LIGO-T666--R Ec sampling n rquncy dmain analysis David P. Nrwd W rviw h wll-knwn cs digial sampling n h rquncy dmain analysis an analg signal, wih mphasis n h cs upn ur masurmns. This discussin llws h nain Gaskill. Th signal b sampld is assumd b a harmnically varying signal, dampd in a way ha can b characrizd by a im cnsan,. Tha is, w assum h riginal analg signal is h rm: ( ( < > (N ha h rquncy includs any shi du h assumd damping. This signal is sampld N 4 ims a a im inrval.3 s, rsuling in a al sampling im N 4 s. s w will s, i is signiican ha his is much largr han h damping im, 35-4 s. Th sampld daa can b wrin rmally as: ( ( ( P ( whr ( is rm Eq ( and ( is a sampling uncin, ha w wri as: N ( ( n n ( δ (3 Tha is, h sampling uncin is a rain N Dirac dla uncins, sparad by h sampling im,. Using h s-calld rcangl uncin, dind as:, > rc (.5, (4, < w can rwri h sampling uncin as: n ( δ ( n N rc N N rc N (whr w v dind h uncin, again llwing Gaskill. This will simpliy sm h llwing discussin [I hp ]. Th rquncy dmain analysis hn J. D. Gaskill, Linar ysms, urir Transrms and Opics, Wily, Nw Yrk, 978 (5
LIGO-T666--R prcds by prrming a urir ransrm upn h sampld uncin Eq. (. rmally, his is givn by: ( ( ( P ( N rc N whr dns h cnvluin prain, and h scrip dns h urir ransrm prains. W hav usd h cnvluin hrm, which hlds ha h urir ransrm a prduc uncins is h cnvluin h ransrm ach uncin. This ransrm will giv a pak whs lcain and widh ar drmind, bviusly, by and, bu als by and N, as w shall discuss. I is usul brak h discussin in w pars, n rlad bradning and n rlad h lcain h pak. radning Only h xpnnial dcay and h rcangl uncin cnribu bradning (h ransrm h in and h rsul in Dirac dla uncins, as w discuss lar. W can hn din a bradning uncin by: ( ( N rc N (6 (7 I w d h ransrm xplicily, w hav: ( N i d i i + N (8 + Th ampliud ( is givn by: and h phas by: N N + ( N N + an Φ + ( (9 ( sin( N ( N ( + sin( N N (
LIGO-T666--R Ths xprssins sm rbidding, bu h ssnial pin can b sn i w cnsidr w limis: N >>, h cndiin ha h signal is sampld r a im lng cmpard h dcay im (i.., ha i dcays signiicanly during h acquisiin h daa, and N <<, h cndiin ha h signal is sampld r a im shr cmpard h dcay im (i.., ha i ds n dcay signiicanly during h acquisiin h daa. In h irs cas, h ampliud and phas ak h rm: ( + an Φ and in h lar cas: ( ( N sin ( an Φ an( N Tha hs ar h xpcd rms can b sn ms asily by rurning Eq. 8 and aking h limi ha (N g Eq. ( and g Eq. (. rm Eq. (, w s ha h widh h ransrmd signal is rdr /. Tha is, i h im during which daa is sampld is larg cmpard h damping im, hn h widh h ransrmd daa will b givn by h damping im. Cnvrsly, Eq. ( shws ha in h vn ha h al sampling im is shr cmpard h damping im, hn h widh is givn by h invrs h al sampling im, /N. In gnral, h widh h ransrmd daa will b rlcd by h uncin Eq. (9 and will b rdr (/ + /N. In ur masurmns, PEND 35 sc and PITCH 4 sc, whil N is 4 sc, and s w ar in h rgim dscribd by Eq. (. liasing Th hr w uncins in P ( (Eq. (6 drmin h lcain (in rquncy spac h uncin givn by Eq. (8. Tha is, Eq. (8 drmins h shap h scillar pak and h rmaindr: PEK ( ( ( drmins h lcain. Prrming h ransrms, his bcms: ( [ δ ( + δ ( ] ( ( ] PEK + nd prrming h cnvluin, w hav: ( ( (3 (4
LIGO-T666--R PEK ( n + δ n n + δ + Using his, h ransrm h sampld daa is givn by: ( ( ( P n + n PEK n + + Tha is, h ransrmd daa cnsiss rpad vrsins h bradnd uncin rm Eq. (8, cnrd a +n/ and ( -n/. Thr ar svral cnclusins b drawn. upps ha h rquncy saisis < </( (his is h Nyquis cndiin. Thn clarly saisis (n/( < +n/ <(n+/(. Tha is, h irs rm in Eq. (6 nsurs ha vry hr inrval (-/ /, /, / 3/, and s n includs a cpy h pak crrspnding. urhr, sinc (n-/( < n/ < n/, h scnd rm in Eq. (6 nsurs ha hr is a cpy in h rmaining inrvals (- 3/ /, -/, / /, and s n. Thrr, nly h inrval / nd b cnsidrd and hr will b prcisly n cpy h uncin ( in ha inrval (rm h irs rm in Eq. (6 wih n. Hwvr, supps ha h rquncy saisis / < < /. This is a cndiin undrsampling, in which h sampling ra is shr aihully rprduc h sampld signal. I is again h cas ha vry inrval n/ (n+// cnains h sam inrmain. u in his cas, h pak ha appars in h inrval / is rm h scnd rm in Eq. (6 and ccurs a h rquncy /. Rrring igur: n (5 n (6
LIGO-T666--R W s a pak a.5 Hz, which is h naural pndular min h pndulum and a pak a ~.5 Hz. Th.5 Hz pak is an alias h pich a 3.8 Hz : (3.33 Hz 3.8 Hz.5 Hz. ulipl Oscillains imulanusly Nw supps ha w hav mr han n scillain in h signal b sampld. Equain ( hn bcms: ( > < (7 Th sampling uncin is as givn in Eq. (5 and h sampld daa is sill givn rmally by Eq. (. Th urir ransrm h sampld daa is hn: [ ] [ ] PEK ROD P N N rc N N rc N N rc N N rc,, (8
LIGO-T666--R Tha is, h ransrmd sampl daa will cnsis paks ha shw h sam prpris as discussd abv r h singl pak. Each pak will b bradnd in h sam way as discussd br, ihr by h damping rprsnd by r by h sampling im, N s. ls, ach pak lcain will b drmind by h scillain rquncy and h sampling ra, / s, again as discussd prviusly. ny aliasing will ccur r ach pak indpndnly, dpnding upn h rquncy ha pak,, and h Nyquis sampling ra, /( s. N ha hr ar n paks crrspnding h sum r dirnc any h scillain rquncis, as migh b hugh a irs. u his is xpcd r a linar prcss lik a urir ransrm.