Analog Integrated Circuits and Signal Processing, 40, 205–213, 2004 c  2004 Kluwer Academic Publishers. Manufactured in The Netherlands. Symbolic Techniques for the Selection of Test Frequencies in Analog Fault Diagnosis F. GRASSO, A. LUCHETTA, S. MANETTI AND M.C. PICCIRILLI Department of Electronics and Telecommunications, University of Florence, Via S. Marta 3, 50139, Firenze, Italy E-mail: luchetta@det.unifi.it Received January 3, 2003; Revised March 4, 2003; Accepted March 5, 2003 Abstract. In this work symbolic methods are used for implementing a procedure for the selection of test frequencies in multifrequency parametric fault diagnosis of analog linear circuits. The proposed approach is based on the evaluation of the condition number and the norm of a sensitivity matrix of the circuit under test. This matrix is determined by exploiting the testability and ambiguity group concepts. A Test Error Index (T.E.I.) is obtained which permits to select the set of frequencies which better leads to locate parametric faults in analog linear circuits. A program implementing the proposed procedure has been realized by using symbolic techniques. Examples of application are also included. Key Words: analog system testing, testability analysis, condition number, ambiguity groups 1. Introduction In the past years there has been a large progress in the field of symbolic analysis due to an increased number of researchers and designers working in this area and in its applications. As it is well known, symbolic analysis is a technique for generating formulas in closed form describing the characteristics of a circuit as a function of its parameters and complex frequency. It is a com- plementary technique with respect to numerical anal- ysis for circuit simulation. In fact numerical analysis is a powerful verification tool, very useful for testing and modifying the circuit that the designer is planning, while symbolic analysis strongly reduces the work of determination of the malfunctioning causes. Neverthe- less there are applications in which the symbolic anal- ysis can give noteworthy advantages with respect to numerical one, as, for example, in fault diagnosis of analog circuits. Fault diagnosis of analog circuits is much more com- plex with respect to that of digital circuits, because the analog fault diagnosis has difficulties not present in digital one. In fact, in analog circuits there is a con- tinuum of possible failures, a good component may be “in tolerance” but not nominal, there are complex feed- back structures, simulations are slow and costly, post- fault component characteristics may not be known, a fault in one component may induce an apparent fault in another. All this implies that in analog fault diagno- sis an extremely large and more expensive number of simulations with respect to digital case is required. As a consequence the automation of fault diagnosis pro- cedures has a development level much less advanced with respect to digital case. Symbolic techniques can be very useful for overcoming this disparity if method- ologies of parametric kind are considered for analog fault diagnosis. Parametric fault diagnosis techniques, starting from a series of measurements carried out on a previously selected test point set, given the circuit topology and the nominal values of the components, are aimed at de- termining the effective values of the circuit parameters by solving a set of equations nonlinear with respect to the component values. Because the actual component values are the unknown quantities, symbolic approach results particularly suitable to be used for the automa- tion of parametric fault diagnosis techniques, as shown, for example, in [1–3]. Obviously, all this is applicable to linear analog circuits or to nonlinear circuits suitably linearized. It is important to note that in analog fault diagnosis two phases can be considered: the first one is the phase of testability analysis and the second one is the phase of fault location. The phase of testability analysis consists