Thorcal Ssmology Lcur 9 Sgnal Procssng
Fourr analyss Fourr sudd a h Écol Normal n Pars, augh by Lagrang, who Fourr dscrbd as h frs among Europan mn of scnc, Laplac, who Fourr rad lss hghly, and by Mong. H was appond o h Écol Cnral ds Travau Publqus, h school bng undr h drcon of Carno and Mong, whch was soon o b rnamd Écol Polychnqu. In 1807 h asoundd many of hs conmporary mahmacans and scnss by assrng ha an arbrary funcon could b prssd as a lnar combnaon of sn and cosn funcons, calld Fourr rgnomrc srs, ar appld o h analyss of prodc phnomna ncludng vbraons and wav moon. Jan Baps Josph Fourr Born: 21 March 1768 n Aurr, Bourgogn, Franc Dd: 16 May 1830 n Pars, Franc
Rspons Spcrum Arup
Th Ssmomr Basc prncpl mass aachd o a movabl fram whn fram s shakn by ssmc wavs h nra of h mass causs s moon o lag bhnd rlav moon rcordd on roang drum, on magnc ap or dgally Mass s dampd o prvn connud oscllaon Ths lms h frquncy rspons of h ssmomr Rlav moon amplfd up o 100s of housands of ms Smon Day Schmac of a horzonal moon mchancal ssmomr
Vbraons: Smpl Harmonc Moon 0 2 2 2 + u u Smpls vbrang sysm: s h angular frquncy, f / π Thr ar wo soluons: u A sn and u B cos A and B ar amplud W can chck hs by subsung: cos A u sn B u sn 2 2 2 2 u A u sn 2 2 2 2 u B u Dsplacmn u
Vbraons: Smpl Harmonc Moon 2 u A sn and u B cos Ths ar boh soluons and h only dffrnc s h pon on h cycl a whch moon sars. If w shf h cos plo by a phas shf of ϕ w g h sn plo. ϕ So hs mans anohr soluon s u C cos + ϕ Or h gnral soluon s u A sn + B cos Anohr soluon s u C p[- + ϕ] cos + sn Dffrna hs o chck ha s a soluon s sqr of -1
Fourr dcomposon Hghr ordrs of r f Mans hghr and hghr frquncs a 2 { a cos πr / T + b sn πr T } 0 + r r / r 1 d.c. offs sum of sn & cos Can mak any funcon such as hs squar wav aka bo car funcon ou of h suprposon of sn and cosn wavs harmonc no.
Tm doman Any f can b dscrbd by a Fourr srs of sn and cos rms of dffrn frquncs: Fourr srs T < T f a a r 0 1 T 1 T a 2 T T 0 T T + f r 1 f d { a cos πr / T + b sn πr / T } r cos For vn fns b r 0 [ f f- ] For odd fns a r 0 [ f -f- ] πr / T d b f sn πr / T r r 1 T T T
Fourr dcomposon Squar wav or bo car funcon f 1 -T/2 T/2
Frquncy doman Amplud spcrum cosn harmoncs a r b r sn harmoncs 0 2 4 6 r π r / T r harmonc no. Amplud A r a r2 + b r2 Phas φ r an -1 b r / a r A r φ r r r
Sag pos W hav sn ha vbraons and wavs conan a whol rang of frquncs Thy can b rprsnd as a sum of sns and cosns of dffrn frquncs a Fourr srs Bu gvn a wav or mpuls, how do w fnd ou wha frquncs conans: W us h Fourr ransform Th sudy of frquncy conn of sgnals ssmc, lcronc, lgh, radowavs c., s calld spcral analyss Only a b can b don analycally, h rs rls on compurs Malab uoral
Fourr Transform End nrs from T < T o - < Th Fourr Transform of a funcon f s d F f d f F Invrs Fourr Transform s sn cos sn cos + Rmndr
Fourr Transform of a bo-car funcon d d f F T T T T f 1 -T T f 1 T < T f 0 lswhr Duraon 2T
Fourr Transform of a bo-car funcon 2sn T F Afr som algrbrac manpulaon w can fnd snc funcon 1/ dcay Phas and amplud In gnral F conan Ral and Imagnary componns vn f f s ral Im A 2 R 2 + Im 2 A φ an -1 R/Im φ R
Convoluon Th masurd rspons of h ssmomr s h convoluon of h arhquak sourc-m rspons, h Earh rspons u and h rspons of h ssmomr slf z. Th convoluon of 2 funcons f and g s dfnd mahmacally as: h f g f g d * Dnos convoluon s a dummy varabl of ngraon
Convoluon 2 Th Fourr Transform s: dd g f dd g f d h H Inroduc a facor 1 - d d g f H
Convoluon 3 L u du d f f d g u u g u dud u du Snc u and ar now ndpndn H F G Capals ar Fourr Transforms Now you don nd o rmmbr hs proof, bu you nd rmmbr:
Convoluon 4 So w hav shown ha convoluon n h m doman h f g has bcom a mulplcaon n h frquncy doman H F G Convoluon n h Mulplcaon n h Tm doman Frquncy doman Ths s h bass for all modrn sgnal procssng
Fas Fourr Transform Compur analyss Rplac h nfn sgnal f by fn sampls of h sgnal fnt S Rplac h ngral of h Fourr Transform wh h sum of h dscr Fourr ransform f fnt S Appromaon of a sgnal by samplng: no f you don hav nough sampls you mss ky faurs 0 T/N T S T NT S N sampls Th compuaon procss s calld h fas Fourr ransform
Comparson of Fourr chnqus Fourr srs Fourr ransform Dscr Fourr ransform DFT Fas Fourr ransform FFT f connuous F dscr f connuous F connuous f dscr F dscr f dscr F dscr Analyss of sgnals Frquncy analyss of sgnals Analyss of sampld sgnals Algorhm o compu DFT
Comparson of Fourr chnqus Prodc funcon Fourr srs f Dscr Spcrum f Connuous f Fourr ransform Connuous Spcrum Dscr prodc Dscr f Dscr Fourr ransform Spcrum DFT
Pon of Fas Fourr Transform Fas Fourr ransform nroducd by ssmologss Rapdly sprad o ol ndusry and mor rcnly o all sgnal procssng, ncludng musc Drc calculaon of DFT rqurd N 2 mulplcaons Calculaon of FFT rqurs N log 2 N mulplcaons For small sampl,.g. N4096, DFT rqurs mor han 16 mllon calculaons, whl h FFT rqurs lss han 50,000 So hs s h bass of modrn dgal sgnal procssng