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Mathematical Approach to Computational Networks

IP.com Disclosure Number: IPCOM000128657D
Original Publication Date: 1978-Dec-31
Included in the Prior Art Database: 2005-Sep-16
Document File: 19 page(s) / 51K

Publishing Venue

Software Patent Institute

Related People

Danny Cohen: AUTHOR [+3]

Abstract

1 ?.. The Design Goals 3 3. The FIR-Filter Example 5 The Z Operator 5 The FIR-Filter Implementation 6 Improving this Design 8 About Notation !0 Improving the Operation Rate 11 Another Look 14 And Another Look 15 Applying the Z-Notation to Design Evaluation 19 4. Multiplication of Polynomials 23 The Problem of Multiplication of Polynomials 23 Reversing the Order of X 25 Computing the Sum of Polynomial Products 26 5. Division of Polynomials 29 The Problem of Division of Polynomials 29 Checking the Multiplication and Division 31 Simultaneous Multiplication and Division of Polynomials 33

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

Mathematical Approach to Computational Networks

Danny Cohen

ARPA ORDER NO. 2223

ISIIRR-78-73 November 1978 INFORMATION SCIENCES INSTITUTE 4676 Admiralty Wayl Marina del ReylCali f ornia 90291 UNIVERSITY OF SOUTHERN CALIFORNIA fTT (213) 822-1 S 11

THIS RESEARCH IS SUPPORTED BY THE ADVANCED RESEARCH PROJECTS AGENCY UNDER CONTRACT NO. DAHC15 72 C 0308. ARPA ORDER N0.2223. VIEWS AND CONCLUSIONS CONTAINED IN THIS STUDY ARE THE AUTHOR'S AND SHOULD NOT BE INTERPRETED AS REPRESENTING THE OFFICIAL OPINION OR POLICY OF ARPA, THE U.S. GOVERNMENT OR ANY OTHER PERSON OR AGENCY CONNECTED WITH THEM.

THIS DOCUMENT APPROVED FOR PUBLIC RELEASE AND SALE; DISTRIBUTION IS UNLIMITED.

CONTENTS

Abstract v

Acknowledgments vi

1. Introduction 1

?.. The Design Goals 3

3. The FIR-Filter Example 5 The Z Operator 5 The FIR-Filter Implementation 6 Improving this Design 8 About Notation !0 Improving the Operation Rate 11 Another Look 14 And Another Look 15 Applying the Z-Notation to Design Evaluation 19

4. Multiplication of Polynomials 23 The Problem of Multiplication of Polynomials 23 Reversing the Order of X 25 Computing the Sum of Polynomial Products 26

5. Division of Polynomials 29 The Problem of Division of Polynomials 29 Checking the Multiplication and Division 31 Simultaneous Multiplication and Division of Polynomials 33

G. Synthetic Aperture Radar 35 The STIR Problem 85 The Design of the Network 36

7, Summary and Conclusions 39

ABSTRACT

7"his report deals with design principles for iterative computational networks. Such computational networks are used for performing repetitive computations which typically are not

University of Southern California Page 1 Dec 31, 1978

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Mathematical Approach to Computational Networks

data-dependent. Most of the signal processing algorithms, like FFT and filtering, belong to this class.

The main idea in this report is the development of mathematical notation for expressing such designs. This notation captures the important features and properties of these computational networks, and can be used for analyzing, for designing, and objectively evaluating computational networks.

ACKNOWLEDGMENTS

I am grateful to Bob Sproull (of CMU) for encouraging me to write this report, to Bob Kahn (ARPA) for suggesting the application of this technique to multiplication and division of polynomials, and to Chuck Seitz (Caltech) for his critical review of the draft of this report.

I would like to thank Lisa Moses for her skillful typing of this report, and especially the mathematical formulae, Dehe Hays for hex help, Nelson Lucas for his help to preparing the many drawings which are an integral part of this report, and Jim Melancon for the skillful ,job of putting this report into its present shape.

1. INTRODUCTION

The central point of this report is the application of a precise mathematical notation to express computational networks. This notation captures the concepts...