CHAPTER 4. FREQUENCY ESTIMATION AND TRACKING

Transcription

1 CHPTER 4. FREQUENCY ESTITION ND TRCKING 4.. Itroducto Estmtg mult-frquc susodl sgls burd os hs b th focus of rsrch for qut som tm [68] [58] [46] [64]. ost of th publshd rsrch usd costrd ft mpuls rspos (IIR) dptv otch fltr (NF) to stmt th frqucs. Th costrd IIR structur s dsgd such tht t hs vr rrow bdwdth roud th stmtd frqucs. Ths hghl-tud structur gvs som dvtgs ovr ft mpuls rspos [FIR] structur. Th costrd IIR c hv vrbl or fxd bdwdth [58] [68] [46]. Nhor d Port [57] proposd structur tht c b usd to stmt hrmocll rltd frqucs usg dptv comb fltr. Howvr ths structur s computtoll complx sc t hs to comput cos fucto t vr trto. Hr w propos th us of th CRLS-S lgorthm/structur to stmt th prmtrs of mult-frquc susodl sgl. Ech scto of CRLS-S s th form of costrd IIR NF wth fxd bdwdth. Sc ch scto of th CRLS-S structur s scod ordr scto th frquc of trst c b obtd drctl from th coffcts of tht scto. Ths chptr s orgd s follows. Scto 4. gvs th structur d th lgorthm for CRLS-S. Som smultos to vlut th prformc of th stmtor r coductd ths scto. Scto 4.3 shows th us of dow-smplg tchqu to stmt th frquc of sgl hvg low fudmtl frquc much lowr th th smplg frquc ccompd b hrmocs. Scto 4.4 cocluds ths chptr. 50

2 4.. Structur d lgorthm Th rcvd sgl for whch th frqucs r to b stmtd s: x u (4..) whr u s ro-m ddtv wht Guss os x s susodl sgl: x C ( ) s ω ι θ ι (4..b) C s th mpltud d θ ι s uforml-dstrbutd rdom phs. Th vrc os u dtrms th sgl-to-os rto (SNR) of ch susodl sgl x tht s: σ of th SNR C 0log 0 σ (4..) Th costrd IIR NF s show Fgur 4.. for th cscd d Fgur 4.. for ch scto of th cscd. Th sstm fucto for th costrd IIR s: ( ) ( ( ) ) ( ) ( ) (4..3) whr l ( ) l l (4..3b) l l ( ) l l ( ( ) ) ( ) ( ) (4..3c) ( ) (4..3d) or w c xprss ( ) s: 5

3 ( ) ( ( ) ) ( ) ( ) (4..3) s costrd to so tht th ros of ( ) r loctd o th ut crcl d 0 < (4..3) Th stmtd frquc s rltd to th coffcts s: cosw [ ] (4..3f) ( ) - ( ) - ( ) Fgur 4.. Th Cscd NF. - ( ) ( ) Fgur 4.. Sgl Scto of th NF. Th vrbl dtrms th bdwdth of th NF d lso costrs th pols d ros of ch scto to l o th sm rdl l whl th rd of th pols r lss th th 5

4 rd of th ros. If 0 th th NF bcoms ll-ro fltr. B rstrctg th ros to fll o th ut crcl th NF wll hv flt rspos w from ts ctr frquc d bdwdth. Hc th os wll ot b mplfd. Th ffct of - for d s show Fgur I ths fgur w s tht th closr s to th bro l s for qul to 0.98 th rrowr th bdwdth of th otch fltr. lso for clos to th g outsd th otch bdwdth s vr clos to 0 db. Hc th os outsd th otch bdwdth wll ot b mplfd. Th sold l s for qul to 0.8 whch lds wdr bdwdth d modrt g w from th otch frquc. Fgur 4..3 Frquc Rspos for Dffrt. From (4..3b) d (4..3c) d Fgurs 4.. d 4.. w hv th followg rltos: 53

5 54 (4..4) (4..4b) (4..4c) S Fgur 4.. for dftos of th ottos. Th rror of th fl scto c b wrtt s follows: (4..4d) whr N/ N bg th ordr of th fltr. Th grdt for th -th scto c b computd s: ψ (4..5) whr s gv (4..4b). Th drvtv of th rror of th -th scto wth rspct to th coffct of th -th scto c b drvd s follows. Frst rpt (4..4b)

6 55 (4..5c) Th t th drvtv of (4..5c) wth rspct to ts coffct: (4..5d) From (4..4) (4..5) Substtutg (4..5) to (4..5d) d rrrgg trms w obt: 3 (4..5f) Usg (4..5f) (4..5) w obt: ψ (4..5g) Th CRLS-S lgorthm for frquc stmto s th sm s th orgl CRLS-S lgorthm drvd th prvous chptr xcpt tht th grdt of ch scto s computd s (4..5g).

7 Th rsult of stmtg sgl ormld frquc of 0.5 H s show Fgur 4..4 for dffrt SNR s. W s hr tht th covrgc rt d th std-stt frquc stmt r ot ffctd sgfctl b th SNR. Th SNR sms to bs th std-stt frquc stmt slghtl. Fgur 4..4 Estmtd Sgl Frquc for Dffrt SNR. Fgur 4..5 shows th rsult for stmtg two frqucs t 0.5 H d 0.5 H t dffrt SNR s. W s hr tht CRLS-S c stmt th frqucs wll rgrdlss of th SNR. It sms tht th lowr frquc covrgs slghtl fstr th th hghr frquc. 56

8 Fgur 4..5 Two Frquc Estmts t Dffrt SNR. Fgur 4..6 Estmts for Two Closl-Spcd Frqucs t Dffrt SNR. Fgur 4..6 shows th rsult of stmtg two closl-spcd frqucs t 0.5 H d 0.5 H t dffrt SNR s. g w s tht CRLS-S c stmt th frqucs wll. Sc th 57

9 frqucs r clos w s tht both covrg t lmost th sm tm. Fgurs 4..7 d 4..8 show th rsults for trcg sgl tht hs tm-vrg frquc compot wth th trgt frquc dctd b th stppd ls. W s hr tht th CRLS-S trcs th frquc chg wll. Fgur 4..7 Trcg of Stp Chg Frquc t 0 db SNR. 58

10 Fgur 4..8 Trcg of Lrgr d Fstr Stp Chgs Frquc t 0 db SNR Frquc Estmtor Oprtg o Dow-Smpld Sgl W sw (4..3f) tht th coffcts of ch scto th cscd r rltd to th stmtd frquc s follows: cosω [ ] (4.3.) Not from (4.3.) tht to stmt frquc ω whch s vr smll much smllr th th smplg frquc th coffct wll b vr clos to th boudr of ts ccptbl rgo. Udr stmto codtos ths c cus to jump outsd ts ccptbl boudr 59

11 from tm to tm spcll durg th frst fw trtos. Hc stblt motorg s dd whch c slow th covrgc. O w to ovrcom ths problm s v dow-smplg of th sgl. It s ow tht f w dow-smpl sgl b fctor of D th frquc of th dow-smpld sgl rltd to th frquc of th orgl sgl ω s [64]: ω D s ( π ) ω Dω mod D (4.3.) d th coffct of th cscd ssoctd wth ω D s: ω D cos D (4.3.b Hc for pproprt choc of D D wll ot b so clos to boudr of ts ccptbl rgo. lso f th sgl cossts of svrl closl-spcd frqucs dow-smplg th sgl wll crs th sprto of thos frqucs. Fgur 4.3. shows th PSD of sgl cotg two closl-spcd frqucs d th PSD ftr dow-smplg th sgl b fctor of 5. Th wdr sprto of th hghr frqucs ftr dow-smplg wll m t sr to stmt ths frqucs. Not tht th dow-smplg tchqu s spcll usful for stmtg frqucs tht r much smllr th th smplg frquc. For hghr frqucs th dow smplg tchqu mght ot b dd. 60

12 Fgur 4.3. Spctrum of Orgl d Dow-smpld Sgls. Estmtg frqucs dow-smpld sgl s spcll usful stmtg th fudmtl frquc.. ptch of spch sgl s t c hv ormld fudmtl frquc s low s H. To s th prformc of th frquc stmtor wth th dow-smplg tchqu svrl smultos r coductd. For ll th smultos th sgl modl (4..b) s usd whr th sgl cots compot wth fudmtl frquc F o d two hrmocs. Th mpltuds for th fudmtl frquc d th hrmocs r 0.5 d 0.5 rspctvl d th dow-smplg fctor D s 4. Fgur 4.3. shows th rsults for stmtg th frquc of sgl hvg fudmtl frquc of d two hrmocs t 0 db d 0 db SNR. W s tht th SNR dos ot ffct th rsult sgfctl. Th x-xs s th umbr of smpls ftr dowsmplg. 6

13 Fgur 4.3. Estmtg th Fudmtl Frquc of Sgl wth Thr Hrmocs; F o.0083 H D4. Fgur shows th rsults for stmtg th frquc of sgl whr th frquc of th sgl chgs from frm to frm d cosqutl th hrmocs lso chg. Howvr wth frm of th sgl th frquc s ssumd to b vrt. Ths rsmbls stmtg fudmtl frquc or ptch of spch sgl whr th ptch s ssumd to b costt frm. Th frquc chgs from frm to frm r rdom but th chgs r md such tht from frm to frm thr wll b smll chgs s wll s lrg chgs frquc. Th frm s s 30 smpls ftr dow-smplg b fctor of 4. Th vlu of th frquc stmt from th prvous frm s usd s th tl vlu for th currt frm. Ths s spcll usful wh th fudmtl chg from frm to frm s smll s hpps durg vocd-spch. W s Fgur tht th stmtd frqucs for ch frm dctd b x r vr clos to th orgl frqucs dctd b o. 6

14 Fgur Estmtg th Fudmtl Frquc of Sgl wth Thr Hrmocs; Ech Frm th Fudmtl Frquc Chgs Cocluso Th CRLS-S lgorthm shows good rsults stmtg frqucs whthr sgl frquc or multpl frqucs. Th SNR of th sgl ffcts thr th covrgc or th std stt stmtd vlu sgfct w. To stmt frquc tht s much smllr th th smplg frquc dow-smplg of th sgl bfor stmtg th frquc shows promsg rsults. lso th CRLS-S lgorthm wth dow-smplg c b usd to cotuousl stmt th frquc of sgl whr th frquc s chgg from frm to frm thr b smll or lrg mouts. Thrfor stmtg frquc wth CRLS-S wth dow-smplg c b usful tchqu to stmt th fudmtl frquc or ptch of spch sgl. 63