Analysis of Musical Sounds

Transcription

1 Aalysis of Musical Souds Musical souds are produced by he vibraio of physical sysems, e.g. Srig Isrumes guiars, piaos, violis ec.: Use he aural vibraio of sreched srigs or wires. Wid Isrumes rumpes, saxophoes, flues ec.: Use he aural resoaces of air chambers or orifices Percussive Isrumes seelpa, xylophoe, drums Use he aural vibraios of sreched membraes seel, aimal ski, paper or solid objecs he bars of a xylophoe, he rods of a glockespiel Elecroic Isrumes elecric piaos, orgas, syhesisers use he aural vibraios of cerai elecric circuis ad/or compuer algorihms. The soud is geeraed by feedig he correspodig volage flucuaios o a amplifier ad speaker. Dr. Bria Copelad FE3B Fourier Aalysis, P1/3 Feb 1 1

2 Aalysis of Musical Souds Soud Geeraor Model The soud wave geeraed by hese physical sysems may be show o resul from he e effec of several geeraors each of which has he followig equivale mechaical model: k sprig F1 mass m x F b dashpo Compoe srig wid percussive sprig esed srig air compressio esed membrae aural compliace mass srig mass air mass maerial mass dashpo air drag air drag fricio i maerial fricio a air fricio air resisace suspesio P. Dr. Bria Copelad FE3B Fourier Aalysis, P1/3 Feb 1

3 Aalysis of Musical Souds Differeial Equaio Soluio mass F 1 F sprig mx mx bx kx F 1 kx dashpo x x F bx Dr. Bria Copelad FE3B Fourier Aalysis, P1/3 Feb 1 3

4 Aalysis of Musical Souds The soluio o a iiial displaceme or a displaceme impulse a give by Dampig Frequecy speed σ Toe x Ae si φ Pich Time varyig ampliude: waveform evelope Differeial Equaio Soluio A,σ,,φ may be derived from he sysem parameers. This sysem is said o have a mode of vibraio of frequecy. We should be more ieresed i dx/d i.e. he velociy sice soud pressure developed is direcly proporioal o exciaio velociy. Mos sigifica parameers: evelope shape, sigal frequecy σ v x Be si θ Dr. Bria Copelad FE3B Fourier Aalysis, P1/3 Feb 1 4

5 Aalysis of Musical Souds Differeial Equaio Soluio a expoeially decayig siewave v Expoeial evelope Be - σ σ cosφ ; siφ k m ; φ cos 1 b km High frequecy siewave more ypical of musical souds Low frequecy siewave Dr. Bria Copelad FE3B Fourier Aalysis, P1/3 Feb 1 5

6 Aalysis of Musical Souds Frequecy Characerisics: Overoes & Parials Siusoidal vibraios are heard as PURE TONES I geeral, isrumes coai may vibraio modes i.e. hey do NOT geerae pure oes. However. For good soud he modal frequecies should be harmoically relaed i.e. if he fudameal mode has frequecy f, he oher compoes overoes, parials are a f,,3,4.. The relaive sreghs of he harmoic overoes deermie he isrume imbre. This is wha primarily allows us o disiguish bewee differe isrumes The huma ear has preferece for cerai combiaios of oe frequecies. The preferece is culure based. Huma hearig rage: Hz -- khz Preferred frequecy raios i decreasig order of preferece Weser Music: :1ocaves; 3: perfec 5hs; 4:3 perfec 4hs; 5:3 major 6h 5:4 major 3rd 8:5 mior 6h 6:5 mior 3rd Dr. Bria Copelad FE3B Fourier Aalysis, P1/3 Feb 1 6

7 Aalysis of Musical Souds Music Frequecies Music has a variey of sadard scales ha defie pich frequecies. These have all evolved over ime ad differ largely by culure. All are based o he divisio of a ocave A pich of frequecy f 1 is a ocave above aoher of frequecy f iff f 1 f Popular Weser Music o which mos seelpas are ued uses he jus emperame scale which evolved from he Pyhagorea scale o accommodae he piao The Jus Temperame scale divides he ocave io 1 logarihmically equal pars i.e. adjace oes a ierval of 1 semioe o he scale are i a geomeric raio of 1/1 The referece oe ad frequecy for he JT scale is A4 / 44Hz A i Ocave No. 4. Examples of oher scales: Chiese culures use 5oes/ocave, Arabic culures use 17 oes/ocave Dr. Bria Copelad FE3B Fourier Aalysis, P1/3 Feb 1 7

8 Aalysis of Musical Souds Sample souds ad specra The soud specrum is used o show he compoe frequecies ad relaive iesiies for a paricular soud. A sharp spike i he specrum idicaes he presece of a siusoidal compoe a he correspodig frequecy Specrum for C4 :pure oe Specrum for C4 oe o a flue C4~61Hz Dr. Bria Copelad FE3B Fourier Aalysis, P1/3 Feb 1 8

9 Aalysis of Musical Souds Sample souds ad specra Syhesised Expoeially damped C4 oe. Here σ.13, f C4 61.7Hz Dr. Bria Copelad FE3B Fourier Aalysis, P1/3 Feb 1 9

10 Aalysis of Musical Souds Dissoace ad Beaig Time waveform of resula of C4 ad C4# pure oes. Noe he beaig effec. Power Specrum of C4 C4# Dr. Bria Copelad FE3B Fourier Aalysis, P1/3 Feb 1 1

11 Aalysis of Musical Souds Beas ad oher dissoa sources... The bias o oly cerai combiaios is caused by a pheomeo called beaig which occurs whe oe frequecies are oo close ogeher -- ampliude of he resula sigal beas a a frequecy equal o he differece bewee he wo compoe frequecies. Resulig vibraio is 1 1 cos1 cos cos cos a frequecy w 1 w wih a ime varyig Ampliude modulaio ampliude evelope ses up bea a rae w 1 -w.rad/s For w 1 -w <7Hz he beaig effec is disracig o he ear whe i he rage. A higher differeces he sesaio is heard as a cosoace or dissoace, depedig o he frequecy raios. These will be reaed i our discussio o he hearig mechaism Dr. Bria Copelad FE3B Fourier Aalysis, P1/3 Feb 1 11

12 Aalysis of Musical Souds Beaig Pheomeo Dr. Bria Copelad FE3B Fourier Aalysis, P1/3 Feb 1 1

13 Aalysis of Musical Souds Evelope Characerisics:ADSR A D S R 4 compoes of musical soud evelopes: Aack, Decay, Susai ad Release. Aack ad decay secios deermie how he soud sars ad sops respecively. The susai ad decay secios, if hey exis, occur i he middle porios. The waveform show for he vibraio model is ypical of he seelpa - a very sharp aack secio followed by a almos expoeial release secio. The evelope of he geeraed soud varies grealy depedig o how he vibraory mechaism is excied.. E.g. a violi whe is srigs are bowed vs. whe is srigs are plucked. Dr. Bria Copelad FE3B Fourier Aalysis, P1/3 Feb 1 13

14 Dr. Bria Copelad FE3B Fourier Aalysis, P/3 Feb 1 1 Fourier Techiques for Aalysig Music Sigals Fourier Series Ay fucio wih frequecy f1/t ca be wrie as a ifiie sum of siusoidal waveforms of frequecy f f, where,1,,3 i.e. 1,,... d si 1,,... d cos average, sigal d 1 frequecy fudameal he frequecy of, where si cos / / / / / / T T T T T T f T b f T a b f T a f T b a T f f π Fourier Coefficies

15 Dr. Bria Copelad FE3B Fourier Aalysis, P/3 Feb 1 Fourier Techiques for Aalysig Music Sigals... si7 7 1 si5 5 1 si3 3 1 si, 4, 4 < < < < x x x x x x x g x g π π π π π E.G.Square Wave

16 Fourier Techiques for Aalysig Music Sigals These compoes are show as a discree lie specrum Dr. Bria Copelad FE3B Fourier Aalysis, P/3 Feb 1 3

17 Fourier Techiques for Aalysig Music Sigals The Fourier series herefore allows us o ideify he siusoidal geeraor compoes ad heir correspodig frequecies ad ampliudes i a musical oe susaied o ifiiy!!!!!! The si ad cos compoes ca be combied o give a oal magiude ad phase: acosxbsix a b cosx-θ, θa -1 b/a For music, more ieresed i compoe magiudes sice phase chages have o bee show o affec hearig percepio For music aalysis, he specrum y-axis is usually scaled i db because huma aural sesiiviy follows a logarihmic curve. Mai problem wih he F Series: ONLY ACCOMMODATES INFINITELY LONG PERIODIC SIGNALS ad is urealisic. Soluio: A aperiodic sigal is periodic wih period T 1/T. This leads o he Fourier Iegral represeaio ad a coiuous specrum... Dr. Bria Copelad FE3B Fourier Aalysis, P/3 Feb 1 4

18 Dr. Bria Copelad FE3B Fourier Aalysis, P/3 Feb 1 5 Fourier Techiques for Aalysig Music Sigals As T π π d si 1 d cos 1 d si cos f B f A B A f d 1 where d 1 j j e f F e F f π π Fourier Cosie Trasform of f Fourier Sie Trasform of f Fourier Trasform of f OR i more compac form more commo

19 Fourier Techiques for Aalysig Music Sigals Noes o he Fourier Trasform ad he correspodig ampliude specrum The specrum is a eve fucio of i.e. FF- The area uder he power specrum F gives he oal power i he sigal. A plo of F is called a Power Specral Desiy PSD plo The area uder F over [a,b] gives he oal power i he sigal over compoes i his frequecy rage. Musical isrumes used for playig melodies have sharp peaks i heir F. Each peak idicaes he exisece of a frequecy compoe a he peak value. Musical isrumes such as drums used for rhyhm have wha is called a broadbad specrum wih few, if ay, sharp peaks Dr. Bria Copelad FE3B Fourier Aalysis, P/3 Feb 1 6

20 Fourier Techiques for Aalysig Music Sigals Examples... [1] Fsi <> F1/ δ - δ-, a impulse [] Fe -σ si <> F jσ σ Is ampl specrum is coiuous. I may look like a spike o a liear y-axis. Here is wha i looks like o a db ampliude scale Dr. Bria Copelad FE3B Fourier Aalysis, P/3 Feb 1 7

21 Fourier Techiques for Aalysig Music Sigals Digial Compuaio of he FT... The Discree Fourier Trasform DFT approximaes Fw usig a sampled record of f. NB Every record of f is a sampled record. The DFT resuls i a a evaluaio of he specrum a evely spaced frequecies. For a N-sample record ad sample rae f s, he frequecy resoluio spacig is ff s /N The Fas Fourier Trasform FFT is a special algorihm for doig he DFT ha is less compuaioally expesive i erms of ime ad memory use. The effec of samplig geeraes errors i he compuaio which ca be reduced by icreasig he sample rae, f s. The miimum rae is called he Nyquis rae ad is wice he maximum frequecy compoe i he sigal beig sampled. Dr. Bria Copelad FE3B Fourier Aalysis, P/3 Feb 1 8

22 Fourier Techiques for Aalysig Music Sigals Digial Compuaio of he FT... The sampled record is of fiie legh ad is herefore usually a rucaed versio of he acual sigal. I essece i is obaied by muliplyig he sigal by a recagular pulse of heigh 1 over he ime occupied by he record. DFT algorihms refer o his as a recagular widowig fucio. This ca furher affec he compuaio of he specrum because he sharp rasiios a he begiig ad ed of he record iroduce spurious high frequecy compoes. The effec is reduced by usig a more graceful widowig fucio., he mos commo beig he Hammig widow π cos w N 1 N 1 oherwise 1 recagular Hammig.8 N-1 Dr. Bria Copelad FE3B Fourier Aalysis, P/3 Feb 1 9

23 Fourier Techiques for Aalysig Music Sigals Digial Compuaio of he FT... The N-poi DFT of a legh L sigal is give by: X k x L 1 1 N x e N 1 k jπk N The iverse rasform is X k e jπk N k represes he k h specrum frequecy a fkf s /N Hz. For efficie compuaio we se LN m, m a ieger. If he record legh, L is o a power of, we pad wih zeros so ha he padded sample is of legh m Xk repeas every fs/. For music samples, reduce errors by: Usig a high eough f s. For seelpa, a absolue miimum of 8Hz is recommeded sice he Nyquis frequecy of 4Hz is way above he ocave of he highes oe F6 ad should also accommodae o-musical HF compoes Place a small 5% deadspace before ad afer he soud sample be sure ha i decays adequaely. A recagular widow should iroduce lile error here ad would lead o a more accurae specrum Dr. Bria Copelad FE3B Fourier Aalysis, P/3 Feb 1 1

24 Dr. Bria Copelad FE3B Fourier Aalysis, P/3 Feb 1 11 Fourier Techiques for Aalysig Music Sigals Digial Compuaio of he FT... Example. 5kHz NB : arg arg khz {,,,1}, N f j X e e x X j e e x X x X f x s o j j o j j s π π π π f, khz Xf

25 Fourier Techiques for Aalysig Music Sigals Examples... Specra show i previous slides all geeraed by he MATLAB DFT fucio FFT. A ew example showig a broad bad effec.. GUNSHOT Dr. Bria Copelad FE3B Fourier Aalysis, P/3 Feb 1 1

26 Fourier Techiques for Aalysig Music Sigals The Shor Time Fourier Trasform STFT Cosider he wo fucios f g si π si π si π 4 si π 14 8ms 8ms 16ms oherwise Claim: The ampliude specra will be similar, poss same Reaso: The FT, DFT average he fucio, weighed by a seleced siusoid,over all ime or all samples. The ampliude specrum CANNOT iform o variaios i properies eg ampliude, frequecy of siusoid compoes over he observaio ime!! This ifo is ofe ecessary because he properies of he compoes all real sigals do vary wih ime i.e. hey are o-saioary Dr. Bria Copelad FE3B Fourier Aalysis, P/3 Feb 1 1

27 Fourier Techiques for Aalysig Music Sigals The Shor Time Fourier Trasform STFT Fw Gw Ca ell he sie compoes, bu NOT heir locaio i ime Dr. Bria Copelad FE3B Fourier Aalysis, P/3 Feb 1

28 Fourier Techiques for Aalysig Music Sigals The Shor Time Fourier Trasform STFT The STFT is a modificaio of he DFT process ha helps us rerieve ime varyig iformaio by performig he DFT o successive iervals of he sampled record. x L samples Widow 1 Widow Widow 3 Now we ge a 3-D plo! Xf f /fs Dr. Bria Copelad FE3B Fourier Aalysis, P/3 Feb 1 3

29 Fourier Techiques for Aalysig Music Sigals The Shor Time Fourier Trasform STFT N samples/widow, N < L Overlappig widows help accuracy Frequecy resoluio decreased for N each DFT sice N<L > fs/n>fs/l. Time resoluio icreased as N MUST rade frequecy resoluio for ime resoluio: he uceraiy priciple BUT sill ge a beer picure of how he frequecy compoes chage over he record Dr. Bria Copelad FE3B Fourier Aalysis, P/3 Feb 1 4

30 Fourier Techiques for Aalysig Music Sigals The Shor Time Fourier Trasform STFT, sec, sec db Ampliude Colormap NB: Shif i frequecy a.8sec STFT for f f, Hz STFT for g f, Hz Dr. Bria Copelad FE3B Fourier Aalysis, P/3 Feb 1 5

31 Fourier Techiques for Aalysig Music Sigals The Shor Time Fourier Trasform STFT Nwi51 Novlap64 STFT for C4 o flue Dr. Bria Copelad FE3B Fourier Aalysis, P/3 Feb 1 6

32 Fourier Techiques for Aalysig Music Sigals The Shor Time Fourier Trasform STFT Nwi18 Novlap64 C4 o Flue wih smaller STFT widow Noe loss i frequecy resoluio Dr. Bria Copelad FE3B Fourier Aalysis, P/3 Feb 1 7

33 Fourier Techiques for Aalysig Music Sigals The Shor Time Fourier Trasform STFT C4 o Flue 3D view Dr. Bria Copelad FE3B Fourier Aalysis, P/3 Feb 1 8