Math 345: Applied Mathematics

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1 Math 345: Applied Mathematics Introduction to Fourier Series, I Tones, Harmonics, and Fourier s Theorem Marcus Pendergrass Hampden-Sydney College Fall

2 Sounds as waveforms I ve got a bad feeling about this... 2

3 Sounds as waveforms I ve got a bad feeling about this... 2

4 Sounds as waveforms Botteri s Sparrow 3

5 Sounds as waveforms Botteri s Sparrow 3

6 Sounds as waveforms Phase Modulated Sinusoids 4

7 Sounds as waveforms Phase Modulated Sinusoids 4

8 Sounds as waveforms Algorithmic Composition 5

9 Sounds as waveforms Algorithmic Composition 5

10 Basic Questions 6

11 Basic Questions How do we model these waveforms? 6

12 Basic Questions How do we model these waveforms? How do we analyze these waveforms? 6

13 Basic Questions How do we model these waveforms? How do we analyze these waveforms? How do we approximate these waveforms? 6

14 Basic Questions How do we model these waveforms? How do we analyze these waveforms? How do we approximate these waveforms? How do we synthesize these waveforms? 6

15 Pure Tones and Harmonics Pure Tone s(t) =A cos(2πf 0 t + φ) 7

16 Pure Tones and Harmonics Pure Tone s(t) =A cos(2πf 0 t + φ) - A = amplitude - f 0 = frequency (Hz) - t =time(sec) - φ = phase (rad) 7

17 Pure Tones and Harmonics Pure Tone s(t) =A cos(2πf 0 t + φ) - A = amplitude - f 0 = frequency (Hz) fundamental frequency - t =time(sec) - φ = phase (rad) 7

18 Pure Tones and Harmonics Cosine addition formula A cos(2πf 0 t + φ) =a cos(2πf 0 t)+b sin(2πf 0 t) - a, b are constants Pure tone = linear combination of sines and cosines (no phase shift) 8

19 Pure Tones and Harmonics Harmonics pure tones whose frequencies are integral multiples of the fundamental frequency. s k (t) =A cos(2πkf 0 t + φ) Also can be written in terms of un-shifted sines and cosines. 9

20 Harmonic Analysis Foundational Question: What functions can be expressed as linear combinations of a pure tone and its harmonics? Joseph Fourier (French, ) 10

21 Vector-Space Setting Function space T V = {s :[0,T] R : 0 s(t) dt =0} vector space under ordinary addition and scalar multiplication of functions. Recall the functions s in V must be nice enough so that this integral exists. The technical jargon is that the functions in V must be measurable. 11

22 Vector-Space Setting Function space T V = {s :[0,T] R : 0 s(t) dt =0} vector space under ordinary addition and scalar multiplication of functions. Standard inner product T s, r = 0 s(t) r(t) dt 12

23 Vector-Space Setting Function space T V = {s :[0,T] R : 0 s(t) dt =0} vector space under ordinary addition and scalar multiplication of functions. Norm T s 2 = s, s = 0 s(t) 2 dt 13

24 Model Interpretations Function space = set of sounds V = { reasonable sounds of duration T } Norm squared = energy of a sound T s 2 = 0 s(t) 2 dt = energy of s 14

25 Tone Spaces A tone space is a subspace of V that is spanned by a set of harmonics of some fundamental frequency. What fundamental frequency? Want an integer number of cycles in [0, T] for all harmonics. Can (and usually will) take f 0 = 1 T 15

26 Tone Spaces Basic spanning sets β n = {cos(2πkf 0 t), sin(2πkf 0 t):1 k n} Example: β 3 = { cos(2πf 0 t), sin(2πf 0 t), cos(2π2f 0 t), sin(2π2f 0 t), cos(2π3f 0 t), sin(2π3f 0 t)} 16

27 Tone Spaces Basic spanning sets β n = {cos(2πkf 0 t), sin(2πkf 0 t):1 k n} Example: trigonometric basis β 3 = { cos(2πf 0 t), sin(2πf 0 t), cos(2π2f 0 t), sin(2π2f 0 t), cos(2π3f 0 t), sin(2π3f 0 t)} 16

28 Tone Spaces Basic tone spaces W n = span β n Functions in Wn take the form: n s(t) = a k cos(2πkf 0 t)+b k sin(2πkf 0 t) k=1 17

29 Tone Spaces Example of an s W 3, for f 0 = 440. s(t) = cos(880πt) sin(880πt) sin(1760πt) cos(2640πt) sin(2640πt) 18

30 Tone Spaces Recall: a pair vectors s and r are orthogonal if s, r =0 A set of vectors is orthogonal if every pair from the set is orthogonal Orthogonal sets are nice. For one thing, they re linearly independent... 19

31 Tone Spaces Theorem 1. Let T > 0 and f0 = 1/T. Then βn is an orthogonal basis for Wn. Moreover s 2 = T 2 for all s in βn. 20

32 Orthogonal Projections Definition. The orthogonal projection of s onto W is the vector ŝ in W satisfying s ŝ, w =0 w W s s ŝ W ŝ 0 w 21

33 Approximations and Projections Theorem 2. The orthogonal projection of s onto W is the closest vector in W to s. s ŝ s w w W s W ŝ 0 w 22

34 Calculating Projections Theorem 3. Let W be a subspace of V, and suppose β = {w 1,w 2,...,w n } is an orthogonal basis for W. Then for any vector s in V, the orthogonal projection of s onto W is given by ŝ = n k=1 s, w k w k,w k w k 23

35 Calculating Projections Corollary 4. Let s be in V, let f0 = 1/T, and let Wn = span βn, where β n = {cos(2πkf 0 t), sin(2πkf 0 t):1 k n} Then the projection of s onto Wn is given by where ŝ = n k=1 a k cos(2πkf 0 t)+b k sin(2πkf 0 t) a k = 2 T T 0 s(t) cos(2πkf 0 t) dt, b k = 2 T T 0 s(t) sin(2πkf 0 t) dt 24

36 Example 1 Find the projection of s onto W3: s(t) = 1 if 0 t<1 1 if 1 t 2 0 otherwise 25

37 Example 1 T = 2, f0 = 1/2 26

38 Example 1 T = 2, f0 = 1/2 β 3 = {cos(πkt), sin(πkt) :1 k 3} 27

39 Example 1 T = 2, f0 = 1/2 β 3 = {cos(πkt), sin(πkt) :1 k 3} a k = 2 0 s(t) cos(πkt) dt =0 28

40 Example 1 T = 2, f0 = 1/2 β 3 = {cos(πkt), sin(πkt) :1 k 3} a k = 2 0 s(t) cos(πkt) dt =0 b k = 2 0 s(t) sin(πkt) dt = 4 πk if k odd 0 if k even 29

41 Example 1 ŝ 3 (t) = 4 π sin(πt)+ 4 3π sin(3πt) 30

42 Example 1 Note: ŝ is periodic, with period 2 So it also approximates the periodization of s. ŝ 3 (t) = 4 π sin(πt)+ 4 3π sin(3πt) 31

43 Example 1 ŝ 3 (t) = 4 π sin(πt)+ 4 3π sin(3πt) 32

44 Example 2 General square wave 1 1 T/2 T s(t) = 1 if 0 t<t/2 1 if T/2 t T 0 otherwise 33

45 Example 2 General square wave 1 1 T/2 T f 0 =1/T a k =0 b k = 4 πk if k odd 0 if k even ŝ(t) = 4 π sin(2πf 0t)+ 4 3π sin(2π3f 0t) (2m 1)π sin(2π(2m 1)f 0t) 34

46 Example 2 General square wave ŝ(t) = 4 π sin(2πf 0t)+ 4 3π sin(2π3f 0t) (2m 1)π sin(2π(2m 1)f 0t) 35

47 Example 2 General square wave ŝ(t) = 4 π sin(2πf 0t)+ 4 3π sin(2π3f 0t) (2m 1)π sin(2π(2m 1)f 0t) 36

48 Example 3 General sawtooth wave 1 T/2 T s(t) = 1 2 T t if 0 t T 0 otherwise 1 37

49 Example 3 General sawtooth wave 1 1 T/2 T f 0 =1/T a k =0 b k = 2 πk ŝ(t) = 2 π sin(2πf 0t)+ 2 2π sin(2π2f 0t) nπ sin(2πnt) 38

50 Example 3 General sawtooth wave ŝ(t) = 2 π sin(2πf 0t)+ 2 2π sin(2π2f 0t) nπ sin(2πnt) 39

51 Example 3 General sawtooth wave ŝ(t) = 2 π sin(2πf 0t)+ 2 2π sin(2π2f 0t) nπ sin(2πnt) 40

52 Example 4 General triangular wave T t if T/2 t 0 T/2 T/2 s(t) = 1 4 t if 0 t T/2 T 0 otherwise 1 41

53 Example 4 General triangular wave T/2 1 T/2 f 0 =1/T a k = 8 π 2 k 2 if k is odd 0 if k is even 1 b k =0 ŝ(t) = 8 π 2 cos(2πf 0 t) cos(2π3f 0t)+ + 1 (2m 1) 2 cos(2π(2m 1)f 0t) 42

54 Example 4 General triangular wave ŝ(t) = 8 π 2 cos(2πf 0 t) cos(2π3f 0t)+ + 1 (2m 1) 2 cos(2π(2m 1)f 0t) 43

55 Example 4 General triangular wave ŝ(t) = 8 π 2 cos(2πf 0 t) cos(2π3f 0t)+ + 1 (2m 1) 2 cos(2π(2m 1)f 0t) 44

56 Energy in Wn Theorem 5. If w is an element of Wn, with n w = a k cos(2πkf 0 t)+b k sin(2πkf 0 t) k=1 then n w 2 = T 2 a 2 k + b 2 k k=1 45

57 Energy in Wn Definition. If w is an element of Wn, with n w = a k cos(2πkf 0 t)+b k sin(2πkf 0 t) k=1 then the energy of w at frequency kf0 is T a 2 k + b 2 k 2 46

58 Bessel s Inequality Theorem 6. The Fourier coefficients ak and bk of s satisfy n ŝ 2 = T 2 a 2 k + b 2 k s 2 k=1 s W ŝ 0 w 47

59 Convergence of Fourier Coefficients Corollary 7. If s has finite energy, then k=1 a 2 k + b 2 k < In particular lim k a k = lim k b k =0 48

60 Fourier Series Definition. The Fourier series of a function s is the infinite series s(t) =a 0 + a k cos(2πkf 0 t)+b k sin(2πkf 0 t) k=1 where a0 is the mean value of s, and T T a k = 2 T s(t) cos(2πkf 0 t) dt, b k = 2 T s(t) sin(2πkf 0 t) dt

61 A Fourier-Type Theorem Theorem 8. If s is continuously differentiable at t, then the Fourier series of s converges to s at t. s(t) =a 0 + a k cos(2πkf 0 t)+b k sin(2πkf 0 t) k=1 50

62 Carleson s Theorem Theorem 9. If s is measurable and square-integrable, then the Fourier series of s converges to s almost everywhere. Lennart Carleson (Swedish, present) 51

63 Parseval s Relation Theorem 10. If the Fourier series for s converges to s, then s 2 = Ta 0 + T 2 a 2 k + b 2 k k=1 52

64 Parseval s Relation Definition. If the Fourier series for s converges to s, then the energy of s at frequency kf0 is Ta 0 if k =0 a 2 k + b 2 k if k 1 T 2 53

65 General Square Wave 1 T/2 T 1 s(t) = 4 π k=0 1 (2k + 1) sin(2π(2k + 1)f 0t) 54

66 General Sawtooth Wave 1 T/2 T 1 s(t) = 2 π k=0 1 k sin(2πkf 0t) 55

67 Triangular Wave 1 T/2 T/2 1 s(t) = 8 π 2 k=0 1 (2k + 1) 2 cos(2π(2k + 1)f 0t) 56

68 Summary Sounds are waveforms. Tones and their harmonics are part of the basic vocabulary of music. By Fourier s Theorem, tones and their harmonics can be used as building blocks to produce practically any waveform. Harmonic analysis is central to music theory. 57

MTH 309Y 37. Inner product spaces. = a 1 b 1 + a 2 b a n b n

MTH 309Y 37. Inner product spaces. = a 1 b 1 + a 2 b a n b n MTH 39Y 37. Inner product spaces Recall: ) The dot product in R n : a. a n b. b n = a b + a 2 b 2 +...a n b n 2) Properties of the dot product: a) u v = v u b) (u + v) w = u w + v w c) (cu) v = c(u v)