Exact Analysis of DDS Spurs and SNR du to Phas Truncation and Arbitrary Phas-to-Amplitud Errors Arthur Torosyan Elctrical Enginring Dpartmnt Univrsity of California, os Angls (UCA os Angls, USA torosyan@alum.mit.du Alan N. illson, Jr. Elctrical Enginring Dpartmnt Univrsity of California, os Angls (UCA os Angls, USA willson@.ucla.du Abstract This papr prsnts th thortical basis for an algorithm that prforms an xact analysis of th output spctrum of Dirct Digital Frquncy Synthsizrs (DDS or DDFS in th prsnc of phas accumulator truncation, finit arithmtic prcision and arbitrary approximations and rrors in th sin/cosin mapping function (SCF. Th drivation provids strong insight into spurious frquncy (spur magnitud and spctral location, and mas vidnt that th st of spurs du to phas-word truncation and th st rsulting from SCF imprcision and rrors ar ffctivly disjoint. Phas-truncation spurs ar shown to hav distinct magnituds and thir spctral locations, ordrd from largst to smallst in magnitud, ar asily ascrtaind. ithout gnrating all spur magnituds, th algorithm supportd by this thory is capabl of xactly computing th Signal-to-Nois Ratio (SNR and Spurious-Fr Dynamic Rang (SFDR and xactly computing th magnituds and locations of th N worst (i.., largstmagnitud spurs or all spurs with magnituds largr than ε, du to th combind ffct of phas truncation and arbitrary SCF imprcision (whr N and ε ar usr-spcifid paramtrs. I. INTRODUCTION Th gnral structur of a DDS, dpictd in Fig., was first introducd in 97 by Tirny, Radr, and Gold [3]. Th DDS can b partitiond into two functional units. Th first, th phas accumulator, consisting of an ovrflowing addr and a fdbac rgistr, accumulats th input frquncy control word (fcw to produc a phas angl for th sin/cosin mapping function (SCF, which is th scond functional unit. Th SCF can b viwd as a mappr (.g., looup tabl btwn an input phas angl θ and its corrsponding sinθ and cosθ valus. Various fficint implmntations hav bn proposd for this mapping function,.g., [4], [5], but w nd only considr that, howvr implmntd, it is simply a mapping from th input angl θ to sinθ and/or cos θ. To rduc th SCF implmntation complxity, th phas accumulator output typically is truncatd bfor bing fd to th SCF, as shown in Fig., whr only bits out of bits ar rtaind. This phas truncation causs rrors (dtrministic, priodic rrors, oftn rfrrd to as nois at This rsarch was supportd by Analog Dvics, roadcom, Globspan, and Rocwll Scintific through UC ICRO Grants -4 and - and by an Intl Corporation Ph.D. Fllowship. This articl xtnds th fundamntal thory and corrcts quation ( in Sction 3 of th papr prsntd at th -nd Imag and Signal Procssing and Analysis Confrnc in, []. A mor thorough and complt tratmnt of this subjct can b found in []. -783-95-/5/$. 5 IEEE. 5 fcw Phas Accumulator R g truncat Sin Cosin apping Function (SCF s(n c(n th DDS output which manifst thmslvs as a st of spurious frquncis (output signal componnts at undsird frquncis, oftn rfrrd to as spurs. Furthrmor, practical implmntations of th SCF hav finit prcision outputs and thir implmntation may mploy algorithmic approximations and quantization and othr non-idal oprations that collctivly giv ris to rrors spcific to th SCF implmntation,.g. [4], [5]. Ths SCF rrors ar collctivly rfrrd to as SCF non-idalitis, and thy also gnrat dtrministic nois at th DDS output and hnc contribut anothr st of spurs. Early fforts for dtrmining th magnitud and spctral location of (only phas-truncation spurs wr rportd in 983 by hrgardt [6] and latr in 985 by Nicholas [7], and in 987 by Nicholas and Samuli [8]. Ths publications dscrib similar approachs and bas thir analyss on th phas-rror squnc du to phas truncation and, using proprtis of this rror squnc along with th assistanc of small-angl approximations, driv a complx procdur for th charactrization of phas-truncation spurs. Shortly thraftr, Jnq [9] usd an lgant approach, on for analyzing a class of non-uniformly sampld signals, to modl th phas rror du to phas truncation and, without approximations, drivd an xprssion for magnituds of (only phas-truncation spurs. In 993 Kroupa [] usd an approach similar to that of [6], [7], and [8] and prsntd an algorithm for th stimation of phas-truncation spurs with th introduction of mor approximations. Non of [6], [7], [8], [9], and [] considrs th xact charactrization of spurs du to arbitrary SCF non-idalitis, sinc th thoris dvlopd ar drivd solly from th proprtis of phas-truncation rror. Two DDS with xactly th sam top-lvl dsign paramtrs (i..,,, and D valus, but diffrnt implmntations for th SCF can hav vry diffrnt SCF non-idalitis, hnc, yild vry diffrnt sts of spurs. As w will show subsquntly, th ovrwhlming majority of spurs gnratd by a DDS ar thos causd by SCF non-idalitis. ith th xcption of [] and [], sinc no algorithm to dat has bn rportd for th xact Figur. Th gnral structur for DDS. D D
charactrization of spurs du both to phas-truncation and arbitrary SCF non-idalitis, naturally, thr is as yt no algorithm for th charactrization of th N worst spurs and/or xact computation of SFDR and SNR for an arbitrary DDS xpct for th brut forc approach of gnrating all spurs via simulations and sarching for th worst ons. ithout such an algorithm, th computational and storag rquirmnts to gnrat all of th spurs ar, to say th last, daunting, and lily infasibl []. In this papr w xtnd th fundamntal thory of [] and obtain a supr-fficint algorithm, that can b applid to any DDS having th gnral structur of Fig. (i.., any SCF implmntation, that: accpts all rlvant SCF implmntation dtails via th signatur squnc, i.., th squnc of sampls gnratd by th SCF for th possibl SCF inputs, without computing all spur magnituds, and for all phas-truncation-causing fcw and du to th combind ffct of phas truncation and th rrors of th spcifid SCF, xactly computs th SFDR and SNR, and xactly computs th magnituds and locations of N largst-magnitud spurs (whr N is a usr-spcifid intgr, or all spurs with magnituds gratr than ε (whr ε is a usr-spcifid thrshold. or dtaild and thorough tratmnt of th dvlopmnt and application of this thory can b found in []. Th analysis for th rlativly simplr cas of fcw that do not gnrat phas truncation and th considration of all rlvant initial phass in this cas, ar also rportd in []. II. ASIS SETS FOR A SPURS For an arbitrary fcw and th fcw comprisd of all zros xcpt for th bit at th position corrsponding to th rightmost non-zro bit of fcw, it is shown in [] and [] that th DDS output squncs corrsponding to fcw and fcw hav idntical priod and thy ar simpl rarrangmnts of ach othr. For xampl, if fcw =, thn fcw =, whr th rightmost non-zro bit position is dnotd by and undrlind at = 7 for th abov fcw and fcw (position is countd from S to S bginning at position for th S. Th DDS output has a priod of sampls, hnc for th abov xampl th DDS output has a 7 priod of = 8 sampls. t s ( n and s ( n dnot th DDS output squncs du to any fcw and it s corrsponding singl-nonzro-bit fcw, rspctivly, whr n is th sampl indx. Using th normalizd fcw obtaind from fcw by omitting all of th S zro bits following position and th smallst positiv intgr J satisfying ( J mod fcw =, th rarrangmnts gnrating s from s and s from s ar shown in [] and [] to b (( n (( n J s ( n = s fcw ; s ( n = s ( mod mod whr ( mod dnots a modulo opration. Th intrprtation of ( is that th s squnc is obtaind by picing trms from th s squnc in stps of fcw and wrapping around to th bginning of th s squnc whn raching its nd. Similarly, th s squnc is obtaind by picing trms from th s squnc in stps of J and wrapping around to th bginning of th s squnc whn raching its nd. For th xampl fcw = and th corrsponding fcw = th normalizd fcw = (dcimal valu 45 and J = 37. Applying ( th rarrangmnt of s that yilds s is s( n = s ( ( n 45 8 and th rarrangmnt of s that yilds s is s ( n = s ( n 37. ( 8 Sinc w ar intrstd in th spctra of th DDS outputs, w nd to invstigat th rlationship btwn th Discrt Fourir Transforms (DFT of s (n and s (n. It was shown in Sction.3 of [] that th shortst propr lngth for th DFT is points (i.., on full priod of th DDS output squnc. t S and S dnot th -point DFT vctors of s and s, rspctivly. Thn, S = s and S = whr is th matrix with s, j n ( n, = for, n =,,. To stablish th rlationship btwn th spctra S and S w first obsrv that th matrix H obtaind from by rplacing th n-th column of with column ( n fcw, mod for n =,,, can also b obtaind from by rplacing th -th row of with row ( fcw mod, for =,,. Proof: Th valu of th lmnt in position (, n of H obtaind from column ( rarrangmnt of is j fcw n mod. Sinc th intgr part of ( n fcw contributs nothing to this mod ( ( fcw mod mod xprssion, it can b rwrittn as j n = j ( fcw n mod, which is th valu of th lmnt in position (, n obtaind from row rarrangmnt of. Th last stp rlis on th idntity ( amod cbmod c mod c = ( ab mod c. Hnc, prforming th invrs rarrangmnt on ithr th rows or columns of H, i.. rplacing th n-th column of H with column ( n J mod for n =,,, or rplacing th -th row of H with row ( J mod for =,,, will rproduc th DFT matrix. Now lt us considr th matrix quation v = u, whr v and u ar column vctors whil is a squar matrix. Rarranging th lmnts of vctor u and prforming th sam rarrangmnt on th columns of will lav th 5
rsulting vctor v unchangd. Finally, laving u unchangd and rarranging th rows of will similarly rarrang th lmnts of v. Considr th spctrum S = s and prform th forward rarrangmnt (by fcw on th lmnts of s and th columns of to produc th following xprssion: S = Hs. Nxt, prform th invrs rarrangmnt (by J on th rows of H to chang H bac to and rproduc S = s. Sinc th row rarrangmnt on H rarrangs th lmnts of S w conclud that S is obtaind by prforming an invrs rarrangmnt on th lmnts of S. Consquntly, S is obtaind by prforming a forward rarrangmnt on th lmnts of S. This rsult is also rportd in []. Summarizing: ( n ( n J mod ( J ( s ( n = s ( fcw ; s ( n = s ( mod S ( = S ( ; S ( = S ( fcw mod mod Th st of spur magnituds cratd by fcw is idntical to th st cratd by fcw; th spurs ar simply rarrangd in frquncy. Thrfor, th st of spurs cratd by any frquncy control word is a rarrangmnt of on of th sts of spctra corrsponding to th distinct frquncy control words having a singl nonzro bit. Ths sts of spctra can b viwd as basis sts. III. PHASE-ORD TRUNCATION As shown in Fig., th phas squnc at th output of th phas accumulator is truncatd to bits bfor addrssing th SCF. Consquntly, all input frquncy control words can b groupd into two catgoris. (Rcall that w dnot th S bit position as position. Th first group includs all fcw with thir rightmost nonzro bit position at or bfor position, i... In this cas thr will b no phas truncation and all spurs at th DDS output will b du to only th non-idal SCF. DDS bhavior for such fcw can b found in []. Th scond group includs all fcw with >, and for such fcw thr will b phas truncation. This papr prsnts th analysis of DDS spurs for fcw with >. t us considr, for xampl, a DDS with a 4-bit fcw, and a 5-bit SCF input. In Fig., this corrsponds to = 4 and = 5. To analyz th DDS output spctrum for any fcw with =, for xampl, as discussd in Sction II, w can considr th singl-nonzro-bit frquncy control word fcw =, whr th singl nonzro bit is at position = and th 5 S bit positions rtaind at th input of th SCF ar undrlind. Th rightmost nonzro bit is thrfor = = 5 bits aftr th truncation position = 5. Th four S bits could b omittd sinc thy do not impact th outcom, and aftr such normalization w notic that th squnc q at th output of th phas accumulator will incrmnt by on on vry cycl. Th corrsponding squnc q ' at th output ( 5 of th truncat bloc will stp onc for vry = = 3 cycls. In othr words, it will b for 3 cycls, thn it will b for 3 cycls, thn for 3 cycls, and so forth. Sinc th SCF simply maps th input angl to its corrsponding outputs, th sam rdundant bhavior can b obsrvd at th DDS outputs s(n and c(n. Notic that th SCF will go through all possibl inputs and thir corrsponding outputs, ach rpatd tims, bfor rpating th cycl again. Fig. illustrats ths proprtis. To continu th discussion of phas-word truncation, lt us considr again th cas with = 4, = 5, = 5 and assum that th SCF is idal (with infinit prcision outputs. Onc w gain a firm undrstanding of th spurs gnratd from phas-word truncation, th analysis will b xtndd to includ arbitrary SCF non-idalitis. As illustratd in Sction II, to obtain th basis st of spur magnituds for all fcw with =, w nd to prform a -point DFT on th DDS output corrsponding to th singl-nonzro-bit fcw =. Using s( n and S ( to dnot th DDS output and its corrsponding spctrum obtaind via -point DFT, rspctivly, w obtain j n S ( = sn ( whr n, <. (3 Expanding (3 as 3 (in gnral, summations w gt: 5 5 (3 (3 + j n j n ( 3n ( 3n + S( = s + s + fcw Phas Accumulator 5 5 (3 + 3 (3 + 3 j n j n s( 3n + 3 s( 3n + 3 + + 5 whr <, n< which, by factoring xprssions not dpnding on n out of th summations, and writing 3 = 5, bcoms: R g q truncat Sin Cosin apping Function (SCF s(n c(n Figur. DDS squncs du to phas truncation. q' D D 5
5 5 5 5 j n j j n ( 3n ( 3n+ S ( = s + s 5 3 5 + + s (4 j j n ( 3n + 3 for <. Using s(3n = s(3n + = = s(3n + 3, as discussd prviously (Fig., w can again rwrit (4 as: 3 ( 5 5 j j j n ( 3n S ( = + + + s j j j 3 = ( + + + + S'( for < whr S'( is th 5 -point DFT of th non-rdundant subsqunc s '( n = s( 3n. Thus, by summing th finit gomtric sris, w obtain: whr j 5 j S ( = VS ( '( (5 V( =. Notic that, if th SCF is implmntd as a looup tabl, s '( n corrsponds to th contnts of th tabl at addrss n. For our particular xampl, th SCF would hav 5 ntris sinc w chos = 5. shall rfr to V( as th windowing function. Clarly, th ntir drivation for S( and V( in (5 can b carrid out with th variabls and instad of 5 and 5 as in our xampl, in which cas on obtains th following gnral xprssions for S( and V(: j n + S' ( = s'( n =,,, j j ( + V, ( = + =,,, S ( = V ( S' ( + =,,,,, whr S' ( is priodic in with priod and on priod of V, ( windows ovr priods of S' (. IV. SPURS DUE TO PHASE-ORD TRUNCATION At this point w hav all th ncssary tools to addrss th assssmnt of spurs at th output of th DDS. To bgin th discussion on th maning of th windowing function V, ( lt us ma a fw commnts rgarding S' ( (which is windowd by V. hn mploying th assumption of an idal SCF (with infinit prcision outputs th (6 xprssions for S' ( and S' ( ar quit simpl. Gnrally, DDS can gnrat cosin only, sin only, or cosin and sin outputs and w rfr to thm as cosin DDS, sin DDS, or quadratur DDS, rspctivly. ithout loss in gnrality, assuming an idal sin DDS, s '( n is th sin output of th SCF with frquncy radians pr sampl. n Thrfor, for s'( n = sin w hav: n ( sin S' ( = = j n j n j n j n j j ( n j ( + n ( = j and, using th wll-nown rlationship j nl if l is an intgr multipl of = if l is any othr intgr w hav S' ( ( δ( δ( ( j =. Hnc, for ithr an idal sin or idal cosin DDS: ' ( δ( δ( ( S = + (7 whr δ( = and δ( = for. now rcall (s (3 that S, ( is priodic in with priod + and it is th rsult of multiplying S ' ( with V, ( (s (6. It is asy to vrify that V, ( is + priodic in with priod. From (6: V j + ( + +, ( + = j ( + ( + + j j V, ( j j +. ( + ( + = = Thrfor, on priod of V includs (i.., windows ovr priods of S' (. Notic that in th cas whn =, which mans thr is no phas-word truncation, V, ( = and, of cours, with no phas-word truncation thr is no phas-truncation distortion, hnc S, ( = S' (. hn > howvr, V, ( will assum som nontrivial shap and, by windowing S' (, it will introduc spurious frquncis at th locations of th dltas in S' ( for all. Th dltas at =± will also b attnuatd by th magnitud of V, (. To dmonstrat ths points, considr th cas for =. That is, assum w hav a frquncy 53
control word with its rightmost nonzro bit at position +. Rcall that S' ( and S (, ar priodic in with priods and + rspctivly. Thrfor, S (, can b constructd by considring four priods of S' ( and windowing thm with V (,, as dpictd in Fig. 3. Sinc w now th locations of th dltas in S' ( not only do w now th locations of th spurs cratd from phas truncation, w also now th xact spur magnituds sinc w can valuat th windowing function V for th valus of corrsponding to th locations of th dltas in S' (. or prcisly, th magnitud of th spurious frquncy at, corrsponding to th locations of th dltas, rlativ to that of th dsird sinusoid, is S, ( V, ( V, ( = = That is, using dc S, ( V, (. V, ( notation, th spur magnitud at will b: log V ( log V ( dc. (8,, Clarly, for, th valu of (8 is ngativ. Th spur magnitud is log V, ( log V, ( d down from th main componnts at =±. Notic that th phastruncation spur magnituds in (8 dpnd only on th windowing function V, (. For our particular = cas, w simply valuat (8 at th six points = {( ±, ( ±, (3 ± }. To charactriz th spurs for any frquncy control word, w simply rpat this xrcis for all possibl valus of, (. For an arbitrary th sin or cosin DDS phas truncation spurs will b locatd at {( d ± : for d ( }. Notic that, for quadratur DDS if w considr both outputs (sin and cosin as a complx xponntial squnc j n cs '( n = n n = cos( + j sin( and if w charactriz th DDS output spctrum via th complx-input DFT, thn Dfinition of dc is: d rlativ to th carrir. S ' ( - + - - + V (, - + - + Figur 3. indowing function V (, ovr S' (. CS ' ( = δ ( (rathr than (7 and th absnc of th dlta at = in CS ' ( rducs th st of spurs at th output of th DDS to th valus of {( d + : for d ( }. Th sts of spur magnituds for sin, cosin, and quadratur DDS ar idntical, as xpctd. In th cas of sin or cosin DDS thr ar two corrsponding spurs with idntical magnituds ( S, ( is symmtric around th origin whil in th cas of quadratur DDS thr is a singl spur for a givn magnitud. Th thory w hav dvlopd thus far allows on to idntify th locations of all phas-truncation spurs and comput thir magnituds rlativ to th main componnt with th assistanc of th windowing function V,. Th idntification of th worst (i.., th on having gratst magnitud spur (or spurs is usually th most critical issu whn charactrizing DDS spurs. Although th tchniqus dvlopd in th prcding sctions could b usd to comput all of th spurs and w could thn idntify th worst on or ons by ordring thm, this approach would rquir mor tim and ffort than ncssary. Thr is a much mor powrful tchniqu which dirctly idntifis th locations of worst-cas spurs and, through (8, nabls on to dirctly calculat thir magnituds. Thrfor, by using this mthod, th N worst phas-truncation spur magnituds ar calculatd by valuating (8 only N tims. Hnc, th magnitud of th worst spur, which also yilds th SFDR, can b obtaind by valuating th windowing function only onc. Considring a sin or cosin DDS, w bgin by rfrring to xprssion (8 and noticing that th worst spur is th on from th st {( d ± : for d ( } that maximizs th magnitud of windowing function V, (. Using (6: V, j j j ( j + ( + ( = =. Th rsult is a ratio of absolut valus, ach having th j form θ j. If w viw θ as th magnitud of th diffrnc btwn two unit-lngth vctors, thn it can b jθ θ shown ([] pag 54 that = sin, and w obtain th following convnint form for V, ( : V, - j - sin( sin( ( = = =. (9 - j ( + sin( ( + sin( ( - + 54
sin( Th plots for th xampl with = 5 and = 3 ( = 8 and similarly, sin( ( d = sin( d = sin. 8 Figur 4. Phas-truncation spurs and th windowing function ma th following obsrvation about th dnominator sin( ( + of (9. Sinc sinθ is strictly monoton incrasing for θ and sinc ( = for + ( + = w conclud that th dnominator of (9 is strictly monoton incrasing whn lis within th closd ( + intrval [, ]. Similarly, sinc sinθ is strictly monoton dcrasing for θ and sinc ( ( + + = for = w conclud that th dnominator of (9 is strictly monoton dcrasing whn ( + lis within th closd intrval [,]. Thrfor, th dnominator sin( ( + of (9 incrass as th indx movs away from th origin. Nxt, w prform a similar analysis on th numrator of (9. Sinc sinθ is zro for θ = d, whr d is an arbitrary intgr, sin( is zro for = d. Ths zros in th numrator crat th nulls of our windowing function. Also, sin( is symmtric (vn symmtry around ths nulls. For phas-truncation spurs, what w actually car about ar th points immdiatly to th right and immdiatly to th lft of ths nulls sinc ths ar th locations of th phas-truncation spurs, as indicatd by th st {( d ± : for d ( }. Th following is a simpl proof, showing that th valu of th numrator in (9 is th sam for all whr phastruncation spurs occur. For {( d ± : for d ( }, sin( ( d + = sin( d + = sin Thrfor, th numrator of (9 is constant at th location of all phas truncation spurs and th dnominator is strictly monoton incrasing as movs away from th origin. Hnc, th worst (largst magnitud phas-truncation spur is th on closst to th origin. For a sin or cosin DDS thr ar two largst-magnitud phas-truncation spurs locatd at =± ( and for a quadratur DDS th largst-magnitud phas-truncation spur is locatd at =(. Furthrmor, sinc th dnominator is strictly monoton incrasing as th indx movs away from th origin, it follows that all phas-truncation spurs, in ach of ( th closd intrvals [ + (,] and [, + ], hav distinct magnituds, and that thy arrang thmslvs from largst to smallst as thir location movs away from th origin. This suggsts that th N worst spurs could b calculatd by valuating th windowing function at th N phas truncation spur locations closst to th origin. Fig. 4 illustrats this point for a sin or cosin DDS with = 5 and = 3. Sinc V ( = V( (s (9, th worst phastruncation spur for sin, cosin, or quadratur DDS, rlativ to th main componnt, is thrfor obtaind by using (9 and valuating (8 for =. Th rsult is: log sin( log sin( ( dc. ( ( + ( + V. SPURS DUE TO ARITRARY SCF NON-IDEAITIES IN THE PRESENCE OF PHASE-ORD TRUNCATION can asily xtnd th analysis of Sction IV to account for th spurs rsulting from arbitrary SCF non-idalitis mployd to facilitat fficint SCF implmntation. All argumnts mad in Sction IV still hold whn w hav such a SCF implmntation, xcpt that S' ( will hav spurs btwn th prviously discussd dlta functions. Th magnituds of ths spurs dpnd on th output prcision and implmntation dtails of th SCF. Thrfor, for frquncy control words with > (i.., whr thr is phas-word truncation th spurs in S' ( ar also windowd by th windowing function V, (. Th xprssions s '( n, S' (, and S, ( for an arbitrary non-idal SCF ar: s'( n = sin( + q( n n ' ( = δ( + ( δ( + δ( ( S a a ( - i ( ( ( ( i= + a δ + a δ i + δ i S ( = V ( S' (,, ( 55
- + - + whr q(n is th spcific rror squnc associatd with th non-idal SCF, whr a is th distortion of th main componnts at =± (and ±,, whr a (DC spur and a through a ar th spur magnituds in S' (. Sinc th squnc s '( n in ( capturs all SCF nonidalitis du to th spcific implmntation, w rfr to it as th SCF signatur squnc. For sin or cosin DDS s '( n is a ral-valud squnc, S' ( = S' ( (it is an vn function. Thrfor, th two main componnts S ' ( ± in ( hav idntical distortion a and, in gnral, S' ( ± = S' ( d ± for any intgr d. Consquntly, th spur magnitud at = d ± rlativ to that of th dsird sinusoid is: V, (. V, ( S ' ( & V, ( S, ( V, ( S' ( S, ( V, ( S' ( = = Hnc th magnituds of th spurs gnratd from phas-word truncation, rlativ to th main carrir magnitud, rmain dictatd by (8 (i.., thy ar only a function of V, (. That is, th phas truncation spur magnituds (and locations ar idntical for th idal and non-idal SCF. Effctivly, th st of spurs causd by phas truncation and th st gnratd from a non-idal SCF, rlativ to that of th main carrir componnts, ar disjoint. Th only spur not attnuatd by V, ( is th DC spur, hnc (8 indicats th incras of th nonzro DC spur rlativ to th main componnts. In Sction 5.4 of [], it is shown that, whn th SCF is implmntd to xploit th sin/cosin wav symmtry, th DC spur and all th spurs in vn DFT frquncy bin locations will b zro. Fig. 5 illustrats th gnral situation. Th thory dvlopd thus far can b usd to compltly charactriz all spurs du to phas truncation and non-idal SCF for any fcw with > through th xprssions (6. For fcw with larg valus of (such as = 3 or = 48 th spctrum S, ( will contain componnts. Th charactrization and storag of all componnts may - - + S, ( - - + Figur 5. V (, ovr S' ( in a DDS with a non-idal SCF. rquir a prohibitivly larg amount of mmory and computation. Th computation ncssary to charactriz a singl spur is small (6, but thr ar too many spurs. If all th spurs nd to b charactrizd thn on has no choic but to valuat (6 for all =,,,. If, instad, th worst N spurs (for xampl th worst spurs, or all spurs abov a thrshold (such as dc nd to b charactrizd, thn th strictly monoton proprty of th windowing function, from Sction IV, can b xploitd to crat a vry fast algorithm. Rcall (, which constructs th ntir spctrum S, ( by concatnating copis of S' ( and windowing thm with V, (. Th main componnts of S' ( giv ris to all phas-truncation spurs in S, (, and vry spur in S' ( givs ris to spurs in S, ( (which ar uniformly distributd in frquncy at intrvals of lngth. For th intrval [, ] (quivalnt to th intrval [, ] for th DFT, from Sction IV w now th dnominator of V, ( in (9 is strictly monoton incrasing as movs away from th origin. From Sction IV w also now that th numrator of V, ( is priodic in with priod. Thrfor, sinc th st of spurs in S, ( gnratd from on spur in S' ( ar qually spacd at lngth- intrvals, w conclud that th numrator of V, ( is constant at th positions of ths spurs (s Fig. 4 for rfrnc. Hnc, similar to th phastruncation spurs, th st of spurs cratd from on spur in S' ( arrang thmslvs with dcrasing magnituds as movs away from th origin. This proprty can b usd to crat a fast algorithm for th charactrization of th N worst spurs or all spurs abov a spcifid thrshold valu. For a sin or cosin DDS, th following is th outlin for a two-phas algorithm to accpt an arbitrary SCF signatur squnc s '( n,,, and N or spur magnitud thrshold ε, and to gnrat th magnituds and locations of th largst-magnitud spurs. Phas I. Comput S' ( for <.. Comput S, ( = V, ( S' ( for < and sort in a vctor S_ord in th ordr of dcrasing magnituds and p trac of th indics in a vctor _ord. 3. Th first lmnt of _ord is, corrsponding to th indx of th main componnt, and th first lmnt of S_ord will contain th magnitud of th main componnt. Sav th magnitud of th main componnt and rmov it from both _ord and S_ord. 56
4. Construct a vctor potntial_indx_st (which ps trac of th potntial locations for th nxt worst spur containing only th first lmnt of _ord. Rmov that first lmnt from _ord. If >, thn appnd potntial_indx_st with +. 5. Crat two vctors spurs and locations to stor th magnituds and locations of th worst spurs to b computd in Phas II. Phas II (Rpat ths stps N tims for th N largst-magnitud spurs, or rpat until th obtaind spur is blow th thrshold ε to obtain all spurs abov th thrshold ε.. Comput th spur magnituds for th indics in potntial_indx_st using (6 and stor th magnitud and location of th worst on in th vctors spurs and locations rspctivly.. If (th indx of th worst spur from stp is lss than, thn, if >, incrmnt that indx in potntial_indx_st by. If =, thn rmov that indx from potntial_indx_st. If th vctor _ord is not mpty, rmov its first lmnt and add it to potntial_indx_st as wll. Els If (th indx of th worst spur from stp is gratr than, thn rmov it from potntial_indx_st. Othrwis, incrmnt th indx of th worst spur from stp in potntial_indx_st by. Each spur magnitud in th vctor spurs will appar twic on th DDS output spctrum sinc w considrd a sin or cosin DDS. Th corrsponding lmnt in vctor locations will hav th positiv-indx location and th scond is simply th ngativ of that indx. Ths spur locations, of cours, corrspond to a singl-nonzro-bit fcw. To obtain th spur locations for any fcw with th givn, on may apply th rarrangmnt of ( to th indics in vctor locations. A similar outlin for a quadratur DDS is found in Sction 3.5. of []. VI. DDS SNR DUE TO ARITRARY SCF NON- IDEAITIES IN THE PRESENCE OF PHASE-ORD TRUNCATION It is shown in Sction 4. of [] that two DDS with idntical SFDR can hav vry diffrnt SNR sinc SFDR masurs only th largst-magnitud spur whil SNR collctivly masurs all th spurs. Thrfor, th considration of DDS output SNR in conjunction with SFDR is important. Th total powr for any priodic tim-domain squnc xn ( can b computd dirctly as N N N P = x( n, or from its DFT X( as P = X(, whr N is th priod N = of xn (. For fcw with common rightmost-nonzro-bit position, sinc th DDS outputs and thir spctra ar just th rarrangmnts of ach othr dictatd by (, w conclud that thy hav idntical total powr. Furthrmor, sinc thir signal powr is dictatd solly by th squard magnitud of thir corrsponding main DFT componnts, and th main DFT componnts ar also rarrangmnts of on anothr, w conclud that thir signal powrs ar also idntical. Hnc, thir total nois powrs PNois = PTotal PSignal P must also b idntical. Thrfor, th SNR= Signal for all fcw P Nois with a common rightmost-nonzro-bit position can b computd by considring th singl-nonzro-bit fcw. Considring a sin or cosin DDS, for th singl-nonzro-bit fcw, th signal powr P SIGNA = ( + (+ S (+ S ( + ( rsids in th dltas corrsponding to th main frquncy componnts and th magnituds of ths dltas ar idntical as shown in ( + Sction V (i.., S( = S(. Thrfor, by using (6 w can xprss th signal powr as P = V( S'(. ( SIGNA (+ Using th fact that ( s n = s( n+ = = s( n+ = s'( n, as discussd in Sction III, th total ( + powr PTOTA = ( + s( n can b rwrittn in trms of th signatur squnc s '( n. Th rsult is PTOTA = '( s n and, using Parsval s thorm, it can also b writtn in trms of S'( as = S'(. Subtracting th signal powr ( from th total powr, w obtain th total nois powr (that du to both SCF nonidalitis and phas truncation: or NOISE '( ( + ( '( P = s n V S '( ( + ( '( = PNOISE = S V S. P PNOISE SIGNA Hnc, th SNR = in d can b xprssd as: or SNR = log V( S'( ( + s'( n V( S'( V( S'( ( S'( V( S'( = SNR = log. (3 a (3 b 57
Sinc th windowing function V and th signatur squnc s '( n for an arbitrary SCF ntr into th xprssions (3, th impact of both phas truncation and arbitrary SCF non-idalitis on th DDS output SNR ar considrd in (3. Diffrnt DDS implmntations with idntical phas truncation and idntical output prcision can hav substantially diffrnt SNR sinc diffrnt implmntations of th SCF can giv ris to diffrnt signatur squncs s '( n. Notic that (3 computs th SNR xactly (i.., without us of any approximations, and it corrsponds spcifically to th SCF implmntation giving ris to th signatur squnc '( s n usd in (3. For quadratur DDS, th only diffrnc in th drivation of thir SNR xprssion is that it has a singl main componnt instad of two main componnts as in th cas of a sin or cosin DDS. Th SNR xprssions corrsponding to quadratur DDS ar similarly drivd in Sction 4. of [] to b or SNR = log V( CS'( ( + cs '( n V ( CS '( V( CS'( ( CS '( V ( S '( = SNR = log. (4 a (4 b Thrfor, valuating (3 or (4 for th givn and th givn signatur squnc and all possibl valus of, whr (, yilds th SNR for all fcw rsulting in phas-word truncation. Computation of SNR for fcw rsulting in no phas-word truncation (i.., fcw with is asir du to thir considrably shortr priod of th DDS output. A thorough tratmnt of SNR computation for such fcw and rlvant initial phas conditions can b found in Sction 4.6 of []. A ajor Practical Issu: A point of major importanc is that th SNR xprssions (3 and (4 ar computationally vry fficint. In practic, th squnc at th output of a DDS has a (larg! priod of ( + sampls, thrfor its dirct SNR computation using summations for th powr of ( + th DDS output squnc is proportional to. Sinc th xprssions (3 and (4 ar computd from th windowing function V and th rlatd signatur squnc with a priod of sampls, th computational complxity of (3 and (4 is proportional to. Thrfor, th SNR xprssions (3 and (4 ar mor fficint by a factor of. This fficincy factor is crucial sinc, for practical DDS implmntations, is typically as larg as 8 or mor. This larg fficincy factor mas th xact valuation of th SNR,.g. by xprssions (3 and (4, fasibl. It is also shown in Sction 4.3 of [] that (3 and (4 ar dcrasing functions of. Thrfor, valuating (3 and (4 for = and = will yild th maximum and minimum SNR bounds, rspctivly, for fcw with >. Othr usful SNR bounds ar drivd in Sctions 4.3-4.5 of []. VII. CONCUSIONS A simpl and fast (patnt pnding algorithm is prsntd for th xact computation of DDS output spur magnituds and locations, SFDR, and SNR, in th prsnc of both phas-word truncation and arbitrary implmntation of th DDS sinc/cosin mapping function (SCF. A y windowing function provids th mans to comput an arbitrary numbr of largst-magnitud spurs and th total powr of th DDS output spctrum without gnrating th ntir spctrum. Sctions 3.7 and 4.7 of [] us th prsntd algorithm and SNR xprssions (3 and (4 for phas-truncation causing fcw, along with vn simplr algorithms for non-phas-truncation causing fcw to prform a complt DDS analysis for an arbitrary DDS (i.., arbitrary SCF implmntation having th gnral form of Fig.. This complt DDS analysis on an ordinary prsonal computr tas approximatly on minut. REFERENCES [] A. Torosyan and A. N. illson, Jr., Analysis of th output spctrum for dirct digital frquncy synthsizrs in th prsnc of phas truncation and finit arithmtic prcision, in Proc. -nd Imag and Signal Procssing and Analysis Confrnc,, pp. 458-463. [] A. Torosyan, Dirct Digital Frquncy Synthsizrs: Complt Analysis and Dsign Guidlins, Ph.D. Dissrtation, Univrsity of California, os Angls, 3. [3] J. Tirny, C.. Radr, and. Gold, A digital frquncy synthsizr, IEEE Transactions on Audio and Elctroacoustics, vol. AU-9, pp. 48-57, arch 97. [4] H. T. Nicholas and H. 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