SPECTRUM ESTIMATION (2)

Transcription

1 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 covaranc mthod D. Burg algorthm: E. Slcton of th modl ordr: Movng avrag sctrum stmaton: Autorgrssv movng avrag sctrum stmaton: Mnmum Varanc Sctrum Estmaton (a nonaramtrc mthod): A. Th rlaton of th varanc wth th owr sctrum B. FIR bandass fltr bank and th varancs of th fltrs oututs: C. Th FIR bandass fltrs wth mnmum varanc D. Th mnmum varanc sctral stmat Mamum ntroy mthod (an AR aramtrc mthod): A. Th conct of ntroy B. Etraolaton of th autocorrlaton C. Th mamum ntroy sctral stmat Summary of sctral stmaton mthods: Som usful MATLAB functons 6

2 PARAMETRIC METHODS FOR POWER SPECTRUM ESTIMATION Paramtrc mthods for owr sctrum stmaton ar basd on aramtrc modls, and thy nclud thos of th autorgrssv (AR) sctral stmaton, th movng avrag (MA) sctral stmaton, and th autorgrssv movng avrag (ARMA) sctral stmaton, whch ar, rsctvly, basd on th AR, MA, and ARMA modls. Th mamum ntroy mthod s of th sam form of th AR sctral stmaton. In aramtrc mthods, a aramtrc modl for a random rocss s frst slctd and thn th modl aramtrs ar dtrmnd. Th aramtrc sctral stmators ar lss basd and hav a lowr varanc than th nonaramtrc sctral stmators. Wth aramtrc mthods t s ossbl to sgnfcantly mrov th rsoluton of th sctrum stmaton unlss th modl usd s consstnt wth th random rocss bng analyzd. Othrws naccurat or msladng sctrum stmats may rsult. Gnral consdraton of aramtrc modl sctrum stmaton: As w hav larnt n th rvous lcturs concrnng Sgnal Modlng, a random rocss can b modld wth an ARMA modl, or an AR modl, or an MA modl. Th AR and MA modls ar th scal cass of th ARMA modl. Suosng that a random rocss (n) s modld as an ARMA(, ) rocss wth an ARMA(, ) modl, thn th systm functon of th modl s H ( ) + b k a In ths cas, th owr sctrum of th rocss (n) can b comutd n th followng mannr, P ( ) + b a Altrnatvly, f th autocorrlaton r (k) s gvn, th owr sctrum can b obtand from th Fourr transform of r (k), r P ( ). (45) Es. (44) and (45) dmonstrat two aroachs to comutng th owr sctrum of an ARMA rocss, and also rval two uvalnt rrsntatons of an ARMA random rocss, that s, th rocss can b rrsntd uvalntly thr by a fnt sunc of modl aramtrs a (k) and (k), or by an autocorrlaton sunc r (k). Th uvalnc of th two rrsntatons s bcaus th autocorrlaton and th modl aramtrs ar rlatd wth th Yul-Walkr uatons, l v l k r + a ( l) r ( k l) σ b ( l) h ( l k). (46) In ractc, a random rocss (n) s oftn gvn only ovr a fnt ntrval, n N, and n ths cas th autocorrlaton of (n) must b stmatd n a fnt sum as follows, N k ( n + k) ( n),,,, N. (47) N n b (43) (44) 7

3 Whn th ARMA modl n E. (43) s slctd for modlng rocss (n), th modl aramtrs n ths cas ar dtrmnd from ths stmatd autocorrlaton sunc, and thy ar dffrnt from a (k) and b (k) dtrmnd from r (k) snc s, n gnral, not ual to (k). Such modl aramtrs that ar dtrmnd from ar dnotd by aˆ and ˆ, whch gv an stmat of th owr sctrum, b r Pˆ ( ) + bˆ aˆ (48) E. (48) s a gnral cas of th aramtrc sctral stmaton mthods. In ths cas, all w nd to do for stmatng th owr sctrum s to fnd aˆ and bˆ. Whn aˆ and bˆ ar dtrmnd, ˆ P ( ) s found. Among ths aramtrc sctral stmatons, th AR stmaton s th most oular. Ths s bcaus th AR aramtrs can b found by solvng a st of lnar uatons. Whras, for th ARMA and MA aramtrs, a st of nonlnar uatons nd to b solvd, whch wll b much mor dffcult. Autorgrssv sctrum stmaton: Th autorgrssv sctrum stmaton s basd on th AR modl. In ths cas, a random rocss (n) s modld as an AR() rocss. If th autocorrlaton r (k) of a random rocss (n) s gvn, th AR aramtrs, a (k) and b (), can b dtrmnd from r (k) usng th AR modl. Thn th owr sctrum of th AR rocss s P ( ) + k b() a If a random rocss (n) s gvn ovr a fnt ntrval n N, th autocorrlaton of (n) must b stmatd, and t s dnotd. Th AR aramtrs that ar dtrmnd from th stmatd autocorrlaton ar dfnd as aˆ and ˆb (). Th owr sctrum that s stmatd basd on (n) for n N s Pˆ ( ) + k bˆ() aˆ Svral aroachs ar avalabl for fndng from th fnt data rcord of a rocss (n) for n N, such as autocorrlaton mthod, covaranc mthod, and modfd covaranc mthod, whch ar rsntd blow. Whn ar found, aˆ and ˆb () can b found usng th MYWE mthod or th EYWE mthod, whch ar studd tnsvly n th lcturs on Sgnal Modlng. (49) (5) A. Th autocorrlaton mthod Th AR aramtrs aˆ ar found by solvng th autocorrlaton normal uaton 8

4 () () ( ) () () ( ) ( ) aˆ () () ( ) aˆ () () () aˆ ( ) ( ) whr th autocorrlaton stmat s gvn by N k ( n + k) ( n) ;,,,. (5) N n b ˆ () () + aˆ. (53) k Substtutng aˆ and ˆb () nto E. (5) gvs th stmat of th owr sctrum of th rocss (n). Not that E. (5) s th sam n form as th modfd Yul-Walkr uatons, but th autocorrlaton valus n E. (5) ar th stmatd ons,, from a fnt data rcord,.., (n) for n N. Th autocorrlaton stmat s basd. Ths mthod that stmats th owr sctrum usng th autocorrlaton mthod s also rfrrd to as th Yul-Walkr mthod (YWM). Snc E. (5) gvs a basd stmat, a varaton of E. (5) s of th followng form N k ( n + k) ( n) ;,,,. (5 ) N k n (5) whch may gv an unbasd stmat. Howvr, ths can not guarant th autocorrlaton matr to b ostv dfnt and consuntly, th varanc of ˆ P ( ) tnds to b larg whn th matr s llcondtond or sngular. Thus, th basd stmat n E. (5) s mor rfrabl to th unbasd stmat n E. (5 ). Lk n a rodogram, th autocorrlaton mthod uss th wndowd data to stmat th autocorrlaton and thus has wndow ffct on th sctral stmat. Snc th wndow ffct wll bcom svr for short data rcords, th autocorrlaton mthod s not oftn usd n th cas of short data rcords. An artfact, calld sctral ln slttng that mans th slttng of a sngl sctral ak nto two or mor sarat and dstnct aks, may aar n th autocorrlaton mthod whn (n) s ovrmodld,.., whn th modl ordr s too larg, bcaus a ol n th modl, n gnral, may crat a sctral ak. An aml of such an artfact s shown n Fg Fg Sctral ln slttng of an AR() rocss, (n).9(n ) + w(n). Two all-ol sctrum stmats wr comutd usng th autocorrlaton mthod wth ordr 4 (sold ln) and (dashd-dot ln) 9

5 B. Th covaranc mthod In th covaranc mthod th AR aramtrs aˆ ar dtrmnd by solvng th normal uatons, (,) (,) (, ) (,) (,) (, ) ˆ (,) a () (,) ˆ (,) a () ˆ r (,) (, ) aˆ ( ) (, ) whr th autocorrlaton stmat s gvn by N ( n l) n r ˆ ( k, l) ( n k), (55) whch s dffrnt from th autocorrlaton mthod n that no wndowng of th data s rurd snc th valus of (n) usd for fndng ( k, l) n E. (55) ar all n th ntrval n N and thus no zroaddng s ndd. Ths mans that thr s no wndowng ffct n th varanc mthod. Thrfor, for short data rcords th varanc mthod gnrally gvs hghr rsoluton sctrum stmats than th autocorrlaton mthod. C. Modfd covaranc mthod In th modfd covaranc mthod th AR aramtrs aˆ ar also dtrmnd by solvng th normal uatons n E. (54). But th autocorrlaton stmat s found n a dffrnt way as follows [ ( n l) ( n k) + ( n + l) ( n + k) ] N n r ˆ ( k, l) (56) whch s drvd by mnmzng th sum of th suars of th forward and backward rdcton rrors, that s, N + + ε ( ) ( ) + ( ) ( ) + ( ) n ε n ε n n n (57) whr + k n ( n) ( n) + a ( n k) (58) ( n) ( n ) + a ( n + k) (59) k ar th forward and backward rdcton rrors, rsctvly (s n th Hays book). In contrast to th autocorrlaton and covaranc mthods, th modfd covaranc mthod s obsrvd to gv statstcally stabl sctrum stmats wth hgh rsoluton, and not to b subjct to sctral ln slttng. (54) Eaml 5. Estmaton of th owr sctrum of an AR(4) rocss. Consdr th AR(4) rocss gnratd by th dffrnc uaton ( n).7377( n ) ( n ) +.693( n 3).94( n ) + w( n) whr w(n) s unt varanc wht Gaussan nos. Th fltr gnratng (n) has a ar of ols at ± j.π z.98 and a ar of ols at ± j.3π z.98. Usng th data rcords of lngth N 8, n nsmbl of 5 sctrum stmats wr calculatd usng th Yul-Walkr mthod, th covaranc mthod, and th modfd covaranc mthod, and th Burg s mthod. Th ovrlay lots of th 5 stmats from th four mthods ar shown n art (a) n 8.5 to 8.8, and th nsmbl avrag of th 5 stmats and th tru owr sctrum ar shown n art (b) n 8.5 to 8.8.

6 Fgs to 8.8.

7 E. Slcton of th modl ordr: Th slcton of th modl ordr n th AR sctral stmaton s crtcal n th aramtrc mthods. Fg. 8.4 shows that sctral ln slttng artfact aars n th autocorrlaton mthod whn th modl ordr slctd s too larg, bcaus ach of th ols n th modl, n gnral, may crat on sctral ak. Ths brngs u th uston of how to slct an arorat modl ordr for an AR sctrum stmaton. If th modl ordr usd s too small, thn th rsultng sctrum wll b smoothd bcaus th sctral aks n a tru owr sctrum can not b rrsntd wth an nough numbr of ols. If, on th othr hand, th modl ordr usd s too larg, thn th owr sctrum may contan mor sctral aks than thos n a tru owr sctrum, and n ths cas th so-calld sctral ln slttng artfact s rsnt, as th aml n Fg Ths rvals a ncssty to hav som aroachs to slctng an arorat modl ordr that gvs th aramtrc modl a bst ft of a gvn data rcord. A rlvant da would b to adjust th modl ordr untl a crtan modlng rror bcoms mnmum. Thr ar svral aroachs to slctng modl ordr that wr stablshd basd on such an da. On of thm s th Akak Informaton Crtron rssd as AIC( ) N logε + (6) and th othr s th mnmum dscrton lngth MDL( ) N logε + (log N) (6) Two othr oftn usd crtra ar Akakr s Fnal Prdcton Error N + + FPE( ) ε (6) N and Parzn s Crtron Autorgrssv Transfr functon N k N CAT( ) + (63) N k Nε k Nε In Es. (6) to (63), s th modl ordr to b slctd, N s th lngth of th data rcord, and ε s th rdcton rror. For short data rcords, non of th crtra tnd to work artcularly wll, and thus, ths crtra should only b usd as ndcators of th modl ordr. Es. (6) to (63) show that all ths crtra dnd on ε. Snc dffrnt modlng tchnus,.g., th autocorrlaton and covaranc mthods, may hav dffrnt rdcton rrors, thn th modl ordr may b dffrnt vn for th sam data.

8 Movng avrag sctrum stmaton: Th movng avrag sctrum stmaton s carrd out basd on th MA modl that modls a random rocss (n) as an MA rocss. Th systm functon of th fltr that gnrats th MA rocss by fltrng unt varanc wht nos s of th form, k k H ( z) b z. (64) Thus, th owr sctrum of th rocss (n) s P ( ) b. (65) In trms of th autocorrlaton r (k) th owr sctrum can b wrttn as P ( ) r. (66) whr r (k) and b (k) ar rlatd wth th Yul-Walkr uatons v k l r σ b b ( k) b ( l+ k ) b ( l) (67) If a random rocss (n) s gvn ovr a fnt ntrval n N, w only hav th stmat of th autocorrlaton of (n),. Th MA aramtrs that ar dtrmnd from th stmatd autocorrlaton ar ˆ. Th stmat of th owr sctrum of (n) s Pˆ MA ( b ) bˆ. (68) Euvalntly, drctly usng th autocorrlaton stmat ˆ, w may hav th altrnatv form of th stmat of th owr sctrum, ˆ MA ( ). (69) P Comarng th stmat n E. (69) wth th Black-Tuky stmat n E. (4), w can s that th MA stmat s uvalnt to th Black-Tuky stmat f th wndow w(n) usd tnds from to. Howvr, thr s a subtl dffrnc btwn th two stmats; for th MA sctral stmat n E. (69) th random rocss (n) s modld as an MA rocss of ordr, and thus th autocorrlaton sunc s zro for k >. In ths cas, f th autocorrlaton stmat s unbasd for k, thn { ˆ E PMA ( )} P ( ) so that ˆ P ( ) s unbasd. MA In th Blackman-Tuky mthod, no assumton s mad about (n), and du to th wndowng ffct, thus, th Blackman-Tuky sctral stmat wll b basd unlss (n) s an MA rocss. r Autorgrssv movng avrag sctrum stmaton: Th ARMA sctrum stmaton s rformd basd on th ARMA modl that modls a random rocss (n) as an ARMA rocss. Ths mthod has bn dalt wth n th arlr scton, Gnral consdraton of aramtrc modl sctrum stmaton. 3

9 Mnmum Varanc Sctrum Estmaton (a nonaramtrc mthod): In th mnmum varanc (MV) mthod th owr sctrum s stmatd by fltrng a random rocss wth a bank of narrowband bandass fltrs. Th bandass fltrs ar dsgnd to b otmum by mnmzng th varanc of th outut of a narrowband fltr that adats to th sctral contnt of th nut rocss at ach fruncy of ntrst. A. Th rlaton of th varanc wth th owr sctrum Consdr a zro man WSS rocss y(n). Th varanc of y(n) s y n) { y( ) } σ ( E n, (7) whch s th owr of th rocss y(n). For a gvn autocorrlaton r y (k) w hav { y( ) } r y ( ) E n, (7) y n and thn σ ( ) r () dos not vary wth n, and thus dnot { y(n) } y σ y E. (7) π y P π y ( ) Snc r ) ( / ) ( π dω, th avrag owr of such a WSS rocss y(n) s { y(n) } E σ π π y Py ( ) π whch shows th rlaton of th varanc wth th owr sctrum. dω, (73) B. FIR bandass fltr bank and th varancs of th fltrs oututs: Consdr a bank of FIR bandass fltrs (Fg. 3), all havng ordr and th fruncy rsonss (or th systm functon) of th followng form, G ( ) n g ( n) jnω,,,, L. (74) Fg. 3. A bank of bandass fltrs n th mnmum varanc sctrum stmaton Th nut to th fltrs s (n), and th oututs of th bandass fltrs ar y (n) for,,, L. To us such a fltr bank to stmat th owr sctrum of (n) wth a fnt-lngth data rcord, w should constran all bandass fltrs that, at thr cntr fruncs ω, hav a unt gan, 4

10 G ( ) n g ( n) jnω (75) so that th owr sctra of th fltrs' oututs y (n) ar P y ( ) G ( ) P ( ) P ( ). Usng th vctor notatons, g g (), g (),, g ( ) (76) and [ ] T j [ ] T ω,,, (77) E. (75) can b wrttn as H G ( ) (78a) g or uvalntly as [ ( )] j ω G H g (78b) Snc th autocorrlatons of th outut rocsss y (n) and th nut rocss (n) ar rlatd n th followng mannr (s E. (9) n DISCRETE-TIME RANDOM PROCESS (3)) ry k) r g g ( k) l m ( g ( l) r ( m l + k) g ( m), (79) thn th varanc of th outut rocss y (n), whch s ual to r (), s of th form y ry () l m σ g ( l) r ( m l) g ( m), (8) and ts matr form wll b H σ g R g, (8) y whch shows th rlaton of th varanc of th outut of th th fltr wth th fltr coffcnts g (n), for a gvn random rocss wth th autocorrlaton matr R. y C. Th FIR bandass fltrs wth mnmum varanc Dsgnng a fltr s just dtrmnng th fltr coffcnts basd on a crtan crtron. Th crtron that w us hr s th mnmum varanc of y (n), whch s obtand by mnmzng σ y n E. (8) undr th constrant gvn by E. (78). Th aroach to ths constrand mnmzaton roblm s gvn n Scton.3. n th Hays' ttbook. Usng ths aroach, th coffcnts of th otmum fltr n trms of mnmum varanc ar found as follows R g (8) H R whch obvously satsfs dtrmnd as follows { } H g H y g R g H R, and nsrtng E. (8) nto E. (8) th mnmum varanc s, thus, mn σ (83) whch s th bst stmat of th varanc of th rocss (n) at fruncy ω n trms of mnmum-varanc, that s, ˆ ( σ ω ) mn{ σ }. Snc Es. (8) and (83) ar drvd at an arbtrary fruncy ω, thn ths two y uatons hold for all ω. Thrfor, th otmum fltr and th stmat of th varanc of (n) can b wrttn, rsctvly, as 5

11 R H R g (84) and ˆ σ ( ω) (85) H R whch s fruncy dndnt, and whr r () r () r ( ) r () r () r ( ) H R E{ } r () r () r ( ) (86) r ( ) r ( ) r () and [ g (), g(),, g( ) ] T g (87) [,,, ] T. (88) Tll now w hav found th varanc stmat of th rocss (n) but not th owr sctrum stmat yt. D. Th mnmum varanc sctral stmat To fnd th owr sctrum stmat, lt us look at th bandass fltr bank agan. Snc th bandass fltrs ar narrowband and th bandwdth of th th fltr G ( ) s assumd to b, thn n th bandwdth, that s, ω ω ω +, w may assum G ( ) (du to th gvn constrant n E. (75)), and out of th bandwdth, G ( ). In ths cas, th rlaton of th varanc of y (n) wth th owr sctrum of (n) n E. (73) bcoms π π σ y P ( ) ( ) ( ) y dω G P dω π π π π π ω + ) dω P( π P( ω Snc th stmat of th varanc of (n) s ual to th mnmum varanc of y (n), that s, ) ˆ (89) σ mn{ } σ, thn th (bst) stmat of th owr sctrum of (n) n trms of mnmum varanc can b rssd, from E. (89), as π P( ) mn π { σ } ˆ σ ( ω) ˆ y π H R. (9) In E. (9) th bandwdth s stll unknown. To fnd, w consdr stmatng th owr sctrum of a wht nos rocss wth a zro man and a varanc of σ. Th autocorrlaton matr of th wht nos s R σ I, and thn th bandass fltrs wth mnmum varancs ar R H R g σ I H σ I σ H σ, (9) + whch s fruncy dndnt, and th stmat of th varanc of (n) s σ ˆ σ ( ω) σ H R H σ I H (9) + whch s ndndnt of fruncy. Substtutng E. (9) nto E. (9) ylds th mnmum varanc stmat of th owr sctrum, y 6

12 P ˆ ( π ) ˆ σ ( ω) π σ +. (93) Snc th owr sctrum of a wht nos rocss s ual to ts varanc σ, thn w st th owr sctrum of wht nos to b ual to ts stmat,.., ˆ P ( ) P ( ) σ, (94) whch s actually an assumton w mos hr. Th bandwdth can, thus, b dtrmnd, π. (95) + For a gnral WSS random rocss (n), w adot ths bandwdth for th wht nos cas, and thn th stmat of th owr sctrum of th gnral rocss (n) n E. (9) bcoms ˆ ˆ + P ( ) PMV ( ). (96) H R Usually R s unknown, and thn R may b rlacd wth an stmat, Rˆ. Snc th otmum bandass fltrs ar stablshd basd on th autocorrlaton Rˆ whos valus ar dtrmnd from th data, thn th mnmum varanc sctrum stmaton may b thought of as a dataadatv modfcaton to th rodogram. Gnrally th mnmum varanc sctrum stmaton offrs hghr rsoluton than th rodogram and Blackman-Tuky mthods. It should b notd that, although th MV mthod s stablshd usng a bank of fltrs, th MV sctral stmat dos not nd to us th fltrs n th nd. On of th rasons s that th bandwdth found for wht nos n E. (95) s adotd to a gnral WSS random rocss so that th MV sctral stmat s ndndnt of th fltrs whos bandwdth should b dtrmnd from th gnral random rocss. Snc n th MV sctral stmat no fltr modl and thus no modl aramtrs nd to b found and usd, thn th MV sctral stmaton falls nto th catgory of th nonaramtrc mthods. Not that th nvrs transform of th MV stmat dos not match th autocorrlaton sunc that s usd to crat th MV stmat, unlk th autorgrssv sctrum stmat that dos match. Eaml 5. MV stmat of th owr sctrum of an AR() rocss (Eaml. 8.3.) 7

13 Mamum ntroy mthod (an AR mthod): Th mamum ntroy sctral stmaton s stablshd basd on an lct traolaton of a fnt lngth sunc of a known autocorrlaton of a random rocss (n). Th traolaton should b chosn so that th random rocss charactrzd by th traolatd autocorrlaton sunc has mamum ntroy. Th random rocss tratd hr s assumd to b Gaussan so that th concrnd roblm bcoms solvabl. A. Th conct of ntroy Entroy s a masur of randomnss or uncrtanty. For a Gaussan random rocss (n) wth owr sctrum P ( ), th ntroy of th random varabl (n) s rssd by π H( ) ln P ( ) dω (97) π π B. Etraolaton of th autocorrlaton Gvn th autocorrlaton r (k) of a WSS rocss for k, w want to traolat r (k) for k >. Suosng that th traolatd autocorrlaton s r (k), th owr sctrum of (n) can b wrttn as P ( ) k r + k > r Now th uston s how or what crtron should b usd to dtrmn th traolatd autocorrlaton. As th nam of th mthod ndcats, th mamum ntroy s th crtron for rformng th traolaton. A mamum ntroy traolaton s uvalnt to fndng th sunc of th traolatd autocorrlatons that mak (n) as wht (random) as ossbl. From th owr sctrum ont of vw, ths mamum ntroy traolaton maks th owr sctrum as flat as ossbl. (98) C. Th mamum ntroy sctral stmat If a random rocss (n) s assumd to b a Gaussan rocss wth a gvn sgmnt of th autocorrlaton r (k) for k, thn th traolatd autocorrlaton r (k) that mamzs th ntroy n E. (97) can b found by sttng H ( ) r, scfcally H ( ) r π π P ( π P ( ) r ) dω, k > (99) Usng th conjugat symmtry r r ( k) n E. (99), th drvatv n th ntgral bcoms P ( ) P ( ) r r ( k) Insrtng E. () nto E. (99), w hav π π π P ( ) jkω r ( k) r r ( k) k > r ( k) k > ( k) jkω jkω () dω, k > () Dfnng Q ( ) P ( ), E. () bcoms π jkω Q ( ) dω ( k), k > () π π whch shows that th nvrs Fourr transform of Q ( ), namly (k), s ual to zro for k >. Thrfor, th Fourr transform of (k) s 8

14 Q ( ) j. (3) ω P ( ) k From E. (3), w may dfn th mamum ntroy (MEM) stmat of th owr sctrum P ( ) for a Gaussan rocss, as follows j P ω mm ( ) (4) Notng that ( k) and jkω + + () sctral factorzaton on E. (4) and may hav ˆ P ( ) n th followng form Pˆ mm ( ) + a b() b () + a mm jkω + b() a b() A ( ) A, w can rform (, (5) ) whch s th sam as th AR sctrum stmat. Snc r (k) ar gvn for k, th coffcnts (k) can b found by solvng th followng normal uatons r () r () r ( ) r () () r () r ( ) a ε (6) r ( ) ( ) () a ( ) r r and b () can b dtrmnd from r () + k b( ) a r ε. (7) Dfnng a [, a (),, a ( ) ] T and [,,, ] T P ˆ mm, th MEM may wrttn as ( ε j ω ) (8) H a Th mamum ntroy mthod s uvalnt to th Yul-Walkr mthod (YWM), an AR sctrum stmaton mthod. Th dffrnc btwn th two mthods ls n that n th MEM th random rocss (n) s assumd to b Gaussan, whras n th YWM (n) s assumd to b an AR rocss. a Eaml 6. MEM stmaton of th owr sctrum of a coml onntal n nos (Eaml 8.4.) 9

15 Summary of sctral stmaton mthods: Powr sctrum of a WSS rocss (n): transform of th autocorrlaton sunc r (k). r P ( ), (not < k < ) whch s th Fourr P ( ) can only b stmatd whn th data avalabl for a random rocss (n) ar of fnt-lngth or th data ar contamnatd wth nos. Estmatng th owr sctrum s uvalnt to stmatng th autocorrlaton. Two classs of mthods for owr sctrum stmaton: nonaramtrc mthods and aramtrc mthods In ach class thr ar a st of mthods. Snc P ( ) s stmatd wth ths or that mthod, th rformanc of th stmatng mthod must b valuatd by lookng nto th ctd valu (bas) and th varanc of th stmat P ˆ ( ). Som usful MATLAB functons >> P PYULEAR(X,ORDER) % rturns th PSD stmat of a dscrt-tm sgnal vctor X n % th vctor P, usng Yul-Walkr's mthod. >> P PCOV(X,ORDER) % rturns th PSD stmat of a dscrt-tm sgnal vctor X n th % vctor P, usng th Covaranc mthod. >> P PBURG(X,ORDER) % rturns th PSD stmat of a dscrt-tm sgnal vctor X n th % vctor P usng Burg's mthod. >> P PMCOV(X,ORDER) % rturns th PSD stmat of a dscrt-tm sgnal vctor X n th % vctor P, usng th Modfd Covaranc mthod. 3