Improvement of Radix-2 k Signed-Digit Number for High Speed Circuit

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1、FORMALIZEDMATHEMATICSVolume11,Number2,2003UniversityofBiałystokImprovementofRadix-2kSigned-DigitNumberforHighSpeedCircuitMasaakiNiimuraYasushiFuwaShinshuUniversityShinshuUniversityNaganoNaganoSummary.kkInthisarticle,anewradix-2signed-digitnumber(Radix-2subsigned-digitnumber)isd

2、efinedanditspropertiesforhardwarerealizationarediscussed.kUntilnow,highspeedcalculationmethodwithRadix-2signed-digitnum-bersisproposed,butthismethodused“ComparesWith2”tocalculatecarry.“Compareswith2”isaverysimplemethod,butitneedsverycomplicatedhar-dwareespeciallywhenthevalueofkb

3、ecomeslarge.Inthisarticle,weproposekkasubsetofRadix-2signed-digit,namedRadix-2subsigned-digitnumbers.kRadix-2subsigned-digitwasdesignedsothatthecarrycalculationuse“bitcompare”tohardware-realizationsimplifiesmore.kInthefirstsectionofthisarticle,wedefinedtheconceptofRadix-2subsigned

4、-digitnumbersandprovedsomeoftheirproperties.Inthesecondsec-tion,wedefinedthenewcarrycalculationmethodinconsiderationofhardware-realization,andprovedsomeoftheirproperties.Inthethirdsection,weprovidekksomefunctionsforgeneratingRadix-2subsigned-digitnumbersfromRadix-2signed-digitnu

5、mbers.Inthelastsection,wedefinedsomefunctionsforgenerationknaturalnumbersfromRadix-2subsigned-digit,andweclarifieditscorrectness.MMLIdentifier:RADIX3.Thearticles[11],[14],[8],[12],[1],[4],[3],[13],[10],[7],[2],[9],[5],and[6]providethenotationandterminologyforthispaper.1.Definition

6、forRadix-2kSubSigned-DigitNumberWeadoptthefollowingconvention:i,n,m,k,xarenaturalnumbersandi1,i2areintegers.Nextwestatetheproposition°c2003UniversityofBiałystok133ISSN1426–2630134masaakiniimuraandyasushifuwa(1)((Radixk)n)·Radixk=(Radixk)n+1:NNLetusconsiderk.Thefunctork−SDSubSis

7、definedasfollows:(Def.1)k−SDSubS={e;erangesoverelementsofZ:e−Radix(k−01)∧e¬Radix(k−01)−1}:Letusconsiderk.Thefunctork−SDSubisdefinedby:(Def.2)k−SDSub={e;erangesoverelementsofZ:e−Radix(k−01)−1∧e¬Radix(k−01)}:Thefollowingpropositionsaretrue:(2)Ifi1∈k−SDSub;then−Radix(k−01)−1¬i1andi1

8、¬Radix(k−01):(3)Foreverynaturalnumberkholdsk−SDSubS⊆k−