The Double Rotation CORDIC Algorithm: New Results for VLSI Implementation of Fast Sine/Cosine Generation

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1 he Doble Rotato CORDIC Algorthm: New Reslts for VLSI Implemetato of Fast Se/Cose eerato ze-y Sg * Chch-S Che ** Mg-Cho Shh * * Departmet of Electrcal Egeerg ** Isttte of Egeerg Scece Chg Ha Uerst, Hsch, awa 3- {sg, s, shh}@btc.com.tw Abstract- he COordate Rotato DIgtal Compter (CORDIC algorthm s a arthmetc algorthm to ealate aros elemetar fctos throgh a seres of terate operatos. I ths paper, a hgh-speed se/cose geerator s based o doble rotato of the orgal CORDIC algorthm b predctg all the rotato drectos from the tal pt agle. he proposed archtectre has a smple predcto scheme throgh a effcet determato strateg of rotato drecto. he crtcal dela path s redced b tlzg the carr-sae adder (CSA. hs, the comptato complet of the proposed archtectre s ealated; the proposed archtectre mproes the latec of 37.5% 6-bt operad,.6% -bt operad ad.5% 3-bt operad, respectel. Whle the large mber-bt operad, the speed shold be mproed b 8%. ewords: doble rotato algorth CORDIC, -predcto algorth carr-sae adder, se/ cose geerator.. Itrodcto I drect dgtal freqec stheszer (DDFS sstem [] ad orthogoal freqec dso mltpleer (OFDM sstem [], [3], the ke compoet s the se/cose fcto geerator to comptes sθ ad cosθ to a precso of N fracto bts. I ths paper, a hgh-speed se/cose geerator based o the CORDIC algorthm s proposed. CORDIC (COordate Rotato DIgtal Compter s a algorthm for performg a seqece of terato comptatos sg coordate rotato [], [5]. It ca geerate some powerfl elemetar fcto realzed ol b a smple set of adders ad shfters. he basc CORDIC terato eqatos are s( m ( s( z z ( (3 m detfes crclar ( lear ( ad hperbolc (m- coordate sstems,,,,.,-,,,,3,,5,... s (,,3,,5,6,... m m,,3,,,5,... m a hperbolc coordate sste the teratos are repeated at3. / s( m ta [ m ] ( the rotato for rotato mode ( z s sg( z, whle for ectorg mode (, t s sg( sg(. For the -th terato, a scale factor becomes s(, k m m. After -teratos, the prodct of all the scale factors s m k s ( m, s ( m, m, m m (5 the rotato drectos are defed to {, }.. Doble Rotato CORDIC Algorthm he basc cocept of the accelerated CORDIC algorthm s to redce the teratos. he doble rotato CORDIC algorthm s deeloped to redce the teratos or comptato tme [6]. he doble rotato CORDIC terato eqatos shold be dered ad the comptato complet shold be also ealated. he CORDIC terato eqatos crclar coordate sstem are also wrtte the form of matr mltplcatos. (6 Accordg to eqs.(6, we obta ( ( (7! the eq. (7 s a terato eqato of the doble rotato CORDIC algorthm. hs, the doble rotato CORDIC terato eqato crclar coordate sstem s modfed as show below ( ( ( ( (8

2 ( ( ( ( ( z z ta ta (9 ( he comptato complet of parallel processg s creased to two carr-sae addtos ((3,CSAs ad oe shft for each terato [7]. I -bt operad sste whle, eqs.(8 ad (9 becomes ( ( ( ( ( ( hs, the comptato complet of parallel processg s oe (3,CSA ad oe shft for each terato. 3. A Noel -redcto Algorthm he basc teto to realze the doble rotato CORDIC algorthm s to geerate more ales each step. Now, the proposed archtectre reqres two ales each step. he -ale predcto algorthm s descrbed as below: I ths algorth the ad are geerated followg steps, the s determed b sg of z (. he seres of ew costats ca be defed as ( z ( z( (ta ta (3 3( z( ta ( z ( z( (ta ta ( z (5 hree eqatos for determg ad z( are defed as z( z( ( (6 z z ( 3( 3 z( ( (7 z( ( (8 he determato strateg of, } ad { z( s llstrated Fgs., ad 3. he flowchart for the -predcto ad z( determato algorthm s llstrated Fg., detaled flowcharts for specfc cases are llstrated Fg. ad 3, respectel. Now, the -predcto ad z ( determato algorthm s aalzed ad deeloped, ths algorthm s smple ad eas to mplemet o hardware. hs, the algorthm s er sted to VLSI mplemetato. he determato crct of ad z( s show Fg... he Accelerated CORDIC Archtectre for Se/Cose eerator he proposed archtectre has -bt word legth, so t makes -terato to compte the crclar coordate sstem. I ths archtectre, the (, carr-sae adder (CSA ad carr-propagato adder (CA cossts of two three-pt, two-otpt (3, carr-sae adders/sbtractors ad oe carr-look-ahead adder [8]. Fg. 5 shows the proposed accelerated archtectre wth the rotato mode a crclar coordate sstem. hs, the comptato complet s two CA comptatos, a CLA comptato ad a shft for each terato at frst teratos, ad the comptato complet s a CA comptato, a CLA comptato ad a shft for each terato at last teratos. he Se/Cose geerator s mplemeted b the accelerated CORDIC archtectre wth the rotato mode the crclar coordate sstem. he pt of s ad pt of s, the pt of z s a agle for se ad cose fcto. I ths archtectre, the shft seqece { s(, } s pre-defed, so that the s a costat. 5. erformace Aalses of the Accelerated CORDIC Archtectre Sce the comptato complet of accelerated CORDIC archtectre s two addtos ad oe shft for each terato at frst -terato. At last -terato, the comptato complet of the archtectre s a addto ad shft for each terato. he total comptato complet of the accelerated CORDIC archtectre s ( CSA CLA shft (CSA CLA shft ( ( 6 8 ( ( (5 CSA (operato tme of CSA FA (dela of fll-adder (dela of sgle gate, shft (operato tme of hardwred shft ad CLA (operato tme of CLA ( 3 [7]. ( 8 ad ( 6 are comptato tme for frst -terato ad last -terato, respectel. he total comptato complet of coetoal CORDIC s CLA comptatos ad shfts. he comptato complet s represeted

3 as 8 (. Accordg to Fg. 3, the ad ( z determato crct cossts of three sbtractors, ad oe mltpleer. he determato tme of ad ( z s MUX CLA 5 ( 3 ( (6 MUX. hs, the determato tme of ad ( z s less tha comptato tme of [ ], whch s 8 ( or 6 (, ad the process of the ad z( determato ca ot redce the throghpt. he percetage of latec mproemet erss mber of bt each operad s show Fg Nmercal Aalses of the Doble Rotato CORDIC Algorthm he mercal aalss of the doble rotato CORDIC algorthm s dscssed ths secto. Seeral error aalss researches of the CORDIC algorthm hae bee doe [9], [], []. he dfferece betwee the doble rotato CORDIC algorthm ad the coetoal CORDIC algorthm s the term dropped eqs. (8, (9, ad (. Now, the mamm error of the doble rotato CORDIC algorthm related to the coetoal CORDIC algorthm s aalzed ad dered as heorem [6], [9], []: heorem : he pper bod of the doble rotato CORDIC algorthm s,, s the error of ad s the mber of bts. roof:,,,,,.,,, ( (,,, (7, ad, are errors of,, ad, respectel. Here, the error aalss of s beod ths paper [6], [], so we assme that, ad, s the error trodced b doble rotato. Accordg to eq. (7, we obta, (8,, (9 Accordg to eq. (3, for all, we obta s( cos( k k (3 k ad ( k. It makes matr of be chaged as (3 (3 Hece, s( cos( (33 he error pper bod wold be dcated, ad the etra bts ad teratos to ga same accrac of the coetoal CORDIC algorthm cold be estmated. Whe 3, we hae ad the errors are smaller tha the pper bod, whch s also llstrated Fg Coclso hs paper presets doble rotato archtectre

4 ad a oel -predcto algorthm of CORDIC terato applg them to the se/cose geerator. he proposed archtectre does ot reqre etra ROM or complcated determato hardware. he speed s mproed b sg carr-sae adder (CSA wth redce the dela tme of the crtcal path. he doble rotato CORDIC archtectre wth a oel -predcto algorthm mproes the latec of the coetoal CORDIC algorthm at least 37.5%, the effcec of the CORDIC comptato s creased b bts ad teratos, ad t makes the latec of the coetoal CORDIC algorthm mproe at most 8%. 8. Refereces [] A. Madsett, Y.. Wets, ad A. N. Wlso, Jr., A -MHz, 6-bt, Drect Dgtal Freqec Stheszer wth a -dbc Spros-Free Damc Rage, IEEE J. Sold-State Crcts, Vol. 3, Ag. 999, pp [] S. Y. ark, N. I. Cho, S. U. Lee,. Desg of //8-ot FF rocessor Based o CORDIC Algorthm OFDM Receer, IEEE acfc Rm Coferece o Commcatos, Compters ad Sgal rocessg (-ACRIM, Vol., Ag., pp [3]. Mowak, R. Nota, M. Bekoo, ad E. Deprettere, A Effcet Implemetato of a 56-ot FF rocessor wth CORDIC for OFDM Sstems, roc. rorisc, 998. [] J. E. Volder, he CORDIC rgoometrc Comptg echqe, IRE rasactos o Electroc Compters, Vol. EC-8, 959, pp [5] J. S. Walther, A Ufed Algorthm for Elemetar Fctos, Sprg Jot Compter Coferece roceedgs, Vol.38, 97, pp [6] S. Wag, E. E. Swartzlader Jr., Merged CORDIC Algorth roc. It l Smp. Crcts ad Sstems, 995, pp [7] I. ore, Compter Arthmetc Algorthms, Secod Edto, A.. eters, Natck, MA,, Chapter 5. [8] J.. wak, J. H. Cho, E. E. Swartzlader, Jr., Hgh-Speed CORDIC Based o a Oerlapped Archtectre ad a Noel -redcto Method, Joral of VLSI Sgal rocessg, Vol. 5,, pp [9] Y. H. H, he Qatzato Effects of the CORDIC Algorthm, IEEE rasactos o Sgal rocessg, Vol., No., 99, pp []. ota, J. R. Callaro, Nmercal Accrac ad Hardware rade-offs for CORDIC Arthmetc for Specal-rpose rocessors, IEEE rasactos o Compters, Vol., No. 7, 993, pp []. Y. Sg, Y. H. Sg, he Qatzato Effects of CORDIC Arthmetc for Dgtal Sgal rocessg Applcatos, he st Workshop o Combatoral Mathematcs ad Comptato heor, achg Healthcare ad Maagemet Uerst, achg, awa, Ma ~,, pp Beg z( For ( ; ; Sg(z ( Ealate sg( z( Sg(z ( Mltpleer Yes Brachg No Flowchart Fg. Flowchart Fg. 3 z ( z ( Fg.. (a Determato crct of z ( Fg.. Flowchart for the -predcto ad z ( determato algorthm. Detaled flowcharts for specfc cases whe sg(z( ealato retrs, -, ad whe the algorthm s a brachg are llstrated Fgs. ad 3, respectel.

5 sg( z( erform parallel f sg( z ( " " ( sg( z ( " " (, z( z( the f sg ( z ( " " ( sg( z ( " " (, z( z( the f sg ( z ( " " ( sg( z ( " " ( sg( z ( " " ( 3 the, z( z ( f sg ( z ( " " ( sg( z ( " " ( sg( z ( " " ( 3 the, z( z ( Fg.. Flowchart for -terato for the case whe sg( z( ealato retrs. sg( z( erform parallel f sg( z ( " " ( sg( z ( " " (, z( z( the f sg ( z ( " " ( sg( z ( " " (, z( z( the f sg( z ( " " ( sg( z ( " " ( sg( z ( " " ( 3 the, z( z ( f sg ( z ( " " ( sg( z ( " " ( sg( z ( " " ( 3 the, z( z ( Fg. 3. Flowchart for -terato for the case whe sg( z( ealato retrs -. z ( ± z ( Sg(z ( z ( ± z ( Sg(z ( Determato Crct (: Mltpleer z( z ( Fg. (b, ad z ( geerator ±

6 Hardwre shft -( Hardwre shft - Hardwre shft -( Coter- (,CSA (3,CSA CLA Hardwre shft -( Hardwre shft - Hardwre shft -( Coter- (,CSA (3,CSA CLA Fg. 5. he accelerated CORDIC archtectre wth the rotato mode the crclar coordate sstem % bt Fg. 6. he percetage of latec mproemet erss mber of bt each operad Fg. 7. he erss of pt agles

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