TY - GEN

T1 - Fixed Polarity Pascal Transforms with Symbolic Computer Algebra Applications

AU - Smith, Kaitlin N.

AU - Thornton, Mitchell A.

N1 - Publisher Copyright:
© 2019 IEEE.

PY - 2019/8

Y1 - 2019/8

N2 - The fixed polarity forms of the Reed-Muller (RM) transform exist in 2n different polarities. The integer-valued Pascal transform is related to the binary-valued RM transform through the Sierpinski fractal, calculated by performing the modulo-2 operation on Pascal's triangle, as it appears in the lower triangular portion of the positive-polarity RM transform. We generalize the relationship between the fixed-polarity forms of the RM transform and introduce associated forms of the Pascal transform that are characterized by a polarity value allowing for a family of fixed-polarity Pascal (FPP) transform matrices to be defined. We observe and prove several properties of the FPP transforms and their inverses. An application of FPP transforms in the area of symbolic computer algebra that enables very fast decomposition of real-valued polynomials as weighted sums of different binomials raised to a power as compared to manual symbolic manipulation is described. The decomposition weights can be considered to be the inverse FPP spectrum with respect to a real-valued polynomial since they are computed using one of the linear orthogonal FPP transformation matrices.

AB - The fixed polarity forms of the Reed-Muller (RM) transform exist in 2n different polarities. The integer-valued Pascal transform is related to the binary-valued RM transform through the Sierpinski fractal, calculated by performing the modulo-2 operation on Pascal's triangle, as it appears in the lower triangular portion of the positive-polarity RM transform. We generalize the relationship between the fixed-polarity forms of the RM transform and introduce associated forms of the Pascal transform that are characterized by a polarity value allowing for a family of fixed-polarity Pascal (FPP) transform matrices to be defined. We observe and prove several properties of the FPP transforms and their inverses. An application of FPP transforms in the area of symbolic computer algebra that enables very fast decomposition of real-valued polynomials as weighted sums of different binomials raised to a power as compared to manual symbolic manipulation is described. The decomposition weights can be considered to be the inverse FPP spectrum with respect to a real-valued polynomial since they are computed using one of the linear orthogonal FPP transformation matrices.

UR - http://www.scopus.com/inward/record.url?scp=85084356867&partnerID=8YFLogxK

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U2 - 10.1109/PACRIM47961.2019.8985082

DO - 10.1109/PACRIM47961.2019.8985082

M3 - Conference contribution

AN - SCOPUS:85084356867

T3 - 2019 IEEE Pacific Rim Conference on Communications, Computers and Signal Processing, PACRIM 2019 - Proceedings

BT - 2019 IEEE Pacific Rim Conference on Communications, Computers and Signal Processing, PACRIM 2019 - Proceedings

PB - Institute of Electrical and Electronics Engineers Inc.

T2 - 2019 IEEE Pacific Rim Conference on Communications, Computers and Signal Processing, PACRIM 2019

Y2 - 21 August 2019 through 23 August 2019

ER -