Fourier Series: main points
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- Hilary Hodge
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1 BIOEN 3 Lcur 6 Fourir rasforms Novmbr 9, Fourir Sris: mai pois Ifii sum of sis, cosis, or boh + a a cos( + b si( All frqucis ar igr mulipls of a fudamal frqucy, o F.S. ca rprs ay priodic fucio ha w ca physically produc
2 Udrlyig pricipl: suprposiio + Fourir cofficis: rig form + si( cos( ( b a a f k d k f a d f a cos( ( ( + k d k f b si( (
3 + + + Sourc of h Fourir cofficis cos( m si( d cos( m cos( d si( m si( d + + cos ( m d, all m ad, m, m si ( m d Symmry of fucios Ev symmry: f( f(. 8 si(x x^4-*x^ Ev, Apriodic Ev, Priodic Odd symmry: f( f( 5.5 x^5-5*x^3+x Odd, Apriodic Odd, Priodic 3
4 Fourir cofficis: Complx xp. form j f ( C whr C + j Exampl: for priodic puls rai C f ( Vm τ si( τ / τ / si( x sic( x x Magiud ad phas plos Magiud plo shows C ( Phas shows a (Im{C }/R{C } Plo xiss a oly 4
5 Magiud ad phas plo - xampl Fourir Sris: scalig propry f ( C j Complx cofficis: +, C f ( j Magiud ad vary ivrsly wih Cofficis bcom smallr ad mor closly spacd as priod icrass 5
6 Fourir Sris: scalig propry Full-wav rcifid si.5 im domai -4-4 im, sc. Frqucy domai C frqucy, rad/sc Fourir Sris: scalig propry.5 Full-wav rcifid si, / im domai -4-4 im, sc. Frqucy domai C frqucy, rad/sc 6
7 Fourir Sris: scalig propry.5 Full-wav rcifid si, -4-4 im, sc im domai. Frqucy domai C C frqucy, rad/sc Fourir Sris: scalig propry.5 Full-wav rcifid si, im, sc im domai. Frqucy domai C C frqucy, rad/sc ad so o... 7
8 higs o o: Fourir Sris: scalig As,, i.. h sris bcoms coiuous As, C bu h sum of h cofficis ovr a irval o rmais fii hrfor, w ca calcula C, which is a fii, coiuous fucio of h rsulig fucio is... Radia form Hrz form h Fourir rasform G( g( j d g( G π ( G( f g( g( G( f j d jπf jπf df d 8
9 h Fourir rasform f ( higs o o: j d f ( π j d h F is a wighig fucio for siusoidal (or complx xpoial co i sigal h F rasforms a coiuous, apriodic fucio i im...io io a apriodic, coiuous fucio i frqucy Bcaus boh h F ad IF coai complx xpoials, hr ar may cass of dualiy amog F pairs A fw Fourir pairs f ( j d f ( π f( (or ay cosa j d his fucio has zro frqucy (or ifii priod h ara udr h F curv mus b fii for h ampliud of h im-domai sigal o b o-zro hrfor, F{} δ( 9
10 f( cos( A fw Fourir pairs his fucio has a spcific frqucy Ngaiv ad posiiv frqucis ar boh prs o cacl h imagiary par hrfor, F{cos( } (π/[δ(+ + δ( ] No ha F{cos( } is ral ad v R{} Im{} f( si( A fw Fourir pairs his fucio has a spcific frqucy A9 phas shif i a complx xpoial mas muliplicaio by j hrfor, F{si( } j π [δ(+ δ( ] No ha F{si( } is imagiary ad odd R{} Im{}
11 f( squar puls of high A ad widh b, crd a Ab si(b/b Ab sic(b h ampliud is h ara udr h puls is ral ad v if h puls is crd o. is complx if h puls is o crd. h IF of a puls i frqucy is a sic fucio i im rasfr fucios i h s-pla H(s / (s + 3s
12 Magiud plo G( 4/( (jw^ + 3(jw G omga For imag aalysis: wo-dimsioal Fourir rasformaio F ( u, v + f ( x, y jπ ( xu+ yv dxdy Igral form Opras o coiuous imag Dos o assum imag is priodic
13 For imag procssig: wo-dimsioal Fourir rasformaio F ( m, M N x y f ( x, y jπ ( xm+ y Summaio form Usually calld Discr Fourir rasform Opras o sampld imag (o coiuous Assums h imag is priodic i x ad y Opraioal rasforms f ( j f ( π raslaio i h im domai d j d Equival o muliplicaio i h frqucy domai raslaio i h frqucy domai Equival o muliplicaio by complx xpoial i h im domai No a smar hig o do is ral ad v if h puls is crd o. 3
14 Scal chag Opraioal rasforms f ( j d f ( π j d Wh o domai is srchd ou, h ohr domai is comprssd Exampl: icrass, dcrass Widr i im mas arrowr ad allr i frqucy Opraioal rasforms f ( j d f ( π Modulaio (.g. AM radio j d Ampliud of high-frqucy carrir is modifid by ampliud of low-frqucy sigal F{f(cos( } ½ + + ½ h origial sigal would hav crd aroud ; h modulad sigal would hav duplicad ad shifd alog h axis. Irsig... Wha was F{cos( }? 4
15 Covoluio ol Opraioal rasforms f ( j d f ( π j d h oupu y( from a sysm wih ui impuls rspos h(, wh h ipu is x(, ca b rprsd i wo ways: by covoluio i h im domai by muliplicaio i h frqucy domai Y( X(H( Similar o V OU (s V IN (sh(s i Laplac domai Laplac vs. Fourir f ( j f ( π Laplac rasforms ar br d j d For sysms aalysis (covrgc for widr variy of fucios For corol sysms aalysis Fourir rasforms ar br Easir o udrsad j axis ha s pla Basis for FF for discr daa Widly usd i sigal procssig 5
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