Arithmetic with the Two-Dimensional Logarithmic Number System

180 Pages·2017·3.02 MB·English

by  Mahzad Azarmehr

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Electronic Theses and Dissertations Theses, Dissertations, and Major Papers
2011
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Mahzad Azarmehr
University of Windsor
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Arithmetic with the Two-Dimensional
Logarithmic Number System (2DLNS)
by
Mahzad Azarmehr
A Dissertation
Submitted to the Faculty of Graduate Studies through
the Department of Electrical and Computer Engineering in Partial Ful(cid:2)llment
of the Requirements for the Degree of Doctor of Philosophy at the
University of Windsor
Windsor,Ontario,Canada
2011
c 2011MahzadAzarmehr
(cid:13)
All Rights Reserved. No Part of this document may be reproduced, stored or otherwise
retained in a retreival system or transmitted in any form, on any medium by any means
withoutpriorwrittenpermissionoftheauthor.
ArithmeticwiththeTwo-DimensionalLogarithmicNumberSystem(2DLNS)
by
MahzadAzarmehr
APPROVEDBY:
D.Al-Khalili
DepartmentofElectricalandComputerEngineering, RoyalMilitaryCollege
A.Edrisy
DepartmentofMechanical, AutomotiveandMaterialsEngineering, UniversityofWindsor
R.Muscedere
DepartmentofElectricalandComputerEngineering,University ofWindsor
M.Khalid
DepartmentofElectricalandComputerEngineering,University ofWindsor
M.Ahmadi
DepartmentofElectricalandComputerEngineering,University ofWindsor
A.T.Alpas
DepartmentofMechanical, AutomotiveandMaterialsEngineering, UniversityofWindsor
September23,2011
Declaration of Previous Publication
This thesis includes 2 original papers that have been previously published/submitted for
publicationinpeerreviewedjournals,asfollows:
ThesisChapter Publicationtitle Publicationstatus
Chapters4and5 High-SpeedandLow-PowerRecon(cid:2)gurableArchitectureof2-digit2DLNS-basedRecursiveMultipliers Published
Chapter6 Low-PowerFiniteImpulseResponse(FIR)FilterDesignusingTwo-DimensionLogarithmicNumberSystem(2DLNS)Representations Submitted
IcertifythatIhaveobtainedawrittenpermissionfromthecopyrightowner(s)toinclude
the above published material(s) in my thesis. I certify that the above material describes
work completedduringmyregistrationas graduate studentat the Universityof Windsor. I
declarethat,tothebestofmyknowledge,mythesisdoesnotinfringeuponanyone’scopy-
right nor violate any proprietary rights and that any ideas, techniques, quotations, or any
othermaterialfromtheworkofotherpeopleincludedinmythesis,publishedorotherwise,
are fully acknowledged in accordance with the standard referencing practices. Further-
more, to the extent that I have included copyrighted material that surpasses the bounds of
fairdealingwithinthemeaningoftheCanadaCopyrightAct,IcertifythatIhaveobtained
a written permissionfrom the copyrightowner(s) to include such material(s) in my thesis.
I declare that this is a true copy of my thesis, including any (cid:2)nal revisions, as approved
by my thesis committee and the Graduate Studies of(cid:2)ce, and that this thesis has not been
iv
DECLARATIONOFPREVIOUSPUBLICATION
submittedforahigherdegreetoanyotherUniversityorInstitution.
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Abstract
The ever increasing demand for low power DSP applications has directed researchers to
contemplate a variety of potential approaches in different contexts. Since DSP algorithms
heavilyrelyonmultiplication,therearegrowingdemandsformoreef(cid:2)cientmultiplication
structures. In this regard, using some alternative number systems, which inherently are
capable of reducing the hardware complexity, have been studied. The Multi-Dimensional
LogarithmicNumberSystem(MDLNS),amulti-digitandmulti-baseextensiontotheLog-
arithmic Number System (LNS), is considered as an alternative to the traditional binary
representation for selected applications. The MDLNS provides a reduction in the size of
the number representation with a non-linear mapping and promises a lower cost realiza-
tion of arithmetic operations with a reduced hardware complexity. In addition, using the
recursive multiplication technique, which refers to the published multiplication algorithm
that uses smaller multipliersto implementa larger operation, reduces the size of operands
and corresponding partial additions. As part of this research, 2DLNS-based multiplica-
tion architectures with two different levels of recursion are presented. These architectures
combinesomeof the(cid:3)exibilityof software withthe highperformance of hardware by im-
plementingthe recursive multiplicationschemes on a 2DLNS processing structure. These
implementationsdemonstratethe ef(cid:2)ciency of 2DLNS in DSP applicationsand show out-
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ABSTRACT
standing results in terms of operation delay and dynamic power consumption. We also
demonstrate the application of recursive 2DLNS multipliers to recon(cid:2)gurable multiplica-
tion architectures. These architectures are able to perform single and double precision
multiplication, as well as fault tolerant and dual throughput single precision operations.
Modern DSP processors, such as those used in hand-held devices, may (cid:2)nd considerable
bene(cid:2)t from these high-performance, low-power, and high-speed recon(cid:2)gurable architec-
tures. In the (cid:2)nal partof thisresearch work,recursive2DLNS multiplicationarchitectures
have been applied to a FIR (cid:2)lter structure. These implementationsshow considerable im-
provementtotheirbinarycounterpartsintermsofVLSI areaandpowerconsumption.
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Towhommademebelievethatwhere thereisawill,thereisaway.
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Acknowledgments
Thereare severalpeoplewhodeservetobeacknowledgedfor theirgenerouscontributions
to this project. I would (cid:2)rst like to express my sincere gratitude and appreciation to Dr.
Majid Ahmadi, my supervisor, for his invaluable guidance throughout the course of this
thesis work. I would also like to thank my committee members: Dr. Afsaneh Edrisy, Dr.
Roberto Muscedere, Dr. Mohammed Khalid, and Dr. Dhamin Al-Khalili from the Royal
MilitaryCollege for reviewing my thesisand their constructivecomments. Special thanks
to Dr. Roberto Muscedere for his expert guidance and constant support throughout my
study.
I am deeply grateful to my friend and partner Hassan Haftbaradaran for his love and con-
stant support and encouragement. I also sincerely appreciate my family and friends for
theirsupport,helpandfriendship.
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