Lesson 7. Chapter 8. Frequency estimation. Bengt Mandersson LTH. October Nonparametric methods: lesson 6. Parametric methods:

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1 Otmal Sgnal Procssng Lsson 7 Otmal Sgnal Procssng Chatr 8, Sctrum stmaton onaramtrc mthods: lsson 6 Chatr 8. Frquncy stmaton Th rodogram Th modfd Prodogram (ndong Aragng rodogram Bartltt Wlch Th nmum aranc mthod Th Blacman-Tuy mthod Paramtrc mthods: LT Octobr 0 Bngt andrsson Dartmnt of Elctrcal and Informaton Tchnology, Lund Unrsty Lund Unrsty Dscrbd n chatr 4 Pad Prony All-ol modl Lattc structurs n chatr 6 Frquncy stmaton (Estmaton of snusods, lsson 7 Th ll non mthods l Psarnco armonc Dcomoston and th USIC algorthm ar rsntd hr. Ths mthods ar basd on th gnctors of th corrlaton matr. Psarnco armonc Dcomoston Th USIC algorthm Th Egnctor mthod (EV (nmum norm Prncal comonnts Blacman-Tuy frquncy stmaton nmum aranc Frquncy stmaton 9 0 Otmal Sgnal Procssng Frquncy stmaton Th modl s that ha snusods n ht nos. Otmal Sgnal Procssng Frquncy stmaton, Eamls on gnctors and gnalus of th corrlaton matr. Snusod n ht nos ( n sn(* π * 0.* n + ( n ( n A th th coml amltud A A n + ( n jφ Th has s randomly dstrbutd n th ntral π φ π W ant to stmat Ur: Waform of a snusod n ht nos ddl: Sctrum from DFT Lor: Th gnalus of th corrlaton matr. I: Th amltuds II: Th frquncy A ; A A ω, f jφ III: umbr of snusods Th frst 5 gnctors (lft and thr sctra (rght

2 Otmal Sgnal Procssng Frquncy stmaton, Eamls on gnctors and gnalus of th corrlaton matr. Vol. Otmal Sgnal Procssng Frquncy stmaton, corrlaton matr. W assum frst that, n ( n A + ( n n 0,..., Ur: Waform of a ol. ddl: Sctrum from DFT Lor: Th gnalus of th corrlaton matr. or th A + [ ( 0, (,..., ( ] T j j ( [ ] T ω, ω,,,... [ ( 0, (,..., ( ] T Th corrlaton matr s E{ E{( A + ( A + } P } I Th frst 5 gnctors (lft and thr sctra (rght Th or of th snusods s P. A 3 4 Otmal Sgnal Procssng Frquncy stmaton, gnalus and gnctors. Egnalus and gnctors for snusods n ht nos. Otmal Sgnal Procssng Frquncy stmaton, gnalus and gnctors. Th othr gnctors must b orthogonal to gnctor. ultly th, ( P ( P ( P I λ λ... λ ( P 3 σ I σ,3,..., ( λ mn W no dntfy on gnalu and corrsondng gnctor ( P Th sgnal subsac s dtrmnd by λ P ( only ( λ ma f Th nos subsac s dtrmnd by,,..., 5 6

3 Otmal Sgnal Procssng Frquncy stmaton. A: Estmat and dtrmn th gnalus and gnctors. σ λ B: Estmat th aranc of th nos as mn. Otmal Sgnal Procssng Frquncy stmaton, Frquncy stmaton functon Th gnctors to ar orthogonal to. C: Estmat th sgnal or as ot that λ ma λ λ P mn D: Estmat th frquncy from th gnctor. ω arg{ (} (scond nd But V ( 0, For ω ω ha V (,..., ( 0 ( 0 0 Ths s ald for all gnctors to. 7 8 Otmal Sgnal Procssng W dfn th frquncy stmaton functon as Otmal Sgnal Procssng Frquncy stmaton. Aragng or all nos gnctors yld ( j V ( ω 0 ( ( α Eaml from th ttboo ag 455 Ur fgur: Aragng or th nos gnctors th th ght qual to on. Lor fgur: Orlay lot or th frquncy stmaton functon from ach of th nos gnctors W can also comut th Z-transform V ( z ( z 0 and dtrmn th zros of V (z 9 30

4 Otmal Sgnal Procssng Frquncy stmaton. Sral snusods n ht nos Otmal Sgnal Procssng Th frquncy stmaton functon s no and for ( n A n + ( n ( + α E{ E{( A + A P + P I + A } } + ( A +σ + W ll no loo at som mthods usng th frquncy stmaton functon abo. Th frst s calld th Psarnco Dcomoston mthod. Ths mthod s ry snst to th nos but dscrb th rncl for th mthods. A ll non mthod s th USIC algorthm. Egnauls υ sgnal subsac λ σ nos subsac υ ar th gnalus n th sgnal subsac Th frquncy stmaton functon s somtms calld th sudosctrum or gnsctrum. 3 3 Otmal Sgnal Procssng Frquncy stmaton: Psarnco : Assum coml snusods n ht nos : Assum th dmnson of (+*(+,.. only on nos gnctor. Ths assumtons mans that only on gnctor corrsonds to th nos subsac. Otmal Sgnal Procssng Frquncy stmaton: USIC Pag 464, 465 USIC: Ultl SIgnal Charactrzaton Th frquncy stmaton s achd by aragng th sudosctra or th nos gnctors. Thn λ λ σ mn + and th frquncy stmaton functon (sudosctrum s dfnd U ( + PD ( 0 mn mn ( P ( Thn stmat th oston of th as n U ˆ V mn ( z mn 0 ( z 33 34

5 Otmal Sgnal Procssng Prncal Comonnts Sctrum Estmaton. Ths mthods us th sgnal subsac. (ag Th Blacman-Tuy or sctrum as dtrmnd from a ndod autocorrlaton squnc ˆ + ( r ( ( BT Otmal Sgnal Procssng o, us only th gnctors corrsondng to th snusods. Thn th Blacman-Tuy rncal frquncy stmaton s dfnd by PC BT ( λ If ( s a Bartltt ndo, th Blacman-Tuy stmat can b rttn n trms of th autocorrlaton matr P ˆ j ω j ω BT ( ( rˆ ( In trms of gnctors (gndcomoston ths s Th mnmum aranc or sctrum stmat as dfnd by P ( V BT ( λ rt ths n trms of gnctors and only us gnctors corrsondng to th snusods gs th mnmum aranc frquncy stmaton P ( PC V λ Otmal Sgnal Procssng Otmal Sgnal Procssng Eaml of snusods n ht nos Por sctrum stmaton (n sn ( π 0. 0 n + 0.5sn ( π 0. 5 n + (n th (n ht nos th araanc o : Waform of nut sgnal (n o FFT of (n, 04 (Prodogram o 3 Aragng th th Wlch mthod (0 subntrals, rctangular tm ndo o 4 Blacman-Tuy stmat th 0 (hammng ndo o 5 nmum aranc mthod th 0, o 6 All-ol modl of ordr 0 (Lnson Durbn algorthm. (All sctra n 04 frquncy onts. y as n db 37 38

6 Otmal Sgnal Procssng Eaml of snusods n ht nos Frquncy stmaton mthods (n sn ( π 0. 0 n + 0.5sn ( π 0. 5 n + (n th (n ht nos th araanc o FFT of (n, 04 (Prodogram. o Psarnco armonc Dcomoston 4, 5. o 3 Th USIC algorthm 4, 30. o 4 Th Egnctor mthod (EV 4, 30. o 5 Prncal comonnts Blacman-Tuy frquncy stmaton (PC-BT 4, 30. (All sctra n 04 frquncy onts, y as n db 39

7 Otmal Sgnal Procssng 0 A brf r W ha focusd on mthods usd n ractcal alcatons All-ol modlng n chatr 4 (LPC, Prdcton Error Fltr Lnson Durbn curson usng th rflcton aramtrs Γ n chatr 5. Lattc structur n chatr 6, Burgs algorthm Wnr FI Fltrs n chatr 7 Por Sctrum Estmaton usng th Prodogram n chatr 8 Frquncy stmaton (USIC Chatr 3 Fltrng random rocsss (ral sgnals nut: (n outut: y (n y( n ( n h( n ( h( n r ( r ( h( h( r y y ( h( r ( r ( h( r ( y P ( y P ( P ( P ( y P ( y P ( ( ( ( ( P ( z P ( z ( z ( z y P ( z P ( z ( z y P ( z P ( z ( z y Chatr 4 Systm modlng All-ol modlng Ths can b dscrbd by th follong fgur. nmz muls δ(n A ( z a ( z + 0 ( z + a ( z ε n ( th n 0 gs th normal quatons l our sgnal (n 0 ( z A ( z n ( n ( + a ( n ( a ( l r ( l r (,..., 0 z ( A( z muls δ(n Chatr 5 Lnson-Durbn rcurson Th Lnson-Durbn algorthm dtrmn rlaton btn autocorrlaton r(,olynomal a( and th rflcton coffcnts. Th matr quaton can b rttn r(0 r ( r ( r( ε r( r(0 r ( r ( a( 0 r( r( r(0 r ( 3 a( 0. r( r( r( 3 r(0 a ( { a ε u a ε u Ths can b summarzd n th fgur blo and n th tabl Tabl r(, r(,..., r(, r(0 r ( r ( r( a( r ( r( r(0 r ( r ( a( r ( r( r( r(0 r ( 3 a(3 r (3. r( r( r( 3 r(0 a( r ( a ε,mn ε r (0 + a( r ( c Tabl Γ, Γ,..., Γ Tabl a(, a(,..., a( Lnson-Durbn Tabl 5

8 Chatr 6 Lattc Structur Chatr 7. Wnr fltrs FI (za (z (n g(n G(z s(n nos (n rcd (n h(n (z Estmatd dn ˆ( dsrd d(n rror (n (n 0 + (n + (n + (n Γ Γ W ll mnmz th outut rror (n, hch dscrb as th dffrnc btn th dsrd outut d(n and th stmatd outut. Γ z - z - Γ 0 - (n Burgs algorthm Lattc FI Lattc II - (n - (n Alcatons. Fltrng s(n: Smoothng: Prdcton: Equalzaton: Th dsrd sgnal s s(n and ll dtrmn th otmum fltr for nos rducton. L fltrng but allo an tra dlay n th outut sgnal (scally mag rocssng. Th outut s a rdcton of futur alus of s(n. On st rdctor. rdct nt alu s(n+. Th dsrd sgnal s (n and ll dtrmn th otmum fltr for htnng th outut sctrum (nrs fltrng, dconoluton. Othr alcatons: Echo cancllaton. os cancllaton. Puls shang. nmzng ˆ ξ E [ ( n] E[( d( n d( n ] gs th nr-of quatons hlr ( ( l rd( l ξmn rd(0 h( l rd( l l FI Wnr fltr l 0 hlr ( ( l r ( 0,,..., on-causal II Wnr fltr Pd ( ( P ( Causal II Wnr fltr ( z d P ( z d σ Qz ( Qz ( Tath causal art of ths Chatr 8 Por Sctrum Estmaton Th Fourr Transform of rˆ ( s calld th rodogram ˆ P ( ˆ r r ( + W s that t also can b rttn ˆ Pr ( X ( X ( X ( Usng DFT (FFT, th rodogram ll b ˆ P ( X ( X ( X ( j π / r FI-fltr for nos rducton h ( + r ot s s on-causal II-fltr for nos rducton Ps ( ( ; P( + P( s Aragng rodogram. Bartltt s thod Wlch s mthod Blacman-Tuy mthod nmum aranc mthod Frquncy stmaton (Estmaton of snusods Psarnco armonc Dcomoston Th USIC algorthm Th Egnctor mthod (EV Prncal comonnts Blacman-Tuy frquncy stmaton

SPECTRUM ESTIMATION (2)

SPECTRUM ESTIMATION (2) SPECTRUM ESTIMATION () PARAMETRIC METHODS FOR POWER SPECTRUM ESTIMATION Gnral consdraton of aramtrc modl sctrum stmaton: Autorgrssv sctrum stmaton: A. Th autocorrlaton mthod B. Th covaranc mthod C. Modfd