Complex Number Representation in RCBNS Form for Arithmetic Operations and Conversion of the Result into Standard Binary Form


 Lawrence Higgins
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1 Complex Number epresentaton n CBNS Form for Arthmetc Operatons and Converson of the esult nto Standard Bnary Form Hatm Zan and. G. Deshmukh Florda Insttute of Technology ABSTACT Ths paper ntroduces a novel method for complex number representaton. The proposed edundant Complex Bnary Number System (CBNS) s developed by combnng a edundant Bnary Number and a complex number n base (. Donald [] and Walter Penny [,3] represented complex numbers usng base j and ( n the classfed algorthmc models. A edundant Complex Bnary Number System conssts of both real and magnaryradx number systems that form a redundant nteger dgt set. Ths system s formed by usng complex radx of ( and a dgt set of α = 3, where α assumes a value of 3, , ,,,, 3. The arthmetc operatons of complex numbers wth ths system treat the real and magnary parts as one unt. The carryfree addton has the advantage of edundancy n number representaton n the arthmetc operatons. esults of the arthmetc operatons are n the CBNS form. The two methods for converson from the CBNS form to the standard bnary number form have been presented. In ths paper the CBNS reduces the number of steps reured to perform complex number arthmetc operatons, thus enhancng the speed. Keywords: Complex Bnary number, edundant Bnary Number, edundant Complex Bnary Number System, Addton and Subtracton.. INTODUCTION Algorthms n complex orthogonal transformatons, correlatons, and fltratons are nvolved n arthmetc computatons (e.g. geometrc analyss n graphcs or mage processng). These applcatons reure effcent representatons and treatment of complex numbers. In today s computers nvolvng complex numbers, the complex operatons use the real and magnary parts separately and then accumulate ther ndvdual results to obtan the fnal result. For example, the natural way of multplcaton of two complex numbers reures four real multplcatons and one real addton and one real subtracton. To convert complex number n the CBNS form, the real part and the magnary part of the gven number must be treated as one unt. Ths wll ncrease the speed of arthmetc operatons due to a decrease n the number of steps nvolved n these operatons. The complex Bnary Number System (CBNS) method uses a base of ( nstead of a base normally used n the Standard Bnary Number (SBN). There s a unue representaton for each complex number n the new base of (. The value of an nbt bnary number n the CBNS form wth the base ( can be wrtten n the form of a power seres as shown below.... a ( a ( a ( j () ) The coeffcents a assume a value of or. The powers of ( are grouped n a row of 4bt postons as shown n Table. Each row labeled as s generated by multplyng the current row by 4. In other words, the followng expresson can be used to generate a row of 4 bts: ( = = ( j) ( j) ( ) ),,,3,,, j, =, *( 4 where ). For example, multplyng row by 4 generates the st row and multplyng the st row by 4 generates the nd row and so on. In 988, T. T. Dao attempted to defne specfc arthmetc operatons n radces / 4 by mprovng an algorthm to generate several bts [4]. The focus s on the CBNS form where ( s the mnmum possble radx n the defnton of CBNS. Table shows the weghts of ( n a complex plane and ths s closely related to the radx 4 Sgn Dgt (SD) number system. Therefore, 4 s an element of the set of the CBNS of some power and t s effcent to work wth the edundant Bnary Sgn Dgt numbers. The CBNS s a postonal number system that has a complex radx and uses a dgt set { α to α} that allows for carry free addtons. The combnaton of the two algorthms, the redundant bnary numbers system [4, 5] and the bnary number system for the complex numbers [, ] results n the edundant Complex Bnary Number System (CBNS). 78 SYSTEMICS, CYBENETICS AND INFOMATICS VOLUME  NUMBE 6
2 ( ( j ) = ( 3 6 ( j ) = 9 ( ( j ) = 8 ( 6 4 ( j ) =,3,,, ( j ) = ( j ) = ( j ) = ( j ) =,3,,,   ow = ow * (4) ow = ow * (4) ow = In ths paper the representaton of CBSN s ntroduced for the radx r = ( and a dgt set, D α Table. eal and magnary values wth radx ( D α where α = 3 and has the dgt set of (3, , ,,,, 3). Alpha ( α ) s restrcted to [( r ) / ] α ( r ) where r = 4. edundancy provdes carryfree addton n arthmetc operatons. The value of a number, X = ( xn, xn,..., x, x ) n the CBNS of base ( and α =3 can be shown by the followng expresson. X n = = x ( X = { α,..., α} () Fgure shows a block dagram for the complex number converson, followed by the arthmetc operatons (addtons and subtractons only), and then obtanng the results n the CBNS form and fnally convertng the CBNS result nto a regular bnary number. the bnary form for the real and magnary parts and then by usng the followng steps: Step : Check the sgn of both the real and magnary parts. If each s postve or negatve, then for the postve numbers place a n front of each of bt dgt (e.g. for 9 =, then and ), and smlarly for the negatve numbers place n front of each of bt dgt (e.g. for 9 = , then and ). Step : Get 3 bts of each part and combne them together wth the LSB 3bt of the magnary part catenated to the 3 bts of real part and then put them n the order of. st Complex Number nd Complex Number In the next secton, the basc propertes of the CBNS epresentaton are ntroduced wth ther basc arthmetc operatons.. EDUNDANT COMPLEX BINAY NUMBE SYSTEM EPESENTATION The redundancy n the CBNS representaton usng radx ( allows the converson of any complex number wthout any restrctons. Any complex number s represented by a row of 4 bts n the radx 4. To smplfy the converson from a bnary number to a base 4 CBNS form, a range of 3j3 through 3j3 has been used for the representaton. A porton of the representaton s shown n Table. 3. COMPLEX NUMBE CONVESION IN THE TEM OF CBNS FOM Conversons of the complex numbers to the CBNS form s acheved by convertng the complex number nto Bnary code to CBNS Converson form st n n n = row s nd CBNS Converson form t Bnary form Fgure. Block dagram for the proposed algorthm s SYSTEMICS, CYBENETICS AND INFOMATICS VOLUME  NUMBE 6 79
3 ( 3 ( Table. Bt representatons for a row n 4 bts ( ( No j  j  j 3 j3   j j j j3  j33j j j j3  j3  j3  j j j 3  j  j  j j j 3j 3 Step 3: Combne the rght three bts of the magnary part to the rght three bts of the real part. epeat the same for the next three bts to form (see example ). There are four tables for the combnatons of the sgn of the real part and the sgn of the magnary part (/, /, /, /). Table 3 shows only the postve sgn for both the magnary and real parts. Step 4: Label a group of 4 bts n the frst row as. To generate the successve rows, multply the current row by 4 and so on. Ths results n the change of sgn and magntude from one row to the next. Therefore, even rows have postve and odd rows have negatve sgns as shown below. (...,,,, ) , The example below llustrates the converson of a complex bnary number to the CBNS form. The converson of the complex number, (j7) to the CBNS form s as follow. Step : Express numbers n the bnary form for magnary and real parts. Part () 7 = 7 = Part () = = Step : Get 3 bts of each part and combne them together startng wth LSB as 3 bts of and put them together wth 3 bts of. & Step 3: Fnd the euvalent values for each group of and from Table Step 4: Multply the postve sgn to even rows and the negatve sgn to the odd rows. ,  Fnally combne them agan n the followng order....,,,,, Use the term [( ) * ] 5 4 3, for each n the 4bt row. 3. educton of Converson tables 4 The four tables generated for the four possble combnatons of the sgn of real and magnary parts of complex numbers can be reduced to only two tables. The / table can be obtaned by changng the sgn of the 4bt rows of / table, and vce versa. The same s vald for / table to / table and vce versa. ( ) base CBNS (,3 j j j S 3 S 3 S 3 S 3 Table 3. Table of converson from Bnary form to CBNS form n / case 4. AITHMETIC OPEATION Complex numbers expressed n the CBNS form are used as operands to perform arthmetc operatons. The results of the operaton wll be n the CBNS form. An example of the addton operaton s shown below. In the subtracton operaton the sgn of real and magnary parts 8 SYSTEMICS, CYBENETICS AND INFOMATICS VOLUME  NUMBE 6
4 of the subtrahend are complemented and then added to the mnuend. An example of the addton of two complex numbers, (9 and (35j5) expressed n the CBNS form s llustrated below. *,3 *, *, 3bt 3bt 3bt *,3 *, *, 3bt 3bt 3bt Frst, two complex numbers (9 and (35j5) are converted nto the CBNS form. Then the addton operaton s performed. The complex number (9 s euvalent to ( ) n the CBNS form, and smlarly the complex number (35j5) s euvalent to (  ). X Y S In some cases the result may have numbers rangng from 6 to 6; n such cases normalzaton s necessary. The process of normalzaton s as follows: Normalze the ntermedate result to be n the set of {3, , ,,,, and 3}. Get the fnal result S = Normalzaton Carry, or S= S, f S n the range of 3 to 3. The above result wll be n the CBNS form. The next secton dscusses the converson of the CBNS result to an euvalent bnary number. 5. CONVESION FOM CBNS FOM TO BINAY FOM The result of the arthmetc operaton that s n the CBNS form should be converted back to the standard bnary number (SBN) form. Two methods are presented for the converson. Both methods produce the fnal result n the s complement form. Method : In ths method splt the real and magnary parts of each group (one row) and forward t to two dedcated regsters. The real number porton s moved nto regster and the magnary number porton s moved nto regster as shown n Fgure (a), (b).,3,,, j j j n n (a) 5bt 5bt (b) Fgure. Converson dagram from CBNS to SBN The calculaton for a row s gvng by the followng expressons. = ( *,3 ) (*, ) ( *, ) (*, ) = * ) ( * ) (* ) (* ) (,3,,, The maxmum value of the eal part for one row of the CBNS form s between  and ([ 3 *( j)] [ 3*( ] (3*) ), and the maxmum value of the Imagnary part for one row of CBNS form s between 5 and 5 ([ 3*( j )] [ 3*( j)] [3*( ] ). Ths method reures 5 bts for the output n the adder unt. For example, n ths method, convert the CBNS number ( ) ( s used for  and for  for convenence) to SBN form. The result s eual to row ( )  4* row ( ) 6* row ( ) = 44j36 where row = [ * * *] j[* * *] and so on for the rest of the rows. Method : In ths method, separate the real and magnary numbers of each dgt n a row to have two parts, one for real and the other for magnary. Ths method modfes each row n the form shown, =,3 =, =,, =, and the entre real number part goes nto regster. For each row, has 5 bts and have 3 bts, and the magnary number part goes nto regster. For each row, has 3 bts and has 5 bts. The values for the real and the magnary parts go nto the output regsters and for each row of,,, and where: =, = =, =, SYSTEMICS, CYBENETICS AND INFOMATICS VOLUME  NUMBE 6 8
5 Fgure 3 shows the block dagram for mplementaton of Method for converson from the CBNS form to the SBN form. An example of ths converson s shown n Fgure CONCLUSION A method to convert complex numbers to ther euvalent CBNS form has been presented. Combnng two algorthms, the redundant bnary numbers system and the bnary number system for complex numbers, results n the edundant Complex Bnary Number System. Operands are receved n the CBNS form as nputs to an adder for the addton (wth the sgn changed to the subtrahend operaton) and the sum results n the CBNS form. Two methods have been presented for the converson of the CBNS result nto the SBN form. The entre process wll ncrease the speed of arthmetc operatons due to a decrease n the number of steps nvolved. A crcut dagram mplementaton of the arthmetc unt based on CBNS form usng FPGA s under progress. Load, 3 5bt 3bt ,,, Normalzaton to be n SBN 3bt 5bt Fgure 3. Converson dagram from CBNS to SBN,3,,,,3,,,,3,,, = = = =  = = =  = 4 = = 4 = The fnal result s = [6*()4*() 4] j[6*()4*(4) (4)] = 44 j36 Fgure 4. Converson of the fnal result back to SBN 7. EFEENCES [] D. E. Knuth, An Imagnary Number System, Communcatons of the ACM, Vol. 3, pp , 96. [] W. Penney, A Number System Wth a Negatve Base, Mathematc Student ournal, Vol., No. 4, pp. , May 964. [3] W. Penney, A Bnary System For a Complex Numbers, ournal of the ACM, Vol., No., pp , Aprl 965. [4] Y. Oh, T. Aok, and T. Hguch, edundant Complex Number System, 5 th IEEE Int. Symp. on Multple Valued Logc, pp. 49,Bloomngton,IN,USA, May 995. [5] A. Avzens, Sgned Dgt Number epresentatons for Fast Parallel Arthmetc, IE Trans. Electronc Computers, Vol. EC, pp , September 96. [6] T. aml, N. Holmes, and D. Blest, Towards Implementaton of a Bnary Number System for Complex Numbers, Proceedng of the IEEE SouthEastCon, pp ,. [7] T. aml, The Complex Bnary Number System, Proceedng of the IEEE potentals on, pp [8] T. Aok, K. Hosh and T. Hguch, edundant Complex Arthmetc and Its Applcaton to Complex 8 SYSTEMICS, CYBENETICS AND INFOMATICS VOLUME  NUMBE 6
6 Multpler desgn, 9 th IEEE Int. Symp. on Multple Valued Logc, pp. 7, Freburg, 999. [9]. Mcllhenny, and M. D. Ercegovac, OnLne Algorthem for Complex Number Arthmetc, Proceedng of the IEEE 998, 3 nd Aslomar Conference on Sgnals, Systems & Computers, part (of ), Pacfc Grove, CA, USA, pp. 776, Nov [] T. T. Dao, Specfc Arthmetc n adces /4, 8 th IEEE Int. Symp. on MultpleValued Logc, pp , 988. SYSTEMICS, CYBENETICS AND INFOMATICS VOLUME  NUMBE 6 83
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