SCALING OF NUMBERS IN RESIDUE ARITHMETIC WITH THE FLEXIBLE SELECTION OF SCALING FACTOR

Transcription

1 POZNAN UNIVE RSITY OF TE CHNOLOGY ACADE MIC JOURNALS No 76 Electrical Egieerig 203 Zeo ULMAN* Macie CZYŻAK* Robert SMYK* SCALING OF NUMBERS IN RESIDUE ARITHMETIC WITH THE FLEXIBLE SELECTION OF SCALING FACTOR A scalig techique of umbers i residue arithmetic ith the flexible selectio of the scalig factor is preseted. The required scalig factor ca be selected from the set of moduli products of the Residue Number System (RNS) base. By permutatio of moduli of the umber system base it is possible to create may auxiliary Mixed-Radix Systems (MRS). They serve as the itermediate systems i the scalig process. All MRS s are associated ith the give RNS ith respect to the base, but they have differet sets of eights. For the scalig factor value resultig from the requiremets of the give sigal processig algorithm, the suitable MRS ca be chose that allos to obtai the scalig result i most simple maer.. INTRODUCTION I may digital sigal processig applicatios the multiplicatio of sigal samples by a costat factor is a commo task. Such applicatios may use various kids of arithmetics. Typically the distributed arithmetic [, 2] is used. A alterative ca be the Residue Number System (RNS) [3-5], that allos to decompose operatios o large itegers ito sets of operatios carried out i small iteger rigs. This advatage pertais first of all to multiplicatio. Whe the RNS base is composed of relatively small moduli ith the biary size of 5-6- bits, multiplicatio by a costat ca be carried out by memory look-up usig look-up tables ith 5-or 6-address or by the logic blocks ith the respective umber of variables. The RNS permits to attai high pipeliig rates because the fie graulatio of the circuit becomes possible. The mai draback he performig the multiplicatio i the RNS is the ecessity to scale the product by a fixed costat after multiplicatio. This results from the fact that the RNS is a iteger umber system but i the sigal processig algorithms coefficiets are fractioal or real umbers, therefore the algorithm coefficiets have to trasformed to itegers by premultiplicatio by a suitable costat, termed the scalig factor, that ill provide for the sufficietly accurate represetatio of the coefficiets. The scalig factor may also iclude the compesatio of the dyamic of the sigal * Gdańsk Uiversity of Techology.

2 76 Zeo Ulma, Macie Czyżak, Robert Smyk groth due to summatios of terms i the give algorithm. The scalig algorithms have bee cosidered sice the pioeerig ork of Szabo ad Taaka [3] by may authors. The revie of the scalig techiques ad their properties as preseted i [6, 0]. The recet orks o scalig has bee preseted by Brade ad Philips [7], Dasygieis et al. [8 ], Ma [9]. The scalig algorithms use the Chiese Remaider Theorem (CRT) [6, 0], the MRS [] or the core fuctio Burgess [2]. I this ork the use of the MRS for scalig performed i the FPGA eviromet is cosidered. The MRS may be advatageous for these applicatios he scalig of siged umbers is eeded. The MRS allos for relatively easy detectio of the sig of the RNS umber for certai RNS bases ith the specific orderig of the moduli of the base. Moreover, certai orderigs of the moduli of the base may allo to obtai the required scalig factor ad may also lead to the simplificatio of the scaler structure. The RNS is created for the system base B = { m, m2,..., m }, here m, =,2,...,, are the pairise relatively prime. The umber X from the rage [0, M-], here M m, is oe-to-oe represeted by the vector { r, r 2,..., r }, here = = r = X ad X is the oegative residue of divisio of X by m m m. I the MRS associated ith the RNS ith respect to the system base B, the umber X from the rage [0, M-] is calculated as here = E = E0+ Em + E2mm E mm 2... m = X () X E =, = = m k 0 m + ad 0 E < m + The umber rage of the MRS is the same as i the RNS. The MRS forard coversio ca be easily performed usig residue arithmetic. For this reaso, the MRS is ofte used as the auxiliary eighted umber system that supports scalig, sig detectio, base extesio ad other residue arithmetic operatios cosidered to be difficult. 3. SCALING IN THE RNS The scalig result i the RNS for usiged umbers ca be calculated as: X X K Y =, (2) K here the scalig factor K is ay atural umber. The scalig operatio ca be easier he K is the product certai moduli from the RNS base hich

3 Scalig of umbers i residue arithmetic ith the flexible selectio of scalig 77 simultaeously form the eight of the MRS, k. After the coversio of the RNS umber X to the MRS, the scalig operatio ca be performed as follos: X X E E Ek E kk Y = = = K (3) E0 Em Ek + Ek E here every products, for > k is a iteger. The scalig error is smaller tha, because X E0 + E Ek k k = < (4) The biggest draback here is that the scalig factor has to be a product of sequetial moduli of the umber system base ad o the other had it has to provide the ecessary accuracy of the represetatio of coefficiets of the DSP algorithm. 4. EXTENSION OF THE SCALING FACTORS SET It ill be assumed that scalig is performed ith the use of coversio to the MRS. The scalig factor K is the product of the moduli of the umber system. If all possible permutatios of the moduli from the base B are ko, the umber of the scalig coefficiets ca be computed. The umber of permutatios of -elemet set is!, this determies the umber of differet MRS s that ca be created. Each permutatio determies oe MRS. Every MRS is associated ith the RNS ith respect to the RNS base. The MRS s have the same the umber rage M, but differet eights ad digits. Assume that the scalig factor is a product of s< moduli. We may cosider to optios: Optio : The scalig factor is a product of s moduli, here s =,2,...,. The the umber of differet coefficiets is equal to the umber of the differet MRS s eights created for all possible permutatios. For example, e ca take the RNS ith the base B = {2,3,5,7 }. I a associated MRS e have x = E0 + E 2+ E E , that shos that the otrivial coefficiets ca be equal to the eights 2,6, 30. I fact, the umber of the base

4 78 Zeo Ulma, Macie Czyżak, Robert Smyk permutatio! = 4! = 24 gives 24 MRS s, hich meas that e obtai 4 differet eights that ca be used as the coefficiets: A = {2,3,5,6,7,0,4,5,2,30,35,42,70,05}. The set of MRS s required for the creatio of A ca be reduced based o the folloig permutatios of the base { 2,3,5,7},{3,7,5,2},{5,3,2,7},{5,2,7,3},{7,5,3,2},{7,2,5,3} Optio 2: The scalig factor is a product of s from moduli, here s = 2,3,...,. I this case, the coefficiet alays ca be created as a product of at most 2 moduli. The quatity of possible coefficiets is loer tha i Optio. It shall be assumed that the real umber of possible coefficiets does ot exceed = 8 ad 2 s 5. The umber of all possible coefficiets ca be calculated! based o C =. The results are preseted i Table. ( s)! s! Table. The umber of possible scalig coefficiets for 8 ad 2 s 5 K SUMMARY Scalig operatio i the RNS ca be based o the coversio to the MRS, subsequet scalig of idividual terms of X ad forard coversio to the RNS. The residue umber ca be scaled by a coefficiet equal to oe of the eights of MRS. The mai draback of the solutio is small amout of differet values that is equal to. I this ork a techique of extesio of the set of scalig has bee preseted. The optimal coefficiet ca be selected from the set of differet eights of MRS s, created ith the use of all possible permutatios of the moduli.. This approach exteds the possibility of selectig a appropriate scalig factor ad demostrates ho it ca be attaied.

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