Power Spectrum Estimation of Stochastic Stationary Signals

Transcription

1 ag of 6 or Spctru stato of Stochastc Statoary Sgas Lt s cosr a obsrvato of a stochastc procss (). Ay obsrvato s a ft rcor of th ra procss. Thrfor, ca say: <,... ˆ hr s,... a a rctag o. o t s s hat s th ffct of ths tato or trucato of th sga (), o ts SD. By fto () FFT SD ˆ. Thrfor, ) (. { } { } o o Lt Rag of : [, ] rag of [ ] : Rag of : [, ] { } () () but ()

2 ag of 6 a thrfor { () } { } o 3 WB ( ) ( ) () { } () () () Tru SD of () Thrfor, th ffct of trucato o th por spctru of th sga s that gt a asyptotcay ubas stator of th tru SD of th sga. ( ) s ca roogra. As sa, t s a asyptotcay ubas stator but ts varac s ot cosstt or o bcaus var ˆ ( ) { } ( ) a t os t go to zro. That s hy roogra s rfrr as a osy stator of th tru SD of (). Thr ar a f thos to ry ths varac prob. Bartt (Avragg roogra) tho I orr to ruc th varac of SD stators, Bartt sgt th ata to sgts ach th gth. (, hr s th ubr of ata saps), coput th roogra of ach sgt a th got th avrag of th as th SD stator. ata of ach sgt: () ( ),,., # of sgts. sgt,.., gth of ach roogra of ach sgt:

3 ag 3 of 6 ( ) B Th th avrag roogra s: p ( ) Lts s ts statstca proprts: B { ( )} p ( ) ( { } { p ( )}., hch s th sa as bas of roogra () ) a hc, asyptotcay ubas stator. 3 WB Th bas s, hch go to f W () B s ca Bartt o, hch s a hrt o as a s rsut of trucato. Lt's oo at th varac of Bartt tho: var B { ( )} var { } var{ ( )} Thrfor, th varac s ruc by th factor. So, t ss as cras th ubr of sgts (), ruc th varac of stator or. But, ths s at th cost of osg frqucy rsouto. Wch tho Wch tho s fact th of Bartt tho by th foog to ofcatos:. Sgts hav a ovrap. Appyg a o rathr tha a Rctag o to ach sgt. Thrfor, () ata saps ach sgt ( D), hr,.., - gth of ach sgt,.., - ubr of sgts To o sgtato atab: For :

4 ag 4 of 6 : ; th 5% ovrap D/ ; If D, th thr s o ovrap. If, D thr s a 5% ovrap bt th succssv sgts. Aso, thr s a o for sgtato ca Data Wo () s a o such that f f W. Thr ar ay ffrt chocs for () such as Hag, Hag, Baca, arz, tc., os. Th por spctru stato Wch tho s th f as. hr,...,, U a U s th orazato factor to rsut f f W o, { } { } { } a { } { } U U but, Γ { } U Γ But S ( ) SD of a U S { } W W S S Γ Γ

5 ag 5 of 6 So th bas of SD th th Wch tho ps o th shap of ata o. You ca pct th arror th a ob of S W (), th sar th bas. But aga as usua, th ssr bas ay caus a argr varac of th stator. var { } { ( )} var Γ 9 Γ 8 ( ) ( ) th o ovrap th 5% ovrap ot that th cas of o ovrap abov s ot th sa as th o th cas of 5% ovrap. Suary I suary a goo ay to stat th por spctru of a statoary stochastc sga s to sgt t to qua gth ovrappg (5%) sgts a th cacuat th FFT of ach sgt, ta ts agtu to th por of a f th avrag of t bt th sgts. Spctrogra of Stochastc o-statoary Sgas by Short-T Fourr Trasfor (STFT) O tho to stat por spctru of a rao sga that s o-statoary gra ( ay auo sga) s to v th sga to f gth short-urato sgts a cacuat th FFT of ach sgt a sho th spctrogra. I ths tchqu, th urato of th sgts ust b short ough to sur that th sga ras statoary th that urato. I orr to hy o ust o sgtato for o-statoary sgas (h for statoary sgas t s bttr to o but o t hav to), t's o th foog prts. Dooa th av f "sash.av" a typ th foog co a ru atab. %% Aaptv sgtato

6 ag 6 of 6 [y fs]avra('sash'); sousc(y,5) %% stg to th sou of sga y(5:63); %% ths s th part that sous "sh"! fgur subpot(,,) pot(), ho, bar([ ],[- - ]); %% pottg th bouars (:5); (5:35); 3(35:gth()); subpot(,,), tt(' spctru') ps(,gth(),5,[],) fgur subpot(,,), tt(' spctru') ps(,gth(),5,[],) subpot(,,), tt('3 spctru') ps(3,gth(3),5,[],) Obsrv that th thr spctra hav ffrt frqucy charactrstcs. o, t's prt th f sgtato tchqu. Ra th hp of atab for th coa "spcgra". It s a fucto to pot th spctrogra of a sga D. Th agtu s bg sho by coor. You ca choos th gth of sgt a aso choos to put ovrap bt aact sgts or ot. Ru th foog co a obsrv th pots. %% f sgtato fgur spcgra(,56,5,[],) tt('urato of ach sgt56 saps3 sc, ovrap bt th sgts') fgur spcgra(,56,5,[],8) tt('urato of ach sgt56 saps3 sc, ovrap bt th sgts5%') Thrfor, for th o-statoary sgas ay boogca sgas or ay auo sga, shou tr th STFT hch s cacuat by th foog stps: - Sgt th ata to qua gth ovrappg sgts (5% ovrap). - Cacuat th FFT of th sga ach sgt. 3- Ta thr th og (abs(fft(sga))) or abs(fft(sga)).^ a that s th por spctru of vry sgt. Of cours you oy haf of th gth of th FFT.