Frequency Measurement in Noise

Transcription

1 Frqucy Masurmt i ois Porat Sctio 6.5 /4

2 Frqucy Mas. i ois Problm Wat to o look at th ct o ois o usig th DFT to masur th rqucy o a siusoid. Cosidr sigl complx siusoid cas: j y +, ssum Complx Whit ois Gaussia, Zro-Ma Variac: σ γ Di: Iput Sigal-to-ois Ratio (SR): SR i sigal por ois por σ I db: log σ Modl or Widod DTFT o Rcid Sigal: Y ( ) W ( ) + V ( ) Y ( ) π P W ( ) ois Floor π /4

3 Impact o ois. Maks it diicult to s th sigal pak d sigal pak ll abo th ois loor I ot. Might ot dtct prsc o sigal. ois prturbs th pak locatio Dgrads accuracy o th rqucy stimat So Procssig ds To. First, Dtct th Sigal Look or paks i th DFT Th, Estimat th Frqucy (ad amplitud/phas) Sam as bor d to do aalysis to dtrmi th prormac o ths to procssig tasks. (Us DTFT i aalysis rathr tha DFT) W ll oly cosidr Dtctio Prormac (s Porat s Book or EE5 or Estimatio). 3/4

4 Sigal Dtctio alysis Goal: alyz rlatioships bt pak ll i DTFT du to sigal ad th ois loor hight to asr: Q: What paramtrs dtrmi ho high th sigal s pak is abo th ois loor? DTFT o Widod oisy Sigal: Y ( ) { j DTFT ( + )} Sigal Part j( ) + ois Part j 4/4

5 Sigal Dtctio alysis (pt. ) Sigal part paks at, so look thr: Y + ( ) Pak Hight Boostd by Σ For Rct. Wido this Boost is: R j Q: What is th boost or othr idos? Compar Σ or othr idos to that or th Rct ido: CG R Di Cohrt Gai o Wido Boost Lost du to usig a o-rct Wido ot: CG ( or Rct. Wido) CG arly idpdt o 5/4

6 Sigal Dtctio alysis (pt. 3) R-rit DTFT Pak Usig CG: Y CG) + ( ) ( j Impact o Sigal o Pak Hight Impact o Procssig Lgth Lgth Pak Impact o Wido Shap Mor Rct Pak Output Pak (Iput mplitud) ( CG) Hor, th ois loor also icrass. So d a ay to masur Impromt. Output SR 6/4

7 Sigal Dtctio alysis (pt. 4) Output SR SR o Por o DTFT' s Sigal Pak DTFT ois Por at Pak DTFT Por at Pak: Y ( ) Y ( ) Y ( ) FOIL Th Trms ( CG) + CG R + m m m j j ( m) Sigal Part ois Part 7/4

8 Sigal Dtctio alysis (pt. 5) o d to look at th arag output por: Expctd Valu o st ois trm is zro bcaus E{} E Y ( ) Sigal Pak s Por: ( CG) + m E{ m } m δ m j ( m) Us Sitig Prop. ( CG) σ σ utocorr. o Whit ois ois Pak: σ 8/4

9 9/4 Sigal Dtctio alysis (pt. 6) o Ca rit xprssio or Output SR: ( ) ( ) ( ) PG i o CG SR CG CG SR σ σ SR i o To simpliy di Procssig Gai PG: ( ) CG PG PG o SR i PG SR Masurs Ect o Sigal Eiromt Masurs Ect o Wido Typ (i.., Shap) Masurs Ect o Procssig Lgth (Do t Cout Zro-Pads!!!)

10 Sigal Dtctio alysis (pt. 7) Commts Grally d SR o 4 db to sur rliabl dtctio! PG ( ith or Rct Wido) Cohrt Gai (CG) s. Procssig Gai (PG) CG rlats Pak Ll to Sigal mp: Pak Ll CG PG rlats Pak s SR to Sigal SR: SR o PG SR i CG ad PG ar usually Spciid i db CG i db: log (CG) PG i db: log PG Squard! *Bcaus CG is a mplitud Gai* ot Squard! *Bcaus PG is a Por Gai* /4

11 Sigal Dtctio alysis (pt. 8) othr Vi o Output SR Rcall a arlir quatio or output SR: SR o SR o SRi ( CG) σ Cosidr (or as) th Rct Wido (CG ad Σ ) so SR o σ (Iput Sigal Por) (Iput ois Por) Sigal Por Boostd by ois Por Boostd oly by Sic th Sigal is Boostd Mor Tha th ois, gt a Boost i SR: (rcall : PG or Rct) /4

12 /4 Sigal Dtctio alysis (pt. 9) Yt othr Vi o Output SR Rcall this orm or th DTFT at th pak: + + () ) ( ) ( j j j j Y R Im R Im Sigal Trms dd Cohrtly Sum Gros Fast Sigal Trms dd Icohrtly Sum Dos t Gro s Fast

13 Sigal Dtctio alysis (pt. 9) Impact o ctually Usig DFT rathr tha DTFT lthough did our aalysis usig th DTFT, th actual procssig is do usig th DFT. Q: What Impact Dos This Ha? Rcall: DFT is DTFT computd o a grid DTFT Pak May ot Fall O th Grid Worst Cas: Pak Halay Bt Grid Poits Do t Gt th Full CG Hr o th Grid!!! Δ 3/4

14 Sigal Dtctio alysis (pt. ) Impact o ctually Usig DFT rathr tha DTFT (cot.) Lads to Diig Worst-Cas Gais: Do t d to djust Dom b/c it accouts or th ois Ect (hich is Flat, ot Pakd) CG PG j.5( Δ ) j.5( Δ ) um. i PG coms rom CG Us Worst-Cas Gais: h you d to b cosrati i prdictig dtctio prormac!! 4/4