#### Date of Award

1993

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Electrical and Computer Engineering

#### First Advisor

Alexander Skavantzos

#### Abstract

This dissertation focuses on a new approach for a hardware implementation of the cyclic convolution operation. The cyclic convolution operation is the core of several functions used in applications related to digital signal processing and error control. Since the operation is multiplication intensive and the cost of a multiplication operation is very high, most of the present research effort attempts to reduce the number of multiplications. Our approach, however, aims at obtaining an efficient implementation by relying on the properties of the special case of multiplication, namely, the squaring operation. Due to the properties exhibited by the squaring operation the hardware cost and time delay of a squarer unit is both cheaper and faster than that of a multiplication unit. This is true for both memory and non-memory based implementations. In this dissertation we have developed all the necessary theory required to express the cyclic convolution of two n-point sequences, where n is a power of 2, in terms of the elementary arithmetic operations add, square, and subtract. Our algorithms require fewer squaring operations than multiplication operations required by a traditional implementation of the cyclic convolution operation, do not introduce any round-off errors, place no restriction on word length, and are valid when the number of points to be convolved is a power of two. We then clearly demonstrate that our algorithms are also more hardware efficient for both memory and non-memory based implementations. Further, schemes to multiply two numbers based on the cyclic convolution operation are presented. Finally, efficient ways of computing the squaring operation when arithmetic is performed in modular rings are developed.

#### Recommended Citation

Rao, Poornachandra Bellamkonda, "Squared Law Algorithms: Theory and Applications." (1993). *LSU Historical Dissertations and Theses*. 5591.

https://digitalcommons.lsu.edu/gradschool_disstheses/5591

#### Pages

175