Symbolic analysis has many applications in the design of analog circuits. Existing approaches rely on two forms of symbolic-expression representation: expanded sum-of-product form and arbitrarily nested form. Expanded form suffers the problem that the number of product terms grows exponentially with the size of a circuit. Nested form is neither canonical nor amenable to symbolic manipulation. In this paper, we present a new approach to exact and canonical symbolic analysis by exploiting the sparsity and sharing of product terms. It consists of representing the symbolic determinant of a circuit matrix by a graph-called a determinant decision diagram (DDD)-and performing symbolic analysis by graph manipulations. We show that DDD construction, as well as many symbolic analysis algorithms, takes time almost linear in the number of DDD vertices. We describe an efficient DDD-vertex ordering heuristic and prove that it is optimum for ladder-structured circuits. For practical analog circuits, the numbers of DDD vertices are several orders of magnitude less than the numbers of product terms. The algorithms have been implemented and compared respectively to symbolic analyzers ISAAC and Maple-V in generating the expanded sum-of-product expressions, and SCAPP in generating the nested sequences of expressions.
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