1 T T 1 = T ν 1 = ν fundamental or 1 st harmonic. 2 T T 2 = T/2 ν 2 = 2 ν 2 nd harmonic. 3 T T 3 = T/3 ν 3 = 3 ν 3 rd harmonic

Transcription

1 PAR 4 Fourier Series ad Fourier rasform By the use of the famous Fourier Series, a periodic fuctio is expressed as a sum of harmoics. I the case of o-periodic fuctios a geeralisatio of the Fourier series is used i.e. the Fourier rasform where the sum is replaced by a itegral ad as a result the o-periodic fuctio is expressed as a itegral where also o-harmoics cotribute to the fuctio represetatio. Fourier Series: ay periodic fuctio ca be expressed as a sum of harmoic fuctios i.e. cos ad si (mathematical glossary) or harmoics (musical glossary). Mathematically this is expressed i the followig: HEOREM 1: If a cotiuous fuctio f(t) is periodic with period, i.e. with frequecy ν = 1/, the it may be approximated arbitrarily well 1 by a Fourier series : a 0 πt πt f (t) = + a cos + si, (1) = 1 where the coefficiets are give by: ad / πt = f (t) cos dt, = 0,1,,3, (a) / / πt = f (t)si dt. = 1,,3, (b) / he siusoidal fuctios (cosies ad siuses) which are added i Eq. (1) are called Fourier compoets. he coefficiets ad are called Fourier coefficiets. Let us ow characterize the various Fourier compoets to gai some isight. Please have i mid that a fuctio cos(at) or si(at) has period a = π/a. period, frequecy, ν commet a ay costat backgroud a 1 t t b 1 1 = ν 1 = ν fudametal or 1 st harmoic a 4 t 4 t b = / ν = ν d harmoic a 3 6 t 6 t b 3 3 = /3 ν 3 = 3 ν 3 rd harmoic a 4 8 t 8 t b 4 4 = /4 ν 4 = 4 ν 4 th harmoic As ca be see from the last colum of this table the oomatology characterizes fudametal the Fourier compoet which has the same period ad frequecy as the origial fuctio. All other Fourier compoets are called harmoics ad the Fourier compoet with frequecy ν = ν is characterized as «th harmoic». 1 Ουσιαστικώς, το θεώρηµα λεει ότι το όριο / lim (f (t) s k (t)) dt = k / 0, όπου s k (t) η αντίστοιχη πεπερασµένη σειρά. Το κατά πόσο η σειρά συγκλίνει σηµειακώς και αναπαριστά την f(t) προσδιορίζεται από τις συνθήκες Dirichlet. Αν λοιπόν (α ) η συνάρτηση είναι τµηµατικά συνεχής και µονότονη, και (β ) σε κάθε σηµείο ασυνέχειας υπάρχει το όριο από δεξιά και από αριστερά, τότε: k a πt πt f (t), όπου f συνεχής 0 lim + a cos + si = = 1 /, στα σηµεία ασυνέχειας k ( f (t ) + f (t + ))

2 Oe ca easily show that if the fuctio is eve (odd), the oly the ( ) Fourier coefficiets which multiply cosies (siusus) survive. Example 1 Oe of the simplest examples is the fuctio f(t) = 1 + cos(πt/), which has period. Accordig to Eqs. (), a 0 = ad a 1 = 1 i.e. i this case we have oly the costat backgroud ad the fudametal. period, frequecy, ν commet a 0 = 1 0 ay costat backgroud a 1 =1 t 1 = ν 1 = ν fudametal or 1 st harmoic he picture o the right shows a example of a composite curve (gree) which is the algebric sum of the other two curves: (a) the black oe i.e. the fudametal with frequecy 0.5 Hz, ad (b) the red oe i.e. the 3 rd harmoic with frequecy 1.5 Hz. he process of addig the black ad the red curve to costruct the gree oe is the harmoic sythesis. he opposite process of fidig which compoets are eeded to costruct the gree curve is called harmoic aalysis. he frequecy of the composite gree curve is 0.5Hz i.e. the greatest commo divisor (GCD) of (0.5 Hz, 1.5 Hz). I this case it coicides with the fudametal frequecy. b b 1 = t b 3 =0.5 6 t Example period, frequecy, ν commet 1 = ν 1 = ν fudametal or 1 st harmoic 3 = /3 ν 3 = 3 ν 3 rd harmoic

3 Almost the same example (Joos Fig.7) of a composite curve (gree) which is the algebric sum of the other two curves: (a) the black oe i.e. the fudametal with frequecy 100 Hz, ad (b) the red oe i.e. the 3 rd harmoic with frequecy 300 Hz. he frequecy of the composite gree curve ( pitch ) is 100 Hz i.e. the GCD of (100 Hz, 300 Hz). I this case it coicides with the fudametal frequecy. b 1 = b 3 =0.5 si( 00 πt ) si( 600 πt ) period, frequecy, ν commet 1 = 1/100 s ν 1 = 100 Hz fudametal or 1 st harmoic 3 = 1/300 s ν 3 = 300 Hz 3 rd harmoic Example 3(Joos Figs. 9-10) A composite wave (blue) whose compoets do ot iclude the fudametal frequecy i.e. the frequecy of the wave is ν but its compoets are ν, 3ν, 4ν ( the fudametal compoet ν is missig from the eergy spectrum ). Specifically, the blue curve (100 Hz) is made up from the sum of the black (00 Hz), the red (300 Hz) ad the gree (400 Hz) curve. he frequecy of the composite curve ( pitch ) is the GCD of (00 Hz, 300 Hz, 400 Hz). Although oe of its compoets is 100 Hz, the resultig composite wave has a pitch of 100 Hz!

4 Now, if we remove the 300 Hz compoet, the resultig blue curve will have a frequecy of 00 Hz i.e., agai, the pitch is the GCD of the compoets frequecies (00 Hz, 400 Hz). Now, the pitch of the composite curve coicides with the fudametal frequecy (00Hz). he correspodig fact of perceptio is that by removig the 300 Hz compoet we have trasformed a wave of pitch 100 Hz, ito a wave of pitch 00Hz! Πρέπει να το δοκιµάσω αυτό. Να δοκιµάσω επίσης το παρακάτω. Listeig to a composite soud wave with compoets 600 Hz, 800 Hz ad 1000 Hz, a pitch of 00 Hz will be heard. he, if to these a fairly strog compoet at 300 Hz is added, the pitch will be heard to drop oe octave to 100 Hz. hus, the pitch ca be defied as the greatest commo divisor (GCD) of the frequecies actually preset. Notice Joos paragraph 1.3 referrig to Figs. 6,8. It might seem that ot oly the siusoid but equally well some other shape, e.g. Fig. 8A or 8B, could be take as basic, so that a arbitrary exactly repetitive wave could tha be aalysed ito compoets of that chose basic shape, each compoet havig the same shape as the other compoets though with a differet period ad presumably a differet amplitude too. I IS NO POSSIBLE O DO SO. As soo as it is decided that the compoets shall all have the same shape ad shall be exactly repetitive, that settles it: the compoets will be siusoids. he siusoid is as basic to harmoic aalysis as the itegers 1,,3 are to ordiary arithmetic. Είναι έτσι; ες επίσης τι ακριβώς είναι τα wavelets. (Μήνυµα Σταυρινού) Example 3

5 A secod famous example is a pulse fuctio with a period = π, i.e. f (t) α, = + α, π < t < 0 0 < t < π ad likewise for ay we obtai: t R. Usig Eqs. () f (t) 4α si 3t si 5t = si t π 3 5 he followig two figures depict how addig the siusoidal fuctios with the correct amplitudes we obtai i the limit the pulse fuctio.. fuctios we add result after havig added each fuctio Aother form of heorem 1 is a 0 πt f (t) = + A si + φ, where = 1 A = a + b ad a ta = φ. (3) Yet, aother form of heorem 1 is obtaied usig complex otatio. Suppose that a fuctio f(t) is periodic with period, i.e. with frequecy ν = 1/, the it may be approximated arbitrarily well by a complex Fourier series : + πt i f (t) = c e (4) = where a 0, = 0 / πt 1 i c = f (t) ( i ) dt e =, > 0 (5) / (a + ib ), < 0 his form of heorem 1 via Eqs. (4-5) is the basis for the extremely importat Fourier trasform, which is obtaied by trasformig c from a discrete variable to a cotiuous oe as the period. I the case of o-periodic fuctios a geeralisatio of the Fourier series is used i.e. the Fourier rasform where the sum is replaced by a itegral ad as a result the o-periodic fuctio is expressed as a itegral where also o-harmoics cotribute to the fuctio represetatio. Next talk for Fourier rasform:

6 - Παραδείγµατα επιτυχούς και µη αναλύσεως Fourier. - Fourier aalysis of MEG sigal (α, µ rhythm or geeral). - Παραδείγµατα µετασχηµατισµών t f (useful for MEG ad MRI) και x k (useful for MRI). - he recogitio of differet vowel souds of the huma voice is largely accomplished by aalysis of the harmoic cotet by the ier ear. Refereces [1] Brostei Semedjajew, achebuch der Mathematik [] Fourier Series: ad Fourier rasform: Created by C. Simserides for Soud Properties: part3. Commets should be addressed to csimseri@if-magdeburg.de Next-Rest Bibliography: Fourier aalysis ad sythesis: Harmoic cotet differeces i vowel souds: Formig the vowel souds, vocal formats, vowel formats: =========== By the use of the famous Fourier Series, a periodic fuctio is expressed as a sum of harmoics. I the case of o-periodic fuctios a geeralisatio of the Fourier series is used i.e. the Fourier rasform where the sum is replaced by a itegral ad as a result the o-periodic fuctio is expressed as a itegral where also o-harmoics cotribute to the fuctio represetatio. I the case of o-periodic fuctios a geeralisatio of the Fourier series is used i.e. the Fourier rasform where the sum is replaced by a itegral ad as a result the o-periodic fuctio is expressed as a itegral where also o-harmoics cotribute to the fuctio represetatio. Next talk for Fourier rasform : - Παραδείγµατα επιτυχούς και µη αναλύσεως Fourier. - Fourier aalysis of MEG sigal (α, µ rhythm or geeral). - Παραδείγµατα µετασχηµατισµών t f (useful for MEG ad MRI) και x k (useful for MRI). - he recogitio of differet vowel souds of the huma voice is largely accomplished by aalysis of the harmoic cotet by the ier ear. Refereces [1] Brostei Semedjajew, achebuch der Mathematik [] Fourier Series: ad Fourier rasform: Next-Rest Bibliography: Fourier aalysis ad sythesis: Harmoic cotet differeces i vowel souds: Formig the vowel souds, vocal formats, vowel formats: last bookmark Created by C. Simserides for Soud Properties: part4. Commets should be addressed to csimseri@if-magdeburg.de