Direct Reed-Muller Transform of Digital Logic Netlists

A method for computing the Reed-Muller spectrum of a digital logic circuit based on the circuit topology is developed. The technique is based on the definition and use of a transfer matrix to characterize the circuit of interest. The transfer matrix is computed through the use of the structure of the circuit and the transfer matrices of the individual logic gates. The resulting transfer matrix describes the Boolean domain response of the circuit. We next formulate the transfer matrix in the Reed-Muller domain that is also capable of being similarly formulated directly from the structure of the circuit. The result is that the Reed-Muller spectrum of a circuit, or subcircuit partition, can be computed directly thus alleviating the need to first determine the switching function followed by subsequent application of the RM transform. 1.0 INTRODUCTION Many applications for spectral analysis of digital logic circuits have been developed including the use of the Reed-Muller family of spectra. Central to these applications is the computation of the spectra in an efficient manner since resulting applications are often hampered by the initial large computational expense of the spectrum. Past methods rely upon transforming the switching functions representing the circuit behavior. Techniques to address the spectral computation problem have been devised that utilize cube list and decision diagram representations, however, all of these approaches are exponentially complex in the worst case since the entire switching function must be represented in some form prior to transformation. The concept of a transfer function describing the input-output relation of a linear system is a mature idea and is heavily used in electrical circuit analysis. Such transfer functions have the property of allowing for their construction based on the transfer functions of individual system components. This hierarchical construction of transfer functions is useful for dealing with large systems allowing the overall system characterization to be achieved in a bottom-up manner. Furthermore, such transfer functions can be transformed to frequency domain representations allowing for design and analysis techniques to be employed in alternative basis domains. We describe a technique that enables frequency domain analysis of digital logic circuits through the use of the Reed-Muller transform applied directly upon a logic circuit netlist. The technique is based on formulating a linear algebraic model of the switching function allowing for the overall circuit switching function to be represented by a characteristic linear transfer matrix in the Boolean domain. The transfer matrix is analogous to a transfer function or matrix as is commonly used in classical linear system analysis. Linear system theory is a mature approach and many excellent references are available, an example of which is [1]. Preliminary ideas used to formulate these results are briefly outlined in [2] but they were only discussed in the context of a single logic gate in the Boolean domain and they utilized a different form of the transfer matrix than that described here. Because the circuit transfer matrix can be derived from a hierarchical construction of the individual logic gate transfer matrices, the technique is applicable to the use of the structure of a netlist or interconnection of individual logic gates. Furthermore, the RM transform may be applied at the single gate or larger subcircuit level avoiding the memory explosion that can occur when the entire circuit model switching function is transformed. A similar formulation for the Walsh transform is described in [3] although the transfer matrices were formulated as transposes of those used here. An alternative method for the computation of single Walsh spectral coefficients was described in [4], however, this method required structural modification of the circuit netlist before it was traversed for the computation of each coefficient. The Reed-Muller transfer function response of the entire circuit can also be derived in a hierarchical manner. These results lead to a method for computing the Reed-Muller transform of a circuit or subcircuit directly from a netlist representation. An application of this technique is that the Reed-Muller transform of partitioned subcircuits can be computed directly from the netlist without resorting to first computing the representation of the overall switching function. The organization of the paper is as follows. A review of the mathematical properties of the Reed-Muller transform and its definition are first reviewed. Next, we will describe a formulation of classical digital logic circuits utilizing linear algebra rather than Boolean algebra analogous to time domain analysis of linear electrical circuits. Following these introductory concepts, the ReedMuller transform of digital circuit netlists is formulated using the concepts of linear algebraic analysis. Finally, conclusions and future work are outlined. 2.0 REED-MULLER TRANSFORM The Reed-Muller transform has been previously described and written about extensively [5, 6]. In this paper, we will provide a brief overview of the transform for the sake of completeness and to emphasize its relationship with other transforms. One way to conceptualize discrete spectral transforms is to consider the set of function range values to be transformed as a set of coefficients over which to evaluate a particular polynomial. The particular polynomial that is selected defines both the transform and the discrete function undergoing transformation. For example, if the polynomial is based on various powers of the roots of unity with coefficients representing the discrete range space of a function, f(x), the discrete Fourier transform results. The form of the polynomial is shown in Equation (1).