6.003: Signals and Systems

Transcription

1 6.003: Signals and Sysems Fourier Series November 1,

2 Las Time: Describing Signals by Frequency Conen Harmonic conen is naural way o describe some kinds of signals. Ex: musical insrumens (hp://heremin.music.uiowa.edu/mis.hml) piano piano violin violin k bassoon bassoon k 2 k

3 3 Las Time: Fourier Series Deermining harmonic componens of a periodic signal. a k = 1 j 2π k x()e T T T d ( analysis equaion) 0 x()= x( + T ) = k= a k e j 2π T k ( synhesis equaion) We can hink of Fourier series as an orhogonal decomposiion.

4 4 Orhogonal Decomposiions Vecor represenaion of 3-space: le r represen a vecor wih componens {x, y, and z} in he {xˆ, ŷ, and ẑ} direcions, respecively. x = r xˆ y = r ŷ z = r ẑ r = xxˆ + yŷ + zẑ ( analysis equaions) ( synhesis equaion) Fourier series: le x() represen a signal wih harmonic componens 2 j0 j 2 T π j π k {a 0, a 1,..., a k } for harmonics {e, e,..., e T } respecively. a k = 1 j 2π k x()e T T T d ( analysis equaion) 0 j 2 x()= x( + T ) = a k e T π k k= ( synhesis equaion)

5 5 Orhogonal Decomposiions Vecor represenaion of 3-space: le r represen a vecor wih componens {x, y, and z} in he {xˆ, ŷ, and ẑ} direcions, respecively. x = r xˆ y = r ŷ z = r ẑ ( analysis equaions) r = xˆ x + yŷ + zˆ z ( synhesis equaion) Fourier series: le x() represen a signal wih harmonic componens 2 j0 j 2 T π j π k {a 0, a 1,..., a k } for harmonics {e, e,..., e T } respecively. a k = 1 j 2π k x()e T T T d ( analysis equaion) x()= x( + T ) = 0 k= j a k e 2 T π k ( synhesis equaion)

6 6 Orhogonal Decomposiions Vecor represenaion of 3-space: le r represen a vecor wih componens {x, y, and z} in he {xˆ, ŷ, and ẑ} direcions, respecively. x = r ˆx y = r ŷ z = r ẑ r = xxˆ + yŷ + zẑ ( analysis equaions) ( synhesis equaion) Fourier series: le x() represen a signal wih harmonic componens 2 j0 j 2 T π j π k {a 0, a 1,..., a k } for harmonics {e, e,..., e T } respecively. a k = 1 j 2π x()e T k d T T ( analysis equaion) 0 j 2 x()= x( + T ) = a k e T π k k= ( synhesis equaion)

7 7 Orhogonal Decomposiions Inegraing over a period sifs ou he k h componen of he series. Sifing as a do produc: x = r xˆ r xˆ cos θ Sifing as an inner produc: j 2π k 1 j 2π k a k = e T x() x()e T d T T where 1 a() b() = a ()b()d. T T The complex conjugae ( ) makes he inner produc of he k h and m h componens equal o 1 iff k = m: 1 j 2π k j 2π m 1 j 2π k j 2π m 1 if k = m e T e T d = e T e T d = T T T T 0 oherwise

8 Check Yourself How many of he following pairs of funcions are orhogonal ( ) in T = 3? 1. cos 2π sin 2π? 2. cos 2π cos 4π? 3. cos 2π sin π? 4. cos 2π e j2π? 8

9 Check Yourself How many of he following are orhogonal ( ) in T = 3? cos 2π sin 2π? cos 2π sin 2π cos 2π sin 2π = 1 2 sin 4π 3 d = 0 herefore YES

10 Check Yourself How many of he following are orhogonal ( ) in T = 3? cos 2π cos 4π? cos 2π cos 4π cos 2π cos 4π = 1 2 cos 6π cos 2π 3 d = 0 herefore YES

11 Check Yourself How many of he following are orhogonal ( ) in T = 3? cos 2π sin π? cos 2π sin π cos 2π sin π = 1 2 sin 3π 1 2 sin π 3 d = 0 herefore NO

12 12 Check Yourself How many of he following are orhogonal ( ) in T = 3? cos 2π e 2π? e 2π = cos 2π + j sin 2π cos 2π sin 2π bu no cos 2π Therefore NO

13 Check Yourself How many of he following pairs of funcions are orhogonal ( ) in T = 3? 2 1. cos 2π sin 2π? 2. cos 2π cos 4π? 3. cos 2π sin π? X 4. cos 2π e j2π? X 13

14 14 Speech Vowel sounds are quasi-periodic. ba bai be bee bi bie bough boa bu boo

15 15 Speech Harmonic conen is naural way o describe vowel sounds. ba bai be bee bi k bie k bough k boa k k bu k boo k k k k

16 16 Speech Harmonic conen is naural way o describe vowel sounds. ba ba bee bee k boo boo k k

17 Speech Producion Speech is generaed by he passage of air from he lungs, hrough he vocal cords, mouh, and nasal caviy. Nasal caviy Hard palae Sof palae (velum) Pharynx Epiglois Tongue Larynx Lips Esophogus Vocal cords (glois) Trachea Somach Lungs 17 Adaped from T.F. Weiss

18 Speech Producion Conrolled by complicaed muscles, vocal cords are se in vibraion by he passage of air from he lungs. Looking down he hroa: Vocal cords open Glois Vocal cords closed Vocal cords Gray's Anaomy Adaped from T.F. Weiss 18

19 Speech Producion Vibraions of he vocal cords are filered by he mouh and nasal caviies o generae speech. 19

20 Filering Noion of a filer. LTI sysems canno creae new frequencies. can only scale magniudes & shif phases of exising componens. Example: Low-Pass Filering wih an RC circui R + v + i v o C 20

21 Lowpass Filer Calculae he frequency response of an RC circui. R + v + i v o C 1 KVL: v i () = Ri() + v o () C: i() = Cv o() Solving: v i () = RC v o() + v o () V i (s) = (1 + src)v o (s) H(s) V o (s) 1 = = V i (s) 1 + src H(jω) H(jω) π ω 1/RC ω 1/RC

22 Lowpass Filering Le he inpu be a square wave jω x() = 0 k e ; jπk k odd 1 0 T 1 2 2π ω 0 = T X(jω) X(jω) π ω 1/RC ω 1/RC

23 Lowpass Filering Low frequency square wave: ω 0 << 1/RC jω x() = 0 k e ; jπk k odd 1 0 T 1 2 2π ω 0 = T H(jω) H(jω) π ω 1/RC ω 1/RC

24 Lowpass Filering Higher frequency square wave: ω 0 < 1/RC jω x() = 0 k e ; jπk k odd 1 0 T 1 2 2π ω 0 = T H(jω) H(jω) π ω 1/RC ω 1/RC

25 Lowpass Filering Sill higher frequency square wave: ω 0 = 1/RC jω x() = 0 k e ; jπk k odd 1 0 T 1 2 2π ω 0 = T H(jω) H(jω) π ω 1/RC ω 1/RC

26 Lowpass Filering High frequency square wave: ω 0 > 1/RC jω x() = 0 k e ; jπk k odd 1 0 T 1 2 2π ω 0 = T H(jω) H(jω) π ω 1/RC ω 1/RC

27 Source-Filer Model of Speech Producion Vibraions of he vocal cords are filered by he mouh and nasal caviies o generae speech. buzz from vocal cords hroa and nasal caviies 27 speech

28 28 Speech Producion X-ray movie showing speech in producion. Couresy of Kenneh N. Sevens. Used wih permission.

29 Demonsraion Arificial speech. buzz from vocal cords hroa and nasal caviies 29 speech

30 hp:// 30 Formans Resonan frequencies of he vocal rac. F1 F2 ampliude F3 frequency Forman heed head had hod haw d who d Men F F F Women F F F Children F F F

31 We deec changes in he filer funcion o recognize vowels. 31 Speech Producion Same glois signal + differen formans differen vowels. glois signal vocal rac filer vowel sound a k b k a k b k

32 Singing We deec changes in he filer funcion o recognize vowels... a leas someimes. Demonsraion. la scale. lore scale. loo scale. ler scale. lee scale. Low Frequency: la lore loo ler lee. High Frequency: la lore loo ler lee. hp:// 32

33 33 Speech Producion We deec changes in he filer funcion o recognize vowels. low inermediae high

34 34 Speech Producion We deec changes in he filer funcion o recognize vowels. low inermediae high

35 Coninuous-Time Fourier Series: Summary Fourier series represen signals by heir frequency conen. Represening a signal by is frequency conen is useful for many signals, e.g., music. Fourier series moivae a new represenaion of a sysem as a filer. Represening a sysem as a filer is useful for many sysems, e.g., speech synhesis. 35

36 MIT OpenCourseWare hp://ocw.mi.edu Signals and Sysems Fall 2011 For informaion abou ciing hese maerials or our Terms of Use, visi: hp://ocw.mi.edu/erms.