DCSP-5: Fourier Transform II

Transcription

1 DCSP-5: Fourier Transform II Jianfeng Feng Department of Computer Science Warwick Univ., UK

2 Assignment:2018 Q1: you should be able to do it after last week seminar Q2: need a bit reading (my lecture notes) Q3: standard Q4: standard

3 Assignment:2018 Q5: standard Q6: standard Q7: after today s lecture Q8: load jazz, load Tunejazz load NoiseJazz plot soundsc plot plot

4 Fourier Theorem This representation is quite general. In fact we have the following theorem due to Fourier. Any signal x(t) of period T can be represented as the sum of a set of cosinusoidal and sinusoidal waves of different frequencies

5 This week s summary Introduce FT in layman s language (today) Applications (tomorrow)

6 This week s summary Introduce FT in layman s language (today) Applications (tomorrow) You might find the world is different!

7 100m-for-medical-imaging-tech/

8 This week s summary Introduce FT in layman s language (today) Talk about continuous FT since it is clean and simple Come back to it on how to numerically calculate it later on Intuition Fourier theorem Examples Bandwidth

9 Intuition of FT Two dimensional space (all points) (a, b) = a (1,0) + b (0,1) Signal space (all functions of t) x (t) = a sin(ω t) + b sin (2 ω t)

10 Intuition of FT Two a point in it (a, b) = a (1,0) + b (0,1)

11 Intuition of FT Two a point in it (a, b) = a (1,0) + b (0,1) Signal = coef basis coef basis

12 Intuition of FT Two a point in it (a, b) = a (1,0) + b (0,1) x(t) =? Signal = coef basis coef basis

13 Intuition of FT Two a point in it (a, b) = a (1,0) + b (0,1) x(t) = a F1(t) +b F2(t) Signal = coef basis coef basis

14 Intuition of FT It is a branch of mathematics called functional analysis: treat each function as a point in a functional space It is the starting pointe of modern mathematics It is also the actual power of mathematics Branch space, Hilbert space etc.

15 Intuition of FT How to calculate these coefficients? o (a, b) = a (1,0) + b (0,1) Functional x (t) = a cos(ω t) + b cos (2 ω t) w = 2 pi /T

16 Intuition of FT 1. Coefficient is obtained via Inner product <(x, y), (m,n)> = xm+yn 2. All bases are orthogonal <(0, 1), (1,0)> = 0, (0,1) (1, 0) (a, b) = a (1,0) + b (0,1) Therefore <(a, b), (1,0)> = a <(1,0), (1,0)> + b <(0,1), (1,0)> = a <(a, b), (0,1)> = a <(1,0), (0,1)> + b <(0,1), (0,1)> = b

17 Intuition of FT Two dimensi coefficient is obtained via Inner product (a, b) = a (1,0) + b (0,1) <(a, b), (1,0)> = a <(1,0), (1,0)> + b <(0,1), (1,0)> = a Functional coefficient is obtained via inner product x (t) = a cos(ω t) + b cos (2 ω t) <x (t), cos(ω t) > = <a cos(ω t), cos(ω t) > + <b cos (2 ω t), cos(ω t) > = a (?) If cos(ω t) and cos(2 ω t) are orthogonal

18 Intuition of FT Two dimension orthogonal basis (a, b) = a (1,0) + b (0,1) Functional space basis x (t) = a cos(ω t) + b cos (2 ω t)

19 Intuition of FT It turns out that we can define the inner product in the signal space < x(t), y( t) >= 2 T T / 2 T / 2 x( t) y( t) dt < 2 cos( ωt ),cos(2ωt ) >= cos( ωt)cos(2ωt ) dt T T T

20 Intuition of FT Cos ( 2 p t) Cos ( 2*2 p t) Cos ( 2 p t) * Cos ( 2*2 p t) < 2 cos( ωt ),cos(2ωt ) >= cos( ωt)cos(2ωt ) dt T T T

21 Intuition of FT Cos ( 2 p t) Cos ( 2*2 p t) Cos ( 2 p t) * Cos ( 2*2 p t) { cos (n ω t ), n=1,2 } are orthogonal they form orthogonal bases

22 Intuition of FT It turns out that we can define the inner product in the signal space < cos( mωt),cos( nωt) In particular, we have >= 2 / T, n = 0, n m m x (t) = a cos(ω t) + b cos (2 ω t) a = <X(t), cos(ω t)> = b =? 2 T T /2 ò -T /2 x(t) cos( wt)dt

23 Intuition of FT X 1 X 6 ( ) X 2 X 3 x 4 X5 ( ) ( ) ( ) ( ) ( ) In an n-dim Euclidean space, we decompose any vector X in terms of orthogonal bases Coefficient is obtained by inner product between a basis and X We sometime call x i weight X = x 1 ( ) + x 2 ( ) + + x 6 ( )

24 Intuition of FT X(t) A 1 A 6. cos (ω t ) A 2 A 3 A 4 A5 x ( t) = A1 cos( ωt) + more

25 Intuition of FT X(t) A 1 A 6. cos (ω t ) A 2 A 3 A 4 A5 cos (2 ω t ) x t) = A cos( ωt) + A cos(2 t) + more ( 1 2 ω

26 Intuition of FT X(t) A 1 A 6. cos (ω t ) A 2 A 3 A 4 A5 cos (3 ω t ) cos (2 ω t ) x t) = A cos( ωt) + A cos(2ωt ) + A cos(3 t) + ( ω more

27 Intuition of FT X(t) A 1 A 6. cos (ω t ) A 2 A 3 A 4 A5 cos (3 ω t ) cos (4 ω t ) cos (2 ω t ) x t) = A cos( ωt) + A cos(2ωt ) + A cos(3ωt ) + A cos(4 t) + more ( ω

28 Intuition of FT X(t) A 1 A 6. cos (ω t ) A 2 A 3 A 4 0 cos (3 ω t ) cos (4 ω t ) cos (2 ω t ) x t) = A cos( ωt) + A cos(2ωt ) + A cos(3ωt ) + A cos(4 t) + more ( ω

29 Intuition of FT X(t) A 1 A 6 cos (ω t ) A 2 A 3 A 4 0 cos (3 ω t ) cos (4 ω t ) cos (2 ω t ) x t) = A cos( ω t) + A cos(2ωt ) + A cos(3ωt ) + A cos(4ωt) + A cos(6ω ) ( t

30 Intuition of FT X(t) A 1 A 6. cos (ω t ) A 2 A 3 A 4 0 cos (3 ω t ) cos (4 ω t ) cos (2 ω t ) In an n-dim Euclidean space, we decompose any vector X in terms of orthogonal bases Coefficient is obtained by inner product between a basis and X We sometime call x i weight

31 FT: Fourier Thm in terms of Sin and Cos simply a generalization of common knowledge of the Euclidean space { 1, cos (nωt), sin(nwt), n=1,2 } are orthogonal and complete they form orthogonal bases

32 FT: Fourier Thm in terms of Sin and Cos simply a generalization of common knowledge of the Euclidean space { 1, cos (nωt), sin(nwt), n=1,2 } are orthogonal and complete they form orthogonal bases

33 FT: Fourier Thm in terms of Sin and Cos simply a generalization of common knowledge of the Euclidean space { 1, cos (nωt), sin(nwt), n=1,2 } are orthogonal and complete they form orthogonal bases x(t) = A 0 + ì í î å n=1 A n cos(nwt)+ A 0 =<1, x(t) >= 1 T T /2 -T /2 B n sin(nwt) where A 0 is the d.c. term, and T is the period of the waveform. ò å n=1 x(t) dt A n =< cos(nwt), x(t) >= 2 T B n =< sin(nwt), x(t) >= 2 T w = 2p T T /2 ò -T /2 T /2 ò -T /2 x(t)cos(nwt)dt x(t)sin(nwt)dt

34 FT: Fourier Thm in terms of Sin and Cos simply a generalization of common knowledge of the Euclidean space { 1, cos (nωt), sin(nwt), n=1,2 } are orthogonal and complete they form orthogonal bases x(t) = A 0 + ì í î å n=1 A n cos(nwt)+ A 0 =<1, x(t) >= 1 T T /2 -T /2 B n sin(nwt) where A 0 is the d.c. term, and T is the period of the waveform. ò å n=1 x(t) dt A n =< cos(nwt), x(t) >= 2 T B n =< sin(nwt), x(t) >= 2 T w = 2p T T /2 ò -T /2 T /2 ò -T /2 x(t)cos(nwt)dt x(t)sin(nwt)dt

35 FT: Fourier Thm in terms of Sin and Cos simply a generalization of common knowledge of the Euclidean space { 1, cos (nωt), sin(nwt), n=1,2 } are orthogonal and complete they form orthogonal bases x(t) = A 0 + ì í î å n=1 A n cos(nwt)+ A 0 =<1, x(t) >= 1 T T /2 -T /2 B n sin(nwt) where A 0 is the d.c. term, and T is the period of the waveform. ò å n=1 x(t) dt A n =< cos(nwt), x(t) >= 2 T B n =< sin(nwt), x(t) >= 2 T w = 2p T T /2 ò -T /2 T /2 ò -T /2 x(t)cos(nwt)dt x(t)sin(nwt)dt

36 FT: Fourier Thm in terms of Sin and Cos simply a generalization of common knowledge of the Euclidean space { 1, cos (nωt), sin(nwt), n=1,2 } are orthogonal and complete they form orthogonal bases x(t) = A 0 + ì í î å n=1 A n cos(nwt)+ A 0 =<1, x(t) >= 1 T T /2 -T /2 B n sin(nwt) where A 0 is the d.c. term, and T is the period of the waveform. ò å n=1 x(t) dt A n =< cos(nwt), x(t) >= 2 T B n =< sin(nwt), x(t) >= 2 T w = 2p T T /2 ò -T /2 T /2 ò -T /2 x(t)cos(nwt)dt x(t)sin(nwt)dt

37 Example I x(t)=1, 0< t < p, 2p < t < 3p, 0 otherwise Hence x(t) is a signal with a period of 2p p 2p 3p 4p

38 Example I ì A 0 = 1 p ò dt = 1 2p 0 2 A n = 2 p í ò cos(nt)dt = 1 sin(np ) = 0, n =1, 2,... 2p 0 np B n = 2 p ò sin(nt)dt = 1 (1- cos(np )), n =1, 2,... î 2p np 0 Finally, we have x(t) = p [sin(t)+ 1 3 sin(3t)+ 1 5 sin(5t)+...]

39 Example I X(F) 2/p Frequency domain x(t) = p [sin(t)+ 1 3 sin(3t)+ 1 5 sin(5t)+...] The description of a signal in terms of its constituent frequencies is called its frequency (power) spectrum.

40 Example I X(F) 2/p = Time domain Frequency domain

41 Example I A periodic signal is uniquely determined by its coefficients {A n, B n }. If we truncated the series into finite term, the signal can be approximated by a finite sines as shown below (compression, MP3, MP4, JPG, ) one term Two terms Three terms Five terms

42 Example II (understanding music)

43 Example II a. Pure tone: Script1_1.m This confirms our earlier believe that it is a signal with a finite bandwidth (N-S sampling Thm)

44 Example II b. Different waveforms Script1_2.m This confirms our earlier believe that it is a signal without bandlimit (N-S sampling Thm)

45 Bandwidth can be properly defined Bandwidth is the difference between the upper and lower frequencies in a set of frequencies. It is typically measured in hertz

46 Bandwidth can be properly defined Power 0 B Frequency Bandwidth is the difference between the upper and lower frequencies in a continuous set of frequencies. It is typically measured in hertz

47 Example II C. Approximation (compression) Script1_3.m Without doing much, we can compress the original data now, as in Example I.

48