Fourier Series. Combination of Waves: Any PERIODIC function f(t) can be written: How to calculate the coefficients?

Transcription

1 Fourier Series Combination of Waves: Any PERIODIC function f(t) can be written: How to calculate the coefficients?

2 What is the Fourier Transform It is the representation of an arbitrary NON-periodic signal as a combination of waves. The formula can be directly derived from the Fourier series.

3 Fourier Transform Laws Transform Pair Addition Rule Multiplication With Constant Factor Similarity Rule

4 Fourier Transform Laws Integration Rule Differentiation Rule Translation Rule

5 Correspondence Of Fourier Transform Constant Component Dirac Pulse Gauss Pulse Square Pulse Dirac Pulse

6 Pulse Forms Square Pulse for for for otherwise

7 The -Function A -Function at x=x 0, written as (x-x 0 ), has a value of at x 0 and is 0 for all other x. + (x-x 0 ) dx = 1 Lattice function l (r) = all h,k,l (r-[ha+kb+lc]) 1D: - + f (x) (x-x 0 ) dx = f(x 0 ) 3D: - + f (r) (r-r 0 ) dr = f(r 0 ) -function is useful to represent points or arrays of points, such as lattices

8 Pulse Forms Gauss Pulse

9 Fourier and all that

10 Any periodic function can be considered to be a sum of sinusoidal waves Fourier Series f(x)= n A n cos(nx) + B n sin(nx) Calculate A n and B n A n = period f(x) cos (nx) dx B n = period f(x) sin (nx) dx

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12 Fourier Analysis of a Violin From C.A. Culver, Musical Acoustics

13 Wave Characteristics Wavelength ( ) Amplitude Phase ( ) Branden & Tooze

14 Spatial Frequencies

15 Spatial Frequencies (Waves)

16 Spectrum of the Signal

17 Fourier Domain Amplitudes Phases

18 Demo With Instruments Show how different sounds are represented with similar frequencies

19 Fourier Transform of a Very Very Very Very Small Box 0 X x F(X)= (X) f(x)=1 D. DeRosier, Brandeis

20 Fourier Transform of a Box -a/2 a/2 f(x)=1 if a/2<x<a/2 f(x)=0 otherwise 1/a F(X)={sin( ax)/( X) X D. DeRosier, Brandeis

21 Fourier Transform of a Lattice -3a -2a -a 0 a 2a 3a x f(x)= (x+a)+ (x)+ (x-a) -3/a -2/a -1/a 0 1/a 2/a 3/a X F(X)= (X+1/a)+ (X)+ (X-1/a) D. DeRosier, Brandeis

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26 Convolution What s that? c (u) = f (r) g(r) = all r f(r) g(u-r) dr 1. Generate g(u-r) 2. Multiply f(r) by g(u-r) 3. Integrate over all r (in 1D calculate area under graph) 4. Return to step 1. using u=u i and repeat for all u. This generates the entire function.

27 Convolution Why That??? An example: =???

28 Convolution Why That??? inflect displace Now carry out product f(x)g(u-x)

29 Convolution - Execution

30 Convolutions with arrays of -Functions

31 Other Convolutions

32 2D Crystal

33 A helical tubes of virus head protein The protein subunits can be seen clearly in some places but not others. Although we see some regularities, they are not everywhere.

34 Photographic image superposition (averaging) by Roy Markham. The image was shifted and added to the original. He saw subunit

35 Superimposed images using Adobe Photoshop. Markham s lattice was used to determine how much to shift by and in which directions.

36 How would we figure out the distance and direction to shift if there weren't a guide? We could guess and pick the answer (image) that we liked best. We could try all possible shifts and pick out the image with the strongest features (measured objectively rather than subjectively)

37 The positions and directions of the reflections tells us how much to shift by and in which directions to shift. EM of catalase Optical diffraction pattern weak and strong exposure. (Erikson and Klug, 1971)

38 II. What the Fourier transform tells you.

39 What you see. Spots Excited What you get

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41 What you see. Spots Spot positions Excited What you get Unit cell size and shap

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43 What you see. Spots Spot positions Spot size What you get Excited Unit cell size and shap Size of coherent domains

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45 What you see. Spots Spot positions Spot size Intensity relative to background What you get Excited Unit cell size and shap Size of coherent domains Signal/noise ratio

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47 What you see. Spots Spot positions Spot size Intensity relative to background Distance to farthest spot What you get Excited Unit cell size and shap Size of coherent domains Signal/noise ratio Resolution

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49 What you see. Spots Spot positions Spot size Intensity relative to background Distance to farthest spot Amplitude and phases of spots What you get Excited Unit cell size and shap Size of coherent domains Signal/noise ratio Resolution Structure of molecules

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51 What you see. Spots Spot positions Spot size Intensity relative to background Distance to farthest spot Amplitude and phases of spots Positions of Thon rings What you get Excited Unit cell size and shap Size of coherent domains Signal/noise ratio Resolution Structure of molecules Amount of defocus

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53 What you see. Spots Spot positions Spot size Intensity relative to background Distance to farthest spot Amplitude and phases of spots Positions of Thon rings Ellipticity of Thon rings What you get Excited Unit cell size and shap Size of coherent domains Signal/noise ratio Resolution Structure of molecules Amount of defocus Amount of astigmatism

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55 What you see. Spots Spot positions Spot size Intensity relative to background Distance to farthest spot Amplitude and phases of spots Positions of Thon rings Ellipticity of Thon rings What you get Excited Unit cell size and shap Size of coherent domains Signal/noise ratio Resolution Structure of molecules Amount of defocus Amount of astigmatism Asymmetric intensity of Thon rings Amount of instability

56 Weaker here Weaker here Stronger here Stronger here

57 What you see. Spots Spot positions Spot size Intensity relative to background Distance to farthest spot Amplitude and phases of spots Positions of Thon rings Ellipticity of Thon rings What you get Excited Unit cell size and shap Size of coherent domains Signal/noise ratio Resolution Structure of molecules Amount of defocus Amount of astigmatism Asymmetric intensity of Thon rings Amount of instability Direction of asymmetry Direction of instability

58 value discrete value discrete value continuous value continuous Classification Of Signals value time time continuous time discrete value time time continuous time discrete analog signal samp- ling sampled signal digita lising digitalised signal digital signal spec. binary signal

59 vlue continous value discrete Classification Of Signals spatial continuous spatial discrete

60 vlue continous value discrete Classification Of Signals spatial continuous spatial discrete

61 Signal Deterministic Random

62 Signal/ Fourier Domain Deterministic Random

63 Different types Of Noise

64 Different types Of Noise

65 Different types Of Noise

66 Different types of Barbara

67 Sampling Of Signals time signal with limited bandwidth related spectra Fourier transform Ideal Sampling Sampling Sampling Operator

68 Sampling Of Signals ideal sampling ideal reconstruction

69 Oversampling arbitrary for for for

70 Undersampling

71 General Sampling Deterministic Fourier Domain

72 Oversampling

73 Undersampling

74 How Do You Sample in Real Life What do you want to see? Magnification Pixel Size (How do this things relate to each other) Capabilities of the recording camera What do you want to do with the data Here we need a number of Examples