . 1549-7747/$26.00 © 2010 IEEE Authorized licensed use limited to: Universita di Genova. Downloaded on March 02,2010 at 03:22:28 EST from IEEE Xplore. Restrictions apply.