Digitaalne signaal Diskreetimine ja Dirac Delta Digitaalfiltrid. Digitaalne heli. Hendrik Nigul. Mathematics of Sound and Music.
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1 Mathematics of Sound and Music Aprill 2007
2 Outline 1 Digitaalne signaal 2 3 z-teisendus
3 Mis on heli? Digitaalne signaal Heli on elastses keskkonnas lainena leviv mehaaniline võnkumine. amplituud heli tugevus sagedus heli kõrgus
4 Outline Digitaalne signaal 1 Digitaalne signaal 2 3 z-teisendus
5 Heli digitaliseerimine Analoogsignaali ei saa arvutis salvestada. Signaal tuleb digitaliseerida. 2 etappi: diskreetimine (sampling) kvantimine
6 Heli digitaliseerimine Samplite valimine Veel on võimalik signaali tagasi teisendada Diskreetimine
7 Heli digitaliseerimine Samplite valimine Veel on võimalik signaali tagasi teisendada Diskreetimine
8 Heli digitaliseerimine Samplite valimine Veel on võimalik signaali tagasi teisendada Diskreetimine
9 Helilaine (hall joon) digitaalsel kujul (punane). Samplite valimine Digitaalne signaal Heli digitaliseerimine
10 Outline Digitaalne signaal 1 Digitaalne signaal 2 3 z-teisendus
11 Digitaalne signaal Virvendamise (dithering) eesmärk on vähendada helimoonutusi, mis on tekkinud digitaliseerimise tõttu. Väikese juhusliku müra lisamine suurendab signaali teravust (resolution). Diskreetimise sagedus peab olema suurem, kui signaali muutumise sagedus.
12 Digitaalne signaal Enne virvendamist: Pärast virvendamist:
13 Digitaalne signaal Ka pilditöötluses on vaja kasutada virvendamist. Pilt 256 värviga Koos virvendusega
14 Outline Digitaalne signaal 1 Digitaalne signaal 2 3 z-teisendus
15 WAV file struktuur $ od -N 64 -t x1 file.wav cut -d " " -f c e0 0a d AC B e0 0a Riff tükk (Resource Interchange File Format). Format tükk Andmed
16 WAV file struktuur $ od -N 64 -t x1 file.wav cut -d " " -f c e0 0a d AC B e0 0a RIFF ülejäänud faili suurus 000AE06C 16 = WAVE
17 WAV file struktuur $ od -N 64 -t x1 file.wav cut -d " " -f c e0 0a d AC B e0 0a fmt formaat tüki suurus baitide arv 1 diskreetimissammu kohta mono; stereo diskreetimissagedus 0000AC44 16 = 44100Hz, = 22050Hz. baite sekundis baite sammus bit mono; bit mono/8-bit stereo; bit stereo bitte sammus = 32
18 WAV file struktuur $ od -N 64 -t x1 file.wav cut -d " " -f c e0 0a d AC B e0 0a DATA ülajäänud faili suurus andmed min..-1,0..max on 8-biti korral unsigned: 00..7F,80..FF, 16-biti korral signed: FFFF, FFF
19 Outline Digitaalne signaal 1 Digitaalne signaal 2 3 z-teisendus
20 formaat Digitaalne signaal MPEG I/II Layer 3 Motion Picture Experts Group Kadudega pakkimine Mõned tehnikad: Arvestatakse inimkõrva eripäraga (Fletcher-Munson kõverad) The phenomenon of masking means that some sounds will be present but will not be perceived because of the existence of some other component of the sound. Keerukamad muusikalõigud võtavad rohkem baite. Stereo Huffmani koodid.
21 Outline Digitaalne signaal 1 Digitaalne signaal 2 3 z-teisendus
22 Digitaalne signaal Musical Instrument Digital Interface Andmeedastusprotokoll Midi sõnumite tüübid: Noodid: algus- ja lõppaeg, helikõrgus ja tugevus. Juhtumissõnumid: parameetrid nagu koor, kaja, panoraam (?) üldine helitugevus. Sõnumid konkreetetele seadmetele: nt. süntesaatori seadistamine Midi modulatsioonikiirus on KBaud (kiloboodi bitti sekundis).
23 Outline 1 Digitaalne signaal 2 3 z-teisendus
24 Kroneckeri delta δ i,j = { 1, kui i = j 0, kui i j Dirac delta Põhiomadus δ(t) = 0, kui t 0 f(t)δ(t t 0 )dt = δ(t) = 1 f(t 0 )δ(t t 0 )dt = f(t 0 )
25 Kui f(t) = δ(t t 0 ), siis Fourier teisendus on ˆf(ν) = δ(t t 0 )e 2πiνt dt = e 2πiνt 0
26 Diskreetimine Üks viis diskreetimise esitamiseks. Diskreetimise funktsioon δ s (t) = δ(t n t), n= kus t on ajavahemik 2 sample vahel (nt t = 1/44100 s). Kui f(t) on analoogsignaal, siis f(t)δ s (t) = f(n t)δ(t n t) n=
27 Diskreeditud signaali Fourier teisendus Selleks on 2 võimalust: Theorem f.δ s (ν) = n= f(n t)e 2πiνn t Theorem f.δ s (ν) = 1 t n= ˆf ( ν n ) t
28 Outline 1 Digitaalne signaal 2 3 z-teisendus
29 Theorem Maksimaalne sagedus, mida saab digitaalse signaaliga edastada, on pool diskreetimise sagedusest. Vaatleme lihtsat signaali f(t) = A cos(2πνt). Diskreetimissagedus N = 1/ t. Kui ν = N/2 + α, siis f(m/n) = A cos(2π(n/2 + α)m/n) = A cos(mπ + 2αMπ/N) = ( 1) M A cos(2αmπ/n) Seega on α ja α korral digitaalne signaal samasugune!
30 Theorem Maksimaalne sagedus, mida saab digitaalse signaaliga edastada, on pool diskreetimise sagedusest.
31 Aliasing Kaks erinevat signaali võivad anda sama valimi aliasing.
32 Aliasing Kaks erinevat signaali võivad anda sama valimi aliasing. Õigesti diskreeditud. Ruumiline alias Moire mustri näol.
33 Aliasing Kaks erinevat signaali võivad anda sama valimi aliasing.
34 Mis juhtub signaaliga, mille sagedus on täpselt pool diskreetimise sagedust? Sõltub faasist. Koosinuslained ei muutu. Siinuslained kaovad ära.
35 Diskreetimisteoreem Kui analoogsignaali sagedusspekter asub allpool Nyquisti sagedust, siis saab teda taastada (aja suhtes) diskreeditud signaalist. f.δ s (ν) = 1 t n= ˆf ( ν n ) t
36 Filtreerimine Et vähendada aliaste mõju, tuleb signaal enne diskreetimist töödelda. Lõigata ära sagedused, mis on suuremad, kui pool diskreetimise sagedust. Selleks kasutatakse madalpääsfiltreid (low-pass filter)
37 z-teisendus Outline 1 Digitaalne signaal 2 3 z-teisendus
38 z-teisendus z-teisendus Diskreeditud signaali Fourier teisendus: f.δ s (ν) = n= f(n t)e 2πiνn t See on perioodiline. Asendame z = e 2πiν t = e 2πiν/N. Diskreeditud signaali z-teisendus F (z) = f(n t)z n Omavaheline seos n= f.δ s (ν) = F (e 2πiν t )
39 z-teisendus z-teisendus
40 z-teisendus Outline 1 Digitaalne signaal 2 3 z-teisendus
41 z-teisendus wiki: digitaalfiltritega saab põhimõtteliselt teha suvalist efekti, mis on algoritmiliselt kirjeldatav. Piiravad tegurid on riistvara kiirus, raha. Sagedust muuta ei saa.
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