Fast Minimization of Polynomial Decomposition using Fixed-Polarity Pascal Transforms Kaitlin N. Smith 1 and Mitchell A. Thornton Department of Electrical and Computer Engineering Southern Methodist University Dallas, Texas, U.S.A. {knsmith,mitch}@smu.edu D. Michael Miller Department of Computer Science University of Victoria Victoria, British Columbia, Canada mmiller@uvic.ca Abstract—Polynomials can be represented as a weighted sum of various powers of binomials where the weights are spectral coefficients and the associated powers of binomials are basis functions. The minimization problem is concerned with finding a set of basis functions that result in a maximum number of zero-valued spectral coefficients, or alternatively, wherein the spectral vector has a norm that is as close to zero as possible. One application of this minimization problem includes compression of signals that are represented with fitted polynomials. This problem can be considered to be a higher-radix generalization of the fixed-polarity Reed-Muller minimization problem since poly- nomial decomposition can be efficiently accomplished using the properties of the fixed-polarity family of Pascal transforms. We devise an efficient decomposition technique based on properties of the Pascal transform and we formulate a heuristic minimization algorithm to search for the minimal decomposition. Experimental results are provided that compare the quality and performance of the heuristic search to an exhaustive search for the minimal decomposition. The experimental results indicate that finding such a minimal decomposition can be achieved in a practical amount of runtime. I. I NTRODUCTION The Pascal transform can be represented by a square matrix with component values that are either zero or a binomial coefficient denoted as ( n k ) and referred to as “n choose k.” In general, the Pascal transform is a matrix of infinite dimen- sion since the integer n increases without bound. However, practical uses of the transform limit n to some upper value resulting in a truncated, finite-dimensioned (n + 1) × (n + 1) square transformation matrix. We shall refer to the truncated Pascal transform as the “Pascal transform.” The Pascal transform has several applications. In terms of filter theory, the transform has found uses that include continuous s- to discrete z-domain conversion [1], various image processing filters [2], transformation of analog lowpass to digital bandpass filters [3], and in defining a new class of filters entirely [4]. With regard to topics in mathematics, there are many applications and relationships based on the transform including those described in [5]–[10]. With respect to computational algorithms, the recursive definition of the Pascal transform matrix as well as its orthogonality have enabled the development of efficient or “fast” algorithms 1 K. N. Smith was with Southern Methodist University. She is now at the Department of Computer Science at the University of Chicago (email: kns@uchicago.edu). [11]–[14]. Pascal transforms are also of interest in the binary and MVL switching theory communities for a variety of reasons including the selected examples disclosed in [10], [15]. The set of k th -degree polynomials p(x) that result from the expansion and simplification of the k th power of the binomial of the form (x + 1) k are explicitly represented as row or column vectors in the Pascal transform matrix for all k ≤ n since the components in the k th row or column vector are the coefficients of p(x). It should be noted that the index, k, in reference to the row and column vectors of the transformation matrices discussed in this paper have values 0 ≤ k ≤ n. The generalized family of Pascal transforms disclosed in [16] is based on concepts in binary-valued switching theory due to the observation that the radix-2 Zhegalkin polynomial is related to the positive-polarity Reed-Muller (PPRM) transfor- mation matrix in the same way that polynomials of the form p(x)=(x + 1) k are related to the Pascal transform [17]. In fact, the PPRM transformation matrix can be computed by replacing the Pascal matrix components with their modulo-2 value. Thus, the concept of a polarity number can be used to determine whether each row or column vector comprising the Pascal matrix is related to the (x +1) k or the (x − 1) k term. In this manner, the polarity number is formed as a binary value where “1” represents a (x − 1) k term and “0” represents a (x +1) k term. Thus, there exist a total of 2 n different matrices based upon the polarity value. We are interested in a family of Pascal transform matrices that can be further extended from the form described in [16]. In particular, we are interested in the family of transforms where the form of the binomial associated with each row or column vector is generalized and of the form, (x ± d k ) k , where d k is any value, d k ∈ Z. In this case, rather than using a polarity number to characterize the Pascal transform matrix, we use a polarity vector, d, of dimension n with associated components, d k . A family of matrices comprise the inverses of generalized Pascal matrices associated with each polarity vector d. They are of interest as they are used to find the representation of an n th -degree polynomial, p(x), as a weighted sum of binomials that are raised to the power k where 0 ≤ k ≤ n. We refer to this form of representation of p(x) as a “polynomial decomposition.” The computation of this decomposition form can be accomplished by assuming a particular polarity vector,