DCSP-3: Fourier Transform II

Transcription

1 DCSP-3: Fourier Transform II Jianfeng Feng Department of Computer Science Warwick Univ., UK

2 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

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

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

5 100m-for-medical-imaging-tech/

6 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

7 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)

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

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

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

11 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

12 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.

13 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

14 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

15 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

16 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)

17 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( wt ),cos(2wt ) cos( wt)cos(2wt ) dt T T T

18 Intuition of FT Cos ( 2 p t) Cos ( 2*2 p t) Cos ( 2 p t) * Cos ( 2*2 p t) 2 cos( wt ),cos(2wt ) cos( wt)cos(2wt ) dt T T T

19 they form orthogonal bases 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

20 Intuition of FT It turns out that we can define the inner product in the signal space cos( mwt),cos( nwt) 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

21 Intuition of FT X 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 ( )

22 Intuition of FT X(t) A 1 A 6. cos (ω t ) A 2 A 3 A 4 A5 x( t) A1 cos( wt) more

23 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( wt) A cos(2 t) more ( 1 2 w

24 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( wt) A cos(2wt ) A cos(3 t) more ( w

25 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( wt) A cos(2wt ) A cos(3wt ) A cos(4 t) more ( w

26 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( wt) A cos(2wt ) A cos(3wt ) A cos(4 t) more ( w

27 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( wt) A cos(2wt ) A cos(3wt ) A cos(4wt ) A cos(6w ) ( t

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 ) 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

29 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

30 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

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 x(t) = A 0 + å A n cos(nwt)+ n=1 n=1 ì ï A 0 =<1, x(t) >= 1 ïï T ï í ï ïï ï ï î 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 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

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 x(t) = A 0 + å A n cos(nwt)+ n=1 n=1 ì ï A 0 =<1, x(t) >= 1 ïï T ï í ï ïï ï ï î 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 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

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 + å A n cos(nwt)+ n=1 n=1 ì ï A 0 =<1, x(t) >= 1 ïï T ï í ï ïï ï ï î 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 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 + å A n cos(nwt)+ n=1 n=1 ì ï A 0 =<1, x(t) >= 1 ïï T ï í ï ïï ï ï î 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 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 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

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

37 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.

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

39 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

40 Example II (understanding music)

41 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)

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

43 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

44 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

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

46

47 DCSP-4: Applications Jianfeng Feng Department of Computer Science Warwick Univ., UK

48 Recap In general, a signal with T = 2p / w can be represented as follows x(t) = A 0 + å n=1 N å A n cos(wnt)+ ~ A 0 + A n cos(wnt)+ n=1 å n=1 N å n=1 B n sin(wnt) B n sin(wnt)

49 Recap In general, a signal with T=2p/w can be represented as follows T 2T X(t) ~ A 0 + A 1 cos(w t)

50 Recap In general, a signal with T=2p/w can be represented as follows T 2T X(t) ~ A 0 + A 1 cos(w t)+a 2 cos(2wt)

51 Recap In general, a signal with T=2p/w can be represented as follows T 2T X(t) ~ A 0 + A 1 cos(w t)+a 2 cos(2wt)+a 3 cos(3wt)

52 Recap In general, a signal with T=2p/w can be represented as follows T 2T X(t) ~ A 0 + A 1 cos(w t)+a 2 cos(2wt)+a 3 cos(3wt) + A 4 cos(4wt)+a 5 cos(5wt)+a 6 cos(6wt) + A 7 cos(7wt)+a 8 cos(8wt)+a 9 cos(9wt)

53 Recap In general, a signal with T=2p/w can be represented as follows T 2T X(t) = A 0 + A 1 cos(w t)+a 2 cos(2wt)+a 3 cos(3wt) + A 4 cos(4wt)+a 5 cos(5wt)+a 6 cos(6wt) + A 7 cos(7wt)+a 8 cos(8wt)+a 9 cos(9wt) +

54 Recap In general, a signal with T=2p/w can be represented as follows A 3 A 1 A 2 T 2T X(t) = A 0 + A 1 cos(w t)+a 2 cos(2wt)+a 3 cos(3wt) + A 4 cos(4wt)+a 5 cos(5wt)+a 6 cos(6wt) + A 7 cos(7wt)+a 8 cos(8wt)+a 9 cos(9wt) +

55 Recap In general, a signal with T=2p/w can be represented as follows A3 A 1 FREQUENCY TIME A 2 T 2T X(t) = A 0 + A 1 cos(w t)+a 2 cos(2wt)+a 3 cos(3wt) + A 4 cos(4wt)+a 5 cos(5wt)+a 6 cos(6wt) + A 7 cos(7wt)+a 8 cos(8wt)+a 9 cos(9wt) +

56 Recap In general, a signal with T=2p/w can be represented as follows x(t) = A 0 + å n=1 N å A n cos(wnt)+ ~ A 0 + A n cos(wnt)+ n=1 å n=1 N å n=1 B n sin(wnt) B n sin(wnt) The more terms we have, the more accurate: compression The plot of {A n, B n } against { n} is called frequency spectrum

57 Today s Summary General form of FT A few applications (6 more precisely) Noise

58 In general, a signal can be represented as follows FT: Neat Form; to remember easily ) )]exp( /(2 2 / [ ) )]exp( /(2 2 / [ ) )]/(2 exp( ) [exp( 2 )]/ exp( ) [exp( ) sin( ) cos( ) ( n n n n n n n n n n n n n n nt j j B A nt j j B A A j nt j nt j B nt j nt j A A nt B nt A A t x w w w w w w w w

59 In general, a signal can be represented as follows FT: Neat Form ) exp( ) )]exp( /(2 2 / [ ) )]exp( /(2 2 / [ ) )]/(2 exp( ) [exp( 2 )]/ exp( ) [exp( ) sin( ) cos( ) ( nt j c nt j j B A nt j j B A A j nt j nt j B nt j nt j A A nt B nt A A t x n n n n n n n n n n n n n n n n w w w w w w w w w C -n C n

60 FT : Neat Form Which is the exponential form of the Fourier series. In this expression the values C n are complex number, we have FT x(t) {C n, n=,-1,0,1,2,, } C C 2 ( A B1 )/ 4 C C 2 ( A B2 )/ 4 TIME T 2T FREQUENCY

61 FT : Neat Form Which is the exponential form of the Fourier series. In this expression the values C n are complex number, we have FT x(t) {C n, n=,-1,0,1,2,, } IFT X( nw) x( t) X( F ) F c n 1 T X( F )exp( jft) T / 2 T / 2 x( t)exp( jnwt) x( t) dt where w = (2 p) / T

62 FT in complex exponential If the periodic signal is replace with an aperiodic signal (general case) FT is given by ì ïx(f) = FT(x) = í ïx(t) = FT -1 î (X) = ò - ò - x(t)exp(-2p jft)dt X(F)exp(2p jft)df Is that deadly simple?

63 Fourier s song Integrate your function times a complex exponential It's really not so hard you can do it with your pencil yes, I agree

64 Example I Consider the case of a rectangular pulse. In particular, let us define ì ï x(t) = í îï 1, if < t < 0.5 0, otherwise Its FT is given by X(F) = 0.5 ò -0.5 x(t)exp(- j2pft)dt = sin(pf) pf Which is called the Sinc function.

65 Example I There are a number of features to note: 1. The bandwidth of the signal is only approximately finite. power Most of the energy is contained in a limited regions called the main-lobe. However, some energy is found at all frequencies. 2. The spectrum has positive and negative frequencies. These are symmetric about the origin. This may seem non-intuitive, but can be seen from equations in periodic case Frequency 3. Note that the spectrum is continuous now: X(F)= Sinc function

66 Fourier's Song Integrate your function times a complex exponential It's really not so hard you can do it with your pencil And when you're done with this calculation You've got a brand new function - the Fourier Transformation What a prism does to sunlight, what the ear does to sound Fourier does to signals, it's the coolest trick around Now filtering is easy, you don't need to convolve All you do is multiply in order to solve. From time into frequency - from frequency to time Every operation in the time domain Has a Fourier analog - that's what I claim Think of a delay, a simple shift in time It becomes a phase rotation - now that's truly sublime! And to differentiate, here's a simple trick Just multiply by J omega, ain't that slick? Integration is the inverse, what you gonna do? Divide instead of multiply - you can do it too. Or make the pulse wide, and the sinc grows dense, The uncertainty principle is just common sense. From time into frequency - from frequency to time Let's do some examples... consider a sine It's mapped to a delta, in frequency - not time Now take that same delta as a function of time Mapped into frequency - of course - it's a sine! Sine x on x is handy, let's call it a sinc. Its Fourier Transform is simpler than you think. You get a pulse that's shaped just like a top hat... Squeeze the pulse thin, and the sinc grows fat.

67 The world has changed Frequency eye This module and a few following ones will make use of the frequency information

68 This module is about practical applications Carpenter: saw, screw driver, hammer. Can you make a table for me? I doubt. How ì to use this idea? ïx(f) = FT(x) = í ï îx(t) = FT -1 (X) = ò - ò - x(t)exp(-2p jft)dt X(F)exp(2p jft)df

69 App I Time domain Frequency domain S 1 (t)= 10 cos(2pt) + cos(10*2pt) for i=1:10000 x(i)=10*cos(i*0.01)+cos(i*10*0.01); End Sound(x)

70 App I Time domain Frequency domain S 2 (t)= cos(2pt) + 10 cos(10*2pt) for i=1:10000 x(i)=1*cos(i*0.01)+10*cos(i*10*0.01); End Sound(s)

71 App I In frequency domain, the height of the spectrum indicates the energy of the corresponding signal. For example, for S 1, energy concentrating on the signal cos(2pt), a low frequency signal for S 2, energy concentrating on the signal cos(10*2pt), a high frequency signal

72 App II: Touch-tone dialing

73 App II: Touch-tone dialing Freqs 1209 Hz 1336 Hz 1477 Hz 1633 Hz 697 Hz A 770 Hz B 852 Hz C 941 Hz * 0 # D touch_tone.m (1)=A n1 cos(n1 w 0 t) +A n2 cos(n2w 0 t) 1 Hz = w 0 Your phone is a section (two terms) of a FT

74 APP III: Spread spectrum techniques In bluetooth, for example, to improve resistance to radio frequency interference by avoiding using crowded frequencies in the hopping sequence. In military use, it is a simple idea (radio guided torpedo, for example)

75 App IV: instruments We all know about pitch It is really about frequency, or cycles per seconds How about harmonics? This is what gives the timbre of an instrument!

76 App IV: instruments The played note is the frequency of the first peak: 220Hz (note A) in this case Other peaks are called harmonics: they define the typical sound of the instrument Without Fourier, we could have been lost When you play an instrument, It is a section of a FT

77 App V: MP3+MP4 How can an MP3 song sound so well, while being so compressed? Compression in a sense introduces noise. a combination of our hearing system + FT

78 App V: MP3+MP4 Original noiseless music Original musie + noise 2 SNR =13 db Same amount of noise Impressive difference of sound What is the secret? >> r = audiorecorder(22050, 16, 1); >> record(r) >> stop(r) >> myspeech = getaudiodata(r, 'int16'); % get data as int16 array >> audiowrite('compress_3.wav',myspeech,fs); load handel.mat filename = 'handel.wav'; audiowrite(filename,y,fs); clear y Fs

79 App V: MP3+MP4

80 App V: MP3+MP4 MP3: complex compression algorithm that introduces errors MP3 minimizes the perceived qaulity decay by shaping the compression errors

81 App VI: Mind Reading MRI is a medical imaging technique to visualize your internal body structure and/or activity in detail Using a physics term: it measures the oscillations of hydrogen nuclei when stimulated by different radio-frequency magnetic fields Reading you mind with FT My brain activity during one scan

82 App VI: Mind Reading MRI is a medical imaging technique to visualize your internal body structure and/or activity in detail Using a physics term: it measures the oscillations of hydrogen nuclei when stimulated by different radio-frequency magnetic fields Reading you mind with FT My brain activity during one scan It is not about sex

83 App VI: Mind Reading Using a DCSP term: it measures the FT of the internal structure and/or activity of interest How does the collected data looks like? img=(t1_image(:,:,50)); img = fftshift(img(:,:)); F = fft2(img); figure; imagesc(100*log(1+abs(fftshift(f)))); colormap(gray); title('magnitude spectrum'); figure; imagesc(angle(f)); colormap(gray); title('phase spectrum');

84 App VI: Mind Reading Little information about the subject can be gathered in the original picture An inverse Fourier transform of the data reveals the slice of a human head activity

85 App VI: Mind Reading Human Brain Project was one of the two projects with a budge of 1B Euros for ten years each (another is graphene) in 2013 The main purpose is to simulate the Brain = USA matched with a 4.5 B US Dollars in 2014 China will do as well