Spctral Proprtis and Intrpolation Error Analysis for Variabl Sampl Rat Convrsion Systms Andr Tkacnko Signal Procssing Rsarch Group Jt Propulsion Laboratory, California Institut of Tchnology 48 Oak Grov Driv, Pasadna, CA 99 E-mail: Andr.Tkacnko@jpl.nasa.gov Abstract Th problm of variabl sampl rat convrsion SRC has rcivd much attntion on account of its applications in softwar dfind radios SDRs that must support a wid varity of data rats. In this papr, w invstigat th spctral proprtis of variabl SRC and focus on th intrpolation rror obtaind using any two intrpolation krnls. W show that SRC is a gnralization of dcimation for both rational and irrational convrsion ratios. In addition, a frquncy domain xprssion for th man-squard intrpolation rror is drivd and simplifid. Simulations prsntd show th dgradation ffcts of using practical picwis polynomial basd intrpolants as opposd to th undrlying bandlimitd sinc function for svral input signals. I. INTRODUCTION A primary challng facing th dvlopmnt of softwar dfind radios SDRs is th ability to accommodat a varity of data rats subjct to fixd systm architcturs [], []. Exampls of ths architcturs ar th analog-to-digital convrtr ADC usd at th front nd and th tracking loops carrir synchronization, symbol timing, tc. usd subsquntly. To conform to both of ths typs of fixd systms pragmatically, th sampl rat must b convrtd digitally. This can b achivd using a variabl sampl rat convrsion SRC systm []. An intrmdiat frquncy IF SDR rcivr mploying variabl SRC is shown in Fig.. In this systm, an analog datamodulatd signal at data rat R is sampld at a fixd rat F s to produc a ral digital IF signal. Upon quadratur convrsion, a complx basband signal sampld at rat F s = Fs is formd. Th signal of intrst is thn cntrd at zro frquncy using a tunabl numrically controlld oscillator NCO. At this point, th sampl rat must b altrd to fit th fixd rdundancy factor # of sampls/symbol stipulatd for th tracking loops. For xampl, if th loops rquir a rdundancy of K as in Fig., th sampl rat coming into thm should b F s = KR. Th purpos of th variabl SRC systm is to adjust this rat from F s to F s. Prior to this, th SRC systm should also rmov out-of-band artifacts to prvnt aliasing. In this papr, w first focus on th spctral proprtis of variabl SRC for th intrpolation krnl signal modl typically assumd []. Though this modl has rcivd much attntion in th litratur i.., s [3], [4], [5], most of th considration has consistd of tim domain analysis. With th drivd Th rsarch dscribd in this publication was carrid out at th Jt Propulsion Laboratory, California Institut of Tchnology, undr a contract with th National Aronautics and Spac Administration. IF analog input data rat = R ADC Fs Fig.. fixd Fs Quadratur Convrsion F s = Fs NCO variabl/rconfigurabl f LO Variabl F s SRC F s = KR fixd Tracking Loops K Block diagram of an IF SDR rcivr mploying variabl SRC. spctral charactrization of th variabl SRC output, w show th rlation btwn SRC and dcimation [6]. Spcifically, w show that variabl SRC is a gnralization of dcimation for both rational and irrational rat convrsion ratios. W thn turn our attntion to intrpolation rror analysis for th variabl SRC signal modl. Using th spctral proprtis of this modl, w driv a simplifid frquncy domain xprssion for th man-squard intrpolation rror for any two intrpolation krnls. Simulation rsults prsntd lucidat th dgradation associatd with using practical picwis polynomial basd intrpolation krnls instad of th undrlying bandlimitd sinc function. In lin with intuition, w show that as th output rat convrsion factor incrass, so that th input sampls appar lss rdundant, th rror incrass. It is also shown that for a fixd rat convrsion factor, highr ordr intrpolation krnls always outprform lowr ordr ons. A. Outlin In Sc. II, w rviw th intrpolation krnl signal modl assumd throughout th papr. Th variabl SRC problm is xplord in Sc. III. Spctral proprtis of th variabl SRC output ar drivd in Sc. IV, whr th th connction btwn variabl SRC and dcimation is stablishd. In Sc. V, w driv a simplifid frquncy domain xprssion for th th man-squard intrpolation rror. Simulation rsults of th intrpolation rror ar prsntd in Sc. VI for svral practical scnarios. Concluding rmarks ar mad in Sc. VII. B. Notations All notations usd ar as in [7]. In particular, continuoustim analog and discrt-tim digital normalizd frquncis ar dnotd as F and f, rspctivly. Parnthss and squar brackts ar rspctivly usd for continuous-tim and discrt-tim function argumnts. For xampl, xt would dnot a continuous-tim function for t R, whras y[n] would dnot a discrt-tim function for n Z.
II. INTERPOLATION KERNEL SIGNAL MODEL All quivalnt complx basband analog signals input to th systm of Fig. ar assumd to b of th following form []. xt = c d [k] h T k Th signal modl of is rfrrd to as th intrpolation krnl signal modl [] whr ht dnots th dilatd intrpolation krnl [], T rprsnts th krnl spacing intrval, and c d [k] is th basis cofficints squnc []. Svral intrsting classs of signals ar gnratd using th modl givn in [8]. For xampl, from th Nyquist sampling thorm [6], if xt is bandlimitd to F BL maning XjπF =for F F BL, thn xt is of th form, xt = xnt s sinc n T s whr th sinc function is dfind as sinc x sinπx πx and T s is givn by T s = F BL. Comparing to, it can b sn that all bandlimitd signals can b xprssd in trms of th intrpolation krnl modl whr w hav, ht = sinc t, c d [k] =xkt s, T = T s 3 For this rason, any signal bandlimitd to F BL can b rcovrd from uniform sampling at a rat of F s = T s =F BL calld th Nyquist rat [7], [6], as this will automatically yild th basis cofficints squnc c d [k] as sn from 3. Aftr uniform sampling of th signal xt from, which is attaind hr aftr quadratur convrsion of th ADC output, w will hav accss to th squnc x d [n] xnt s, whr T s dnots th sampling intrval. For th rmaindr of this papr, w will assum that T s = T. Thn, from, w hav, x d [n] =xnt s = c d [k]hn k =c d [n] h d [n] 4 whr th discrt-tim squnc h d [n] is dfind as h d [n] hn. In othr words, from 4, x d [n] is simply th discrttim convolution of th basis cofficints squnc c d [n] with th sampld intrpolation krnl squnc h d [n]. To rconstruct th original analog signal xt from at som prscribd valu of tim t = t, which is ncssary for variabl SRC, th only quantity ndd is th basis cofficints squnc c d [n], assuming that th intrpolation krnl ht is a known function. From th sampld squnc x d [n] from 4, it can b sn that c d [n] can b obtaind as follows. c d [n] =g d [n] x d [n] 5 whr g d [n] is th convolutional invrs of h d [n] [7]. In th z-domain, 5 bcoms, C d z =G d zx d z, whr G d z = 6 H d z Hr, C d z, X d z, G d z, and H d z dnot th z- transforms of c d [n], x d [n], g d [n], and h d [n], rspctivly. Thus, c d [n] can b obtaind using th systm shown in Fig. [8]. x d [n] sampld analog signal G d z = Hdz convolutional invrs of sampld intrpolation krnl c d [n] basis cofficints Fig.. Discrt-tim systm usd to obtain th intrpolation basis cofficints squnc c d [n] from th sampld continuous-tim signal xnt s=x d [n]. For th spcial cas in which x d [n] =c d [n], w say that ht is intrpolating [8]. Othrwis, it is non-intrpolating. It can b asily shown that ht is intrpolating iff w hav, hn =δ[n] n Z h d [n] =δ[n] 7 whr δ[n] is th Kronckr dlta function [7]. Combining 5 with, it follows that xt can b xpandd in trms of an ffctiv intrpolating krnl h ff t as follows. xt = x d [k]h ff T k 8 Hr, h ff t is dfind as, h ff t g d [l]ht l 9 From 9, w s that h ff t satisfis 7 and is intrpolating. If HjπF and H ff jπf dnot th Fourir transforms of ht and h ff t, rspctivly, thn from 9 and 6, w hav, H ff jπf =G d jπf HjπF = HjπF H d jπf Plots of th impuls and magnitud rsponss of svral ffctiv intrpolating krnls ar shown in Fig. 3 a and b, rspctivly. All krnls hr wr intrpolating, xcpt for th, for which H d z = 6 z + 3 + 6 z [9]. As can b sn, highr ordr mthods yildd bhavior closr to th bandlimitd sinc intrpolant of 3 than lowr ordr ons. Spcifically, th yildd th closst fit to th sinc intrpolant of all of th krnls considrd. III. THE VARIABLE SRC PROBLEM From, th quadratur analog input xt to th systm of Fig. is assumd to b bandlimitd and of th form, xt = x d [m]sinc m T in whr dnots th quadratur input sampling intrval and x d [m] xm dnots th discrt-tim sampld signal obtaind aftr basband down-convrsion of th ADC output. To altr th sampl intrval to say T out, w nd to gnrat th squnc y d [n] x n + ɛt out, whr ɛ is a fractional offst satisfying ɛ<. From, y d [n] is calculatd as, y d [n] = x d [m]sincn + ɛ m Evry intrpolating krnl h ff t considrd hr is ral and vn, maning that H ff jπf is also ral and vn [7]. As such, all impuls and magnitud rsponss ar shown for t and F, rspctivly.
hfft HffjπF db.8.6.4.. sinc.4.5.5.5 3 3.5 4 t 3 4 5 6 7 8 9 a sinc.5.5.5 3 3.5 4 4.5 5 F b Fig. 3. Effctiv intrpolating krnl rsponss: a tim domain impuls rsponss and b frquncy domain magnitud rsponss. Hr, is dfind as Tout.IfF in and F out T out dnot, rspctivly, th input and output sampl rats, thn = Fin F out. Hnc, is calld th input rat convrsion ratio. Similarly, R = Fout F in is th output rat convrsion ratio. To calculat vn a singl sampl of y d [n], th sum in must b computd ovr th ntir st of input sampls {x d [m]} in gnral. For variabl SRC in which is adjustabl, th sinc trms in must also b changd accordingly. As this cannot b don practically, on altrnativ could b to truncat most of th trms in. Howvr, as th sinc function dcays invrs ly i.., sincx dcays as x, a larg numbr of trms must b kpt to minimiz truncation rror ffcts []. Instad, what is typically don to achiv variabl SRC is to modl th continuous-tim input as coming from a diffrnt krnl than th sinc function [3], [4], [5]. Spcifically, th input analog signal vt is modld using as follows. vt = c d [m] h m 3 To altr th sampling intrval to T out, w comput y d [n] v n + ɛt out as bfor. From 3, w gt th following. y d [n] = c d [m] hn + ɛ m 4 Through carful choic of th krnl ht, w can achiv variabl SRC in 4 fficintly and with littl loss compard to th undrlying sinc modl from. For xampl, for picwis polynomial krnls, such as th,,, and intrpolants, y d [n] can b computd fficintly using th Farrow structur [], []. In trms of th ffctiv intrpolating krnl h ff t, th variabl SRC output y d [n] is givn as follows from 8. y d [n] = x d [m] h ff n + ɛ m 5 IV. VARIABLE SRC OUTPUT SPECTRAL PROPERTIES It is insightful to considr frquncy domain rprsntations of th variabl SRC output y d [n] from 4 and 5. Focusing on 4, if Y d jπf dnots th discrt-tim Fourir transform of y d [n], thn w hav th following [7]. jπf y d [n] jπfn = c d [m] hn + ɛ m jπfn 6 In trms of its invrs Fourir transform [7], ht is givn by, ht = HjπF jπft df 7 Using 7 in 6 yilds th following aftr som work. jπf = c d [m] HjπF jπfɛ m jπnf f df 8 Rcall th Dirac impuls train [7] Fourir sris xpansion is, jπn F T = T δ F + k T F, T 9 Hr, δt is th Dirac dlta function [7]. Using 9 in 8 with F = F f and T = yilds th following aftr simplification and xploiting th sifting proprty of δt [7]. jπf = f k c d [m] H jπ = jπ ɛ m H jπ f k c d [m] jπ m jπɛ }{{} C d jπ Thus, from, Y d jπf simplifis to th following form. jπf = f k H jπ C d jπ jπɛ
Xd jπf fbl a fbl f fbl Yd jπf b fbl f Fig. 4. Illustration of gnralizd alias-fr dcimation via variabl SRC: a input spctrum bfor SRC and b output spctrum aftr SRC. Using 6 and in yilds a spctral rprsntation of th variabl SRC output from 5 as follows. jπf = f k H ff jπ jπ jπɛ A. Rlation to Dcimation Suppos ɛ =and h ff t = sinc t. Furthrmor, suppos x d [n] is bandlimitd to f = f BL, maning X d jπf =for f f BL for som f BL with f BL < and f [,. If f BL <, whr w assum hr, thn from, jπf = X πf d j [, f, 3 Thus, Y d jπf is a scald and zoomd in vrsion of X d jπf, whr th zoom magnification factor is th input rat convrsion ratio. Not that 3 is a gnralization of alias-fr dcimation [6] for both rational and irrational convrsion ratios. An illustration of this is shown in Fig. 4. Anothr rlation to dcimation ariss whn w st ɛ =, tak h ff t = sinc t, and choos = M, whr M is any positiv intgr. Using th division thorm [6] to st k = Mq + r in whr q = k M and r = k mod M yilds, Y d jπf = M M r= f r jπ M 4 Not that 4 is th spctrum obtaind by dcimating x d [n] by M [6]. Hnc, variabl SRC bcoms intgr dcimation through propr choic of th krnl and convrsion paramtrs. V. INTERPOLATION ERROR ANALYSIS FOR VARIABLE SRC To gaug th diffrnc btwn using any two krnls, w focus on th man-squard intrpolation rror ζ dfind as, [ ] ζ E ɛ γ[n], whr γ[n] y d [n] ŷ d [n] 5 Hr, y d [n] and ŷ d [n] ar SRC outputs as in 5 using th sam input x d [m] but diffrnt krnls h ff t and ĥfft, rspctivly. Th xpctation in 5 is ovr th offst ɛ, which isassumdtobuniform [] ovr [, i.., ɛ U[,. From 5, th rror squnc γ[n] can b xprssd as, γ[n] = x d [m] dn + ɛ m 6 Hr, dt h ff t ĥfft is th krnl diffrnc signal. As 6 is of th sam form as 5, w can us to say, Γ jπf = f k D jπ jπ jπɛ 7 Hr, Γ jπf and DjπF dnot th Fourir transforms of γ[n] and dt, rspctivly. Rturning to 5, from Parsval s thorm [7], ζ satisfis, ζ = E ɛ [ Γ jπf df ] 8 Using 7 in 8, w gt, aftr som algbraic manipulation, ζ = f k D jπ jπ f l D jπ X f l d jπ { E ɛ [ jπk lɛ]} df 9 As ɛ U[,, it can b asily shown that w hav, E ɛ [ jπk lɛ] = δ[k l] 3 Substituting 3 into 9 yilds th following simplifications. ζ = f k jπ D jπ df = = k k m+ m Djπλ Xd jπλ dλ 3 Djπλ X d jπλ dλ 3 = Djπλ Xd jπλ dλ 33 Hr, 3 follows from th substitution λ =, 3 from th summation indx m = k, and 33 from th fact that th intgration intrvals in 3 ar nonovrlapping and span R. To simplify ζ furthr, not that from 33, w hav, ζ = = l+ l Djπλ X d jπλ dλ 34 Djπf + l Xd jπf+l df 35 Hr, 34 follows from partitioning th intgration intrval of 33, whil 35 follows from th substitution f = λ l. Continuing furthr, w hav th following simplifid formula for th man-squard intrpolation rror ζ. ζ = Djπf k Xd jπf df 36 Hr, 36 follows from th chang of summation indx k = l and th fact that X d jπf is priodic with priod.
5 5 ξ db 5 ξ db 5 5 5 3 3 R 3 35 3 R Fig. 5. Normalizd man-squard intrpolation rror ξ as a function of th output rat convrsion factor R for th input of 39. VI. SIMULATION RESULTS For simulation purposs, w will opt to comput th normalizd man-squard intrpolation rror ξ dfind blow. [ E ɛ y d[n] ŷ d [n] ] ξ [ 37 E ɛ y ] d[n] Hr, th krnl h ff t for y d [n] is th sinc function i.., h ff t = sinc t and ĥfft for ŷ d [n] is som othr krnl. From 36, 37 has th following simplifid form. A ξ = Djπf k jπf df 38 Hr, A jπf is a normalizd form of X d jπf satisfying A jπf = Xd jπf / Xd jπλ dλ. To tst th intrpolation rror in an xtrmal cas, in th spirit of Fig. 4, suppos that Xd jπf is givn by, Xd jπf {, f < [ =, f, f, 39 Variabl SRC in this cas corrsponds to maximal dcimation [6], as thr will b no rdundancy prsnt in th output signal. Aplotofξ from 38 for th input of 39 is shown in Fig. 5 as a function of th output convrsion factor R = for various krnls. For all mthods, as R incrasd, ξ incrasd as wll. This is bcaus for largr R, th input sampls appar lss rdundant, and so any givn mthod is mor likly to yild output sampls that ar lss consistnt with th undrlying sinc intrpolant. Anothr obsrvation is that for a fixd R, highr ordr mthods always outprformd lowr ordr ons. To tst th rror in a mor practical scnario, suppos th input to th systm of Fig. is a data stram with a squarroot raisd-cosin puls shap []. Thn, th spctral dnsity Xd jπf for f [, is a raisd-cosin puls with, X d jπf = K, Kcos πk α [ f α K ] f < α K α +α, K f < K, f +α K 4 Fig. 6. Normalizd man-squard intrpolation rror ξ as a function of th output rat convrsion factor R for th input of 4 with K =4and α =.35. Hr, K is th rdundancy factor for th tracking loops of Fig., whil α is th roll-off factor for th raisd-cosin puls []. As a practical cas, suppos K =4and α =.35 []. Thn, aplotofξ vrsus R for th input of 4 is shown in Fig. 6 for various krnls. As with Fig. 5, whn R incrasd, ξ incrasd as wll. Also, highr ordr mthods outprformd lowr ordr ons for a fixd R. Comparing Fig. 6 with Fig. 5, th rror was lowr for th input of 4 than for that of 39. This is bcaus SRC for 4 corrsponds to non-maximal dcimation du to th xtra rdundancy rquird by th tracking loops. VII. CONCLUDING REMARKS In this papr, w drivd frquncy domain xprssions for th variabl SRC output and th man-squard intrpolation rror. Simulations providd rvald th dgradation ffcts of using a practical intrpolation krnl as opposd to th undrlying sinc function. Extnsions of this analysis to mor gnral intrpolation modls ar th subjcts of futur rsarch. REFERENCES [] J. Hamkins and M. K. Simon, Eds., Autonomous Softwar-Dfind Radio Rcivrs for Dp Spac Applications. Hobokn, NJ: John Wily & Sons, Inc., 6. [] A. Tkacnko, Variabl sampl rat convrsion tchniqus for th Advancd Rcivr, Intrplantary Ntwork IPN Progrss Rport, vol. 4-68, Fb. 5, 7. [3] T. A. Ramstad, Digital mthods for convrsion btwn arbitrary sampling frquncis, IEEE Trans. Acoust., Spch, Signal Procssing, vol. ASSP-3, no. 3, pp. 577 59, Jun 984. [4] F. M. Gardnr, Intrpolation in digital modms - Part I: Fundamntals, IEEE Trans. Commun., vol. 4, no. 3, pp. 5 57, Mar. 993. [5] L. Erup, F. M. Gardnr, and R. A. Harris, Intrpolation in digital modms - Part II: Implmntation and prformanc, IEEE Trans. Commun., vol. 4, no. 6, pp. 998 8, Jun 993. [6] P. P. Vaidyanathan, Multirat Systms and Filtr Banks. Englwood Cliffs, NJ: Prntic Hall PTR, 993. [7] A. V. Oppnhim, R. W. Schafr, and J. R. Buck, Discrt-Tim Signal Procssing, nd d. Uppr Saddl Rivr, NJ: Prntic-Hall, Inc., 999. [8] P. Thévnaz, T. Blu, and M. Unsr, Intrpolation rvisitd, IEEE Trans. Md. Imag., vol. 9, no. 7, pp. 739 758, July. [9] M. Unsr, A. Aldroubi, and M. Edn, Fast B-splin transforms for continuous imag rprsntation and intrpolation, IEEE Trans. Pattrn Anal. Machin Intll., vol. 3, no. 3, pp. 77 85, Mar. 99. [] C. W. Farrow, A continuously variabl digital dlay lmnt, in Proc. IEEE Intrnational Symposium on Circuits and Systms ISCAS 988, Espoo, Finland, Jun 6 9, 988, pp. 64 645. [] M. K. Simon, S. M. Hindi, and W. C. Lindsy, Digital Communications Tchniqus: Signal Dsign and Dtction. Uppr Saddl Rivr, NJ: Prntic Hall PTR, 994.