Performance Evaluation of Filter-bank based Spectrum Estimator

Transcription

1 Intrnatonal Journal of Modrn Engnrng Rsarch (IJMER Vol., Issu., pp-47- I: Prformanc Evaluaton of Fltr-bank basd pctrum Estmator M.Vnakatanarayana, Dr.T.Jayachandra Prasad Assoc. Prof, Dpt. of ECE, KRM Collg of Engg., Kadapa. Profssor, Dpt. of ECE, RGMCET, andyal, Abstract In ths papr on attmpt has bn mad to study th prformanc of Fltr-bank basd nonparamtrc spctral stmaton. vral mthods ar avalabl to stmat non paramtrc powr spctrum. Th pass fltr, whch swps through th frquncy ntrval of ntrst, s man lmnt n powr spctrum stmaton stup. Th fltrbank basd spctrum stmaton s dvlopd and s appld to mult ton sgnal. Th spctrum stmatd basd on fltr-bank approach has bn compard wth convntonal nonparamtrc spctrum stmaton tchnqus such as Prodogram, Wlch and Blackman- Tuky. It s obsrvd that th fltr-bank mthod gvs bttr frquncy rsoluton and low statstcal varablty. It s also found thr s a tradoff btwn rsoluton and statstcal varablty. Indx Trms Corrlogram, Fltr-bank, Frquncy rsoluton, Prodogram, pctral lakag tc., I. ITRODUCTIO In gnal procssng, th nonparamtrc spctrum stmaton plays an mportant rol n dtrmnng prodtcty n random sgnals and thus a comprhnsv laboraton of fltr-bank basd spctrum stmaton tchnqus has bn prsntd. In gnral, spctrum stmaton can b catgorzd nto drct and ndrct mthods. In drct mthod (usually rcognzd as frquncy doman approach, th powr spctrum s stmatd drctly from sgnal bng stmatd x (. On th othr hand, n ndrct mthod, also known as tm doman approach, th autocorrlaton functon of th sgnal bng stmatd s calculatd. From ths autocorrlaton valu, th powr spctrum dnsty can b found by applyng th Dscrt Fourr Transform on. Anothr way to catgorz spctrum stmaton mthods s by classfyng thm nto paramtrc or non-paramtrc mthods. Paramtrc mthod s bascally modl basd approach []. In ths mthod, a sgnal s modld by Auto Rgrssv (AR, Movng Avrag (MA or Auto Rgrssv Movng Avrag (ARMA procss. Onc th sgnal s modld, all paramtrs of th undrlyng modl can b stmatd from th obsrvd sgnal. Estmator basd on paramtrc mthod provds hghr dgr of dtal. Th dsadvantag of paramtrc mthod s that f th sgnal s not suffcntly and accuratly dscrbd by th modl, th rsult s lss manngful. on Paramtrc mthods, on th othr hand, do not hav any assumpton about th shap of th powr spctrum and try to fnd accptabl stmat of th powr spctrum wthout pror knowldg about th undrlyng stochastc approach. Th followng sub-sctons gv rvw on som of th spctrum stmaton mthods. A. Prodogram Th most commonly known spctrum stmaton tchnqu s prodogram, whch s classfd as a non paramtrc stmator. Th procdur starts by calculatng th Dscrt Fourr Transform (DFT of th random sgnal bng stmatd, followd by takng th squar of t and thn dvdng th rsult wth th numbr of sampls. Thr ar fv common nonparamtrc PE avalabl n th ltratur: th prodogram, th modfd prodogram, Bartltt s mthod, Blackman Tuky, and Wlch s mthod. Howvr, all ths nonparamtrc PE, s ar modfcatons of th classcal prodogram mthod ntroducd by chustr. Th prodogram s dfnd as [] n jkn ˆ k ( xx It s known that th prodogram s asymptotcally unbasd but nconsstnt bcaus th varanc dos not tnd to zro for larg rcord lngths. On can show [], that th Varanc on ˆ th prodogram xx of an rgodc wakly statonary sgnal x ( for n : s asymptotcally proportonal to xx, th squar of th tru powr at frquncy bn. Th prodogram uss a rctangular tm-wndow, a wghtng functon to rstrct th nfnt tm sgnal to a fnt tm horzon, th modfd prodogram uss a nonrctangular tm wndow [3]. A way to nforc a dcras of th varanc s avragng. Bartltt s mthod dvds th sgnal of lngth 47 P a g

2 Intrnatonal Journal of Modrn Engnrng Rsarch (IJMER Vol., Issu., pp-47- I: nto K sgmnts of lngth L ach. Th K prodogram mthod s thn appld to ach of th K sgmnts. Th avrag of th rsultng stmatd powr spctra s takn as th stmatd powr spctrum. On can show that th varanc s rducd by a factor K, but th spctral rsoluton s also dcrasd by a factor K, []. Th Wlch mthod lmnats th tradoff btwn spctral rsoluton and varanc n th Bartltt mthod by allowng th sgmnts to ovrlap [4]. B. Blackman-Tuky mthod (Wndowd Corrlogram Blackman-Tuky mthod s a varant of corrlogram that computs th approxmatd autocorrlaton and latr appls a sutabl wndow functon w[k]. Th powr spctra dnsty s thn obtand by computng th Fourr Transform of wndowd auto-corrlaton squnc []. Blackman-Tuky mthod s gnrally dscrbd as follows ( And ts frquncy doman rprsntaton s gvn by (3 from quaton (3 th Blackman-Tuky mthod can actually b vwd as a procss of smoothng th corrlogram by convolvng th corrlogram wth th krnl of slctd wndow. Ths smoothng procss plays an mportant rol to rduc th bas of stmatd PD but ths convoluton procss would rduc th frquncy rsoluton. Th amount of frquncy rsoluton rducton s strongly rlatd to th sz of th man lob of th wndow krnl. cton II talks about fltr bank approach to spctrum stmaton tchnqu. cton III focuss th prformanc analyss of th proposd tchnqus n comparson wth th convntonal tchnqus. II. PECTRUM ETIMATIO A A FILTER BAK AALYI From th prspctv of spctrum stmaton, a fltr bank can b consdrd as an array of pass fltrs that sparats th nput sgnal nto svral frquncy componnts, ach on carryng a sngl frquncy sub- [6]. Th fltr banks ar usually mplmntd basd on sngl prototyp fltr, whch s a low pass fltr. Ths low pass fltr s normally usd to ralz th zro-th of th fltr bank whl fltrs n th othr s ar formd through th modulaton of th prototyp fltr [9]. Fgur.6 llustrats th man da of fltr bank concpt. Ths scton bascally trs to xplor th fltr bank paradgm n spctrum stmaton. Fgur. Th fltr bank concpt. A. Prodogram spctral stmator ralzaton through fltr banks pctrum stmaton s about fndng th powr spctrum dnsty (PD of a fnt sampl st {, n,,.... } for frquncy. Th classcal approach to spctrum stmaton s to us Fourr transforms to obtan a Prodogram, gvn as [7]: p jf x n for any gvn frquncy p xx ( jf n n jfn f, (4 s wrttn as: jf n jf ( t should b notd that ( s possbl ( j f snc. By ntroducng nw varabl H ( f p xx ( k n jf Prototyp fltr ( th st, quaton ( s wrttn as: k k k jf n h k ( jf k whr h for k,,,... by obsrvng th summaton wthn th magntud opraton n (6 and th summaton xprssd as: nd y( h k (7 k (7 s actually th truncatd convoluton sum at partcular pont, whch s agan wrttn as gnral convoluton sum at th sam pont assocatd wth a lnar causal systm by paddng th f (4 ( (6 48 P a g

3 Intrnatonal Journal of Modrn Engnrng Rsarch (IJMER Vol., Issu., pp-47- I: h (k wth zros[43]. Thn (7 s rwrttn as wth w( k h ya ( k ( \ jf k h k fork,,,... othrws and wndow functon w /. It s clar that (8 rprsnts as passng sampls through a fltr havng mpuls rspons h (k and thn takng only sngl sampl of th fltrd sgnal at pont. Basd on ths prspctv, th frquncy rspons of th lnar fltr havng mpuls rspons h (k s H ( k h jk k (8 (9 ( fnally gvs sn[ ( / ] H ( xp[ ( ] ( sn[ / ] If w (k n (9 s takn to b a prototyp FIR (Fnt Impuls Rspons low pass fltr, thn h (k s wll consttut a bank of pass fltrs cntrd at frquncs f s. Ths fltr bank s constructd by modulatng th prototyp fltr. By consdrng (4-(, th prodogram stmat at partcular frquncy pont f s obtand by passng th rcvd sampls through th pass fltr cntrd at f. Th powr calculaton of ths stmat s prformd basd only on a sngl sampl of th output of th fltr from (8. Th ntr prodogram stmats can thn b rlatd to th output of svral fltrs n th fltr bank constructd by modulatng a sngl prototyp fltr w (k. For th cas of smpl prodogram, th wndow functon w (k s rctangular wth w /. As t s clar from (, th frquncy rspons of fltr basd on prototyp fltr havng rctangular wndow as ts mpuls rspons would hav sgnfcant lvl of sd lobs. Ths s actually th man rason why th prodogram stmats hav hgh sd lob or larg lakags. Ths problm can b allvatd by rplacng th rctangular wndow wth a wndow functon wth a tapr that smoothly dcays on both sds to obtan a prototyp fltr wth much smallr sd lobs. A fw popular wndows ar Hannng, Kasr and Blackman [8]. Fgur. Th dmodulaton of rcvd sgnal From th abov, th mplmntaton of a spctrum stmator usng fltr bank for sgnal analyss s clar, namly by passng an nput sgnal through a bank of fltrs. Th output powr of ach fltr s a masur of th stmatd powr ovr th corrspondng sub-. Hnc th powr spctral dnsty (PD stmat of -th sub of th fltr bank s rprsntd as [6]: ( In (, avg [] dscrbs tm avrag oprator whl y ( s th output sgnal of th sub fltr. III. REULT AD AALYI Consdr a sgnal havng two closd frquncs mbddd n wht nos.., whr f., f. and s wht nos havng zro man and unt varanc. Th prformanc of th stmaton tchnqus s manly valuatd wth rspct to thr dffrnt paramtrs: Frquncy rsoluton Varanc of th stmatd powr spctrum dnsty (PD Fg. 6 dpcts Prodogram, Blackman-Tuky, and Wlch approach as wll as fltr-bank basd stmats for th cas of a sgnal havng two closd frquncs. In ths fgurs, th numbr of sampls usd n th xprmnt s =8,6 and. For th purpos of ths xprmnt, th Wlch approach dvds th rcvd sampls nto M=4 sgmnts of K=/M sampls. Two conscutv sgmnts ovrlap to on anothr by %. Bfor prformng th avragng procss, Hammng wndow s appld on ach sgmnt. As n th cas of Blackman-Tuky approach, a trangular wndow s usd havng ts lngth K=/. Th numbr of pass fltr n fltr-bank mplmntaton s dnotd by K. In Fg.6, assumd K=, whr both th fltr-bank basd spctrum stmat and prodogram basd spctrum stmat hav good frquncy rsoluton wth hgh statstcal varablty. In Fg. 7 and Fg.8, assumd K=4, whr th fltr-bank spctrum stmat s much bttr than prodogram havng low varablty whl mantanng accptabl frquncy rsoluton. It s also obsrvd that th lvl of stmatd powr n unoccupd for Wlch approach s hghr than for smpl prodogram manng that th Wlch approach offrs poorr rjcton n th unoccupd. Ths s undrstandabl snc 49 P a g

4 powr n db powr n db powr n db powr n db powr n db Intrnatonal Journal of Modrn Engnrng Rsarch (IJMER Vol., Issu., pp-47- I: Wlch approach dvds th rcvd sampls nto svral sgmnts wth lowr numbr of sampls bfor stmatng ach sgmnt. Hnc Wlch approach offrs vry low varablty and poor frquncy rsoluton. Th Blackman- Tuky mthod s also good stmats to powr spctrum havng low varablty and modrat frquncy rsoluton. Fnally, w notcd that whn K=4 and =6 and, wth rspct to varanc and frquncy rsoluton, th fltr bank approach most prfrabl to stmat powr spctrum among th abov tchnqus. 3 3 prodogram and fltrbank spctrum program FBE 3 3 prodogram and fltrbank spctrum program FBE normalzd frquncy f Hz 3 3 (a Blackman spctrum and Wlch spctrum BTE Wlch normalzd frquncy f Hz 3 3 (a Blackman spctrum and Wlch spctrum BTE Wlch normalzd frquncy f Hz (b Fgur.7. (a Prodogram and fltrbank basd spctrum (b blackman tuky an Wlch basd spctrum (K=4, =8 4 4 prodogram and fltrbank spctrum program FBE normalzd frquncy f Hz (b Fgur. 6 (a Prodogram and fltrbank basd spctrum (b blackman tuky an Wlch basd spctrum (K=, = normalzd frquncy f Hz (a P a g

5 powr n db Intrnatonal Journal of Modrn Engnrng Rsarch (IJMER Vol., Issu., pp-47- I: Blackman spctrum and Wlch spctrum BTE Wlch normalzd frquncy f Hz (b Fgur.8. (a Prodogram and fltrbank basd spctrum (b blackman tuky an Wlch basd spctrum (K=4, = Tabl I: th numbr of pass fltrs K=, In Wlch approach (hammng wndow, % ovrlap and /4 sgmnts and n Black-man Tuky (hammng wndow of lngth / Varanc =8 =6 = Prodogram Fltr-bank approach Black-man tuky Wlch Tabl II: th dtals ar sam as n Tabl I, xcpt chang n th numbr of pass fltrs K=4 Varanc =8 =6 = Prodogram Fltr-bank approach Black-man tuky Wlch Th varanc analyss among ths tchnqus as follows From Tabl I and Tabl II, for lngth of sampls =8,6 and, th Blackman-Tuky approach offrs low varablty than othr mthods. From Tabl II, for a rcord lngth of =6 and, th FBA offrs low varablty wthout scarfyng frquncy rsoluton than othr mthods. Th frquncy rsoluton among ths tchnqus as follows From Fgurs, for rcord lngth of =6, th Wlch approach dos not rsolv two closd frquncs. Its rsolvng capablty ncrass wth data rcord lngth. From Fgurs, whn K=4, =6 and =, th Fltr-bank approach rsolv two closd frquncs. IV. COCLUIO In ths papr on attmpt has bn mad to dvlop and mplmnt fltr bank basd nonparamtrc spctral stmaton tchnqu. Th proposd tchnqu has bn subjctd to mult ton sgnal, and stmatd th spctral componnts. Th prformanc of proposd tchnqu has bn compard wth th convntonal mthods of nonparamtrc spctrum stmaton such as prodogram, Wlch and Blackman-Tuky. It s obsrvd that th FBA mthod produc spctral stmats wth hgh rsoluton and low statstcal varablty at xpns of ncrasd th numbr of pass fltrs. Hnc, thr s tradoff btwn rsoluton and statstcal varablty. Th studs show that th Fltr bank basd spctrum stmaton s smpl, offrs grat flxblty, rconfgurablty and adaptablty. REFERECE []. Haykn, Cogntv Rado: Bran-mpowrd Wrlss Communcatons, IEEE [] J.G. Proaks and D. G. Manolaks, Dgtal gnal Procssng: Prncpls, Algorthms, and Applcatons, Fourth Edton. Uppr addl Rvr, J: Prntc Hall, Inc, 7. [3] M.. Bartltt, moothng Prodograms from Tm rs wth Contnuous pctra, atur (Londo, Vol.6, May 948. [4] P. D. Wlch, Th Us of Fast Fourr Transform for th Estmaton of Powr pctra: A Mthod Basd on Tm Avragng Ovr hort, Modfd Prodograms, IEEETransactons on Audo and Elctroacoustcs, vol. AU-, no., pp. 7-73, Jun 967. [] F. J. Harrs, Multrat gnal Procssng for Communcaton ystms, w Jrsy: Prntc-Hall PTR, 4. [6] B. Farhang-Boroujny, Fltr Banks pctrum nsng for Cogntv Rados, IEEE Transacton on gnal Procssng, Vol. 6, pp. 8-8, May 8. [7] J. Lm and A.V. Oppnhm, Advancd Topcs n gnal Procssng, Englwood Clffs, J: Prntc Hall, 998. [8] D. J. Thomson, pctrum Estmaton and Harmonc Analyss, Procdng of IEEE, vol. 7, no. 9, pp. -96, ptmbr 98 [9] B. Farhang-Boroujny, A quar-root yqust (M Fltr Dsgn for Dgtal Communcaton ystms, IEEE Transacton on gnal Procssng, vol. 6, no., pp. 7-3, May 8. [] P. toca and R.L. Moss, Introducton to pctral Analyss. Uppr addl Rvr, J: Prntc Hall, Inc, P a g