L13: Spectrum estimation nonparametric and parametric

Transcription

1 L13: Spctrum stimation nonparamtric and paramtric Lnnart Svnsson Dpartmnt of Signals and Systms Chalmrs Univrsity of Tchnology

2 Problm formulation Larning objctivs Aftr today s lctur you should b abl to Dscrib what th priodogram is and list its most important proprtis. Explain why th modifid priodogram is somtims an important improvmnt. Dscrib why Blackman-Tuky s mthod has lowr varianc than th priodogram. Driv how to stimat AR-paramtrs from data. abilitis that will hlp you to undrstand laboration 2! Chalmrs Univrsity of Tchnology Lnnart Svnsson 2/16

3 Problm formulation Notation and objctiv Notation and objctiv A first xampl Lt x[n] b a zro man, wid sns stationary procss. Objctiv: stimat P x jω ) from a masurmnt squnc x[], x[1],..., x[n 1]. Application xampls: spch procssing, mdical diagnosis, control systm dsign, radar, tc. Chalmrs Univrsity of Tchnology Lnnart Svnsson 3/16

4 Problm formulation Today s main xampl Notation and objctiv A first xampl Exampl 1: x[n] is an AR2)-procss and thus a WSS-procss 15 PSD in db: 1 log 1 P x ) 4 Masurmnt squnc : x [ n ] f, Normalizd frquncy n Task: stimat P x jω ) from x[n]. Chalmrs Univrsity of Tchnology Lnnart Svnsson 4/16

5 Problm formulation Th priodogram: dfinition Th priodogram Th modifid priodogram Blackman-Tuky s mthod: smoothd priod. Th priodogram is dfind as whr ˆP pr jω ) = 1 N X N jω ) = N 1 n= X N jω ) 2 x[n] jωn. Not: 1 Efficintly computd using DFT FFT). 2 Asymptotically unbiasd! Chalmrs Univrsity of Tchnology Lnnart Svnsson 5/16

6 Problm formulation Th priodogram vs th PSD Th priodogram Th modifid priodogram Blackman-Tuky s mthod: smoothd priod. Th PSD can b dfind as P x jω ) 1 { = lim N N E X N jω ) 2} whr X N jω ) is as abov. Two practical difficultis 1 W ar only givn on squnc can t do xpctd valus. 2 Givn a finit squnc N obsrvations) can t lt N. Th priodogram ignors both of ths! Chalmrs Univrsity of Tchnology Lnnart Svnsson 6/16

7 Problm formulation Th priodogram: bias for finit N Th priodogram Th modifid priodogram Blackman-Tuky s mthod: smoothd priod. Th DFT X N jω ) can b writtn as X N jω ) = N 1 n= x[n] jωn = n= w R [n]x[n] jωn = DTFTw R [n]x[n]) = W R jω ) X jω). W R jω ) changs th PSD and crats two typs of bias: 1 Th mainlob smars out th paks frquncy rsolution is limitd by ω 2π/N. 2 Sidlobs caus frquncy masking and laking. As N, W R jω ) X jω) X jω) which mans that th bias disappars. Chalmrs Univrsity of Tchnology Lnnart Svnsson 7/16

8 Problm formulation Th modifid priodogram Th priodogram Th modifid priodogram Blackman-Tuky s mthod: smoothd priod. Ida: rplac w R [n] by a diffrnt window Among all windows with th sam width N, th rctangular window has th narrowst mainlob which is a bnfit). Exampls of othr windows ar Bartltt triangular), Hamming and Chbyshv. All of ths hav lowr sidlobs but widr mainlobs. Chalmrs Univrsity of Tchnology Lnnart Svnsson 8/16

9 Problm formulation Th modifid priodogram xampl 2 Th priodogram Th modifid priodogram Blackman-Tuky s mthod: smoothd priod. Hr x[n] is a dtrministic function: x[n] = A 1 cos.2 2πn) + A 1 sin.21 2πn) + A 2 cos πn). whr A 2 < A 1 2 log 1 A 1 /A 2 ) = 26). Masurmnt squnc : x [ n ] n Not: x[n] is not a WSS signal. Still usful to illustrat situations whr rsolution is important. Chalmrs Univrsity of Tchnology Lnnart Svnsson 9/16

10 Problm formulation Th modifid priodogram xampl 2 Th priodogram Th modifid priodogram Blackman-Tuky s mthod: smoothd priod. Four common windows valuatd for xampl 2: 2 a) Rctangular window 1 b) Bartltt triangular) window 1 1 Priodogram [db] 1 2 Modifid Priodogram [db] Normalizd frquncy [pr sampl] Normalizd frquncy [pr sampl] 1 c) Hamming window 1 d) Chbyshv window, 4 db attnuation 1 Modifid Priodogram [db] 2 3 Modifid Priodogram [db] Normalizd frquncy [pr sampl] Normalizd frquncy [pr sampl] Chalmrs Univrsity of Tchnology Lnnart Svnsson 1/16

11 Problm formulation Blackman-Tuky s mthod Th priodogram Th modifid priodogram Blackman-Tuky s mthod: smoothd priod. For larg N it holds that VarˆP pr jω )) P 2 jω ) which mans that ˆP pr jω ) is vry noisy. 2 Priodogram in db: 2 log1 X ), N = 4 Priodogram Tru PSD 3 Priodogram in db: 2 log1 X ), N = 1 Priodogram Tru PSD f, Normalizd frquncy f, Normalizd frquncy W would lik to prform avraging in ordr to rduc varianc! Chalmrs Univrsity of Tchnology Lnnart Svnsson 11/16

12 Problm formulation Blackman-Tuky s mthod Th priodogram Th modifid priodogram Blackman-Tuky s mthod: smoothd priod. Hr w only discuss Blackman-Tuky s mthod which is oftn a good choic. s lctur nots for Bartltt s and Wlch s mthods. Important obsrvation: X N jω 1 ) and XN jω 2 ) ar approximatly uncorrlatd whnvr N ω 1 ω 2 is larg. Ida: prform local avraging of th priodogram, ˆP BT jω ) = 1 2π W lag jω ) ˆP pr jω ) whr W lag jω ) is cntrd around ω. Loosly spaking, W lag jω ) is a rctangular box. Chalmrs Univrsity of Tchnology Lnnart Svnsson 12/16

13 Problm formulation Blackman-Tuky s mthod Th priodogram Th modifid priodogram Blackman-Tuky s mthod: smoothd priod. BT s mthod can also b xplaind from ˆP pr jω ) = Fˆr x [n]). Taking th invrs transform of ˆP BT jω 1 ) givs ˆr BT [n] = w lag [n]ˆr x [n] stimats ˆr x [n] which ar lss rliabl ar givn a smallr wight w lag [n]! A simpl wight function usd blow): w lag [n] = { M n M if k M othrwis. Chalmrs Univrsity of Tchnology Lnnart Svnsson 13/16

14 Problm formulation Blackman-Tuky s mthod Th priodogram Th modifid priodogram Blackman-Tuky s mthod: smoothd priod. On can show that BT rducs variancs but incrass bias. 15 Priodogram Tru spctrum Priodogram 6 5 Blackman Tuky Spctrum Estimation Tru spctrum M=2 M=5 M=1 1 4 Spctrum Spctrum Normalizd Frquncy Normalizd Frquncy Small M: 1) givs a wid window W lag jω ) 2) maks P BT jω ) smooth 3) givs a bias in P BT jω ), spcially whr P x jω ) varis quickly. Chalmrs Univrsity of Tchnology Lnnart Svnsson 14/16

15 An illustration Problm formulation AR stimation Th AR-paramtrs can b stimatd using th normal quations. Works vry wll for abov xampl: 6 Tru spctrum AR spctral stimat 5 4 Spctrum Normalizd Frquncy low varianc sinc w only hav two paramtrs to stimat! bias may b larg if spctrum is not AR. Chalmrs Univrsity of Tchnology Lnnart Svnsson 15/16

16 Problm formulation Larning objctivs AR stimation Aftr today s lctur you should b abl to Dscrib what th priodogram is and list its most important proprtis. Explain why th modifid priodogram is somtims an important improvmnt. Dscrib why Blackman-Tuky s mthod has lowr varianc than th priodogram. Driv how to stimat AR-paramtrs from data. abilitis that will hlp you to undrstand laboration 2! Chalmrs Univrsity of Tchnology Lnnart Svnsson 16/16